Perspectives of Polarons

PERSPECTIVES OF

POLARONS

PERSPECTIVES OF

POLARONS

Editors

G. N. Chuev V. D. Lakhno Russian Academy of Sciences

World Scientific Singapore • New Jersey • London • Hong Kong

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Library of Congress Cataloging-in-Publication Data Perspectives of polarons / editors G. N. Chuev, V. D. Lakhno. p. cm. ISBN 9810227787 1. Polarons. I. Chuev, G. N. (Gennady N.), 1961II. Lakhno, V. D. (Viktor D.) QC176.8.P62P47 1996 530.4'16~dc20 96-35326 CBP British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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The monograph "Perspectives of polarons" presents further development of the polaron theory and its applications in various fields of physics. Originally polaron arose in condensed media physics to explain peculiarities of electron states in semiconductors, ionic crystals and other polar media. Subsequently methods of the polaron theory were used to describe electron properties in other condensed media. The states of polaron type in nonpolar media were named as condenson, in magnetoordered media as magnet polaron, in the electrolytes as ionon, etc. Nowadays methods of the polaron theory are widely used in such sciences as biology, chemistry, nuclear physics, etc. Such wide applications result from the fact that polaron is one of the first substantial examples of a nonlinear quasiparticle. This monograph deals with only a small part of the problems indicated. It is based on the talks, given at the Workshops which were held in Pushchino in 1993 and 1994. Our book includes selected articles, which are representative of the exciting scientific presentations that marked the workshops. The first Chapter is devoted to theory and applications of large polaron. The combination of polaron theory with modern quantum chemical calculations is presented in Chapter 2. The polaron problem includes issues which are being studied intensively in the quantum field theory and nonperturbative models of elementary particles. We include three works (Chapters 3, 4, and 13) devoted to applications of the polaron in quantum field theory of particles. The polaron behavior in magnetic systems and magnetic fields are considered in Chapters 6, 10, and 12. Effects of disorder and order of the structure on self-trapped electron states are treated in Chapters 7 and 8. We include the review devoted to quasi-one-dimensional polaron (Chapter 5) in biopolymers, due to vast number of applications of polaron model in one-dimensional structures. The study of high-temperature superconductivity stimulates the bipolaron theory. A generalization of the bipolaron problem to the anisotropic case is considered in Chapter 9. We hope our book will be of interest to a broad range of readers, which can be able to sense the excitment of the articles. The monograph will hopely surve further development and contribute to the introduction of the polaron theory methods into various fields of physics.

Gennady N.Chuev Victor D.Lakhno

v

CONTENTS

PART 1. Papers based on the workshop "Perspectives in Polarons" (Pushchino, 1993). Theory and Applications of Strongly Coupled Large Polaron

1

G. N. Chuev and V. D. Lakhno Theoretical Investigation of Self-Trapped Hole in Alkali Halides.

Long-Range Effects Within the Model Hamiltonian Approach P. B. Zapol and L. N. Kantorovich

38

The Quantumfield Theory of Binucleon V. D. Lakhno

63

Generalized Functional Approach in the Theory Meson-Nucleon Interaction

85

V. D. Lakhno Nonlinear Mathematical Models in Biopolymer Science L. V. Yakushevich

The Ground State of Frohlich ID Fermi-Polaron E. A. Kochetov

94

144

PART 2. Papers based on the workshop "Autolocalized Electron States in Ordered and Disordered Systems" (Pushchino, 1994).

Mean-Field Theory of the Solvated Electron and Dielectron States

150

G. N. Chuev Resonance, Localized and Polaron-Type Electron States in Elastic Materials with Topological Defects V. A. Osipov

188

Bipolarons in Anisotropic Crystals N. I. Kashirina, E. V. Mozdor, E. A. Pashitskij, and V. I. Sheka

202

SU(2) Path Integral for the Heisenberg Ferromagnetic E. A. Kochetov

210

vii

Low-Temperature Electron Mobility of Acoustical Polaron B. A. Kotiya and V. F. Los

Polaron Effects on Magnitoresistance in Quasi-Two-Dimensional Conductors L. S. Kukushkin

216

223

Numerical Investigation of a Quantumfield Model for Strong-Coupled

Binucleon /. V. Amirkhanov, I. V. Puzynin, T. P. Puzynina, T. A. Strizh, E. V. Zemlyanaya and V. D. Lakhno

viii

229

AUTHOR'S ADDRESSES Dr. I.V. Amirkhanov

Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research, Dubna, 141980 Russia

Prof. G.N.Chuev Department of Quantum-Mechanical Systems, Institute of Mathematical Problem of Biology of Russian Academy of Sciences, Pushchino, 142292 Russia

Dr. L.N.Kantorovich Department of Physics, University of Keele, Keele,

Staffordshire, ST5 5BG.UK

Dr. N.I.Kashirina Institute of Physics of Semiconductors of National Ukraine Academy of Science, Kiev 25202, Ukraine

Dr.E.A.Kochetov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research Dubna 141980, Russia

Prof. B.A.Kotiya Georgian Technical University, Tbilisi, Georgia

Prof. L.S.Kukushkin Physical and Technical Institute

of Low Temperature, Kharkov, Ukraine

ix

Prof. V.D.Lakhno Department of Quantum-Mechanical Systems, Institute of Mathematical Problem of Biology of Russian Academy of Sciences, Pushchino, 142292 Russia

Prof. V.F.Los Institute of Metal Physics, Kiev, Ukraine

Dr. E.V. Mozdor Institute of Physics of Semiconductors

of National Ukraine Academy of Science, Kiev 25202, Ukraine

Dr. V.A.Osipov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research Dubna 141980, Russia

Prof. E.A.Pashitskij Institute of Physics of Semiconductors of National Ukraine Academy of Science, Kiev 25202, Ukraine Prof. I.V.Puzynin

Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research, Dubna, 141980 Russia

Dr. T.P. Puzynina, Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research, Dubna, 141980 Russia Prof. V.I. Sheka Institute of Physics of Semiconductors

of National Ukraine Academy of Science, Kiev 25202, Ukraine x

Dr. T.A.Strizh Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research,

Dubna, 141980 Russia Dr. L.V. Yakushevich Institute of Cell Biophysics, Russian Academy of Sciences,

142292 Pushchino, Moscow Region, Russia Dr. E.V. Zemlyanaya Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research,

Dubna, 141980 Russia Dr. P.Zapol Department of Physics,

Michigan Technological University, Houghton, MI, 49931, USA

xi

THEORY AND APPLICATIONS OF STRONGLY COUPLED LARGE POLARON G.N.CHUEV and V.D.LAKHNO Department of Quantum-Mechanical Systems, Institute of Mathematical Problems of Biology,

Russian Academy of Sciences, Pushchino, Moscow Region, 142292 Using the Bogolubov-Tyablikov method, we set forth a pol&ron theory. Taking into account the translational invariance of the free polaron, we consider the autolocaliaed electron states for different types of interaction. It is shown that the structure of these states is strongly related to the structure and peculiar features of the local phonon spectrum. We calculate the phonon spectrum in the strong-coupling limit for electron centers. We discuss the applications of the large pol&ron model and possibilities of its experimental verification. We generalize the BogolubovTyablikov treatment to the strongly coupled bipolaron and propose the criteria of stability and formation for the bipolaron states.

1

Introduction

A polaron concept being the simplest example of a quantum field theory has a huge number of interesting applications in condensed matter physics. We note that the polaron problem was originally formulated as a problem of an autolocalized electron state in an ionic crystal. l In this treatment, the polaron was assumed to correspond to the strong-coupling limit. Figure 1 plots schematically the Pekar polaron treatment. But for a long time the central problem in the polaron theory was to generalize the method to arbitrary coupling, since the strong-coupling criterion

is not satisfied for most of ionic crystals. Significant effort was made to develop various approaches to the calculation of the dependence of the ground electron state on the electron-phonon coupling constant a,3'4 to evaluate the effective mass at various a, to extend the treatment to finite temperatures, 6 to study the polaron transport problem. 4>5 ' 6 ' '8 On the other hand, there is a number of physical examples when the electron can be autolocalized and the strong coupling takes place. Among such examples are the magnetically ordered crystals, where magnetic polaron states are possible, 9'10'11 the polar liquids, where the autolocalized states are the solvated electrons12'13, and other systems. Moreover, even in ionic crystals where polaron states are treated by the weak or intermediate coupling description, the strong coupling is possible for the bipolaron states. We also note that in the important case of the bound 1

polaron arising in the .F-center formation, the strong-coupling criterion is much weaker and can be fulfilled for the ionic crystals. Thus, the study of the strongly coupled polaron is of interest to condensed matter physics, though being a limiting case. Besides, we show in the review that the concept of the strongly coupled polaron can be successfully used for a number of related problems, such as the meson theory of nuclei interaction, 14,15,16 tne evaluation of the mobility of ions implanted into liquid helium ir and so on.

Figure 1: The Pekar polaron treatment in NaCl crystal. Dashed circles denote displaced equilibrium positions of atoms, dotted area is the region of electron localization. In most works on the strongly coupled polaron, the main attention is attached to the study of the ground state. At the same time, the question of the possible existence of polaron states distinct from the ground state is basic to the study of processes related to electron excitation in polar media, for instance, to photoexcitation of ^-centers and other lattice defects. At present, the problem of the excited polaron states, besides being of

theoretical interest, is gaining attention due to the problem of the electron transfer in a wide variety of condensed media (solutions,18 biomacromolecules, 19,20,21,22

et<;

)

2

We note that there are two types of strongly coupled polarons: the small polaron, when an electron is localized on a cite of the medium, 4 ' 23 and the large polaron, when the polaron state is rather diffusive. There is a vast amount of books and reviews devoted to the polaron problem. 3.4,5,6,7,24,25,26,27,28,29 But in our opinion, the unified treatment of the polaron states is absent at present. We make an attempt at such a unified approach with the use of the Bogolubov-Tyablikov method developed to describe a nonrelativistic particle interacting with a quantum field. This enables us to obtain the equations which are asymptotically exact in the strong-coupling limit for different types of quantum fields, and play an important role in solid state theory. In the cases of physical interest, the solutions of these equations give an idea of the complicated structure of autolocalized electron states. Recently, the results have attracted attention due to the development of a number of nonperturbative methods in physics of elementary particles. This review is devoted to the study of the structure of a strongly coupled large polaron, we consider a number of applications of this theory to the problems of condensed matter physics. In Sec. 2, we present the main results of the Bogolubov-Tyablikov method for a nonrelativistic particle strongly interacting with a quantum field. Sec. 3 concerns with the investigation of the electron states determined by different solutions of the nonlinear Shrodinger equation with the self-consistent potential. We present calculations of various parameters of these state and discuss the application of the large polaron treatment. In Sec. 4 we consider the spectrum and characteristics of local phonons caused by the strong electron-phonon coupling. In Sec. 5 we generalize the theory to the bipolaron case. The conclusions are given at the end of the review.

2 2.1

Translational invariant theory of a strongly coupled polaron Basic relations

A starting point in the description of the motion of a nonrelativistic particle in a quantum field is Hamiltonian '31

where fj, is the effective mass of the particle, f is its coordinate, 6 and 6+

are the operators of annihilation and creation of the field quanta with energy huk, Cjt are the constants for the particle-field interaction defined in terms of 3

the macroscopic parameters of the medium. We note that they are proportional to vce, where a is the coupling constant. Hamilton ian Eq. 1 describes the electron-phonon interaction and has been actively investigated in the quantum field theory. 33 In general, in the presence of impurities Hamiltonian Eq. 1 must contain the potential of the particleimpurity interaction. The further investigation of the Hamiltonian depends on the presence of small terms in Eq. 1. If the last term of Eq. 1 is small, the problem is reduced to the weak polaron treatment. 34-35 The other limiting case is that of the strong coupling. In this case there is a small parameter e2 =
where D^ are the equilibrium positions of the lattice coordinates, while Qk and PI denote the deviations from these positions and the corresponding momenta of the coordinates. Then, we separate the variables related to the motion of the particle as a whole where rc is the coordinate of the translational motion of the particle, and r corresponds to the fluctuating part of the motion caused by the particle-field interaction. Eq. 3 determines the canonical Bogolubov-Tyablikov transformation introducing new variables r, rc, Q^ instead of r, Qt, whose number has increased by 3. Since the number of the Bogolubov-Tyablikov variables is larger than the that before the transformation, some additional restrictions are required

where i/t are complex numbers satisfying the normalizing conditions

4

Eqs. 3-5 are a closed set of equations determining the canonical transformation. Having performed the canonical transformation and separated the translational motion, we use the standard perturbation method over parameter e, which enables us to obtain the corresponding wave equations for Hamiltonian Eq. 1

Then, we present the wave function $ of the system as

The equilibrium lattice coordinates are determined by the condition EI = 0, (which means the form linear over displacements to be equal to zero, i.e.

{#! > - 0 ) and take the form

The canonical transformation Eqs. 3-5 eliminates the translational degeneracy of the initial Hamiltonian, and, as a result, enables us to use the conventional perturbation theory. Using the standard methods of perturbations for the wave function o(r)i we obtain the wave equation determining the state of the quantum particle

In Eq. 6, E0 and Wo are the total and electron energies of the system in the zero approximation, U(r) is the self-consistent polaron potential. In the slow-velocity limit, we can evaluate the effective mass of the particle by expanding Eq. 9 over the velocity v

Thus, in the strong-coupling limit, the strong electron-medium interaction gives rise to an autolocalized electron state with the energy E0 the effective mass M, and the wave function determined by the solution of Eq. 9. 5

In turn, the state of the phonon subsystem in an adiabatic approximation corresponds to harmonica! vibrations of the nuclei moving in a parabolic well

where Mk\ is the quadratic matrix defined by the electron density distribution. In Sec. 4 we obtain a relation describing the matrix in terms of states of the electron subsystem.

Eqs. 8—11 completely determine the problem of a slow (nonrelativistic) motion of a particle in a quantum field in the strong-coupling limit. We note that the statement of the problem implies that the medium interacting with the electron is considered to be continual (we treat the effective mass, the

dependence of the coupling constant on the macroscopic medium parameters and so on). This assumption is correct when a characteristic dimension of the electron density distribution described by Eq. 9, greatly exceeds the lattice constant. Otherwise it is necessary to use a small polaroji treatment.23 In the general case, the characteristic dimension of the polaron is specified by the type of the electron-medium interaction. In the case of the local shortrange interaction, when c<. oc k ' and o/j. oc k, the polaron state is unstable in the strong-coupling limit,36'37 and if we do not impose special limitations caused by the discrete structure of the medium, the dimension of the polaron state will tend to zero. In the case of long-range coulomb interaction, when cfc oc k ~ * , (jj oc u0, the characteristic polaron dimension rp oc (/iw) 1 '' 2 a~ 1 can exceed greatly the lattice constant. Thus, if the particle-field interaction can be approximated in terms of the strong coupling, Eq. 9 is fundamental to describe the behavior of an excess electron in the medium.

S.2

Poiaron Hamiltonian for different types of the interaction

Hamiltonian Eq. 1 is widely used to treat the coupling of a quantum parti-

cle with a condensed medium. For instance, Hamiltonian Eq. 1 describes the motion of an electron in an ionic crystal if (see 1>34 )

where e is the electron charge, V is the volume, c = £~l = e^1 — e~l. The ^"-center model describing the interaction between an electron and

an ion vacancy is very important as an application of the polaron model in 6

solid-state physics. In this case we must add the coulomb vacancy attraction in the interaction potential

where z is the charge of the vacancy. In piezoelectric semiconductors '

'

'

where s is the sound velocity, c is the elastic constant, (e,--j.) is the averaged

square of the piezoelectric tenser. It leads to the same potential as in ionic crystals. In homopolar crystals 42 ' 43

where K and p, are the Young's and the shear moduli respectively, G is the

constant of the deformation potential. In the case of nuclear matter 14 > 44

where g is the constant of the nucleon-meson coupling, JJ,Q is the meson mass, c is the light velocity.

In the case of the continuum exciton 29'45

where up is the plasma frequency, e(k) is the dielectric constant of a doped semiconductor, /CD is the inverse Debye radius. In a number of cases, Hamiltonian Eq. 1 describes the behavior of electrons in magnetically ordered media. 24 > 46 7

3 3.1

States of the electron subsystem The ground slate

Eq. 9 is the nonlinear Shrodinger equation. It has been repeatedly investigated for the ground electron state for various self-consistent potentials. For the polaron Eq. 12 and the .F-center Eq. 13 the ground spherically symmetrical state Vo( r ) was found in 1 by the variation method. The best numerical solution for the ground state was obtained in 4 7 . The results of these calculations were confirmed in 4 8 . The solution of the equation with the short-range potential Eq. 15, corresponding to the ground state was actively investigated in nonlinear optics, ' in physics of disordered systems, 5 1 > 5 2 > 5 3 in nonlinear field theory 54 either by different variation methods or by numerical iterative calculations. 5S In 42 it was investigated by variation methods, as applied to the problem of an autolocalized electron state. The results were revised in 56 by the direct numerical calculation of the problem Eq. 15. The numerical calculations of the ground state for the self-consistent potential Eq. 17 are given in 57 ' 58 .

3.2

Excited states

The detailed study of the nonlinear Shrodinger equation shows that in the general case its solution is not unique, it has a discrete set of solutions, the proper self-consistent potential corresponds to each of them. This is most clearly cut for a .F-center, where the solutions can be put into correspondence to the solutions of the linear problem with the hydrogen-like potential. Then, the ground state corresponds to the la-type solution, the first excited state with spherical symmetry corresponds to the 2s-type, and the state asymmetric about rotation corresponds to the 2p-type and so on. We note that the excited states are less investigated than the ground state. It was proved in 59 ' 60 ' 61 that there is a countable number of solutions (modes) for the self-consistent polaron potential Eq. 12 and for the potential Eq. 15. The spherically symmetrical polaron states were numerically found in 6 , and the ones for the .F-center in 6 3 . In 5O ' 58 , the excited states were calculated for a homopolar crystal with the short-range potential Eq. 15. Some of the first solutions and the corresponding self-consistent potentials are shown in Fig. 2. Generally, for the (n + l)-st spherically symmetrical mode the solution intersects n times the x axis. Table 1 lists the numerical values of the radii, effective masses and total energies for various spherically symmetrical polaron states. Note that in es the quasiclassical asymptotic behavior (when 8

n —» oo) was found for the electron energy of the polaron:

The data in Table 1 suggest that the radius of the state grows rapidly with the number of the solution, while the energy and the effective mass decrease. Thus, to observe self-consistent states with the energy higher than the ground state energy, one must use crystals with a coupling constant much greater than a > 10 (a = e2ch~ v/i/2ftw). Typical values of the coupling constants for the majority of crystals are in the range a ~ 2-3. So, a = 3.97 in KC1, a = 2.0 in AgCl, a = 1.69 in AgBr, a = 0.85 in ZnO.2 For this reason the calculation of self-consistent states of bound polarons is more interesting to be compared with the experimental data.

Table 1: Energies (in units e*fj,c2/h ), radii (in units h /^tce 2 ) and effective masses (in units /*a4) for different self-consistent polaron states.

The self-consistent states have the same origin for different types of interactions considered in Sec. 2, but their physical characteristics can be distinct for various types of interactions. For instance, the total and the electron energies are negative in the case of the polaron in an ionic crystal, so they are below the bottom of the conduction band. In the case of an electron in homopolar crystals Eq. 15, the total energy of each self-consistent state is positive, while the electron energy (equal in absolute value to the total energy) is negative.

9

Figure 2: Wave functions j/(z) and self-consistent potential z(x) for the ground, first, second, and third excited polaron states. 10

Hence, in homopolar crystals the electron energy of the excited state is lower than the energy of the ground state (nodeless state). There is also a great difference between effective masses of self-consistent states for these two cases. In the case of an ionic crystal, the effective mass rapidly decreases with the number n. On the contrary, for a homopolar crystal the effective mass fast increases with the number. In the case of the nonlocal screened potential (deuteron problem), the existence of solutions to the nonlinear Shrodinger Eq. 16 depends on the value of the coupling constant g. The nodeless solution exists if values of g are greater than the critical value
where Yni(-&, 6) are the spherical harmonics. Table 2 lists the total energies of the excited polaron states obtained by some authors. Figure 3 presents the self-consistent polaron states of 2p- and

3p-type, numerically calculated in 7 1 ' 7 2 . As is seen from Fig. 3, the electron 11

density distribution has a peanut shape typical for the p-states. Some of the solutions were tabulated in 70 .

Table 2: Total energies (in units a fuj) for different polaron states.

The calculations also revealed a surprising fact that the solutions to the .F-center problem branch, and there exist the excited states with mixed symmetry. (Below we show that the formation of these states relates to phonon instability). The new solutions may be classified in terms of «- and ti-states.

Figure 3: Polaron wave functions for the self-consistent excited states of

2p- and 3p-type. 12

Figure 4 presents the electron density distribution for the a and d states. The sign " +" (\if>2\ > 103) indicates regions of the density condensation, while the sign " — " (IV" 2 1 < 103) corresponds to the regions with the low electron density. The hybrid states of 2s + 3d or 2s — 3d-type (depending on the sign

of the combination) result from the combination of these two states.

Figure 4: Electron density distribution for the self-consistent excited states with mixed symmetry 2s ± 3d.

These nonspherical states are similar in the shape to the hybrid 2s ± 3dstates of the linear Shrodinger equation with the Coulomb potential. The nonlinearity of Eq. 10 leads to the solutions, which are qualitatively different from solutions to a linear problem, especially for r —> 0. However, similarity to the linear problem remains in the shape and the asymptotic behavior for T —> oo, since asymptotics of linear and nonlinear problems only differ in

constant. 13

Nonlinear polaron-type equations have a remarkable feature: simple scaling transformations allow one to obtain an equivalent problem, but without the eigen value. When searching for a solution, we can introduce a parameter («r) into the latter problem. More precisely, in the right-hand side of our equations

we can write down f(r) • a (instead of 0), where f(r) is a certain prescribed function. In this problem, the procedure of continuation of the parameter
passing through the "reverse point" corresponding to confluence bifurcations of two solutions, the direction of motion along
Figure 5: Typical dependence of the squared norm of solution on parameter.

14

3.3

Application of the large polaron model

The model of the strong coupled polaron was the starting point to explain the absorption spectrum in the F-center problem.1 For a long time, the model was very attractive, since it allowed one to account for the qualitative correlation between spectrum parameters and dielectric constants of crystals. However, the subsequent cyclotron-resonance experiments revealed that the measured effective masses do not agree with the strongly coupled polaron model.2

Besides, according to the experiments on the electron spin resonance,

74

in the majority of ionic crystals the electron ground state has a characteristic dimension comparable with the lattice constant, and the large polaron model can hardly be used in this case. As a rule, to study the electron ground state in the F-center, the quantum-chemical calculations are used at present (see, for example, 75 ' 76 ). At the same time, most objections against the large polaron model disappear if we consider the excited electron states in the F-center. The effect of the relaxed excited state on the absorption process in F-centers is discussed in a lot of reviews. ™,rr,T8,79 The approximated solutions of Eq. 13 corresponding to the relaxed excited states of 2p-type obtained in 6 7 were used by Pekar 80 to describe the photoexcitation of F-centers, by Perlin81 to evaluate the life-time of excited states, by Moskalenko to describe the heat excitation. The exact solutions of Eq. 13 corresponding to the extended excited 2s-like states were used in to calculate the life-time of photoexcited states of the F-center and to evaluate the photoconductivity in crystals with color centers. The discussion of the nature of the relaxed excited state is still under way at present. ' This is due to the fact that in many cases experiments on photoabsorption can be described without any knowledge of the relaxation state itself. Thus, to explain the temperature dependence of the maximum of the F-center photoabsorption, there is no need for the electron wave functions, and the data on the so-called 5-factors, which can be used as fitting constants 84 are sufficient. Another problem arising in the application of the excited states is that there are a number of absorption lines corresponding to transitions between various self-consistent states, and these lines are usually wide and overlap. Each line can have a complicated shape due to the multiphonon nature of the transition, besides the main contribution in the transition can be caused by the local phonons (see Sec. 4). It follows that the question whether the large polaron model can be applied to the description of the relaxed excited state in the F-center is still to be answered.

15

The large polaron can also be used to treat an autolocalized electron state in a polar liquid (solvated electron). The absorption spectrum of the solvated electron is more simple than that of the /""-center and consists of one or several broaden lines. First attempts to consider the spectrum by the large polaron model 85 allowed one to account for details of the spectrum. However, subsequent ex-

periments and calculations have shown that the model should be modified to consider short-range forces with regard to the chemical structure of the environment (the size of the environment molecules, the coordination number, etc).

At present, there are well-developed semi-phenomenological methods based on the semicontinual polaron model (see review 86 ), which treat both the effect of the cavity formation caused by short-range interactions and the influence of polaron tails. These models enable one to explain a lot of the experimental data on the absorption spectrum (the effect of temperature, density and pressure on the width and maximum of an absorption line and so on). We note that though being simple the continual model of the strong coupled polaron is used to account for a number of solvated electron phenomena. Thus, the effect of salt concentrations on the absorption line was investigated by this model in 87. The generalization of the large polaron model to disordered systems was used in 88. This model enables one to explain a lot of experimental data on correlations between the thermodynamic behavior of the medium (density, pressure and the Kirkwood factor effect) and the shape of

the absorption spectrum. A special problem is the use of the large polaron model for the long-range transfer. We emphasize that the large polaron model is the starting point for

the theoretical study. 89 A vast amount of works is devoted to the question (see, for instance, '91' ). In recent years this question has taken on great significance due to the 'superexchange' problem. Available experimental data reveal that the rate of the electron transfer is anomalously great for globular proteins and for a number of organic molecules. The electron transfer is assumed to be realized by the excited electron states (see ). The nature of these states remains to be explored. 93 In 2 0 ' 2 2 , an attempt was made to explain the high rate of the electron transfer with the use of the large polaron model modified

for the protein globule.19 Figure 6 plots the electron density distribution in the protein globule, which is calculated by this model. According to the model, results of the calculations agree with experimental data.

16

Figure 6: Electron density distribution in the protein globule 20 ' 22 for the ground and first excited symmetrical states.

As a rule, the application of the polaron theory in semiconductors is limited by a weak electron-phonon coupling region (a < 1). The experimentally observed doublet structure results from accounting of the polaron effect in the absorption spectrum, when frequencies are in resonance with the optical vibrations of the semiconductor. The polaron effect is also very important to explain the behavior of charge carriers and other phenomena. The strong coupling case can be realized in heavily doped semiconductors. ' In a heavily doped semiconductor of the hole type, an electron is surrounded with a hole atmosphere much as a polaron in an ionic crystal. Similar phenomena happen to a hole in a semiconductor of n-type. The states of this type with the total charge equal to zero were called continuum excitons. 45 Figure 7 shows schematically the continual exciton in a semiconductor and 29 45

metal. 29,45

The ground and excited states of continuum excitons were investigated in Thggg arising continuum excitons narrow the gap in the semiconductor,

that explains the dependence of the gap on the impurity concentration.

17

The usual exciton states do not take place in metals due to the strong screening of an attractive electron-hole potential. The singularities in the Xray absorption spectrum (the so-called edge singularities) can result from the formation of the continuum excitons in metals. The positron annihilation in

metals can also be considered as the annihilation of the continuum excitons, where the positron plays the same role as a hole.

Figure 7: Continual exciton in a semiconductor (a) and metal (b), symbols "+" and "-" denote holes and electrons respectively, dashed region is the region of the exciton localization.

Intimate idea connection between the many-particle theory in solid state and nuclear physics enables us to use the developed methods in both cases.14'15'16 The basic problem in nuclear physics, i.e. that for a nucleon in a meson field is similar to the polaron problem, or electron-phonon interaction. The ultraviolet divergence in the use of the perturbation theory is the principal difficulty

arising in the simplest case of an interaction between a nonrelativistic nucleon and a meson field. A consistent treatment without any divergencies is carried out in

14

for

a strongly coupled nucleon in a meson field. In the case of two nucleons the problem is shown to be similar to the bipolaron problem, where Coulomb 18

repulsion is absent (see details in Sec. 5) and is reduced to the one-particle problem Eq. 9. The solution of nonlinear Eq. 16 gives the values of the bound

energy and the deuteron radius close to the experimental data. A new field, in which the concept of strongly coupled polaron has been intensively developed, is physics of clusters.94'95'96 The modern technology enables one to study the clusters from polar molecules, which contain a fixed

number of molecules.97 The theory of strongly coupled large polaron has been successfully used to describe experimental data on photoionization of clusters. It yields the correct estimation of the critical sizes of the clusters. 9 4 > 9 5 > 9 6 Figure 8 depicts two possible polaron states in clusters: the surface state, for which the electron is dominantly localized at interphase, and the internal state for which the electron is localized inside the cluster.

Figure 8: Internal (a) and surface (b) polaron states in a cluster. Arrows denote the direction of polarization, dots denote the electron density.

4 4-1

Localized phonons Spectrum of the phonon frequencies

We considered above the electron density behavior for the polaron-like states in the strong coupling limit. This electron density distribution can be responsible for a number of important effects and manifested in various experiments (see 19

Sec. 3). Equally significant is the fact that at the strong interaction not only the electron state is localized, but also the phonon subsystem state changes essentially.

The presence of the localized density distribution of an excess electron can be considered in terms of the lattice's dynamics as a special defect. As a result of the strong electron-phonon interaction, the phonon spectrum significantly changes, and the bound phonon states are formed 98 much as it takes place in an imperfect lattice. " While the localization of the latter is due to the changes in either the atom mass or the lattice force constant, the localization of the former is due to that of the electron state proper. In the case of translational symmetry, the localized phonons form a cloud which is coupled to an electron and moves together with it. Note that the bound phonon modes can also arise in the case of weak coupling, where they were originally studied and became known as dielectric modes. °° In the earlier theoretical and experimental works on this theme are reviewed. The strong coupling case is considered in 1O1. Here we present a short description of the results obtained. As was noted in Sec. 2, the characteristics of the phonon spectrum are described in the harmonic approximation and given by the matrix M^i- If we

introduce the eigen vectors A,^ and eigen values CD,, such as

then relation Eq. 11 for the phonon subsystem takes a diagonal form

where C« = J~^L A.,kQk are the normal vibrations of the lattice. Therefore the state of the phonon subsystem in the adiabatic approximation is described as a product of wave functions ©(£) = l I j Q j C C j ) and characterized by a set of quantum numbers [**i> 12> • • • i n »] f°r noninteracting phonons with renormalized frequencies ui,, whose squares are eigenvalues of the matrix M^iThe energy E^ is due to the second order correction of the total energy of the system and expressed as:

Thus, the state of the phonon subsystem is given by the matrix Mjtj and its eigenvalues u>*.

20

In the strong-coupling case the matrix Mki is denned by the electron subsystem state. Expressing the electron wave function of the first order in the form
Eq. 24 determines the local-phonon frequency spectrum as the eigenvalue spectrum of a linear integro-differential problem. This problem was studied for the electron ground state, 3 where a set of the first frequencies were calculated. For the polaron ground state Eq. 24 was investigated in 1O4 in the variation form. In 105 values of w, were found analytically for the one-dimensional problem. The influence of dispersion on the local phonon spectrum was considered in 106 . For excited spherically-symmetric states of the polaron and the ^"-center Eq. 24 was studied in 1O6 > lor . Table 3 lists dimensionless phonon frequencies for the ground and the first self-consistent polaron states. Figure 9 shows the phonon spectrum approximation.

Table 3. Squares of phonon frequencies ^(n, I) for the ground and the first

excited self-consistent polaron states.

21

Figure 9: Approximation of the local phonon spectrum for the ground (a) and the first (b) excited states of the .F-center in KC1. 22

4.2

Phonon instability

If all the eigenvalues of the matrix MM are positive, then the amplitudes of phonon oscillations are small (0(C«) oc exp (—aj,C 2 /2)) and the system remains stable. If any of the eigenvalues is small or negative, the amplitude in this

approximation becomes of the order of the crystal's dimension, and the system looses its stability. Thus the problem of stability with respect to the motion

of phonons implies the study of the eigenvalues of the matrix Mfcj. In 1 0 6 > l o r ) we found numerically that for excited states of the polaron and the .F-center there are critical values of the effective F-center charge vcr = Z/(e0c) — 0.21, when the square of renormalized frequency becomes negative for one or several modes, and the phonon instability of the system takes place. As is shown in 102 , this phonon instability is due to the bifurcation of solutions to nonlinear problem Eq. 13. In r2 Gabdoulline found the critical values i/, at which the self-consistent

states of Is =p 3d type appear in addition to the excited self-consistent 2s state. Figure 10 shows the dependencies of the square frequency w^, for the phonon mode n = 2, / = 3 and the electron energy of self-consistent states E on the

effective .F-center charge. The arising phonon instability is similar to the Jahn-Teller effect. For the polaron in addition to self-consistent states, each of them corresponds to its proper potential, there is an infinite set of non-selfconsistent electron levels available for each potential. Each of these levels is characterized by its
where (r') is the self-consistent solution defined by Eq. 8. If the non-selfconsistent electron levels are not degenerate, then, according to the standard perturbation theory, the first order correction X, can be presented by expanding over these functions

Substituting Eq. 26 into the frequency Eq. 24, we obtain the relation between the renormalized phonon frequencies and the spectrum of the electron states

While E, — EO < ^iiftfoifafo^ \ r —r' I" 1 )), i.e. electron levels are sufficiently close, the frequency square becomes negative UI^/UQ < 0. This results in the 23

instability leading to branching of the solutions. In that way, the self-consistent electron state has mixed symmetry of these two close levels. Figure 10 presents schematically the 2«- and 3e2-states and the arising states of the Is ± Sd-type.

Figure 10: Dependencies of square of the local phonon frequency (a) and the electron energy E (b) on v. Half-width of each line is a = 0.0025 u>0.

The study of the phonon instability of excited polaron states makes it clear that branching of solutions to Eq. 25 takes place as stability loses, that gives rise to a new self-consistent states with mixed symmetry. This effect is due to two main reasons. The polaron state is given by nonlinear Eq. 9, which includes two types of potentials: the self-consistent polarization potential and the Coulomb one. Symmetries of these two potentials are different. As the contribution of these two interactions changes quantitatively, the symmetry of some solutions changes. This effect is probably typical not only for the .F-center, but for some other similar quasiparticles (piezopolaron, fluctuon, etc.) bounded on impurities.

24

Note some important consequences of this phenomenon. Unlike problem Eq. 25, the stability problem Eq. 24 is linear. Thus it can be expected to be easier for calculations. From this standpoint, it would be reasonable first to study the linear problem, to find the symmetry of solutions and bifurcation points, and then to analyze self-consistent solutions of Eq. 25. The described instability may be observed experimentally. It will lead to decay of excited states for time l^"1!- Note that for LiCl (y = 0.35), LiBr (j/ = 0.29), and fluorides of alkali-earth metals the effective charge v = Z/(foc) is close to the critical charge i/cr = 0.21. Apparently, this explains the absence of luminescence 8 in experiments on photoexcitation in these crystals.

5 5.1

Strongly coupled bipolaron History of the bipolaron problem

We note in the Introduction that bipolarons provide another example when the strong coupling takes place. The intensive study given to bipolarons in recent years is due not only to general theoretical interest in the problem but also to its important applications, such as a treatment of high-temperature superconductivity. 1O9'110 The history of bipolaron study is rather dramatic. For a long time, calculation errors have been calling into the question of the possibility of bipolaron existence. The problem was first recognized by Ogg, 11J who suggested the superconductivity at temperatures higher than the boiling point of nitrogen, when he observed the abnormally high conductivity in metal-ammonia solutions, (as early as forty years before the discovery of high-temperature superconductivity in metal-oxide ceramics 112 ). At present there is a reliable experimental evidence that electron spinpairing states are dominant in metal-ammonia solutions at concentrations about 0.1 promille. 113>114 Some authors relate this to the bipolaron formation. But the mechanism of the phenomenon is not fully understood yet. In recent years, some works have been done,115'116'117'118 which have revealed the bipolaron states at this concentration with the use of the Kohn-Sham theory and molecular dynamics methods, and studied the characteristics of these states (electron density distribution, effective potential, etc.). The bipolaron problem was first considered theoretically in 1951. 11 In the description of the bipolaron state, the starting point is the Pekar-Frohlich Hamiltonian of two electrons interacting with a phonon field

25

where r^ and r 2 are the coordinates of the first and the second electrons respectively, ro is an arbitrary reference point, U(r) is the electron interaction potential. It is usually assumed that r0 = 0, and in this form the Hamiltonian of Eq. 28 is initial in the bipolaron study. The physical meaning of the varibale ro is discussed in Chapter 3 of this book. Next most authors turn to variational calculations of bipolaron wave functions. In the semiclassic bipolaron theory, 1 the wave functions were chosen in the multiplicative form, which failed to find the bound bipolaron state. It was Vinetsky and Gitterman who first obtained the bound bipolaron state in the frame of the semiclassic treatment. Best evaluations of the bound bipolaron energy were obtained in , where the trial wave functions were chosen with regard to the electron correlation. In particular, the condition of the bipolaron state formation was obtained 77 > 0.14, where rj = e^ /eo was the ion coupling parameter. The current state of research is presented in reviews. 122 > 123 The treatment extended to the case of the presence of short-range interactions is given in 124 ' 125 .

Note that exact solutions have not been obtained in the bipolaron theory by now in contrast with the polaron theory, where asymptotically exact solutions are known in the limits of both the weak and the strong coupling. Moreover, in the bipolaron case there are no solutions at all for small and intermediate values of the coupling constant a. According to, 12a the bound polaron state is possible only at sufficiently large values of the coupling (a > 5, 2). In the adiabatic limit, both the electrons are believed to move in a single potential well, induced by their fast oscillations. For this reason the interaction of the electrons with the polarization (ri,r2) takes the form:

As a result, the problem is reduced to the calculation of a two-particle bipolaron wave function in a self-consistent potential. Since there are presently no methods to find asymptotically exact solutions of the problem, some additional assumptions are needed. As is mentioned above, the form of the approximation of the wave function can significantly affect the result. 26

5.2

Adiabatic theory of the bipolaron

An alternative approach can be exemplified by 127, where the BogolubovTyablikov method '31 was used to develop a consistent adiabatic translational invariant theory. Following this approximation the bipolaron motion is separated in the adiabatic limit, and the translational motion of the centre of the bipolaron mass is presented as a plane wave. Relative electron coordinates describe fast oscillations in a potential well, which has the form of the electron effective interaction:

The potential well is not fixed in space, but follows adiabatically the centre of the electron mass. The interaction Eq. 30 is clearly translational-invariant unlike the usual phenomenological approach Eq. 29, which does not possess translational invariance with a spatially fixed potential well. Within this approach we can treat the problem in a way similar to Sec.,2 of the paper. As a result, the problem becomes one-particle and is reduced to the study of the Shrodinger equation for the relative motion of the electron pair:

Then, we determine the bipolaron mass similarly to Eq. 10:

5.3

Results of calculations

In 127 calculations were performed for an ion crystal (uk = ui, cj. oc k~l), and the interaction U(r) between two electrons was the Coulomb repulsion screened by the high-frequency dielectric permittivity:

The solutions obtained depend significantly on the parameter ae = 0.125(1 — e

oo/ e o)- Figure 11 shows the particle-like solutions of the boundary problem

27

Eqs. 31 and 32 for some as values. It is evident from the figure, that the probability of the electrons occurrence at the same point decreases as ae grows, while the maximum of the electron density distribution displaces to the right and goes to infinity at the critical value aecr = 0-5. This results from the fact that at sufficiently large r the asymptotic of the potential Eqs. 31 and 32 has the form:

Figure 11: Quasiparticle solutions of Eq. 32 at various ae. The localized solution of Eq. 31 exists only on the condition that the righthand side of Eq. 35 is positive, i.e. for as < 0.5. Then, we have 7jCr = 0.75, for the ion coupling parameter r) = «oo/eo = (8ae — l)/(8ae), that greatly exceeds the value of this parameter obtained by the phenomenological theory. The critical value of the parameter rjs, at which the bipolaron is stable, is given by the condition of energetic advantage of the bipolaron state with respect to its decay into two independent polaron states

28

where Epo\ is the energy of a single polaron state. Calculations show that inequality Eq. 36 is valid if r\ < j/ s , where rj, = 0.56. Table 4 lists values of parameters for crystals satisfying the condition 77 < TJS and the calculated values of bipolaron energies, radii, effective masses. When the experimental data on the effective electron mass ra is not available, we present results depending only on the ratio m/mo, where mo is the mass of the free electron.

Table 4: Values * of energies W, total energies E, radii R and effective masses of bipolarons M.

The condition of the strong adiabatic coupling is that the frequency of the electron oscillations in the polaron well should be much greater than the

29

frequency of lattice oscillations. It follows from Table 4 that this condition is met for reasonable values of the electron effective mass. Thus, while a single polaron meets the condition of weak or intermediate coupling, a bipolaron follows the strong coupling condition in the case of the studied crystals. This enables us to evaluate critical values of electron-phonon coupling constants a,, when the bound bipolaron state is possible. It follows that the energetic requirement for the bipolaron formation is

Since E oc a2, the condition Eq. 37 allows us to estimate values of critical coupling constants. Table 5 lists these values.

Table 5: Critical values of electron-phonon coupling constants a8 for different values of r>.

Note that critical values of coupling constants, obtained from the numerical solution to the bipolaron equations, are much smaller than those evaluated by trial variational functions. Thus, according to l28 , the critical values are as w 5.4 for 77 = 0, as ss 7.2 for 77 = 0.1, they appear to be three times as large as those obtained from the exact solution to the bipolaron problem. Note also that at the polaron-bipolaron transition the solution symmetry changes, which can lead to significant rearrangement of the local phonon spectrum near the critical value of the parameter 77 due to the phonon instability

of the solution in the same way as it takes place in polaron excited states (see Sec. 4). Some of the results obtained can be easily extended to the two-dimensional case. The two-dimensional bipolaron problem has attracted much interest with the discovery of high-temperature superconductivity.

The simplest model used to describe polarons in three-dimensional space is obtained from Eq. 12 with cj, replaced by:

30

The physical parameters (frequencies, dielectric constants, effective masses) were assumed to be the same as in the three-dimensional case. The dependence of 0^,0 on the wave vector is chosen from the requirement that the electron-polarization interaction should be the Coulomb form (as 1/r) in the £>-dimensional case. The numerical factor in Eq. 38 is given by the condition that Ck,D corresponds to cj. at D = 3. To evaluate the energy and the critical constants in the two-dimensional case, the estimates obtained in by the Gauss approximation can be used. These estimates relate bipolaron energies in three- and two-dimensional cases. As a result in the two-dimensional case, we express the bipolaron energy as

where .EaD.bipol is tne bipolaron energy in the three-dimensional case. Hence, aejX) = 8630, 772.0 = TlaD • The critical value of the electron-phonon coupling constant is derived similarly and in the two-dimensional case takes the form:

where values of a, are listed in Table 5. The translational invariant bipolaron theory presented above yields results qualitatively different from those, obtained by the conventional adiabatic method. According to 127, the situation looks as follows. In the bipolaron formation, the electrons are localized in a deep potential well with the electron excitation energy W ~ 1 eV. This energy remains the same up to the critical value of the parameter r)s = 0.56, at which the bipolaron state decays into independent polaron ones. Up to the critical values 77 = 77, the frequency of the electron oscillations in the bipolaron potential well greatly exceeds the frequency of the lattice oscillations, and we can use the adiabatic approximation. The adiabatic criterion fails only for the crystals with very small electron-phonon coupling constants, such as PbSe (a = 0.215), and PbS (a = 0.317), where 77 < 77,, i.e. bipolaron states are conceptually possible. Bipolaron characteristics are best presented in crystals TIBr and T1C1, where continual approximations are fulfilled well; the radii of states are 20 Aand 16 Arespectively. The adiabatic condition is also met with a great safety, despite a relatively small constant a w 2.5. It is significant that in all the cases listed in Table 4, there is a great difference between the electron energy of bipolaron W and the total energy E. The absolute value of the electron energy of the bipolaron is approximately 5 31

times as great as that of the total energy, while for a single polaron in the strong coupling limit this ratio is equal to 3. This distinction can lead, particularly, to a great difference between the energies of photo- and thermodissociation. In some crystals the criterion of stable bipolaron formation is at bound of accuracy. Thus, in Rbl and KI the bipolaron is stable at room temperature and instable at helium temperature. Therefore, in Rbl and KI the cooling from room temperature to helium one leads to the bipolaron dissociation. This phenomenon could be observed on absorption spectra, changes in mobility and cyclotron frequency, etc.

Note also that the considered method for separating out the translational invariant part of the motion and reducing a two-particle problem to the nonlinear Shrodinger equation is rather universal and can be applied to any two-particle problem in the strong-coupling limit, such as biexcitons, electronhole pair, and so on.

6

Conclusions

Despite the long history of the polaron study the interest to the problem of the strongly coupled polaron does not decrease. It seems to be due to the role, which the polaron plays in physics of particle-field interaction. Unlike many other quasiparticles (phonons, magnons, plasmons, etc.) given by the spectrum and type of linear excitations of the system, the strongly

coupled polaron is a 'nonlinear' quasiparticle formed as a result of a nonlinear self-consistent interaction. This, in turn, requires a nonstandard mathematics. 66,132

In our view, the potentialities of modern mathematical methods are little used in the polaron problem. The above example demonstrates that even the generalization of the well-known Bogolubov-Tyablikov method to the case of two-particle self-consistent states yields new results different significantly from the usual variational calculations. The above results show that strongly coupled large polaron possesses a complicated internal structure manifested in a wide variety of self-consistent states for the electron subsystem and in the local phonon spectrum. The peculiar behavior of the local phonon spectrum is closely related to the topology of self-consistent states of the electron subsystem and given by the symmetry of the self-consistent potential in the nonlinear Shrodinger equation. Note that the investigation of such equations has much to do with the development of numerical methods for the solution of nonlinear boundary problems. It seems likely that new interesting results in this field can be obtained with the active use of high-performance parallel computers. 32

The foregoing proves that the experimental test of the possible existence of the strongly coupled large polaron and bipolaron states is rather ambiguous. However, in our opinion, dramatic qualitative results of the theory (such as, the availability of excited self-consistent states, or phonon instability) can be revealed in specially designed experiments.

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112. J. Bednorz and K.A. Muller, Usp. Fiz. Nauk 156 (2), 323 (1988).

113. N. Nicoloso and W. Freyland, J. Phys. Chem. 87, 1997 (1987). 114. Y. Nakamura, J. de Phys. IV 1, C5 (1991).

115. A. Seloni et al, J. Phys. Chem. 91, 4947 (1987). 116. E.S.Fois et al, J. Phys. Chem. 92, 3268 (1988). 117. E.S.Fois et al, Phys. Rev. B 39, 4812 (1989).

118. G.J. Martyna et al, J. Chem. Phys. 98, 555 (1993). 119. S.I. Pekar and O.F. Tomasevich, Zh. Eksp.

Tear.

Fiz. 21 (11), 1219

(1951). 120. V.L. Vinetskii and V.L. Gitterman, Zh. Eksp.

(1957). 36

Tear. Fiz. 33 (3), 730

121. S.G.Suprun and B.Ya. Moizhes, Fiz. Tverd. Tela 24 (5), 1571 (1982). 122. V.I. Vinitskij et al, Ukr. Fiz. Zh. 37 (1), 76 (1992). 123. M.A. Smondyrev and V.M. Fomin in Polarons & Applications, ed. V.D. Lakhno (Wiley, Chisheter, Wiley, 1994). 124. D. Emin and M.S. Hilery, Phys. Rev. B 39, 6575 (1989). 125. H. Hiramoto and Y. Toyozawa, J. Phys. Soc. Jpn 54, 245 (1985). 126. J. Adamowski and S. Bednarek, J. Phys. Condens. Matter. 4, 2845 (1992). 127. V.D. Lakhno, (Preprint Pushchino, Pushchino Research Center, 1993); Phys. Rev. B 51, 3512 (1995). 128. G. ladonisi et al, Phys. Status Solidi (t) 153, 611 (1989). 129. G. Verbist et al, Solid State Commun. 76, 1005 (1990); Phys. Rev. B 43, 2712 (1991). 130. J. Devreese in Polarons & Applications, ed. V.D. Lakhno (Wiley, Chisheter, 1994). 131. P.M. Peeters in Polarons & Applications, ed. V.D. Lakhno (Wiley, Chisheter, 1994). 132. V.E. Zaharov et al in Soliton theory: method of inverse problem (Nauka, Moscow, 1980) [in Russian].

37

THEORETICAL INVESTIGATION OF THE SELF-TRAPPED HOLE IN ALKALI HALIDES. LONG-RANGE EFFECTS WITHIN THE MODEL HAMILTONIAN APPROACH L.N. KANTOROVICH Laboratory of Med.-Phys. Problems, Latvian Medical Academy,

Dzirciema str.16, Riga LV1007, Latvia

P.B. ZAPOL Surface Physics Laboratory, Institute of Physics of Latvian Academy of Sciences,

Salaspils, LVZ169, Latvia A simple small-radiua polaron model of the self-trapped hole (Yk~ center) in alkali halide crystals is presented. Along with the usual contributions like the internal hole part and the long-range hole lattice interaction, used in the Hamiltonian

adopted in this study, the electronic polarization is also carefully included. The latter is treated within the framework of Toyozawa's electronic-polaron model. It is shown that the exact solution of the problem w i t h i n the adopted here LandauPekar approximation really leads to multi-hole q u a n t u m states accompanied by the relevant electronic and lattice polarizations. While solving our equations, the hole wave f u n c t i o n was assumed to be spread over a comparatively large local region of the crystal. The local (^3/1) symmetry of the defect is taken into account to simplify the solution of the corresponding non-linear equations. As an example the KCl crystal is considered, for which the Vfc-center electronic and spatial s t r u c t u r e as well as the self-trapping energy are computed. The results obtained demonstrate quite evidently the importance of taking into account a cAemtcafinteraction within the Vfc-center.

1

Introduction

It was found both experimentally and theoretically 1 that the hole prefers to be self-trapped in alkali halides. The self-trapped hole (STH) represents a point defect in a crystal and is mostly shared between two adjacent anions along the (110) direction forming a quasimolecule X^ , (X stands for the halogen ion). Questions arise why the hole prefers to be self-trapped in the two-site form

in the alkali halides and why the STH is energetically more favorable than a free hole having almost negligible coupling with lattice distortions. To answer these fundamental questions it is necessary to calculate the hole self-trapping (ST) energy. But in order to define the notion of the ST energy more carefully,

the consideration of the STH structure is required. Due to the periodic symmetry of the crystal lattice, all the anions of the 38

perfect crystal are equivalent as traps for a free hole, and the wave functions of both the free (F) hole and STH have the usual Bloch-like form. 2 ' 3 As a result, in both the states we deal with bands of energetic levels having widths Bp and BSTi respectively. As was explained in 4 , at least in the case of alkali halides the

STH can be simulated by a localized state, using a localized variational ansatz for the hole wave function, and, in addition, the STH band can be substituted by a single level. Thus, the Vjt-center can be treated by any method appropriate for point defect calculations. Therefore, the hole ST energy is defined as the energy difference between the middle of the STH band chosen as a single level (narrow STH bands are assumed) and the bottom of the free hole band coinciding in the case of the

alkali halides with the point k = 0 in the Brillouin zone (BZ). 4 ' 5 A number of various theoretical approaches have been developed to study the electronic and spatial structure of the V^-center in alkali halides: atom-atom potentials method, 6 semiclassical static-lattice method, 7 quantum-chemical (QC) methods, based on the molecular cluster approach, both semiempirical4'5'8'9 and ab

«7it
In the series of papers4'5'13'14 a new method has been proposed to calculate the hole ST energy based on the combination of the QC and the model Hamiltonian ( M H ) methods. However, this approach is inconsistent one, since

it contains a number of approximations (the use of different Hamiltonians along some artificial path of the "self-trapping procedure", the introduction of nonstationary intermediate states, etc.) which affect considerably the method precision.

Another approach has been developed in a number of papers15"18 within the framework of the MH method alone. Using the same Hamiltonian of the crystal and taking into account the interaction of the hole with lattice vibrations, the authors calculated the hole ST energy. The results obtained in these papers confirmed the two-site shape of the STH in alkali halides. Besides, the calculated STH structure turned out to agree satisfactorily with experiment. Nevertheless, approximations made there in calculating the STH ground state were quite crude. Indeed, one varying parameter (Aa) was introduced directly

to the hole Hamiltonian. This made it possible to obtain very good fit to the experimental absorption spectrum. However, this parameter was introduced in such a way that it could not be used as a variational one at all. Indeed,

being introduced linearly into the Hamiltonian, it permits one to obtain any desirable results for the hole energies, because its real value (even approxi39

mately) is not known. That is why the authors15 ls obtained quite reasonable results for the hole ST energy even without taking into account the short-range effects associated with the chemical interaction between two Cl atoms of the Cly -molecule. It is a nonsense however, because it was shown1'4'8'19 that just

this interaction mostly causes the hole self-trapping in alkali halides. From our viewpoint this is the main disadvantage of the method. Besides, the inertialless (electronic) polarization was missed there, although it plays a significant role in the evaluation of the hole ST energy.4 At last, the hole wave function

was chosen in the simplest form as the 1 or 2-site hole only (though less severe restrictions were made in 18 ). Therefore, real variational calculations showing the distribution of the hole in the crystal have not been possible by means of the MH approach so far. In this papers we suggest a more rigorous formulation of the MH approach. Our aim is to build up such a formalism on the basis of an appropriate MH, which would be as close to QC methods as possible and would give their quality of description of the local defect region. At the same time, all longrange interactions would be still properly taken into account giving a significant advantage of the method in comparison with usual QC methods. Besides, a problem of the so-called boundary conditions1' >21 inherent to most of the QC methods exploiting the molecular cluster model in treating point defects in crystals does not arise if one uses the MH approach. In contrast to the previous papers in which basic functions (like atomic orbitals, AOs) were ignored completely and unique adjustable parameter was inserted directly into the MH, we have used a set of AOs centered on the crystal sites. Following the philosophy of the quantum chemistry, parameters of the AOs together with corresponding pseudopotentials of the crystal atoms are used as adjustable parameters of the theory. Then, we have not restricted in our method the

variational freedom of the hole wave function, allowing it to be spread over a comparatively large crystal region. In order to make our formalism more

feasible, the complete implementation of the STH point symmetry has been made. Besides, we have taken into account the electronic polarization of the crystal produced by the hole Coulomb field. At last, a more general form of the internal hole Hamiltonian has been used in which a real structure of the

perfect crystal valence band (responsible for the STH electronic structure) was included. In the present paper we give a detailed description of the general formalism including symmetry consideration, absorption spectrum, ions displacements and the STH electronic structure as well. As an example the STH in the KGl

crystal has been considered. For this purpose the simplified form of the holelattice interaction was used. For the sake of simplicity the short-range effects 40

are neglected in this paper. It is postponed for the proceeding papers. The plan of the paper is as follows. In Sec. 2 we consider our general formalism. Frenkel-type excitons are used to simulate the electronic polarization effects within the electronic polaron model of Toyozawa.22"25 In Sec. 3 we show how the point symmetry of the STH allows one to simplify the formalism substantially, making it as feasible as possible. In the next Sections expressions for both the hole absorption spectrum and the ions displacements are derived. A simple model illustrating the theory is analyzed in Sec. 6, where the calculation of the ST energy of the hole is presented. A short discussion and relevant conclusions are given in Sec. 7.

2 &. 1

General formalism for the STH electronic structure The model Hamiltonian

The simplest MH for the hole was used in ' involving only p-like states of an ions. Instead, we suggest to use a general hole Hamiltonian in the form

where F =|| f-p

•*, , || represents the exact Fock matrix of the perfect crystal

in the lattice-site representation; L, L' are lattice vectors, fj,t fit denote occupied AOs centered in the unit cells (UCs) attached to the lattice sites L, I/, respectively;
(CRYSTAL) one.27 The hole-lattice interaction was described in15"1 using a simple model based mainly on the long-range approximation. However, it is not difficult to implement the general expression28 for this part /T^_ p / l of the total Hamiltonian, though a complete knowledge of the perfect crystal lattice dynamics (both eigenvectors and eigenvalues of the lattice dynamic problem) is necessary. Indeed, expanding, as usual, the ion potential, VT (r) = v, (r — L, — u(L,)), produced by the ion of the sort s (= anion, cation) located in the UC L at the point li,, into series over the displacements u(L,) up to the linear term only,

we obtain the following expression for this part of the total MH

41

Here q is a vector in the first BZ, feiy and iq;- are the operators of the phonon creation and annihilation belonging to the branch j, respectively. The factors Vq* in Eq. 2 are

where ve is the UC volume (vc = 2a3 for alkali halides, a being the nearest neighbors distance), e'(qj') and wy(q) are the eigenvectors and eigenvalues

(phonon frequencies) of the perfect crystal lattice dynamic problem, M, is the mass of the 8-type ion, N is the number of the UCs in the main region of

the crystal. At last, v,q is the Fourier image of the full ion potential V,(T) containing in the general case, along with the usual Coulomb term, the shortrange contribution as well. As was already mentioned in the Introduction, the electronic correlation plays a significant role in the self-trapping process. Therefore, we have used the electronic polaron model ~~ in order to describe this kind of effects in terms of the electronic polarization produced by the localized Frenkel-type excitons.

The following contribution to the Hamiltonian responsible for the interaction of the hole with the electronic polarization is used here 24 ' 25

where the operators (7q,, and (?„"„ describe the annihilation and the creation of the i/-type exciton with the wave vector q. It is seen, that the interactions of the hole with both the lattice phonons and the electronic polarization are represented by similar Hamiltonians. The only important difference is in the form of the factor Vq* used in Eq. 4. In the long-wave approximation this factor has the following form 25

where £i,(q) is the exciton energy (it is close to the forbidden gap width Eg) while EOQ is the high-frequency dielectric constant of the crystal, nex the number of the exciton branches involved in the hole-polarization interaction («/ = 1, ..., nex). Completing our Hamiltonian, we must add the usual terms describing the

internal energies of the systems of phonons and excitons to above written 42

contributions

Finally, our Hamiltonian describing the hole interacting with crystal vibrations (including the inertial polarization) and the electronic polarization can be represented as

It is necessary to make some important statements here. First we would like to emphasize that in the above consideration an excitonphonon interaction has been neglected. It means that the alteration of the electronic correlation of the crystal caused by the displacements of ions from their ideal lattice positions is omitted here. This effect is believed to be small comparing with other contributions to the Hamiltonian Eq. 8.

Let us make some comments on the notion of "model Hamiltonian approach" itself. It should be considered in the same manner as semiempirical methods, 30 widely used in quantum chemistry to simplify calculations of the electronic structure of molecules. From this point of view there is no longer

need in a completely non-empirical calculation scheme based on the Hamiltonian Eq. 8. A semiempirical model seems to be more reasonable. 2.2

Basic equations of the method

It is known that an exact solution to the Hamiltonian of the form of Eq. 8 has not been found yet. However, the STH state is considered here as the defect-like one. Therefore, it is sufficient to implement such a method in the framework of which both lattice and electronic polarizations are adequately treated. It was shown in15"17 that the Landau-Pekar approximation appeared suitable for this purpose. Using the Landau-Pekar approximation, we have the total wave function of the crystal containing the STH,

where 9h(R), Xph(Q)

an

d Xcx(&)

are

wave functions of the hole, phonon and

exciton subsystems, respectively, depending only on their own variables. Applying the usual variational procedure as in of 15 , we arrive at the following set 43

of self-consistent equations for these wave functions

Note that the multiplicative ansatz used in Eq. 9 represents a zero-order perturbation term of the Born-Oppenheimer approximation. It is valid only in the case of strongly separated electronic states. Indeed, the function Eq. 9 approximately describes the electron-phonon system in the vicinity of the adiabatic potential energy surface (APES) minima.31 If we have a degenerate electronic state or a state lying nearby others in the energy scale (it means that several APESs are crossed or are close to each other at the point R), the coupling of

vibronic states must be taken into account terms is necessary.

and the sum of the multiplicative

The total energy of the crystal containing the STH is given by

The main advantage of the MH method is in the fact that the last two Eq. 10 can be solved exactly15'16' 2 since these equations coincide with those for displaced quantum oscillators

Here the last equation represents the energy of the phonons, while the STH relaxation energy associated with the hole-phonon interaction is given by

44

and in a similar way,

where the energy Eex of free excitons and the STH relaxation energy, E**lt associated with the electronic polarization, are

In these equations \0)ph and |0)es represent the phonon and the exciton vacuum states (no phonons and no excitons), whereas x°fc = Xph({n(lj}) and X°x = X°x ({"iq,/ }) are the wave functions of free oscillators (belonging to undisplaced phonons and excitons, respectively); nqy = 0, 1, 2, ... and mqv = 0 . 1 being the corresponding occupation numbers of phonons qj and excitons qi/. Note, that the solutions obtained depend on the hole wave function 9^ through the exponential factors f£ = f^[9h] and /q* = /q* [*fc]. This fact reflects the self-consistent character of the set of Eqs. 10: both the displacements of

the crystal ions and their electronic shells "feel" the detailed distribution of the hole wave function, W^, and are changed properly under its the action of field. The self-consistent character of the solution evident for the small polaron problem itseli was ignored in ~~ where 9^ was used in the simplest form with the hole equally shared between two nearest anions only. Substituting the solutions to Eqs. 12 and 13 to the matrix elements in the left side of the first equation of Eq. 10, we obtain the unique equation for the hole wave function in the self-consistent field of both ionic and electronic polarizations

45

where a special notation is used for the sum over the BZ of the product of the plane-wave matrix elements

All the quantities of Eq. 15 are known and their q-dependence is already denned (the detailed form of the i>,q in Eq. 3 depends on the model chosen for the ionhole interaction). Besides, the plane-wave matrix elements can be calculated analytically on the basis of gaussian-type AOs.34 At last, as can be easily seen from Eqs. 3 and 5, the expression in the square brackets in Eq. 15 turns to be proportional to \/N (since both Vq- and V^? contain N~ ' ). Therefore, the

integral over the BZ in Eq. 15 can be calculated numerically using the specialpoints method. 35 A singularity of the type 1/q2 arising at the point q = 0 can be avoided in the manner proposed in . The total energy, Eq. 11, can also be rewritten using the solutions to Eqs. 12 and 13 for the phonon and exciton problems as follows

The second equation of this set is equivalent to the first one, and can be obtained by the definitions of the STH relaxation energies (see Eqs. 12 and 13) Thus the total wave function of the STH, Eq. 9, contains the mean occupation numbers {nqy } and {mq,,}. The latter numbers give actually a set of the hole states, lying above the electronic state in question described by the function ^/h with mq,, = 0. All the other states having nonzero values of { m qj/} are separated from this state by the energy AE ~ Eg £}„„ m qi/> where Eg is the crystal forbidden gap width (typically 10-15 eV for alkali halides). Therefore, considering the lowest solutions of Eq. 14, we can restrict ourselves to the states in which x°x coincides with the exciton vacuum |0)ea. and put all fiqi/ equal to zero. Eq. 14 depends non-linearly on the hole wave function 9^. This nonlinearity is usual for the polaron problem33'35 and comes from the hole-polarization interaction which is taken into account in the self-consistent way. However, in our model this problem does not run into obstacles since the effective hole 46

Hamiltonian Hc?* remains quadratic over the hole operators and depends on

the ~9k itself only through the matrix elements {'Phldi

/^R'v
are c-numbers. Therefore, our effective Hamiltonian Hh exactly.

can be diagonalized

Indeed, making use of the unitary transformation U =|| Ua ^

|| = || E/l

||

we come to a new set of the hole operators

In this new set of operators the Hamiltonian H^ diagonal form

where w^* F =|| Fj^

is an eigenvalue and U' Q ) =|| J/i

given by Eq. 14 has the

|| is an eigenvector of the matrix

£, , || with the elements

and Pa = 0.1 are the quantum numbers. It was assumed here that the AOs

create an orthonormalized set of functions (like the Wannier orbitals) though generalization to the case of overlapping AOs is straightforward.20 Thus, the general solution of the Hamiltonian Eq. 18 can be written as

and the corresponding eigenvalues are

Noteworthy is a complete analogy of the eigenvector and eigenvalue equation, which is solved here,

47

with the usual Hartree-Fock self-consistent field equations. In both cases we have an eigenvalue problem for a matrix like that given by Eq. 19, where the summation over all eigenvectors corresponding to the occupied states (Pa = 1) is really set up. Besides, as it is evident from Eq. 21, the A^ standing for "the hole energy" in the first equation of Eq. 10 is also an equal sum over the "occupied" eigenvalues. At last, the quantities of Eq. 15 play in Eq. 19 the role of the difference between the Coulomb and exchange integrals in the usual Fock matrix while the multiplier in the round brackets in Eq. 19 is an analogy of the bond-order matrix. Now let us discuss the general form of the wave function given by Eq. 20. As usual, the vacuum state |0)fc represents the Slater determinant built up from all occupied states of the perfect crystal. Therefore, the state iP^ appears to be a multi-hole state of the crystal; the actual number of the holes coincides with the number of the non-zero Pa-values, which are the quantum numbers attached to the ty ^ = ^lh{P\ > Pit •••)• The indices <TI,O~?,... run over both the spin values (up and down) while their number coincides with that of holes. The summations over spins
So, the index ao enumerates different one-particle hole states of the crystal. The coefficients £/l standing in Eq. 24 show relative weights of the hole Jj/i distributed over a certain local region in the crystal around the origin (in the zero UC, L = 0). For the sake of simplicity we shall write the superscript a instead of ao in the further consideration. The matrix F =|| F-^ -j^, , ||, Eq. 19, depends only on the product of the eigenvectors associated with the ot-th solution of the problem Eq. 22 since only one term is conserved in the summation over a in Eq. 19

In the second equation of this set, we have supplied the F matrix by the 48

additional superscript a in order to point out the non-linear character of this equation which must be actually solved (cf. Eq. 22). Another important point of Eq. 25 is that the vector U^ a ' =11 E/J II has Inn the unit length

providing the correct normalization {^h I'S'fc) = 1 of the hole wave function IP;,,

Eq. 24. In this respect it is worth noting that in the general case different vectors U'01' are not orthogonal to each other. Indeed, using the second equation of Eq. 25 one obtains a set of solutions, the number of which is equal to the

dimension of the matrix F' a '. All these solutions are orthogonal to each other due to the unitary of the matrix U ( a ) =|| Ua L f j || = || J/' a) ||. However, only one solution having nonvanishing Pa = 1 is chosen and is marked here by the index o; = CXQ. Different solutions of the problem Eq. 25 (if any) arising due to its nonlinearity belong to different U' a '-matrices and must not be orthogonal to each other. Therefore, in the general case one has to solve Eqs. 25 search-

ing for all their non-equivalent solutions. The most feasible (in our opinion) numerical procedure allowing us to solve this problem will be discussed later while considering the results of our model calculations (see Sec. 7). It is evident that the complexity of the problem Eq. 25 grows drastically as the dimension of the F'™'-matrix increases. However, in actual calculations of the STH electronic structure the localized character of the hole can be used. It means that the number of nonvanishing components U^. belonging to any JL/i

solution can be actually reduced to a local region in which the hole is assumed to be really localized. At the end of this Section let us give the expression for the total energy of the a-th state (see first equation of Eq. 16) of the STH

(all raq,, = 0 as was adopted here). Besides, another expression arising from the second equation of Eq. 16 is also useful

49

where the relaxation energies are given by

where indices "ex"

and "ph"

nearby the square brackets mean that the

corresponding contribution from the excitons and the phonon, respectively, should be used in calculating these integrals by Eq. 15.

3

Implementation of the system point symmetry

If the symmetry of the STH (the point group G) is known it can simplify substantially the search for the solutions of Eqs. 25. Indeed in this case the

number of independent components of the vector U'01' =11 Z7.1 II *8 less than L./4 its total dimension; the latter is equal to the number of AOs |L/x) used within the local region. Since only non-degenerate states are considered here, we can restrict the present consideration only to the states associated with onedimensional irreducible representations (IRs) of the group G. For this purpose let us investigate transformation properties of the functions dj |0)fc = v'L <,' They are the Slater determinants deprived of one |LpL a transformed in the way which is complementary to the transformation rule for the one-particle orbitals |L/iu} transforming according to usual rules well-known in quantum chemistry (see, for example 26 ' 37 ). To clarify this point a new notation for the AOs |Ljt
the chain £ by A € £, we have the correspondence: |L/iff) —> \£Amo-), where m = —I, ...,l and / are the AO magnetic and orbital quantum numbers. All AO's

belonging to the same chain £ have the same / (but

all possible m). Thus

acting on any AO \£Amo-) with an element g & G we have

50

while the corresponding complementary transformation

is valid for the (2M — l)-electron hole functions V>(AH
=

dtA

a\Q}k-

Here

(1)

D (a) =|| D$n(g) || is (11+ 1) x ( 2 / + 1) matrix of three-dimensional rotations of usual spherical functions 38 (so-called generalized Vigner spherical functions) of the order / = 0,1,2,... (a,p, d, etc.). In actual calculations the matrices D^l'(g) are real because we use real spherical harmonics Si'm(r). The symmetry adapted hole functions (SAHFs) are denoted as |F£/
where elements of the transformation matrix (TM) W^ r ' =|| W.,

Am

\\ realiz-

37

ing this passage are

where HG = dimG is the dimension of the group G and HI C G is a local group of the first atom of the chain £ (this atom does not move under any

local group operation g €. HI). Besides, x^(9) is tne character of the IR F. Now we can formulate our final equations incorporating the system symmetry as much as possible. Every cx-th solution of Eqs. 25 can be written as: (a) => (F7), where f distinguishes different solutions belonging to the same

IR F. Therefore, instead of Eq. 24 we have

It was assumed here that only functions <pr(fv = |r£/cr) of the same IR F contribute to the ^

. The sum in Eq. 31 runs over all the chains £ (and

51

Since the number of indices (f/) is less than that of indices (£Am), the problem Eqs. 33-36 is much simpler than the initial one, given by Eqs. 25. At last, let us write the expression for the total energy of the state (Ff) (cf. Eq. 27)

or, equivalently,

where the relaxation energies appear to be

In order to obtain all the solutions of the problem Eqs. 33-36, it is necessary to take all possible (one-dimensional) IRs of the group G in question and to find all (7) non-trivial solutions of the system Eqs. 33-36 of the non-linear algebraic equations for each IR F. 4

Crystal ions displacements

An important feature of the STH structure is directions and values of displacements of ions in the local region. They can be obtained from the usual expression32 for the displacement vector operator,

where R^ is the radius-vector of the ion A in the chain £. In order to calculate the displacements ua(li,) we have to make a statistical average of the mean-values {*kX p h.Xex\Ut*(£A)\tyhXphXcx) with respect to all possible phonon

53

corresponding to /) belonging to the local region. Using Eqs. 30 and 31 it is easy to establish the following connection between the new, U^."1 , and the old,

UfjiJit representations of the solutions

It is seen that the number of independent elements U ^ " 1

can be sufficiently

r7

less than the number of elements in the whole vector U^ ^ =11 t/55 ATTl . II, if the " "' group G is rich enough. Making use of the relation Eq. 32 and the unitary of the TM,3T the equations of the present method are easily rewritten in the fully symmetrized form

where

At last, the integrals represented by the square brackets in the first equation of Eq. 34 are defined according to Eq. 15 although spineless SAHFs

} =

F£/) are used instead of the usual AOs \(,Am), as in Eq. 15. These integrals written in the symmetrized form are connected with the ones, given in the

AO-representation (Eq. 15), by means of the 4-fold multiplication on the TM

W (r)

Thus, as is now evident from Eq. 33 the independent set {U^'} 1 7

1

of coef-

ficients in Eq. 31 just represents a 7-th eigenvector U^ " ^ =|| U^" \\ of the problem Eq. 33. As follows from the Eq. 26 and Eq. 32 as well as from the unitary of the TM, it is normalized to unity

52

numbers {nqj } (the electronic wave -function W^ as well as the electronic term are fixed)

since only one term, E°h = •E'f'h({nq; }), in the total energy Eq. 37 depends on the phonon numbers {nqj}, and the product ^hXtx does not affect the operator Eq. 38 deprived of the electron operators (only Xph depends on the phonon variables in the Landau-Pekar approximation). As usual, Zp^ =

Zw/ n

\ exp \~0Eph({n(ij })) and /3 = 1/kT is an inverse temperature.

Using Eq. 12 for Xph and Eq. 31 for
, we have the following expression

^(^ = -?ih<^ v^?e«(qj>
(40)

[^xj^vxw;'']. 5

Vj. center absorption spectrum

The Vjt-center absorption spectrum associated with the transition Tf =*• F'-y' can be easily calculated by the method developed in 15 . The transition probability is defined statistically using Fermi golden rule formula in which the probability of the elementary transition should be averaged over all the phonon states {nq;-} of the initial state F7 and summered up over all the phonon states {n'j } of the final state TV

where the function

was used instead of the 6-function fi(z)in Eq. 41. By means of this substitution a finite lifetime r = 1/F of the final state39 is taken into account, F being the total line width ((A(z) => 8(as) as F -> 0). In Eq. 41 I>r 7 _r' 7 '

54

represents the electronic dipole momentum matrix element which does not depend on the phonon coordinates as it is usually assumed in the Frank-Condon approximation.1 Then, E^"1'

and E^r

7

) are the Vj-center total energies in

the initial and the final states, respectively, Eq. 37. Making a number of formal transformations similar to those used in15, and putting F —» 0 in the final expression,39 we obtain the gaussian-type line shape with the peak position

£r7-rv and the width 2N/2/n2V7 r7 _ r < 7 -, where (cf. 15 ~ 17 )

and

Note that the integrals over the BZ appeared here can be reduced to that given by Eq. 15. In Eq. 43 nqy is Bose-averaged phonon occupation numbers depending on the temperature.

6

Results of model calculations

Though rather complicated, the theory developed can be implemented for actual calculations of STHs in a wide range of crystals as should be clear from the above considerations. In this paper we are going to demonstrate the impor-

tance of the short-range terms in the Hamiltonian. For this purpose a simple model is sufficient. More sophisticated calculations based on a more reliable model will be given in the next paper. Let us start with the description of the simple model, used here. Only Slater-type 3p-orbitals of the Cl~ ions are used as the basic set which is assumed to be orthogonal (the overlap of the AOs centered on neighboring anions is about 0.2 and can be neglected in the present consideration). This approximation is valid in the case of the KCl crystal because the STH is mostly created

from states of the 3p-type of the crystal upper valence bands; 1 ' 4 ' 5 a contribution of potassium ions is small and was dropped here. The simple 4-parameter tight-binding model Hamiltonian of the hole H^ is used.4 Its parameters were 55

approximately fitted to the structure of the perfect KCl 3p upper valence bands and were listed in4. While calculating the integrals (^Am\ett^T\^'A'm'), a decomposition of the Slater-type AO's |£.Am), \£'A'm1) into finite series over

the Gaussian-type AOs were used,34 leading to integrals which were handled analytically. It turned out that already the 3-term decomposition gave quite satisfactory results.

It was shown above (see Eq. 3) that the solution of the perfect crystal lattice dynamic problem is necessary to perform real calculations of the STH

structure with complete account of the hole-phonon interaction.

Since the

lattice dynamic problem itself represents quite a tedious task (though a number of semiempirical models can be applied for this purpose1'20 (see also40 and

references therein)), we have dropped short-range contributions into the Hamiltonian in this paper. Namely, only the longitudinal long-wavelength optical and acoustic lattice phonons have been involved in the present study since they mostly contribute to the hole polarization energy. 20 ' 28 ' 29 The Frohlichtype interaction ' 2 and the deformation potential method have been used for optical and acoustic phonon branches, respectively, leading to the following hole-phonon factors

where £Q and €00 are the static and high-frequency crystal dielectric constants, respectively, huo is the longitudinal lattice phonon energy, 0-^ is the deformation potential constant, u the longitudinal sound velocity, and M = M+ + Mis the total UC mass. The factors Eq. 45 were used in the Hamiltonian Eq. 2 in the real calculations instead of the general ones, Eq. 3. In principal, the value of
from 0 eV to 10 eV in our calculations. It is known also that only one longitudinal excitonic branch dominates in the electronic polarization. 24 ' 25 Hence, we have put ncx = 1 and have omitted the index v in the excitonic part of the Hamiltonian (see Eqs. 4 and 5). While calculating displacements of the crystal ions given by Eq. 40, the

long-wavelength approximation for the unit vectors, e^(q, op) == iQ,\j ~fif~*s~ 56

and e^(q, ac) = ly ~M~*S~>

as

well as for the corresponding frequencies (wop(q) =

<^o, <*>ac(q) = u
Table 1: Parameters16 used in the present calculations.

(*) The exponent of the 3p-Slater-type AO's of the CL~-ion was fitted in order

to reproduce both the electronic and spatial structure of a number of small molecules (K2, C12, KCl, etc.) as well as the KCl perfect crystal.42

The Dih symmetry of the STH was implemented. In the case A it leads to the simplest consideration. Indeed, all six 3p-AO's involved in the local region are splitted into six chains, belonging to different IRs. We have calculated both the ground ( 2 £ ~ , or Sltl in DH group notations) and the tr-type excited ( 2 £jJ~, or Aig) states of the STH. The elements of the transformation matrix for the IR's associated with the STH ground 2 £ ~ , and
case are equivalent to the trial functions taken in16. Then, the matrix F^r' in /P\

Eq. 34 contains only one element, F^ , for any IR F and consequently Eq. 33 has only a trivial solution: U^ r ^ = U^ = 1. Besides, using Eqs. 42 and 43 we have calculated the u-type absorption peak position e = •&Blu-.yi1, and its

width D = 1^/1ln1WB^^A^. The results are presented in Table 3. It is seen that the relaxation energies ETcl are larger than the values -1.46eV (for Vk, 2 E ~ ) and -1.20eV (for V£, S^J") found in . These discrepancies are connected with the fact that we have included the electronic polarization as well. It turned out that the latter contribution gives almost 70 per cent of the total relaxation energy and therefore is very large. However, the contribution from phonons to the Erei 57

appeared to be very small (near -O.TeV) and is twice as small as the value -1.46eV found in16 for the V^-center ground state. Besides, we have found a weak dependence of the results on the acoustic phonon constant,
contribution from this type of the phonons to the Ere\ remains very small (0.5% from the Erei) for both the ground and excited states. This result does not agree with the conclusion made in2 that the acoustic phonons play a dominant

role in the self-trapping phenomenon. Besides, our results for the central ions displacements AX (Cl~) and for the (7-type absorption spectrum (E, D) are far from experimental values and results of other calculations ( &X(Cl~) ~ —1.8

a.u. 4 - 6 , E = 3.39 eV and D = 0.81 eV 18).

fr}

Table 2: The elements of the transformation matrix 11II W,, . . In the case £\,Am

A both chains £ are produced only by 3px, 3pv-AO's of the internal Cl^'molecule, (±a/2, ±a/2.0), while in the case B two additional chains £ of the

same structure but produced by the anions (±3o/2, ±3a/2.0) nearby the molecule are augmented.

The same picture can be drawn in the case B as well (see Table 3). In this case two chains (£ =in, out) contribute into the AIS and Blu (by one of each pair of ions, either internal or external) leading to the 2x2 matrix F^r) in Eq. 34

and to 2 unknown coefficients, {/| and U^ . Consequently, it is necessary to find all the solutions of the system of three nonlinear algebraic equations,

f r*v ^

given by Eqs. 33 and 36. Due to Eq. 36 one of the unknown coefficients, U^ ,

(r**Y \

can be expressed via another one, Uin

, and we arrive at the one-dimensional

optimization problem, which can be solved numerically.

A set of solutions

has been obtained in this way for both IR's under consideration. The results, presented in Table 3, correspond to those having the lowest total energies.

We have found in our calculations, that in both states (Vj. and V£) the hole is preferably distributed on the central anions within the central (internal) region ( U^ ~ 0.996 whereas U^2 ^ 0.1 practically for both IR's T = Alg and 5lu although U^t' is slightly larger than t^,ut , showing a correct trend for the excited state). However, all the energies listed in Table 3 and the displacements 58

practically coincide with those obtained within the model A and are far from the experimental values.

Table 3: Calculated results for the STH in the KCl crystal obtained for both models A and B. All the energies are in eV.

(*) The total alteration of the x-coordinate of the Cl atom (a/2, a/2.0) belonging to the Cl^ -molecule, in a.u.

Besides, we can estimate the hole self-trapping energy, &EgT- As was explained in the introduction, it is denned as the difference between the energy of the STH state EQ and the free hole energy e h (0), calculated at the bottom of the free hole band (F-point). In order to obtain fh(0), we have to drop the phonon part of the Hamiltonian, since in this case the hole-phonon interaction is negligible.36 Due to the failure of the Landau-Pekar approximation in the case of the infinite motion of the hole, we have calculated the free hole energy using the perturbation theory with respect to the excitonic part of the Hamiltonian. 4 ' 24 The following result has been obtained in this way for the total free hole energy at the point T 23'24

59

where tj (q) and Uy (q) =|| f/ w -(q) || are the eigenvalues and the eigenvectors of the bare hole Hamiltonian Hh, Eq. 1; |0/i) is the SP^-AO centered on the anion in the 0-th UC; all three branches j of cy (q) degenerate at the point F (q = 0) yielding identical 6p = ej (0) for any j. In our simple model we have found er = 10.72 eV and € po i(r) = -2.08eV, leading to the total free hole

energy eh(T) = 8.64eV. Using values of the STH energies given in Table 3 we have obtained, however, positive (though rather small) self-trapping energies indicating that the hole is not self-trapped in KCl crystal which is in complete disagreement with

experiment. So we have found an additional argument that the studied model is unable to describe the STH in KCl crystal. Some reasons of this conclusion will be presented in the next Section.

7

Discussion and conclusion

In this paper we have proposed the approach allowing one to treat the STH (the Vfc-center in alkali halides) in a semi-analytical way taking into account self-consistently both the electronic (inertialless) and lattice (inertial) polarizations. We have shown that in the Landau-Pekar approximation our Hamiltonian has a number of exact solutions of multi-hole nature. In the case of the one-hole state, the total hole wave function can be obtained solving the system of nonlinear algebraic equations with respect to the coefficients {/', . They appear in the decomposition of the hole wave function 9^ over the known symmetrized hole states localized within a limited local region. The complete account of the system symmetry results in that the problem can still be handled even for comparatively large local regions. Our approach represents a further development of the method proposed by the authors of15"18. However, in contrast to their method, there are no artificial adjustable parameters in our Hamiltonian (like Aa in 15 ' 16 ). On the contrary, standing on the position closed to the usual QC philosophy, we have introduced a number of variational parameters directly to the hole wave function * fc . In the present paper we have generalized the model developed in15""18 mainly in two ways: 1) inclusion of the inertialless polarization, and (ii) enlarging of the local region (case B). Besides, we had to treat the hole-lattice interaction in a slightly different way, though still within the long-range approximation. Unfortunately, our free parameter approach turned out to be in complete contradiction with the experiment and the results of the papers. ~~ The test performed here indicates an importance of the short-range effects missed in the present paper. Thus, we have shown, that any free-parameter 60

model based only on the long-range approximation must fail and must give unsatisfactory results both for the hole energies and the ions displacements.

Acknowledgments. Authors would like to express their thanks to Prof. E. Kotomin for his interest to this work and a number of useful discussions.

References 1. A.M. Stoneham in Theory of Defects in Solids (Oxford Univ. Press, Lon-

don, 1975). 2. M. Ueda et al, Excitantc Processes in Solids (Springer-Verlag, Berlin-

Heidelberg-N.Y., 1986). 3. H. Haken, Quantenfeldtheorie des festkorpers. (Teubner, Stuttgart, 1973). 4. L. Kantorovich et al, Phys. Rev. B 47, 4875 (1993). 5. A.L.Shluger et al, J. Phys.: Cond. Matter (in print).

6. P.E. Cade et al, Phys. Rev. B 30, 4621 (1984). 7. L.K. Rozarenova et al, Fiz. Tverd. Tela 20, 1836 (1978). 8. W. Hayes and A.M. Stoneham, Defects and Defect Process in NonMetallic Solids (J. Wiley, N.Y., 1985).

9. E. Kotomin and A. Shluger, Phys. Status Solidi (b) 109, 75 (1982). 10. R. Pandey and A.B. Kunz, Phys. Rev. B 38, 10150 (1988). 11. A.L. Shluger et al, Phys. Rev. B 44, 1499 (1991). 12. A.L. Shluger et al, J. Phys.: Cond. Matter 3, 8027 (1991).

13. E.M.Heifets and A.L.Shluger, /. Phys.: Cond. Matter (in print). 14. A.L.Shluger et al, J. Phys.: Cond. Matter 4, 5711 (1992).

15. T. lida and R. Monier, Phys. Status Solidi (b) 74, 91 (1976). 16. T. Higashimura et al, J. Phys. C: Solid State Phys. 17, 4127 (1984). 17. Y. Nakaoka, Phys. Status Solidi (b) 127, 327 (1985). 18. Y. Nakaoka and T. Higashimura, Solid State Comm. 69, 431 (1989).

19. I. Lang, Zh. Eksp. Tear. Fiz. 72, 2152 (1978). 20. J. Zakis et al, in Models of Processes in Wide Gap Solids (Izd. Zinatne, Riga, 1991) [in Russian]. 21. L. Kantorovich, J. Phys. C: Solid State Phys. 21, 5041 (1988); 21, 5057 (1988). 22. Y. Toyozawa, Progr. Theor. Phys. (Kyoto) 12, 421 (1954). 23. A.B. Kunz, Phys. Rev. B 6, 606 (1972); A.B. Kunz and D.J. Mickish, Phys. Rev. B 8, 779 (1973); S.T. Pantelides et al, Phys. Rev. B 10,

5203 (1974). 24. S. Wang et al, Phys. Status Solidi (b) 51, 11 (1972). 25. L.N. Kantorovich, Izv. Akad. Nauk Latv.SSR 5, 18 (1987). 61

26. R.A. Evarestov in Quantum-Chemical Methods of Solid State Theory (Leningrad State Univ., Leningrad, 1982) [in Russian]. 27. C. Pisani et al, Hartree-Fock Ab Initio Tratment of Crystalline Systems. Lecture Notes in Chemistry, 48 (Springer-Verlag, Berlin-HeidelbergN.Y.-London, 1988). 28. O. Madelung, Festkiirpertheorie (Springer-Verlag, Berlin- HeidelbergN.Y., 1972). 29. A.S. Davydov in Solid State Theory (Nauka, Moscow, 1976) [in Russian]. 30. J.M. Pople and D.L. Beverige, Approximate Molecular Orbital Theory (McGraw-Hill, New York, 1970). 31. I.B. Bersuker and V.Z. Polinger in Vibronic Interactions in Molecules and Crystals (Nauka, Moscow, 1983) [in Russian]. 32. J.A.Reissland The Physics of Phonons. John Wiley and Sons Ltd., London-N.Y.-Sydney-Toronto, 1973. 33. T.K.Mitra et al, Phys. Reports 153, 91 (1987). 34. Methods of Electronic Structure calculations of atoms and Molucules, ed. M.G.Veselov (Leningrad State Univ., Leningrad, 1975) [in Russian].

35. R.A. Evarestov and V.P. Smirnov, Phys. Status Solidi (b) 119, 9 (1983). 36. Y. Toyozawa, Progr. of Theor. Phys. 26, 29 (1961).

37. L. Kantorovich and A. Livshits, Phys. Status Solidi (b) (in print). 38. M. Hamermesh, Group Theory and its Application to Physical Problems (Addison-Wesley Pudl. Comp., London, 1964). 39. V.A. Kovarsky et al, in Multiquantum Processes (Energoatomizdat, Moscow, 1985) [in Russian]. 40. V.A. Telezhkin in Theory of Radiative Defects in Semiconductors (Naukova Dumka, Kiev, 1988) [in Russian]. 41. L. Kantorovich, Phys. Status Solidi (b) 123, 325 (1983).

42. E. Stefanovich et al, Phys. Status Solidi (b) 160, 529 (1990).

62

THE QUANTUMFIELD THEORY OF BINUCLEON V.D.LAKHNO Institute of Mathematical Problems of Biology 14229S Pushchino, Motcpw region, Russia We investigate a quantumfield model of a nucleon in the case of a scalai field with spot interaction. The tianslational invariant theory is developed

for a nucleon and binucleon in the strong coupling limit. Using the Bogolubov and Tyablikov canonical transformation, we introduce the collective

cooidinates for binucleon.

The interaction between two nuclei in meson

field is obtained to depend only on the difference in their coordinates. This result is independent of the nucleon-meson coupling. The numerical solutions to nonlinear field equations for the nucleon and binucleon are found in the strong coupling limit. The results obtained for the energy, radius and mass of the binucleon are compared with the experimental data for the deuteron. We propose a scheme eliminating ultra-violet divergencies in the strong coupling theory.

1

Introduction

By now a consistent theory of strong interaction has not been developed. Even the simplest nonrelativistic quantumfield model of nucleon-meson field inter-

action contains divergencies. In the nonrelativistic theory of particle-quantum field interaction, a series of the perturbation theory in the weak coupling limit has a finite radius of convergence. Despite repeated attempts to advance in this direction, the perturbation theory does not seem to be used in the case of weak interaction. Less apparent is the inapplicability of the strong coupling theory. In the nonrelativistic quantum theory of solid states, the simplest model of field is the polaron theory. A consistent quantumfield theory of strong

interaction yields in this case a finite value of the ground state energy, and various variational estimates give a smooth dependence of the ground state energy on the coupling constant over the whole range of its variation.1 In the

case of nucleon-meson interaction whose Hamiltonian is similar to the Frohlich Hamiltonian in the polaron theory,1 the situation is quite different. The perturbation theory leads to the divergence at large meson momenta in the weak coupling limit. For large coupling constants, the theory yields finite values of the ground state energy. But the variational estimates of the ground state of meson-nucleon interaction give the infinite negative energy for all the coupling constants. Since the simplest scalar model of nucleon-meson interaction leads

to the divergence of the energy, it is obvious that no more detailed model, 63

considering all types of interaction, can be free of these shortcomings. Cross concluded that the nonlinear equations having particle-like solutions in the strong coupling limit are not related to this divergence and do not eliminate it, since the divergent part can always be isolated. The classic part of the Hamiltonian diverges, in fact, for large coupling constants. The origin of this divergence is similar to that in the case of interaction between an electron and acoustic phonons in solids. The Hamiltonian of the electron-acoustic-phonon interaction has not a lower bound in the short wave limit. For nucleon-meson interaction, the quark structure of nucleons bounds the wave length of quanta of the meson field. Thus, the divergence of the classic part of the Hamiltonian is related to rough description of nucleus forces by the nucleon-meson interaction. In the case of electron-phonon interaction, the similar rough description is due to the fact that the continual theory is not valid, i.e. the discrete crystal structure is not taken into account. If we do not consider the problems of divergence, the main problem is to develop a translational invariant theory for two nucleons in a meson field. Most general considerations suggest the interaction between two nucleons $(r 1 ,7 > 2 ) in an isotropic and homogeneous space to depend only on the difference of their coordinates

This is just the form of the Yukawa interaction between two nucleons in the weak coupling limit . The situation with the strong coupling limit is rather different. In this case, the meson field is often considered as classical. But in a classical field two nucleons move in a classical potential well. Hence, the nucleon-field interaction can be presented as

This interaction is not translational invariant and differs in form from the Yukawa interaction. It is clear, however, that this interaction should be presented in the form of Eq. 1, if the interaction between nucleons is caused by the exchange the meson field quanta, In Sec. 2 we present the results of the translation ally invariant theory for the nucleon in a meson field. The general expressions for energy, total energy and effective mass of a nucleon are derived in the strong coupling limit. Numerical solutions of nonlinear differential equations are given in Sec. 3 for the nucleon in a scalar meson field. The energy, total energy, and effective radius of the nucleon are calculated. A consequent translational invariant

64

theory is developed in Sec. 4 for two particles in a quantum field. It is based on the method of collective coordinates, which is used in the nonperturbative quantum field theory to quantize near classic solutions. This method was first used in the polaron treatment to develop translational invariant theory of a nonrelativistic particle in a quantum field. The canonical BogolubovTyablikov transformations are used to introduce collective coordinates in Sec. 4. According to general treatment, the interaction between nucleons in a meson field was shown to depend only on the difference in the nucleon coordinates. The obtained result are independent of the nucleon-meson coupling constant. The problem of two particles in the field is reduced in Sec. 5 to the oneparticle problem. Its solution yields the bounding energy and effective radius of the binucleon close to the experimental data for deuteron. We discuss the binucleon effective mass in Sec. 6, and the divergence problem of the theory of meson-nucleon interaction in Sec. 7. As was pointed out above, these divergencies result from the fact that the Hamiltonian of nucleonmeson interaction is roughly determined. The functional of the total energy can have the local minimum in the strong coupling limit. But finite values of calculated parameters can be obtained by the perturbation theory for any power of the inverse coupling constant. In this Section we also present the equation determining the renormalized frequencies of the meson field. In Sec. 8 we use variational approach to calculate the energy of the binucleon ground state. In limit fj,/m —> 0, where p, is the binucleon mass, these variational estimations coincide with those obtained in the strong coupling limit. Sees. 9 and 10 are devoted to the case of pseudoscalar and vector mesonnucleon interactions. These interactions are significant to treat the value of quadrupole moment of deuteron and deuteron stability. The obtained results are discussed in Sec. 11. 2

Interaction of a nucleon with a meson field

In this Section we use the strong coupling theory to consider a nucleon interacting with a meson field. The total Hamiltonian H includes the complex scalar nonrelativistic field 9 and the real scalar meson field
65

where ~mA is the nucleon mass, g is the interaction constant, and A = c = 1, H is the meson mass. If the above fields are classical, Hamiltonian (3) yields a nonlinear equation for the nucleon

where WA is the nucleon energy. The results of solution of Eq. 4 are given in Sec. 2. In quantum treatment, the fields of Eq. 3 are operators

In the considered nonrelativistic problem, the Fermi and Bose statistics for one-nucleon state is out of question and Hamiltonian (3), by Eq. 5, takes the form

where

The complete solution of the nonrelativistic problem in the strong coupling limit in zero approximation was given in3'4. According to,3'4 the nucleon energy corresponding to Hamiltonian (7) is determined by the equation

66

where v is the mean velocity of linear motion of the particle and the field. It is essential that Eq. 9 transforms into Eq. 4 for v = 0 and with account of Eq. 8, i.e. the semiclassical result (4) is obtained as the exact limit of Hamiltonian

(7)In '

the total energy was written as

Expanding Eq. 10 in terms of v in the slow velocity limit, we get the expression for the particle's effective mass fn"A

Eqs.9-11 totally determine the nucleon motion in the meson field in zero

approximation and will be used in our further calculations.

3

Solution of the nucleon equation in meson field

It is convenient to present Eq. 4 as a set of two differential equations

In this Section we do not put ft = 1 and c = 1. Assuming

we get a set of differential equations in dimensionless variables in the sphericallysymmetrical case

67

with the boundary conditions

where

Accordingly, the nucleon energy is

The expression for the total energy, i.e. for the nucleon plus the meson field is equal to

where

The solutions of Eq. 14 provide the function f(k) = Aj/47iT with the dependencies shown in Fig. 1. The form of the curve suggests that there is a minimum g at which solutions to Eq. 4 exist. Still greater values of g are required for the existence of the

nucleon excited state. Since the critical value of k/^TfT is » 0.1, then for the range of the nuclear force K^1 = 1.41 • 10~13 cm the nucleon bound state does not exist even at g/vhc < 10. In this case Eq. 4 admits a solution only for a delocalized nucleon, i.e. * = exp[i'pr](l/\/v). Table 1 lists the values of F, T and R (where R = /o°° Y*(x)x3dx) for various values of the parameter k.

68

Figure 1: 1 - the zero mode, 2 - the first mode.

Table 1: Numerical parameters of quasi-particle solutions

to Eq. 14 for varying parameter k.

69

1

Translations! invariant adiabatic theory

The Hamiltonian of the interaction between two particles and a field takes on the form

where TI and r2 are the coordinates of the first and the second particles respectively, while r0 determines the position of the potential well formed by a meson field. In the isotropic and homogeneous medium, TQ can be arbitrary. If the field is classic, than TQ is fixed by the initial choice. There are two approaches in the quantum theory. In the first approach the position of the well is considered to be fixed in space: r0 = const. Authors of 14 hold this viewpoint. This case corresponds to the break symmetry. In the second approach, the well is not fixed, it fluctuates in space and can occur at any point of the space with equal probability. The translation symmetry of the initial problem remains in this case. The value r0 plays a role of the collective dynamic variable, similarly to variables which are introduced to quantize nonperturbative objects of the quantum field theory such as solitons, instantons, etc.5'6 With the use of Eqs. 5 and 6, the Hamiltonian of two nucleons interacting with the meson field takes the form

It has never been doubted that the interaction between particles in an isotropic and homogeneous space depends only on the distance between the particles. In the quantum field theory, this result immediately follows for the case when the interaction is weak and can be taken into account by the perturbation theory. In the second order perturbation theory, the interaction determined by Eq. 20 takes on the form

70

As it follows from Eq. 22, the interaction caused by exchanging filed quanta

depends only on the difference between the particle coordinates. The interaction of Eq. 22 obtained by the perturbation theory is a basis of many theoretical considerations, in particular, the theory of electron-phonon interaction in

solid-state physics.7 In the other limiting case when g ^ 1, the strong coupling particle-field interaction is more complicated. In the theories dealing with break symmetry (semiclassical theories), the field f in H^nt is considered in zero approximation as a classic field, that corresponds to an intuitive assumption of the particle motion in a fixed potential well. Thus, in this case, the interaction depends on the coordinates of each particle and is not translational invariant. The particles move in the potential well, whose position is fixed in space.

In Sec. 4 we develop a consistent quantum-field theory of interaction between two nucleons and a meson field in the strong coupling limit. Taking into account the translation invariance of the initial Hamiltonian in the strong coupling limit, we find that the two-nucleon potential well is not spatially fixed, as in case of Eq. 2, but conserves the translation invariance of the initial Hamiltonian. As a result, the interaction in the strong coupling limit is presented in the form of Eq. 1. In this Section the problem of two particles in the field is reduced to the one-particle problem. Its solution yields the values of the coupling energy and the binucleon state effective radius close to those obtained

experimentally for deuteron (Sec. 5). Degeneration of Hamiltonian (20) with respect to ro has a dominant role in the development of the translational invariant binucleon theory. Introducing the center mass and relative coordinates of two nucleons r and R by the formulas

and the complex coordinates of the field qj as

where e is e small parameter, we present Hamiltonian (21) in the form

71

According to, 8 ' 7 the radius-vector of the binucleon center mass R can be written as a sum

where r0 is the fluctuating part of the center mass coordinates and q is the translation-invariant part. We can replace qj by new complex field coordinates

Qj using the formulas

These new coordinates must be subject to the three additional conditions

where -Uj are the complex numbers, satisfying the relations of reality v_/ = t/jl, which are chosen so that the orthogonality conditions are fulfilled

where fa,

f& are the components of the vector /.

Relations (29-32) are the canonical Bogolubov-Tyablikov transformation in the quantumfield theory. Turning in Eq. 25 from the variables R, qj to new variables ro, q, Qj and using condition Eq. 31 we rewrite Hamiltonian Eq. 25 as

We take into account that the variable q appears in the transformed Hamil-

tonian only in the form of canonically complicated momentum = —ihd/dq

72

which is the total momentum of the whole system (nucleons plus the meson field). We write the wave function of the system as

As a result, the terms of expansion Eq. 33 do not contain the variable q and take the form

In Eqs. 37 and 38 the vector of the total momentum P is replaced by the vector I = e 2 P. To find the impulse dependence in zero approximation, I is assumed to be of zero order. Taking into account that the form (H±) linear with respect to Qj, Pj is equal to zero

and putting the coefficients before of Qj, Pj equal to zero, we express Uj and Sj as follows 73

where the vector c is given by the condition

It follows from Eq. 37-43 that the coordinates TO and r divide and in the zero approximation the Schrodinger equation takes the form

where c = v/e, v is the mean velocity of the binucleon translational motion.

5

Bound state of two nucleons

Using Eqs. 8 and 6, Eq. 45 may be presented in the form of the variational principle as the equation for the extremum of the functional

where

PJJ is the momentum associated with fluctuation motion. Shrodinger Eq. 44 contains the potential

74

As a result, the

The bounding energy of the binucleon £j,oun
Thus, the equations for the binucleon coincide with the equation for one nucleon but with the replacement: KA —> KA/2, g~* —> 8j 2 , m^ —<• mAf1. Then, relation Eq.,15 for the case of binucleon is written as

The binucleon energy Wg equals

while the total binucleon energy Fg is

Note, that if g/vhc < 10, i.e. when the solution of the one-nucleon problem

yields F = 0 (see Sec. 3) the binucleon bounding energy Eq. 49 is equal to F. The results of solution of Eq. 14 are presented in Table 1, they allow us to calculate the main characteristics of the binucleon. The only constant of nucleon-meson interaction is not specified in this case. Let us choose the value of the coupling constant g from the condition that the binucleon bounding energy would coincide with the experimental value for the deuteron. This corresponds to the following value of the dimensionless coupling constant

For such a choice of the coupling constant we get from Table 1 that: Jb/4*T w 0.072, 47tT a 6.6, 4irT w 4.8. Note that the obtained value of 75

fc/4irF lies within the region where a solution for the binucleon exists and a solution for one nucleon is absent. Using Eq. 49 and 50, we get the nucleon energy of the binucleon Wg w 18.9meV, while the binucleon bounding energy is, by the same formulas and the table values, equal to F ss 2.5 meV. Using relations Eq. 13 and assuming that, g2 = 802 and m^ = m^/2, we express the binucleon radius (r}

Expression Eq. 55 and Table 1 yield for the binucleon radius R fa 10~13cm, which is close to the experimental value for deuteron. 6

Binucleon mass

The expression for the binucleon effective mass is determined similarly to Eq. 11 and has the form

where AM = M — 2m.a, M is the binucleon effective mass. Replacing the summation in Eq. 57 by integration and using Eqs. 6 and 8, we express AM as

The integral in Eq. 58 is of the order of unit. Thus, for the chosen parameter of Eq. 51 the quantity AM is

76

Note that all the expressions contain the nucleon bare mass m^, which in principle differs from the dressed, i.e. measured nucleon mass rn"A. Hence, to calculate the nucleon effective mass M, given by the equation

we should use data on the nucleon bare mass mA. In this case the experimental

value of the deuteron mass would be AAf * < 0. 7

The problem of divergencies in the theory of strong interaction

The calculations of Sees. 2-4 were performed for the zero approximation of the strong coupling theory. They yield finite values of the nucleon energy, nucleon total energy, the binucleon energy, the binucleon total energy and the binucleon size. However the simplest probe wave function

where 4>0 is the vacuum wave function, being applied to Hamiltonian (7) yields infinitely large negative value of the ground state. This has led to the conclusion that divergencies in the strong coupling theory cannot be eliminated in principle. The fact that the contribution causing the divergencies can be isolated does not solve the problem. The decisive point here is that actually the variational estimate is given not to Hamiltonian (7) but to Hamiltonian :H:. Thus, the variational calculation refers only to a part of the total Hamiltonian, i.e. to a part isolated from the infinite positive value of the zero field energy. These procedures cannot be considered as consequent: they may lead to any value of the full energy, including the senseless one. The correct approach consists in a consistent application of the strong coupling theory to Hamiltonian (7). It implies the second order of the perturbation theory with respect to the inverse coupling constant g and yields the following equation for the renormalized frequencies of the meson field u>, in

77

where the functions Xt(T) satisfy the orthogonality condition

Vp(°) corresponds to the solution of nonlinear problem (12) associated with the ground state. Similar equations are derived for the binucleon, but with m^

replaced by ro^/2, g2 replaced 8g2 and KA replaced by Kj^jl. The solution of boundary problem (62) and (63) (boundary conditions result from /X 2 ( r )^ 3 r to be finite) is a nontrivial problem. It allows is to calculate the total energy of the system nucleon plus meson field by the formula

where u>j are given by Eq. 6, and the renormalized meson frequencies by Eq. 62. We cannot guarantee here the finite value of f^E^. But we can expect that &E^ will be finite by analogy with the corresponding quantity calculated in the strong coupling theory for the case of optical phonons. In our case this would correspond to the limit p, —> 0.

Thus, the exact solution of Hamiltonian (7) in the strong coupling limit may give finite values of the nucleon energy and does not require separation of isolated contributions. If we turn back to the perturbation theory where the logarithmic divergence of energy at large momenta is treated as the contribution made by virtual processes of absorption and emission of mesons, we recognize that the idea of invariance of mesons is incorrect at large momenta, i.e. they are not described by Eqs. 6 and 61 in this case.

8

Variational estimates for binucleon

Previously we derived exact equations for the binucleon and found some of their numerical solutions. It seems interesting to compare these results with the variational approach. Unitary transformation of Hamiltonian (21) with TO = 0 by operator

78

eliminates the center mass coordinates from the Hamiltonian. As a result, the Hamiltonian takes the form

To study the ground state of Hamiltonian (67) we choose the Lee, Low, Pines (LLP) probe wave functions in the form

As a result, the mean value of (H1) is

where

Variation of functional (70) under condition of zero total momentum leads to the expression for the coefficients £f

79

Substituting £j from Eq. 72 to Eq. 70 and varying the resultant functional over ^>o( r ii r 2)> we obtain the following Schrodinger equation for the binucleon

given that the probe wave function is normalized. Here

We choose the probe wave function V > o ( r i , r 2 ) in the form

Substituting Eq. 75 into Eq. 74 and taking into account that the meson mass is much less, than the nucleon mass, i.e. n/m^
which exactly coincides with the exact result. Taking into account, that fJ'/rnA ~ 0.1, we conclude that the variational estimate of the binucleon ground state energy for these parameters yields the energy exceeding the exact value by 10%. 9

Interactions with pseudoscalar mesons

It is significant to extend the above theory to the case of the deuteron. The scalar meson field does not consider spine properties of deuteron, in particular the obtained results do not depend on what states we consider , triplet or singlet ones. To describe the triplet deuteron state correctly, should be added by Eq. 5 the interaction with pseudoscalar mesons, which is written as

where O^,T; are spine and isospine operators for t-th nucleon / is the constant of interaction between nucleon and pseudoscalar field. 80

Repeating the above derivation for Eq. 77, we find the modified expression for the functional of the total energy

where F is determined Eq. 46, while the second term in the right side of Eq. 78 determines the effect of the pseudoscalar field on the total energy. Varying Eq. 78 over •, we obtain nonlinear differential equations, however they have

no spherically symmetrical solutions. Their exact solutions is a complicated problem and requires a special consideration.

10

Interaction with vector mesons

Another important case is the generalization of the theory to the case of vector mesons. As in the previous section, we should add in Eq. 78 the interaction between vector mesons and between two nucleons and vector mesons

where HTV describes the field of vector mesons8

and H'T corresponds to the interactions between two nucleons and a vector meson field

where el

are unit orthogonal vectors, i.e. (ej ,e.

parallel to momentum /, i.e. (e™ f) =\ f \, (e™f) = (e

) = Srs, and ek

81

is

f) = 0, [e f] =

( a) f

( l) f

0, [e^f] = - I / I 43), t6/3^] =l / Ie / 2); *i is ihe

vecior

°f spine operator

of j'-th nucleon. Repeating the derivation of previous section, we obtain the functional of total energy JF in the strong-coupling limit

ta

where F is determined by Eq. 78, while FT is expressed as

where V" is the deuteron wave function. The functional FT determines the influence of tensor forces on the interactions between nucleons. Varying f over i/> results in the set of nonlinear differential equations which have spherically symmetrical solutions in contrast to the scalar field. In this case, f can be minimized by the probe deuteron wave function 8

where U(r),u(r) are the radial parts of the wave function, Yfim are spherical harmonics, Xi m are 'ne spin deuteron wave functions.

11

Further discussions

The Bogolubov-Tyablikov theory has been widely used to describe the mesonnucleon interaction. 9 ' 10 In the case of two nucleons interacting with the meson field, this theory was considered in11. There the nucleon mass is supposed to be infinitely large, i.e. the nucleons are assumed to be static sources. The Bogolubov-Tyablikov theory was one of the first translational invariant theories which took into account the symmetry of the Hamiltonian in separating 82

the classic component of the boson field. At present, there are various methods where the collective coordinates are introduced directly in the initial Hamiltonian similarly to r0 in Eqs. 20 and 21. Different ways are possible to impose the

additional conditions retaining the total number of independent variables in the system. For instance, the number of the Fourier components are decreased in 5 ' 6 by the value equal to the number of the introduced collective coordinates

instead of the Bogolubov-Tyablikov condition. In nonrelativistic theory the Bogolubov-Tyablikov method was used in 12 ' 13 . In these works the BogolubovTyablikov transformation was applied to Hamiltonian (2), but the collective variables were not introduced. Zero approximation in these methods yields the same results as in the theories with broken symmetry. As in the polaron case,

these methods use only splitting coordinates, which have no physical meaning. Therefore, these works have not developed the theory which would use the only variables having physical meaning such as relative coordinates r = r\ — r 2 and the potential energy U(TI — r?).

There are serious difficulties in the application of the strong coupling theory to a nucleon moving in the meson field. In this case the criterion of adiabatic approximation consists in the requirement that the characteristic frequency of

nucleon oscillations in the induced potential well should exceed greatly the frequency of the meson field. This requirement can hardly be satisfied at reasonable coupling constants for the nonrenormalized meson frequencies. But it can be satisfied for the "smooth" meson frequencies witch are determined by Eq. 69. In the case of binucleon (or nuclei with greater number of nucleons) the adiabatic condition changes and requires that the nucleon should interact with the meson field induced by the averaged distribution of the second nucleon (other nucleons if n > 2). Since on the average nucleons are spaced from each other in the deuteron (being spot in our model), the time necessary to exchange the meson distanced from the nucleon significantly exceeds the time taken to exchange the meson close to the nucleon. Thus, the potential at a distance from the nucleon is determined by mean distribution of the density of nucleon probability, rather than its instant position, as in the case of nucleon interacting with the induced meson field.

The main reasons of errors arising in the use of the simplest variant of the quantumfield theory are: neglect of relativistic effects (which gives the error of about 10%), nondistinguishing between protons and neutrons, neglecting of nonspot character of the interaction (formfactor), neglecting of noncentral nature of nuclear forces (in particular, the forces depending on the orientation of nucleon spins). Neglecting of the latter fact in our model leads, in particular, to zero quadrupole binucleon moment which is inconsistent with the 83

experiment. In conclusion the author express his gratitude to R.R.Gabdoulline for numerical calculation of problem Eq. 14.

references

1. V.D. Lakhno in Excited Polaron States in Condensed V.D. Lakhno (Manchester Univ. Press, Manchester, 1991).

Media,

ed.

2. E.P. Gross, Annals of Phys. 19, 219 (1962).

3. 4. 5. 6.

N.N.Bogolubov, Ukr. Math. Journ. 2, 3 (1950). S.V.Tyablikov, Zh. Eksp. Tear. Fiz. 21, 377 (1951). N.H.Christ and T.D. Lee, Phys. Rev. D 12, 1606 (1975). R. Rajaraman in Solitons and Instantons (Noth-Holland, Amsterdam, 1982). 7. Ch. Kittel in Introduktion to Solid State Physics (Wiley, New York, 1976); G. Wentzel, Quanttim Theory of Fields (N.Y., 1949). 8. J.M.Blatt and V.E. Weisskopf in Theoretical Nuclear Physics (Wiley,

N.Y., 1952). 9. A.V.Rasumov and O.A. Khrustulev, Tear. Mat. Fiz. 29, 300 (1976). 10. A.N.Tolstenkov et al, Preprint (Ins. Phys. High. Energy 73 - 82, Serpukhov, 1973).

11. V.G. Bornuakov, Tear. Mat. Fiz. 51, 247 (1982). 12. V.A. Moskalenko, Uchenye Zapiski (Kishinev State Univ.) XVll, 103 (1955). 13. E.P. Solodovnikova and A.N. Tavkhelidze, Tear. Mat. Fiz. 21,13(1974). 14. M.A.Smodyrev, J.T.Devreese Phys. Rev. B 1996 (in press).

84

GENERALIZED FUNCTIONAL APPROACH IN THE THEORY OF MESON-NUCLEON INTERACTION V.D.LAKHNO Department of Quantum-Chemical Systems, Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 142292 A generalized Feynman variationat principle for path integrals is applied to the problem of a nucleon in a scalar meson field. It is shown that for the formfactor chosen in the self-consistent form the results of the study correspond to the strong coupling limit. In this case, the nonlinear differential equations for a nucleon in a meson field are obtained. The nucleon ground state energy and effective nucleon radius are found by the variational method for various nucleon—meson coupling constants.

1

Introduction

Variational methods are widely used in physics to obtain approximate non— perturbative solutions. In contrast, the applications of variational principles in the quantum field theory are rather limited (for a review see l > a ) .

One of the central problems in the quantum field theory of meson-nucleon interaction is the ultraviolet divergence of the perturbation theory, which arises even in the simplest case of point nucleon-meson scalar interaction. The use of formfactor leads to finite results of the perturbation theory in the weak coupling limit, but the convergence radius of the perturbation theory remains finite. In the strong coupling limit, the Bogolubov—Tyablikov adiabatic theory results in asymptotically exact nonlinear equations for the particle in a quantum field. ' It still remains unclear how the results obtained in this limit can be applied to real values of nucleon-meson coupling constants.

In this communication

we develop a generalized functional approach based on the Feynman path integrals to describe nucleon—meson interaction. This approach, where the trial potential plays the role of a variation parameter allows us to obtain a functional and, hence, differential equations for strong-to-weak coupling transition. It is shown, that for the form-factor chosen in the self-consistent form, i.e. the nucleon density distribution determined by the square of its wave function, the best results correspond to the strong coupling limit for any coupling constant. In this limit the nonlinear differential equations for the nucleon in a meson

field are obtained. The nucleon ground state energy is found by the variational method for various nucleon—meson coupling constants.

85

2

Application of the generalized functional approach to the problem of a nucleon in a meson field

The main problem in determining the properties of quantum mechanical systems by the method of path integrals is the calculation of a propagator in evolution of the system studied from the state r ' , t ' to the state r",t". The propagator can be presented in the form of path integrals as

In this paper, we apply the generalized functional approach 5 to a nucleon interacting with a scalar meson field. In this case, the action S can be expressed as

where m^ is the nucleon mass, g is the constant of nucleon- meson field interaction. Action S is obtained directly from Lagrangian L for the nucleon interacting with a scalar meson field

The form of S is analogous to S for the electron in a scalar phonon field. 6 Turning in Eq. 1 to complex time t" — t' = —i/SK, we obtain the density matrix p(r" , r ' ; / 3 ) , which yields the expression for the ground state energy of the studied system

where

86

Following,

we choose the trial action So(r,r')

in the form

Below, we use this expression to calculate the ground state energy Eq. 4.

3

Energy inequality

The upper bound of the ground state energy Eq. 4 is given by the Jensen's inequality which can be written by Eqs. 8 and 9 as

where

We are interested in calculating the values

and

4

Calculation of e0 — lim^,*, j(Vo)s

for the nucleon

To calculate Eq. 13, we introduce the coordinate of center mass as

87

Taking into account that Jacobian of transformation Eq. 15 is equal to unity in view of relations Eqs. 8 and 12, eo can be expressed as

where p. = m'/(m1 + m^). It follows from Eq. 16, that «o presents the ground state energy described by the Hamiltonian

where «o( z ) is tne wave function of the ground state of Hamiltonian H0. Using the equality

we get

Finally we have

5

Calculations of lim^oo j(v)s

for the nucleon

Using Eq. 6, we can present Eq. 14 in the form

88

In terms of relations Eq. 15 and definition Eq. 12, the mean value exp(ik[r(T) — r(
where, according t o 7 , the first factor in the right-hand side of Eq. 22 is

Let us denote the second factor in the right-hand side of Eq. 22 by J(k, r — cr). It can be calculated with the use of the Feinman-Kats formula

where G is the Green function satisfying the equation

With regard to Eq. 24 we obtain

89

Similarly using Eq. 25, we get

It follows, that

Using Eqs. 23 and 28, expression Eq. 14 is presented in the form

6

The total energy functional for the nucleon in a meson field

By Eqs. 10, 20 and 29 the total energy functional of the nucleon in a meson field $! is written as

90

The wave functions un(x) are the wave functions of .Ho Eq. 17. To obtain the ground state energy, we should to minimize the functional Eq. 31 with respect to wave functions un. We make inequality Eq. 30 even stronger if we confine the sum in Eq. 31 only by the first term. In this case

The self-consistent expression for the formfactor

allows us to present functional Eq. 32 as

Let us consider the limiting cases of functional Eq. 34. In the strong coupling limit as g —> oo the trial particle's mass rn' also tends to infinity, i.e. H —» 1. Finally, the total energy functional is

In deriving Eq. 35, the scalar meson field frequency is u/i, = VA^ + * 2 . Varying functional Eq. 35 over UQ under conditions that UQ is normalized, we obtain the 91

nonlinear Schrodinger equation

In the weak coupling limit as g —* 0 the trial particle's mass m' also tends to zero. If the delocalized states: u o (x) —> -4= are used, where V is the system's volume then as V —» oo, the total energy functional $1 tends to zero.

Table 1: Nucleon total energies (in nucleon mass units) and effective radius (in Fermi units).

7

Numerical results

To study the minimum of functional Eq. 34, we choose the trial wave function in the Gaussian form for the nucleon

Substitution of Eq. 37 into Eq. 34 leads to the problem of finding the minimum of *!(a,/i)

with respect to parameters o C (0,oo) and /i C (0, 1). Numerical investigation of Eq. 38 demonstrates that min $j(a, /J,) is realized at fj, = 1 for any values of

the coupling constant g. Table 1 lists the values of the nucleon total energies $! (in the units of nuclear mass ma) and effective radius < z >= v2/7r amfn,

92

where o m j n is the parameter at which $j (a, 1) is minimum for various values of the nucleon-meson coupling constant g. Table 1 suggests that there is a critical value gcr ~ 22, at which the bound nucleon state is absent in a meson field. For g < gcr but near to this value, the bound state is metastable.

8

Discussions

The approach developed yields the finite nucleon energy in a meson field for any values of the nucleon-meson coupling constant. The fact that at g < gcr the total nucleon energy tends to zero is due to the self-consistent expression for the nucleon formfactor Eq. 33. But the idea of the nucleon in a meson field, whose state can be described by a wave function il>(r), and, hence, by formfactor Eq. 33 is valid only in the strong coupling limit. The appearance of the total negative nucleon energy at g > gCT gives a qualitative estimate of the nucleon-meson coupling constant, at which the coupling may be considered to be strong. In this paper we have studied only the nucleon ground states. The excited states may be found from Eq. 36. The states like these should contribute to various processes of the nucleon scattering. In particular, the scattering resonances, associated with excited states, may be used in choosing the value of the nucleon-meson interaction constant. The approach may be also useful in calculation of binucleon states, particularly, in generalizing the results of the strong coupling theory, obtained for the binucleon quantumfield model 4 to the case of intermediate values of g and to relativistic field theory of scalar meson—nucleon interaction.

References

1. R.W. Haymaker,.R™. Nuovo dm. 14 (8), 1 (1991). 2. D.I. Diakonov and V.Yu. Petrov, Nucl. Phys. B 245, 259 (1984). 3. V.D. Lakhno in Excited Polaron States in Condensed Media, V.D.Lakhno (Manchester University, N.Y., 1991). 4. V.D.Lakhno, Tear, i Mat. Fiz. 100, 219 (1994) [in Russian].

5. 6. 7. 8.

J.M. Luttinger R.P. Feynman, R.P. Feynman, R. Rosenfelder

and C.Y. Lu, Phys. Rev. B 21 (10), 4251 (1980). Phys. Rev. 97, 660 (1955). Statistical Mechanics (Massachusetts, 1972). et al, Zeit. Phys. A 350, 131 (1994).

93

ed.

NONLINEAR MATHEMATICAL MODELS IN BIOPOLYMER SCIENCE L.V. YAKUSHEVICH Institute of Cell Biophysics, RUssian Academy of Sciences, 14S292 Pushchino, Moscow region, Russia Applications of the nonlinear mathematical models, some of them being known in the polaron theory, to biopolymer science are described. The main nonlinear equations and their soliton-like solutions imitating nonlinear excitations in biopolymers

are presented. Experimental data on the nonlinear properties of biopolymers and examples of nonlinear mechanisms of biopolymer functioning are discussed.

1

Introduction

Nonlinear mathematical models are widely used now in many fields of science.

In this review we describe the main nonlinear mathematical models applied to bipolymer science and especially to the study of nonlinear dynamical properties of DNA and alpha-helical proteins. Nonlinear dynamics of biopolymers is a new and rapidly developing field of biophysical science. It can be considered as a part of general dynamics

which deals with the internal mobility of biopolymers (Fig. 1). Theoreticians define it also as the next (anharmonic or nonlinear) approximation after the first (harmonic or linear) one. The nonlinear approximation is used when the amplitudes of internal motions in biopolymers are large, and the linear approximation which is valid only for small amplitudes, becomes incorrect. Conformational transitions, denaturation processes, formation of opening states in the DNA-protein recognition processes are only some of the well known examples of the large amplitude motions. Mathematical modeling of the nonlinear dynamics of biopolymers made its appearance in 1973 when Davydov and Kislukha published their pioneer work on solitary excitations in one-dimensional chain which was considered as a simple model of alpha-helical fragment of proteins.1 The first work on the DNA nonlinear dynamics was published only in 1980, and the first one on the RNA nonlinear dynamics in 1990. Now the number of articles devoted to the nonlinear dynamics of biopolymers rapidly increases, and specialists in different fields of science make contribution to the problem: mathematicians work out mathematical basis of the nonlinear theory, physicists suggest nonlinear models imitating internal dynamics of biopolymers and propose experimental methods

of detecting the nonlinear excitations, biologists develop various applications. However, the articles devoted to the theme are scattered among many special

94

journals and very often specialists in one field, for example in mathematics, can not understand their colleague-biologist who publishes the articles on the same problem in biochemical journals. Therefore, we decided to try to gather

in this review the results on nonlinear dynamics of biopolymers obtained in different fields: mathematics, physics and biology, and to show the relation

between them.

Figure 1: Scheme illustrating two approximations in the dynamical theory of biopolymers.

2

Internal mobility of biopolymers

As we mentioned above, the nonlinear dynamics is a part of general dynamics. Therefore we begin our review with a brief description of the main features of the general dynamics of biopolymers.

Because the structure of biomolecules is rather complex, it can be expected that the general picture of their internal mobility is also very complex. To describe the picture, different types of classifications of internal motions according to their energies, amplitudes and characteristic times were proposed. Authors of 4 ' 5 ' 6 ' 7 gave examples of the classifications of internal motions in proteins. Analogous examples for DNA presented i n 8,».io,n,l2 gave the detailed

classification for both DNA and proteins. Most of the classifications used are based on the time characteristics of internal motions. The time scale is usually divided into several diapasons: picosecond, nanosecond, microsecond and so on, and internal motions which

belong to these diapasons are described. To illustrate the general picture of 95

the internal mobility, we present here one of the classifications (Table 1). It consists of several diapasons, each being described by the main internal motions, the main structural elements involved in the motions, the energies of activation of the motions, the amplitudes of the motions and the main experimental and theoretical methods of the motion study. To understand when biopolymers can exhibit the nonlinear behavior, let us consider one of the diapasons in detail. Let us choose the nanosecond diapason. This diapason is of special interest because corresponding internal motions are very important in many biophysical phenomena: conformational transitions, gene regulation, DNA-protein recognition, energy transmission, ion conductivity through biomembranes and others which deal with the energies E = several kcal/mol, frequencies v = several cm"1 and r = several nanoseconds. According to Table 1, the nanosecond diapason includes among others the so-called solid-like motions of sugars, phosphates and nitrous bases which are known as the main structural units of DNA. Let us assume that these motions are very important in a biological phenomenon which we study. Let us consider them in detail. In the general case, these motions can be described mathematically by a set of coupled dynamical equations in unknowns which are the amplitudes of the motions and their derivatives. When the amplitudes of the motions are small, the linear approximation can be used, and the nonlinear terms in the corresponding dynamical equations can be neglected. Solutions of the resulting equations will have the form of usual linear waves. However, when the amplitudes of the internal motions are large, the linear approximation is not valid, and we must take into account anharmonic terms and consider the nonlinear mathematical problem. In this case the solutions having the form of solitary waves are possible. This point of view is illustrated by the scheme shown in Fig. 2. Thus, we can conclude that biopolymers could exhibit the nonlinear behavior in the cases when their internal motions have large amplitudes. These large amplitude motions can be generated in different ways. It is supposed, for example, that they can be generated by thermal fluctuations or that they can result from the collisions of "not" molecules of the solutions with biopolymers.13 Some authors suppose that large amplitude motions can be activated due to the chemical energy released in the ATP hydrolysis 14 or due to the local interactions of biopolymers with ligands. In applications we shall consider many examples of large amplitude motions in biopolymers and different sources of their activation.

96

"3 c M

CU -*»

c

o a!

-*9

tS

-o "3 o -u 0) (H O V X ^A

-a § "3

"3
*ce 0)

a x 0)

•s

c o

^J

a o

eg

O i—I

I 01

97

Figure 2: Scheme illustrating a possibility of nonlinear waves in D N A .

3 3.1

Nonlinear mathematical models The sine-Cordon model

We begin our review of theoretical models of nonlinear dynamics with the

sine-Cordon equation

which is well known in mathematics and has been used for many years in

physics.16 Eq. 1 has many wonderful properties including invariance under the 98

Lorenz transformation, infinite set of laws of conservation and soliton-like solutions of kink and antikink types shown in Fig. 3.1T Peculiarity of the sineCordon equation is that it can be modeled by a mechanical system consisting of a set of pendula connected with one another by springs, the pendula being able oscillate in the planes which are perpendicular to the axis of the chain (Fig. 4a). The mechanical model of the sine-Cordon equation was constructed,18 by this

model the visual evidence of soliton-like solutions was obtained. Englander 2 were the first who applied the sine-Cordon equation to biophysics. Their approach was based on the analogy between oscillations of

pendula in the mechanical model and rotational oscillations of bases in DNA (Fig. 4b). In the case of DNA the bases in one of two polynucleotide chains play the role of pendula, the sugar-phosphate backbone — the role of springs, and the influence of the other polynucleotide chain (this chain is shown in Fig. 4b by dotted line) — the role of the gravitational field. However, the approach proposed by Englander and coauthors is too simplified and many attempts to improve it are known. In one of them 19 additional terms describing the effects of damping and external influence are taken into account. So, the modified equation proposed has the form

where f(z,t) is the external "force" and /? is the damping parameter. Several authors 20 ' 21 ' 22 ' 23 improved the model of Englander in other way. They took into account the rotational oscillations of bases in both DNA chains. As a result, they obtained the model equations which can be written in a general form as

where At, A2 , BI , B? are constant parameters. Fundamental solutions of Eq. 3 have not been found yet, but it is known that in two particular cases: 1) when (fl = ? and 2) when (pl = —ipz, the equations have soliton-like solutions of kink and antikink types shown in Fig. 5 (£ = z - vt; v = const). 24 Model Eq. 3 were modified by 25 who proposed to take into account additional terms describing the helicity of the DNA structure. Model Eq. 3 can also be improved by taking into account other internal motions in DNA and their interactions with the rotational oscillations of bases. These modifications of the model will be considered in Sec. 3.5.

99

Figure 3: Kink and antikink solutions of the nonlinear sine-Gordon equations.

Figure 4: Mechanical analog of the sine-Gordon equation (a); two chains of the double helix (b). One of the chains is shown by dotted line.

100

Figure 5: Four types of soliton-like solutions of Eq. 3: (a) kink-kink; (b)

antikink-kink; (c) antikink-antikink; (d) kink-antikink.

3.2

The model of
Another model which is often used to describe the nonlinear internal dynamics of biopolymers is the (f>* equation

The equation takes its name from the term tp* in the Hamiltonian

which corresponds to Eq. 4.

The model of tf>* is well known in mathematics and physics.16 It has many interesting properties, and the existence of soliton-like solutions of kink and antikink types 101

is among them. Here v is the velocity of kink (in the case of"-") or antikink (in the case of "+"); -y = (1 — v 2 )" 1 / 2 . These solutions are shown schematically

in Fig. 6. The first application of the model of * to biophysics was made by 26 . He used this model to describe the changes of sugar puckering in D N A . Indeed, Eq. 4 can be rewritten in the form

where the dimensionless potential V(y>) has the double-well form (Fig. 7)

This form of the potential correlates with the existence of two equilibrium conformations of sugars: "exo" and "endo". Later the model of if>* was used by27 to describe another type of internal

DNA motions named tilting. These motions are associated with the changes of the tilt angle between the base normal and the helical axis. It is necessary to note here that any attempt to describe only one type of internal motions in DNA usually leads to oversimplification of the model. To

avoid it, we need to take into account other degrees of freedom.

Therefore,

in applications the if* equations are often used in combination with other dynamical equations. Some of the combined models will be described in Sec. 3.5.

3.3

The OstTovskii-Sutin model

The equation was derived by28 to describe the dynamics of longitudinal waves in a uniform cylindrical rod with nonlinear elasticity. Here u(z,t) is the longitudinal dis-

placement of a small structural element, which is placed near the point z at the moment t. This equation can be considered as a modification of the Boussinesq equation

which is well known in mathematics and physics. Here q(z, t) is the longitudinal strain which is related to the longitudinal displacement

102

Figure 6: Kink and antikink solutions of the equation of
Figure 7: Double-well potential V(if>).

103

u(z,t) by Eq. 9 can be easily derived from Eq. 10 if we assume that

Eq. 12 is valid, however, only when the effects of dispersion described by the term

and the effects of anharmonicity described by the term

are small. The traveling wave solutions of Eq. 10 were considered by 29 . It was shown that they had the form of choidal waves which in some particular cases took

the form of soliton-like waves. The kink and antikink solutions of initial Eq. 9 were found by integration of Eq. 10 and the integration constant was set equal to zero without loss of generality. Muto et al, proposed to use Eq. 9 and its soliton-like solutions to study internal longitudinal motions in DNA. This model was improved later by many authors. One of the modified models which takes into account the effects of damping and external influence, will be described in Sec. 4.3.

3.4

The Toda model

The model proposed30 consists of N masses (each of mass m) with longitudinal displacements from equidistant positions given by un(t) (n = 1, 2, ...N) as a

function of time (Fig. 8). The masses are connected with nonlinear springs of potential

where a and 6 are constant parameters. The dynamical equations describing longitudinal motions in the model

have the form

where V' means a derivative with respect to the argument. Eq. 16 were used31 to describe internal longitudinal motions in DNA. It was assumed that each 104

mass represented a single base pair, and each nonlinear spring represented the potential between adjacent base pairs. It is necessary to note, however, that the most realistic potential function between the base pairs should be the Van der Waals function

where l,A,B,p and q are constants. But Toda potential Eq. 15 has an advantage, because in this case Eq. 15 are solved exactly and well-known soliton-like solutions are easily obtained.

Figure 8: One-dimensional lattice with longitudinal degrees of freedom.

The approach proposed in was improved by Muto 32 who introduced additional (transversal) degree of freedom. As a result, she obtained the system of dynamical equations

where vn is the transversal displacement of the n-th mass:

In the continuum approximation, Eq. 18 have the form

105

where u = (un+l -un)/l,

v = (vn+l - vn)/l,

p = ml,

/9 = bl.

However, fundamental solutions of modified Eq. 19 have not been found

yet, only some numerical studies were carried out by32. 3.5

Combined models

In the previous paragraphs different simple nonlinear models of the internal DNA dynamics were described. Each of them imitates only one type of internal motions of the nanosecond diapason. However, in a more realistic model we need to take into account all functionally important motions of the diapason and their interactions with one another. To construct the more realistic models,

many authors propose complex models which are combinations of different nonlinear equations considered above. We present below several examples of them. The first one was proposed in

. It consists of six coupled equations

where the functions P^(z,t), Z^(z,t) and ip^(z,t) describe the amplitudes

of the following internal motions: changes of sugar conformations (puckering), longitudinal displacements of nucleotides and rotational oscillations of nitrous bases, respectively; the coefficients A, B, C, K, k, fi, /, M, a, xo, Xi and X2 are the constant parameters of the system; the index t denotes the number of polynucleotide chain. Krumhansl and Alexander succeeded in finding soliton-like solutions of Eq. 20

106

where p. = ^A^/IK^, A^ = A - Xl/(k - Mv2); K^ = K - ftv2 - xl/(k Mv*) — x?/(a — Iv*); v is the velocity of solitary waves. A group of authors 33 have made an attempt to improve model Eq. 20 but the results obtained have not been published yet. Another combined model was proposed by Zhang and Olson34 to describe the B-Z transitions in DNA. These authors chose four types of conformationally significant motions: rotations perpendicular to the z-axis (described by the variables $! and $ 2 ) and flipping processes (described by the variables Xi an
where the coefficients 77, GI , C^, /i, /2 are *ne constant parameters; the indices 1 and 2 denote the number of polynucleotide chain. The solutions of Eq. 22 have not been found yet, but in the particular case when the coupling between the variables ($!,$ 2 ) an d the variables ( x i . X z ) is neglected Eq. 22 become uncoupled and have the form of the sine-Gordon and the double-sine Gordon equations with the solutions in the form of kinks and antikinks. One more example of combined model was proposed by Zhang who studied microwave absorption in DNA. He supposed that the sugar puckering was not very important in microwave absorption, and therefore only longitudinal motions of base pairs (the variable u) and rotations of bases (the variables ViiV?) should be taken into account. To simplify calculations, Zhang considered only one particular case when |y>i| = \f2\' I*1 this case, the model equations take the form

Here v, c, WQ, 8, s, Xi i Xi are constant parameters of the system (see also Sec. 4.3). We can also add short information about another two combined models. The first one was proposed in36 to study the influence of longitudinal vibrations

on the properties of solitary waves in DNA. The second one was proposed in3 to describe the dynamics of B-A transitions in DNA (see Sec. 5.1).

107

As it follows from the material of Sees. 3.1-3.5 the number of simple and combined models describing large amplitude internal motions in DNA is rather large, and sometimes it is difficult to orient between them. Recently however, it has been shown that these models can be arranged in the order of increasing

complexity, and can form the hierarchy.37 In conclusion, we note the combined model proposed by Fernander 3 to describe the internal nonlinear dynamics of RNA. The model consists of two coupled equations in the two unknowns: the pseudorotational angle u(z,t) describing conformational dynamics of sugars, and the torsional angle $(z,i) describing the swinging of the nitrous bases along the sugar-base glycosidic bonds away from the pairing equilibrium positions. The first equation has

the form of the tp* equation, and the second one — the form of usual linear vibration equation. This model of the RNA internal dynamics is the first and the only one in its kind.

3.6

The nonlinear Schrodinger model

We described above several nonlinear equations imitating internal dynamics of DNA and RNA. Here we consider the nonlinear Schrodinger equation and its modifications which are simple models imitating internal dynamics of alphahelical fragments of proteins

and is well known in physics and mathematics as the equation describing various nonlinear phenomena in condensed matter. 16 First, it was applied to biophysics.1 Here < p ( z , t ) is the complex function, K is a constant parameter. Modulus of tf> can be interpreted as a probability to find a local excitation in

the vicinity of the point z at the moment t. Soliton-like solutions of Eq. 24, which have the form of a packet of waves (see Fig. 9), were used to describe the energy transport along the chains of hydrogen bonds in alpha-helical proteins (see also Sec. 5.7). The model was improved later.39'40 They took into account l)the existence of three chains of hydrogen bonds in alpha-helix and 2) interactions between them. According, to40 the improved model takes the form

108

where K, a, b are constant parameters. Three types of soliton-like solutions of Eq. 25, which have the form of the packets of waves, are shown in Fig. 10.

109

Figure 9: Soliton-like solution of the nonlinear Schrodinger equation.

Figure 10: Three types of soliton-like solutions: (a) three (degenerate) solitary waves oscillating in phase; (b) three solitary waves oscillating out of phase by 2?r/3; (c) three solitary waves oscillating out of phase by 4-7T/3.

110

4

4-1

Nonlinear models and experimental data on biopolymer dynamics Experiments on hydrogen-tritium (or hydrogen-deuterium)

exchange

Polar hydrogens of nitrous bases in DNA can exchange with solvent hydrogens given the molecule is ordered and remote from any denaturation transition. This behavior has suggested that an ordered double helix contains small amount of open states in which bases are unpaired. Opening of base pairs is a complex process which includes different types of

internal motions, and we can expect that some of them have large amplitudes. To study the dynamics of open states, methods of hydrogen-tritium and hydrogen-deuterium exchange have been proposed.41'42'43 A simple scheme describing the exchange can be written in the following form

where kop is the rate constant for conformational opening, kci is that for the reverse action (closing), and kchcm denotes the chemical transfer step. Scheme

Eq. 30 is written here for the moment form when a deuteron is exchanged for a proton. The scheme illustrates that the pathway for a hydrogen-deuterium exchange reaction can occur only during a transient conformational open state. Since the exchange of nucleotide hydrogen for deuterium generates a special shift in the ultraviolet, it appears possible to follow hydrogen-deuterium exchange by a spectral difference method using stopped-flow ultraviolet spec-

trophotometry. As a result, proton exchange rates can be measured over a broad range of temperature. From the analysis of the data obtained Englander, 2 concluded that a part of open states which are characterized by low energies and by slow opening and closing rates, can be interpreted as structural deformations which are the

regions consisting of several adjacent unpaired base pairs (Fig. 11). They assumed a mobile character of the deformations, that is a capability to diffuse along the double helix. They suggested at least that the movement of the de-

formations can be modeled mathematically as a propagation of solitary waves which are the soliton-like solutions of Eq. 1 (see Sec. 3.1). This interpretation of experimental data was criticized by both theoreticians and experimenters. The criticism of theoreticians was concerned with the simplicity of model Eq. 1. As a result, many improved models were 111

proposed. 13,20,21,33,44,45,22,35,34,19,23,25

Mogt Qf them were described

jn

Sec8

3X

and 3.5. Criticism of experimenters, especially,46'47 was based on the statement that the estimation of the probability of base-pair opening used by Englander and coauthors, is not correct. Indeed, according to, 2 the probability is rather high (about 10~ 2 ), 43 but according to,46'47'48 the probability of the base opening is only about 10~ 5 . The latter result is considered by opponents as an argument against the suggestion that mobile regions consisting of several unpaired base pairs are possible in DNA. They stated that only single unpaired base pairs can occur. It seems that additional experimental investigations are required to clarify the question. Proton exchange activity of RNA was considered by3 (see Sec. 3.5). He showed that proton exchange catalytic activity of RNA may serve as a probe for solitons. He also supposed that RNA even provides a better study case than its counterpart, DNA. We hope, that these predictions will be confirmed

by further experiments.

Figure 11: Mobile defect within DNA.

4-2

Raman scattering

Much effort has been given in recent years to the Raman studies of bipolymers. The low frequency range of Raman spectra is of much interest because its structure depends on vibrational motions of rigidly bound subgroups which are connected by the weakest bonds and/or nonbonded interactions. As mentioned above (see Sec. 2), these modes are very important in biological activities of biopolymers (some examples will be presented in Sec. 5). However, we must note that there are many difficulties in obtaining and interpretation of spectral lines. In fact, in many cases even the number of 112

Raman active modes is not known. 49 There are also many difficulties in theoretical calculations of Raman intensities. The predictions vary considerably even for "simple" molecules like benzene. Thus, for biopolymers we can expect

that only quantitative assignments of the observed data are possible. In 1980 Webb50 reported observations of laser-Raman scattering from living cells (E. coli).

Among these observations was a set of spectral lines

which indicated internal resonances in far infrared range, between 20 cm"1 and 200 cm"1. The lines of the spectra obtained with synchronized cells varied from appearing as a shoulder on the wings to very strong lines at

The spectra obtained with non-synchronized cells displayed lines below 200cm" 1

which are reproducible but not intensive. These observations were confirmed by similar results.51'52 In 1981-1982 a group of theoreticians53'54 studied numerically internal oscillations of the Davydov soliton in alpha-helical proteins (see details in Sec. 5.7). In the frameworks of nonlinear Davydov model a set of internal

resonances

was obtained. These resonances are almost identical to resonances Eq. 28 reported by Webb. This agreement might seem significant, and it could strongly support the idea that soliton is a possible mechanism for transport and stor-

age of energy in biological materials. However, after publications the authors of50'55 tried to remeasure laser-Raman scattering from metabolically active cells using an optical multichannel analyzer to record spectra. The experiment failed to find Webb's spectral lines in the low-frequency range. So, it can be concluded that this attempt to interpret the resonances in the framework of nonlinear conceptions was not successful.

The low-frequency lines were found, however, in the Raman spectra of proteins, 56 of polynucleotides in solutions57 and of oligonucleotide crystals.58 Interesting results in this field were also obtained by ' ' . It seems that the results on Raman scattering correlate with those on far-infrared studies.62 There is not doubt that further studies of low-frequency Raman modes will lead to a better understanding of the internal dynamics of biopolymers

and a possible role of solitons in the dynamics.63 113

4-3

Resonant microwave absorption

One more useful fool in the research of the nonlinear dynamics of bipolymers is the studies of interaction of biopolymers with microwaves. The most impressive

example is the history of investigations of resonant microwave absorption (in the range of several Gigahertz) by aqueous solutions containing DNA (Fig. 12).

Figure 12: Simplified scheme of the experiment on absorption by DNA.

Experimental evidence of resonant microwave absorption in DNA was received at first 64 and later. 65 ' 66 ' 67 Although their results are still

controversial 68 ' 69 ' ro ' n , they attracted attention of many theoreticians. Different interpretations of the experimental data has been proposed by them. Some interpretations are based on the linear (harmonic) model of the internal DNA dynamics, 72 ' 73 ' 74 the others are based on the nonlinear approach. 29 ' 35 The first nonlinear mathematical model was proposed in 29 . According to the model, the internal DNA dynamics can be described by Ostrovskii-Sutin equation

the solutions of which imitate longitudinal waves in the solid elastic rod with a finite diameter (see also Sec. 3.3). Here u(z,t) is the longitudinal displacement, c is the sound wave velocity, e and S are the dispersion and anharmonic

parameters, respectively. To consider the microwave absorption by aqueous solutions containing DNA, the authors added two terms to Eq. 30. The first

one is the damping term

which presents the viscosity of water. The second one is the driving term

which presents the microwave field. Thus, the resulting equation has the form

Using a special numerical procedure Muto et al calculated the absorption spectrum and compared it with the spectrum calculated earlier72 in the framework of the linear approximation. The results obtained were in close agreement

only for the fourth peak (the third overtone). The most marked differences were observed for the first (fundamental) peak: in the nonlinear approximation, the resonance peak exhibited a multicomponent character (fine structure) and the linewidth of the peak was not simply related to the damping constant as in the case of the linear approximation. The difference between the spectra can be explained by the nonlinear term (u*)z in Eq. 31. This difference might be used as a good test for nonlinearity. The approach proposed by Muto el af9 was improved by Zhang. 35 He

considered a rod-like model of DNA (as Muto and coauthors did), but he took into account both the longitudinal and the torsion al degrees of freedom. As a result, instead of one Eq. 31 he obtained two equations

where u(z,t),
parameters. 115

an

d Xz

are

*he coupling

To consider the microwave absorption by aqueous solutions containing DNA, Zhang added four terms to Eq. 32 (two damping terms and two driving terms) and solved the resulting system of two nonlinear coupled equations by the perturbation method. As a result, he found that 1) the resonant absorption of microwave energy is possible for both longitudinal and torsional modes, 2) the resonance frequencies are in the region of gigahertz and subterahertz, 3) for both modes the so-called subharmonic resonances are possible. It would be very interesting to continue the experimental studies, 65 ' 66 ' 67 and to check the predictions of Zhang.

4-4

Scattering of neutrons

Thermal (or slow) neutrons with de Broglie wavelength between 1 and 2 A are a versatile probe of structure and dynamics of macromolecules. The basic conservation equations governing the interaction between a quantum of incident

radiation and a scattering center are given by

where ftq and Aw are the momentum and energy transfers, respectively (h = Planck's constant /2ir). The initial and final wavevectors and energies are denoted by k, k' and E, E'. Some fraction of radiation is usually scattered elastically. It is the so-called elastic scattering which is characterized by the relation k = k' and produces q-dependent diffraction patterns related by Fourier transformation to the realspace structure. The other fraction of the radiation, however, will be scattered inelastically, i.e. it will exchange energy with thermal vibrations and other excitations in the sample. Thus this component of scattering provides information about internal motions in biopolymers. Most of the currently used spectrometers deal with unpolarized and produce data in the form of a double differential cross-section d which gives the intensity of neutrons with energies between E' and scattered into a solid angle element dd. It can be written in the general form

116

neutrons cr/dfldE' E' + dE' following

Here N is the number of scattering centers; a is the length of scattering; Sco (q, (jj) and S* nc (q, w) are the dynamical factors of coherent and incoherent scattering, respectively. The most interesting is the contribution of inelastic component of coherent scattering. In the first approximation, it is defined by

where Wq is the Debye-Waller factor; R^ is the equilibrium position of the n-th scattering center; un is the displacement of the center from the equilibrium

position. The dynamical factor of coherent inelastic scattering by nitrous bases of DNA was calculated by Fedyanin and Yakushevich. 75 They used both a simple

nonlinear mathematical model based on the sine-Gordon equation (see Sec. 3.1) and its modification where the helical structure of DNA was taken into account. It was predicted that the parameters of the central peak in the scattering spectrum depend on temperature T as follows

where J is the integral intensity of the central peak; Aui is the width of the central peak (Fig. 13); coefficients A and B are independent of temperature. It was also predicted that the helicity of the DNA structure leads to the splitting of the central peak (Fig. 14). These predictions have not been checked by experimenters yet. But one attempt to interpret experimental data on neutron scattering in the framework of nonlinear conception is known. It was made by Baverstock and Cundall. These authors proposed to interpret experimental data on scattering of fast (high energy) neutrons by DNA in the framework of soliton conception. Their interpretation was based on the assumption that the radical yield dependence on the direction of irradiation of oriented fibers of DNA can be explained by long-range energy transfer, and that soliton-like solutions of corresponding dynamical equations could be responsible for this transfer.

117

Figure 13: Temperature dependence of the integral intensity J and the

width Aai of the central peak.

Figure 14: A scheme of the splitting of the central peak due to the helicity of the structure of the scattering system. Figure 12: Simplified scheme of the experiment on absorption by DNA.

118

4.5

Fluorescence depolarization data

The method of fluorescence depolarization is widely used for measuring the torsional constants of bipolymers. In 1992 Selvin et af7 published the results

on measurements of the torsional rigidity of positively, relaxed and negatively supercoiled DNA. For the purpose they used Time Correlated Single Photon Counting of intercalated Ethidium Bromide. The measurements were made in the wide range (from +0.042 to —0.123) of superhelical density with a timeresolution of 75 picoseconds extending from 0.75 nanosecond, the range in which DNA twisting motions dominate over the fluorescence depolarization

signal. The main result of the measurements is that the torsional rigidity is not a constant, as it is usually suggested in the linear mathematical models of DNA, where DNA is modeled by a series of harmonically-coupled torsional

springs. So, it was found that at physiological ionic strength (175 mM), the torsional rigidity monotonically increases from 1.76 X 10~19 erg • cm for the

most positively supercoiled DNA, to 2.28 x 10~19 erg • cm for the most negatively supercoiled DNA. At low ionic strength (7.5mM) the torsional constant rapidly increases from positively supercoiled 1.91 X 10~19 erg • cm to relaxed DNA 2.42 x 10~19 erg • cm, and then levels off at negative superhelical densities (~ 2.3 X 10~ 19 erg - c m . Taking into account these experimental data Selvin and coauthors came

into the conclusion that a more accurate mathematical model of DNA should consist of coupled non-linear torsional springs. They estimated that the anharmonic term in the Hamiltonian should be approximately 15 % (at room temperature). This result can be considered as an evidence of the nonlinear nature of the internal DNA dynamics.

5 5.1

Nonlinear models and biopolymer functioning Conformational transitions

One of the applications is associated with the interpretation of the mechanism of transitions between different conformational forms of the DNA molecule. This idea was first proposed by Krumhansl in 1982 at the workshop in Gysinge.78 Then this approach was developed by many other authors.13'79'80|26'2T'33'22 Three of the main DNA forms: A-DNA, B-DNA and Z-DNA, are shown in Fig. 15. Each of the forms is characterized by a group of parameters including the helix sense (right-handed or left-handed), the number of residues per turn, the diameter, the helix pitch, the tilting of the base pairs relative to the helix 119

axis, the displacement of the base pairs away from the helix axis, the pucker of the furanose rings and others. To explain the idea, let us consider, for example, the transition between

A and B forms of the DNA molecule. This transition can occur due to the change of temperature, pH, hydration or some other parameters of the system. The transition can be easily visualized by ^f-ray diffraction studies of DNA fibers: if the fibers, for example, are allowed to dry, they produce an A-iype diffraction pattern, and if the fibers remain hydrated, the pattern is a 5-type. A schematic picture of A — B transition is shown in Fig. 16a, where one region of the DNA molecule is shown in the A form, and the other — in the B-form. The process of transition is shown as a movement (from the left to the right) of the boundary between the ranges. This process is analogous to that of phase transition in physical systems. It is well known, that phase transition processes in physics are successfully described by kink and antikink solutions of corresponding nonlinear dynamical equations. So, we can expect that the movement of the boundary between two DNA regions can also be described by soliton-like solutions of kink (or antikink) type (Fig. 16b). This suggestion is confirmed by theoretical results. Indeed, according to the model proposed,37 conformational transitions of A — B type are described by the system of two nonlinear differential equations

The first equation describes displacements inside the monomer link (the variable u) and the other describes the displacement of a monomer as a whole (the variable T). Here $(tt) is the potential energy of conformational transition

in a monomer link, and it has the form of a double well (Fig. 17); F(u) is the function which characterizes the structural rotations between subsystems: F(u) = («Q — u2); A, B, alts,c and u0 are constant parameters of the system. Volkov37 obtained the exact solutions of Eq. 39. One of them having the form of kink is shown in Fig. 16b. Another examples of nonlinear models of A — B and B — Z transitions are presented in Sees. 3.2 and 3.5.

120

Figure 15: Skeletal drawing of B,A and Z forms of DNA.

Figure 16: A schematic picture of (a) A — B transition; (b) kink-like solution describing the movement of the boundary between the regions of Atype and fl-type.

121

S.&

Long-range effects

During the 70's and 80's a great deal of experimental work was done on socalled long-range effects. The most impressive results came from the experiments where special regions of DNA molecule (enhancers) were discovered.

To describe these effects in detail, it is enough to consider a simple system consisting of two protein molecules and one DNA molecule (Fig. 18). It is assumed that the first protein molecule can bind to some special range (or

site) of the DNA molecule. Let it be called site 1. It is also assumed that the other protein molecule can bind with another site: site 2. Experimental data (see81) show that the first protein bound at site 1 influences the interaction of the second protein molecule at site 2. The distance between the sites can reach hundreds or thousands (as in the case of enhancers) of base pairs. This

is the reason why the effect is named the long-range one. Many mechanisms of long-range effects have been proposed. Some of them are shown in Fig. 19. According to the first model (Fig. 19a), the DNA molecule

forms a loop and, as a result, sites 1 and 2 become neighbors. In the second model (fig. 19b), the first protein molecule binds to site 1 and then moves (or slides) along the DNA molecule to the vicinity of site 2. The third model (Fig. 19c) suggests that binding of the first protein molecule to site 1 helps binding of another protein to the adjacent site, which in turn helps another protein to bind to the next site and so on. As a result, the protein molecules are sequentially bound and line up along the DNA double chain. In the fourth model (Fig. 19d), it is assumed that binding of the first protein molecule to site 1 is accompanied by a local distortion of the DNA conformation, which can propagate along the double DNA chain. When reaching site 2 it changes

the conformational structure of the site which, in turn, changes the binding constants of the second protein with the site. The fourth model correlates well with the hypothesis according to which nonlinear conformational waves can be excited in DNA and can propagate along the double helix. This model permits

us to construct a simple mechanism of regulation. Indeed, we can suggest that conformational nonlinear waves moving along the DNA molecule can act as regulatory signals in many important biological processes. One of them will

be considered in Sec. 5.3. 5.3

Regulation of transcription

Transcription (or RNA synthesis) is a complex many-stage process which pro-

ceeds in a system of many components, including DNA, RNA-polymerase, regulatory proteins, hormones, ions, water, etc. The simplest scheme of transcription includes three main stages: initiation, elongation and termination

122

(Fig. 20). At the first stage a special ferment (RNA-polymerase) binds with the promoter region of the DNA molecule. When RNA-polymerase takes a correct position and forms several phosphodietheric bounds, the second stage

of the process begins. At this stage a small fragment (subunit a) is released from the RNA-polymerase, and the rest of the molecule (core-ferment) moves along the DNA and elongates step by step the RNA molecule. At the third

stage the process terminates and RNA-polymerase is released from the DNA molecule. A special region of DNA (the terminator) gives a signal to stop the process. Besides, a special block (subunit 6) is required to stop the process and to release RNA-polymerase.

To consider a possible role of nonlinear waves in the regulation of transcription, it is necessary to use an essentially heterogeneous model of DNA, which is shown in Fig. 21.15 It contains the main functional regions necessary for RNA synthesis and regulation: a promoter, P, a coding region, C', several regulatory regions, Ri,R^,R3 and a terminator, T.

Figure 17: A conformational potential.

123

Figure 18: A schematic representation of the DNA molecule (black band) interacting with two protein molecules (small circles). The sites interacting with proteins are shaded.

Figure 19: Models of the action at a long distance: (a) the model explaining the long-range effect by the formation of a loop; (b) the "sliding" model; (c) the

model of sequential binding; (d) the model of propagation of conformational perturbation along the DNA molecule.

124

Figure 20: A generalized scheme of the main stages of the transcription process. Molecules of RNA-polymerase and RNA are shown by small circles and black bands, respectively.

Figure 21: A scheme of a DNA fragment. P, C, T are the promoter, coding region and terminator of the i-th gene, respectively; Rlt Ry, R3 are the regulatory regions of the i-th gene; R[, R'^ are the regulatory regions of the (i + l)-th gene; S is the nonlinear local distortion (soliton). The direction of the soliton movement is denoted by the arrow. 125

As it was mentioned above, the transcription starts with binding of RNApolymerase to a promoter P. The experimental data indicate that binding of RNA-polymerase to a promoter is accompanied by a considerable local distortion of the DNA conformation which can propagate along the DNA double helix (see Ref. 81). From the view point of the nonlinear dynamical theory, these facts can be interpreted as excitation and propagation of a nonlinear conformational wave. When passing through the coding region C, the wave will change the con-

formation of the region. The structural change may give rise to changes in the matrix properties of DNA, which in turn, will change the rate of RNA synthesis at the stage of elongation. Having passed through the (7-region, the wave reaches the terminator T which separates two genes, the i-th and (t + l)-th. The experimental data indicate that the T-region has often rather a heterogeneous structure. For example, the terminator is shown in Fig. 21 as a crusiform region. The passage of the wave through this region may result in a complete absorption of the

wave, or the wave will surmount the T-region, with the velocity and profile being substantially changed. The former case can be interpreted as termination and the latter as preparation for switching to the next gene transcription. The propagation of nonlinear waves through the regulatory regions R(, R'2

(or Ri,Rz,Ra) can be accompanied by changes in the conformation of the regions and hence, in the binding constants of regulatory proteins with these regions. As a consequence, the distribution of ligands throughout the regions will change, which is known to be of crucial importance for the regulation of DNA activity. We have considered transcription process occurring in the vicinity of only one gene. But, nonlinear conformational waves moving along the DNA molecule can also be assumed a suitable model for the regulation and coordination of simultaneous transcription of many genes.

5.4

Direction of transcription process

It is known from numerous biological experiments that the direction of the transcription process is different for different promoter regions and for different living organisms. For some promoters the transcription process moves to the so-called up stream direction, for other ones to the down stream direction. There are also promoters whose transcription process is known to be sometimes directed up stream, and sometimes — down stream. The question is: Can we

predict the direction of the transcription process? In the series of works82'83'84 a positive answer to the question was given. 126

According to Salerno's method we should consider the discrete version of Eq. 3

where / is the moment of inertia of the individual bases; K is the backbone spring constant; r)n is a parameter modeling the strength of hydrogen bonds between complementary bases. The coefficient r)n embodies specific base se-

quence information: ?jn is double (17 = 2/3) for adenine-thymine (A — T) base pair and triple (»/„ = 3/3) for guanine-cytosine (G — C) base pair. As an initial condition a simple kink solution

of the sine-Gordon Eq. 1 is used.

Studying the dynamical behaviour of the

kink in the inhomogeneous double chain described by Eq. 40 Salerno found that the kink can (Eq. 1) move or (Eq. 2) oscillate or (Eq. 3) stay without moving. Which of these three events will happen depends on the region of the DNA molecule with its specific base sequence. For promoter regions it was found that the kink moved in some direction and comparison of the directions for several different promoters with the experimental data was excellent. So, we can consider these results as another example of the successful use of the nonlinear dynamical approach to explain some of the features of DNA functioning.

5.5

Mechanism of long-range effects of carcinogens

At first long-range effects of carcinogens were discussed.85 Later a soliton mechanism of the phenomena was proposed.88 It was assumed that the binding of carcinogen to DNA leads to strong distortion of the DNA structure (Fig. 22). After the removal of carcinogen, the system will not relax immediately and conformational distortion caused by the binding can travel through large distances along the DNA double helix. Mathematically this distortion can be described as a solitary wave. Propagating along the DNA molecule the solitary wave causes long-range interference with the DNA-protein interaction, and in this way it can possibly initiate the activation of oncogene.

Ladik tried to propose a theoretical model of the process. His approach was based on the analogy between the internal dynamics of DNA and that of polyacetelene. As a result, the model hamiltonian was proposed in the form

127

Here Hci describes electronic subsystem, Hconj describes conformational changes, Hci_conf describes coupling between electronic subsystem and conformational changes. However, Ladik did not elaborate the model and did not obtain equations of motions and their solutions.

Figure 22: A scheme of (a) undisturbed and (b) disturbed DNA.
5.6

Open state dynamics and the problem of recognition

Open states are defined as the states where one or several base pairs are unpaired. The most interesting types of open states are those where several adjacent base pairs are unpaired. These open states were investigated in detail by Englander et or (see Sec. 4.1). They analyzed the results of the experiments on hydrogen exchange in DNA and assumed that the low-energy (6 kcal/mol) and long-living (10~ 2 sec) open states can be interpreted as local structural distortions consisting of several adjacent open states (Fig. 23). They also assumed that these distortions can move along the double helix.

128

It is obvious that these open states can play an important role in the process of protein-DNA recognition. Indeed, it is widely accepted now that the binding of small molecules occurs at the sites of transiently open base pairs. This point of view is supported by the fact that nitrous bases are placed inside the double helix, and it is rather difficult for a protein molecule to recognize the base sequence and to find the correct site of binding without local opening of the double helix. If we assume, however, that: 1) the DNA structure is not static, and 2) at room temperature the molecule has one or more local distortions moving along the double helix, the possibility of recognition sharply increases. When modeling the dynamics of open states Englander ei af supposed that rotational motions of nitrous bases make the main contribution to the process of opening. To describe these motions mathematically, they used an analogy between torsional motions of bases in one of the two DNA strands and torsional oscillations of pendula in the well-known, mechanical model of the nonlinear sine-Gordon system18 (see also Sec. 3.1). As a result, they proposed the model Hamiltonian

where
transforms in the continuum approximation to the nonlinear sine-Gordon equation

where a is the distance between neighboring base pairs (a = 3.4 A). Eq. 45 has soliton-like solutions among which are kinks and antikinks shown in Fig. 3. Model Hamiltonian (Eq. 45) was improved later in20.21.22.11.23 and others by taking into account the rotation mobility of nitrous bases in both DNA strands. As a result, instead of one dynamical Eq. 45 they obtained two equations. According, ' these equations have the form

129

where the additional indices 1 and 2 denote the numbers of the DNA strands; k and R are constants. The general solutions of Eq. 44 have not been found yet. However, they can easily be found for two particular cases (when i = — i) Eq. 46 reduces to well known nonlinear equations, i.e. the sine-Gordon and the double sine-Gordon ones. Both the equations have kink and antikink solutions which could imitate open states behaviour.

Figure 23: A schematic picture of (a) undisturbed and (b) disturbed fragment of the double helix. The latter contains one open state.

5.7

DNA denaturation

The nonlinear model for thermal denaturation of the DNA molecule has been recently proposed by the authors of88'89. They assumed that the DNA molecule can be modeled by two discrete chains connected Morse potential Vn (n = 1, 2, 3, ...N) representing the hydrogen bonds (Fig. 24). For each base pair the model includes two degrees of freedom (un and vn) which correspond to the displacements of bases from their equilibrium positions along the direction of hydrogen bonds (Fig. 25). So, the Hamiltonian of the model has the form

with where m is a common mass for bases; k is a coupling constant; D and a are taken equal to 0.33 eV and 1.8 A

, respectively.

130

Figure 24: Two polynucleotide chains connected by hydrogen bonds.

Figure 25: Displacements (un, vn) of the bases in the model.88'89 Hydrogen bonds are shown by dotted lines.

To study denaturation process which includes the strand separation (Fig. 26) it is important to investigate the behaviour of the variables yn

These variables represent the out-of-phase displacements which stretch the hydrogen bonds. The dynamical equation for the variables can be found from Hamiltonian (Eq. 47) in the form

To solve the equation, Peyrard and Bishop restricted themselves to a con-

tinuum approximation and used a multiple-scale expansion which is valid when

131

the precursor phenomena are considered. As a result, they obtained a nonlinear Scrodinger equation whose soliton-like solutions were discussed in Sec. 3.6. This result permits one to suggest that an energy localization analogous to selffocusing phenomena well known in physics initiates the denaturation process in DNJ

Figure 26: A schematic picture of local strand separation.

5.8

Energy transport in alpha-helical proteins

The problem of energy transport along linear-chain molecules has long been posed. In the early 1970's it was already known that the energy (about 0.43 eV or 10 kcal/mol) released in the hydrolysis of adenosine triphosphate molecules (ATP) is an universal energy unit in many biological phenomena. However, the mechanisms of the energy trapping and transport were unknown. 90 In 1973 Davydov and Kislukha1 proposed a nonlinear mechanism for energy transport in one-dimensional chains. Later this mechanism was used to explain the energy transport in alpha-helical proteins.39'91 It was taken into account that proteins consist of chains of hydrogen-bonded peptide (H — N — C = 0) groups, and three such chains in a helical arrangement define the alpha-helix structure. 92 ' The energy released in the hydrolysis of ATP is not sufficient to excite the electronic states of alpha-helical molecule but it is sufficient to

excite the intramolecular C = 0 stretching mode (Amid-I vibration) with the excitation energy 0.21 eV. According to Davydov, the energy released in the hydrolysis of ATP could be trapped and transported in alpha-helical proteins as quanta of the intramolecular C = 0 stretching mode (Amid-I vibration). Indeed, Amide-I

132

vibrational energy might become self-trapped through an interaction with lowfrequency longitudinal acoustic phonons which arises because the Amid-I energy depends on the length of the hydrogen bond between the peptide groups.

The localized spatial region where the energy is trapped can be considered as a combination of intrapeptide and deformation excitations. It can propagate, and thus a soliton-like mechanism for energy transport is possible. Mathematical description of Davydov model was considered in Sec. 3.6. We can add here that many attempts to improve the model are known. The works 94 ' 40 ' 53 ' 95 ' 96 are among them. The model is well known in scientific community, but due to the absence of any reliable experimental basis, the validity of the model is still intensively discussed.97'98'99'100'101

5.9

Energy transfer along the protofilament in the tube of microtubulin

It is well known that neurons and other cells are comprised of protoplasm which in turn consists of membranes, organelles, nuclei, and the bulk interior medium of living cells: cytoplasm. The latter contains extensive arrays of linear polymers responsible for organization and integration of the cell behaviour. The most important of them are networks of microtubules, arrayed in parallel and interconnected by filamentous strands. The microtubules are hollow cylinders 25 nm in diameter whose lengths may span meters in some mammalian neurons 102 (Fig. 27a). The walls of the cylinders are assembles of 13 longitudinal protofilaments. Each of the protofilaments is a series of subunit protein known as tubulin. Each tubulin subunit is a dimer which consists of two slightly different classes of monomers known as a and /3 tubulin (Fig. 27b). Each dimer may be viewed as an electric dipole. The origin of the dipole character is in the fact that one binds 18 calcium ions ((7o2) per dimer. The tubulin dimer subunits are arranged in a hexagonal lattice which is slightly twisted. To explain the process of self-assembly of microtubules, Sataric et al proposed to use the nonlinear model. Taking into account that the process can occur when microtubules are in dynamical equilibrium in aqueous solutions of dissolved monomers, they assumed that in the course of self-assembly, the attachment of one dimer at the (+) end of a protofilament produces one torsion solitonic impulse (Fig. 28). This solitary wave propagates then through the corresponding protofilament. When the soliton reaches the ( —) end, its energy can be employed to detach the furthest dimer in protofilament. Fig. 29 shows that if the microtubule is free in the cytoplasm the process described above will lengthen the (+) end while the (—) end shortens, i.e. the microtubule will move towards the (+) end.

133

Figure 27: A scheme of microtubule (a) and a tubulin dimer subunit (b).

Figure 28: A schematic representation of solitonic role in the process of energy transfer from polymerized to depolymerized dimer.

134

Figure 29: A schematic representation of "treadmilling" assembly from tubulin dimers which could take place at the (+) end and with disassembly at the other extremity with liberation of tubulin dimers.

The dynamics of the model system can be described mathematically by the Hamiltonian

where Bn is the torsion angle of the n-th dimer; Ln is the corresponding an-

gular momentum; A,B,C,k,J are constants.

Hamiltonian Eq. 50 describes

protofilament as a chain of dimers which are capable of electric oscillations. The oscillating segments which form giant dipoles interact among themselves through the forces which are proportional to the inverse cube of the separation

distance between the dipoles. In the continuum limit, this model is reduced to one of the modifications of the model
135

5.10

Proton conductivity in biomembranes

According to the modern point of view, biological membranes are made of a mixture of lipids and proteins.104 The way in which these proteins and lipids are assembled in biological membranes is shown in Fig. 30. These lipids form an "oily" fluid bilayer in which the adherent proteins are free to float at will. Some of these proteins span the thickness of the bilayer. They are named "integral proteins". Other proteins are electrostatically attached to either the inner or outer surfaces of the bilayer; they are referred to as "peripheral proteins". According, to some integral proteins with a suitable fraction of hydrogen

bonding sidegroups, such as the hydroxyls of serine, threonine, tyrosine and the carboxyls of aspartic and glutamic acids, may fold in the hydrophobic lipid environment of the membrane in such a way as to form the chains of about 20 or more hydrogen bonds that span the membrane and the conduct protons across at. A fragment of the chain is shown schematically in Fig. 31. Since the anions always contain the oxygen atoms involved in a H bond with proton, we henceforth designate them as "oxygens" but their real rather complex structure must not be forgotten. The transport of protons in hydrogen-bonded chains is a long-standing problem. An important property of these chains is that the proton potentialenergy curve in each hydrogen bond has the form of a double well with two minima corresponding to the two equilibrium states of a proton between two neighboring oxygen atoms (Fig. 32). It is assumed that protons are transferred by jumps from one minimum to another along hydrogen bonds, i.e. by migration of ionic defects. A nonlinear model describing proton motion in hydrogen-bonded chains has been proposed by Antonchenko et <j/.loe They suggested that the remaining part of the chain, including the oxygen atoms does not behave as a fixed substrate and the lattice distortion around the ionic defects has to be considered. The basic idea of the model is that the coupling between oxygen atoms and protons can provide a mechanism which changes the potential barrier that protons have to overcome to jump from one minimum to another and makes their motion easier. So, the Hamiltonian of the model is the sum of three contributions

The first one is the proton part

136

here un is the displacement of the n-th proton with respect to the center of the oxygen pair; m is the mass of the proton; V(un) = e o (l —un/uo^ *s '^e double-well potential; £0 is the potential barrier; 2uo is the distance between the two minima; the last term represents the harmonic coupling between two neighboring protons with the characteristic frequency Wi. The second contribution is the oxygen part

where pn is the relative displacement between two oxygens in a pair; M is the mass of oxygen; fi0 is *ne frequency of the optical mode which is involved; the last term describes the harmonic coupling between neighboring oxygen pairs.

The third contribution describes the interaction between two subsystems

where x

ls

the strength of the coupling. The equations of motions resulting from the Hamiltonian have the form

In the case of strong coupling, a continuum approximation can be used, and Eq. 55 transforms to

where x = na is the continuum space variable; a is the lattice spacing; Co = au>i is the sound speed in the proton sublattice; Vo = afli. In the general case, solutions of Eq. 56 have not been found yet. However, in the particular case, when the coupling terms are neglected, soliton-like solutions of the equations can be easily found. These solutions could be interpreted as those describing the transport of protons.

137

Figure 30: A schematic representation of the model of biomembrane. It

consists of a lipid bilayer containing peripheral and integral proteins.

Figure 31: A fragment of a chain of hydrogen bonds.

138

Figure 32: A doubly periodic potential-energy curve.

6

Concluding remarks

In this review we presented different aspects of nonlinear dynamics of biopolyrners. First, we described the main theoretical results which led us to the conclusion that bjopolymers are an appropriate medium for excitation and propagation of nonlinear waves. We must note, however, that most of the models used are oversimplified. To construct a more realistic model, we should take into account: 1) a more detailed picture of the internal structure and mobility; 2) the effect of water and ion structure around biopolymers; 3) inhomogeneity of the internal structure; 4) statistical properties of nonlinear excitations (solitary waves) appearing in biopolymers. Second, we described different attempts to interpret experimental data in the framework of the nonlinear conception. We can conclude that this aspect of the problem is not so successful as the theoretical one: some of the results described above cannot be considered as a convincing evidence for the existence of nonlinear conformational waves in biopolymers because they admit alternative interpretations, and other results are required to be checked. Third, we described many possible applications of the nonlinear conception. This aspect of the problem could be considered as very perspective, because it might give an universal mechanism which could explain the role of dynamics in the functioning of biomolecules. However, these results are

considered now only as hypothetical, because they should be confirmed experimentally. 139

Acknowledgments I wish to thank prof. Alwyn Scott (University of Arizona, USA) for useful discussions and valuable comments on the manuscript.

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143

THE GROUND STATE OF FROHLICH ONE-DIMENSIONAL FERMI-POLARONS E.A. KOCHETOV Joint Institute JOT Nuclear Research Bogoliubov Laboratory of Theoretical Physics

141980 Dubna, Moscow region, Russia We study a one—dimensional system of N electrons in an ion crystal described by the Frohlich N—particle H a m i l t o n i a n . The formation of a system of two particles with opposite spins, i.e. Frohlich bipolarons is shown to be the most energetically preferable in the limit of strong electron—phonon coupling.

1

Introduction

Recent attempts at the microscopic interpretation of high— temperature superconductivity (HTSC) have aroused interest in the model of polarons of a large radius, i.e. the Frohlich polarons. A superconducting state in this model is realized as a Bose—condensate of bipolarons.1 Since a polaron arises only in the strong electron-phonon coupling limit, the properties of the polaron collective state resulting in superconductivity can be studied on the basis of data on the ground and weakly excited levels of a set of N electrons strongly correlated by a common phonon cloud. However, in view of obvious technical difficulties, one usually studies the properties of a single bipolaron, assuming its parameters to determine the properties of the collective motion such as the temperature of transition into a superconducting state.1'2 As for the mechanism of bipolarons formation and properties of the spectrum of elementary excitations in the vicinity of a bipolaron condensate, these problems still remain to be solved (waiting for their investigation). The ground state of a strongly coupled one—dimensional polaron system is shown in this note to be a condensate of bipolarons. We prove this statement within the variational approach typical of the problems with strong coupling in which the results are obtained by comparing upper estimations of energies. It should be pointed out that bipolaron clusterization of the wave function arises from strong effective repulsion caused by the Fermi-statistics of particles. Direct Coulomb repulsion is a weaker effect and does not influence the qualitative picture. For the Bose—statistics of polarons, it is just the Coulomb repulsion that provides the energy stability of multipolarons, the latter having a different number of particles. Note also that the determination of the ground state of an n dimensional 144

polaron system for n > 1 is technically a more complicated problem since the degeneration of eigenfunctions of the approximating Hamiltonian grows rapidly with n. The one-dimensional motion is considered as motion in a distinguished direction in a 3 dimensional crystal. Such an isolated direction can occur when a strong electromagnetic field is applied to a sample. 4'5

2

Hamiltonian of the system

The Hamiltonian of a system of charged particles in a quantized scalar field in one dimension is of the form

Here we introduce the following notation:

• V"(£) = V'(r( a! ) i s a quantized Schrodinger field with excitation quanta describing charged Fermi-particles; the z-component of spin a takes the values ±1. The particles are localized in one dimension:

• ak and a*, are quantized amplitudes of the phonon field corresponding to the optical branch of lattice vibrations with frequency uk and wave vector k:

• when linear dimensions L —v oo, we have

In what follows we will use dimensionless units, setting u>k = u> = m — 1. Then, in accordance with Ref. 6, we can put

where a is a dimensionless coupling constant.

145

To take account of the effect of polaron structure in the first approximation, we consider the Lee-Low-Pines transformation that dresses the operators i/> in a phonon cloud:

where

Here /j. and /£ are c-numbers to be determined from the condition of energy minimum. The {/-transformation separates the component representing an effective potential in the first approximation:

from the quantized amplitudes oj. and besides,

where «/ = ds/dx. Since the transformed operators V1 (£) anc' lACO describe the production and annihilation of structure excitations (polarons), it is natural to expand them into the complete set of square—integrable functions on the real axis:

where {i(x)} is the set of functions complete on (—00, oo), and

We take the system {$i} to be a set of eigenfunctions of a harmonic oscillator with frequency Q that will be considered as a variational parameter:

Then operators a\a and a^a are operators of creation and annihilation of excitations from the vacuum state, i.e. the ground state of the Hamiltonian

146

As a trial ground state of the system Eq. 1, we choose the state

where \N > is the ground state of the Hamiltonian Eq. 3 for a fixed number of particles, N, and |{ajb} > is a coherent state of the phonon field; the functions ak will be determined from the condition of energy minimum. Thus, averaging Eq. 1 over the trial state Eq. 4 we obtain the ground-state energy as a function of fi, a/, and /£.

3

Formation of bipolaron Bose-gas

When particles obey the Bose-statistics, the state \N > is a condensate of N particles into the ground state <^ o (z). For Fermi-particles, the state \N > is a result of subsequent filling of states in accordance with the Pauli principle:

where N = 2k and the state |(Ti)i > is the i-th excitation of a one-dimensional oscillator with energy f^ filled with two particles of opposite spin projections.

(For structureless particles, states are filled with momenta starting from zero to the Fermi—momentum pp •) Averaging Eq. 1 over the states Eq.4 and minimizing the resulting expression with respect to o/t and fk, we obtain the upper estimation of the ground—state energy

where we have neglected direct two-particle repulsion and introduced the following notation:

Calculating the averages in Eq. 6 with Eq. 5 and minimizing the result with respect to fi, we obtain in the weak coupling limit

147

which signifies the independence of polarons in the first in a approximation, and in the strong coupling limit

where

and

is a generalized Laguerre polynomial. As an example, consider systems of two and four particles. Taking into account that

we find from Eq. 7

Since the ground—state energy of a one—dimensional polaron, according to Eq. 6, equals

it can be seen that

but

Thus, the state of two uncoupled bipolarons is energetically more preferable than the bound state of four polarons. In the case of interest for physics, when the number of particles is asymptotically large (N —<• oo), using the Trikomi expansion for Laguerre polynomials, we obtain

148

where J\(x) is the first-order Bessel function. Taking into account that

we find from Eqs.

8 and 7

i.e.,

Therefore, the bound state of the system of 2ifc Fermi—polarons is energetically instable with respect to the disintegration into k free bipolarons. Thus, the effect of strong electron-phonon interaction in the one-dimensional Fermi-system of polarons results, in the first approximation, in the formation of the Bose—gas of bipolarons. The spectrum of elementary excitations determining physical (macroscopic) properties of the system should now be determined with respect to a new vacuum, i.e. the condensate of bipolarons.

1. D. Emin, Phys. Rev. Lett. 62, 1544

(1989).

2. D. Emin and M.S. Hillery, Phys. Rev. B 39, 6575 (1989). 3. E.A. Kochetov and M.A. Smondyrev, Tear. Mat. Fiz. 85, 74, (1990) [in Russian].

4. E.A.

Kochetov E.A.

et al, Z. Phys. B 89, 177 (1992).

5. M.A. Smondyrev et al, Europhys. Lett. 19, 519 (1992). 6. F. Peeters and M.A. Smondyrev, Phys. Rev. B 43, 4920 (1991).

149

MEAN-FIELD THEORY OF THE SOLVATED ELECTRON AND DIELECTRON STATES G.N. CHUEV Department of Quantum-Mechanical Systems, Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 14S29S, Russia The self-trapped electron and dielectron states are considered in a disordered medium. The model is based on a variational mean-field estimation of path integrals. As a result, the problem is reduced to evaluation of the Shrodinger-like equation. The asymptotic behavior of the effective potential of this equation is investigated for the aolvated electron as well as for the dielectron (bipolaron) state. The statistical treatment of the environment allows us to consider the effect of environment pressure, temperature and added salts concentration on the electron states. These effects are estimated for the nonadiabatic electron transfer and for the shape of

the absorption spectrum, for its half-width and maximum. The model is compared with other theories of the self-consistent trapped electron states in disordered solidstate and liquid systems. The possibilities of the model are discussed for a wide range of problems of the solvated electron theory.

1

Introduction

The solvated electron problem has been studied for decades. At present, there are techniques allowing one to obtain sufficient experimental data on the solvated electron behavior in different media l. The solvated electron is also the object of intensive theoretical modeling (see reviews 2 ' 3 and references therein). The review is devoted to a solvated electron model, namely, a mean-field theory of the solvated electron and dielectron states in a disordered medium. Note that approaches based on the self- consistent state concept are actively investigated for an excess electron in ionic crystals and polar semiconductors (polaron and bipolaron models) 4 ' 5 , in heavily doped semiconductors 6 ' 7 > 8 ' 9 ) in dense gases 10, in nonpolar fluids ll . 12 > 13 . 14 ] m molten salts 1 5 > 1 6 ) and for the hydrated electron 17. We focus here on the original interpretation of the concept 18 and compare it with other methods. The developed solvated electron models are briefly surveyed in Sec. 2. The Sec. 3 deals with the statistical treatment of the electron state. The problem is formulated in Sec. 3.1 where the terms describing the solvated electron state are defined. The statistical description of the electron environment is examined in Sec. 3.2 where an effective action of the electron is derived by correlation functions over the medium density. In Sec. 4 a key mathematical method based on the reference action method is formulated. The general approach of the

150

method is described in Sec. 4.1. in Sec. 4.2 it is used to derive the Shrodinger equation for the solvated electron state in the molecular fluctuating medium. The physical nature of the effective interaction potential between the electron

and the medium is investigated in Sec. 4.3 where the semicontinual treatment of the potential is given. The mean-field consideration of the dielectron (bipolaron) state is studied in Sec. 5. In Sec.6 the model is used to estimate the electron transfer and the absorption spectrum. Particularly, the shape of the

absorption spectrum is studied in Sec. 6.1. The effect of environment temperature, pressure, and added salts concentration on the solvated electron state is estimated in Sec. 6.2 for the gaussian form of the absorption spectrum. The nonadiabatic electron transfer is examined in Sec. 6.3 on the basis of the mod-

el. In Sec. 7 our theory is compared with other models of the electron state in disordered solid-state systems (Sec. 7.1) and in liquids (Sec. 7.2). The conclusion is given in sec. 8 where the possibilities of the model are briefly viewed. The appendices with deducation of the main formulas are given at the end of the paper.

2

Models of the solvated electron

In terms of theoretical and computer modeling of solvated electron, the following approaches can be highlighted: a.) Polaron models. The main part of these models is based on the Pekar approach 4 . It should be noted, that this model was the first to be used in the russian scientific school 3 > 19 ' 20 . The model is based on the assumption that the self- localized electron state is due to the strong interaction between an excess electron and the polarization field induced by the electron. As a result, the self- consistent

electron state forms. This state is determined by macroscopical parameters of the medium, i.e. by the medium dielectric constants. The advantages of the model are its physical simplicity and clearness. However, its application

to real systems is very difficult, since the model does not consider the detailed molecular structure of the electron environment. The polaron models cannot also be used for the correct estimation of the thermodynamic effects (pressure, density of the medium, etc.) on the electron state. b.) Models based on the quantum- chemical calculations. The models used different modifications of quantum- chemical methods (semiempirical 21, ab initio 22, cluster 2 3 ) to calculate the localized electron state and its molecular environment. The models seem to be good for the detailed molecular structure effects. But the dependence of the electron state on the environment can only be obtained numerically, which does not allow

151

one to define the physical picture of the electron behavior. The calculation of

thermodynamic effects is likely to be impossible for these models. c.) Semicontinual self-consistent models. These models ' are the combination of the two previous types of approaches, they treat the discrete molecular structure of the environment (coordination number, size and charge of the environmental molecules) as well as the self-consistent polarization of the medium as a. whole, induced by the excess charge. Moreover, the short-ranged part of the electron- medium potential has the cavity form, while the long-ranged part is considered in the polaron approximation. As a rule, these models estimate the thermodynamic effects in accordance with the experiment 2 . The shortcoming of these models is a set of semiempirical parameters (radius and depth of the cavity), whose definition is rather arbitrary. d.) Models based on the path integrals treatment.

These models calculate high- dimensional integrals statistically and based on the Monte Carlo method. They are actively investigated at present 25 ' . Recently they have been applied to the bipolaron (dielectron) state 27 . There is a number of the methods allowing one to estimate correctly the expected values determining the state of a quantum particle in the classical environment. Since the environment is considered as a discrete molecular structure, the quantumchemical methods can be incorporated into the calculation scheme. The thermodynamic effects of the environment ( pressure, density) can also be also calculated within the method. But these models require modern high- performance computers and huge volume of calculations. Method does not enable to estimating the influence of numerical discrepancy in the input data on the results. e.) Mean-field models of the solvated electron. The basic idea of the mean-field models is to combine the merits of the two types of the approaches, i. e. the physical simplicity and clearness of the variational estimations for the polaron model and the possibility to consider statistically the detailed molecular structure by the path integrals. The idea was first stated for the solvated electron in 28 as the RISM-polaron theory. This theory was used as a base in numerous works 11'12'14'15'17'29'30'31. The liquid structure is treated by the reference interaction site model (RISM) in these

works. Another type of the mean-field models is connected with the solution of the effective Shrodinger equation 13.18.32'33 for the ground solvated electron state. We present here the original interpretation of the model 18 . The initial formulation of the problem is defined by the pair- wise inter-

action electron-isolated site potential allowing one to treat the environment microscopically. The isomorphism of the path integrals in quantum-mechanical

152

and statistical statement of the problem (see Fig.l) permits one to use the correlation function formalism and to include into consideration the quantum effects as well as the statistical influence of the medium. The simple mean-field variational estimations reduce the problem of the self-consistent free energy functional, which results in the Shrodinger-like equation for the electron wave functions. The application of the model may be wide and include the calculation of the absorption spectrum ' as well as the dynamics of electron mobility 12 . The comparison of the model and other theories of the self-localized electron in disordered medium is given in Sec. 7.

3 3.1

The statement of the problem Formalism of statistical treatment

The quantum particle state is defined statistically by the density matrix /j(r, r',/9) depending on temperature T = I//? and denoting the transfer probability from the state r to the state r' . The density matrix also depends on the time t for non equilibrium states. If we know the density matrix, we can calculate different averaged values

determining the electron state: characteristic size of the electron state, its averaged kinetic and potential energy, the influence of the weak external field, the electron mobility and magnetic susceptibility, the form of the absorption spectrum, etc.. One of the main quantities of the electron state is the partition function, which is a convolution integral of the density matrix

Below we derive a set of formulas for the partition function but not for the density matrix. It is due to awkwardness of the expressions for the density matrix. For quantum systems, the density matrix can be considered as an infinitydimensional path integral 34

153

here S is the action of electron, and symbols denote

In general, the problem of the path integral numerical calculation for a quantum particle can be reduced to the isomorphic many- particle problem of the ring polymer35 (Fig.l). There are effective computational methods for this problem, which perform high-dimensional integration by the Monte- Carlo or quantum molecular dynamic method

Figure 1: Isomorphism of problems of electron and ring polymer in solvent. Black circles denote electron density distribution and the corresponding ring polymer structure, while large circles present surrounding solvent molecules.

When the electron interacts with environment, the density matrix depends on the environment state. If we consider classical fluids as a medium, the partition function Z of the solvated electron in the classical fluid can be written as

154

the usual configurational integral over medium coordinates R n and the functional integral over the electron coordinate T(T). Note that in the general case the environment must be taken into account. We suppose that the degrees are connected with the displacements of the closed electron shells of the molecules. The displacement due to the shell polarization also offered for simplicity and does not restrict the method. Then, the partition function Z can be written

Here Se is the action of the electron in the medium, which can be calculated if we know the electron- medium potential, and M; is the dipole momentum of the i-th particle of the fluid, which is due to the displacement of the electron shell. We divide the interaction between fluid particles into the classical part U0, which is independent of the electron shells displacement, and the action Sjy| attributed to the quantum degrees of freedom

The first and the second terms are the kinetic and potential dipole energy respectively, while the last term is due to the interaction between dipoles. Here, we consider the simplest Drude model of the electron shell, which imply that dipoles oscillate with equal frequency o>o. We assume that the instantaneous part of the dipole equals zero, the inclusion of this part into consideration

does not change the result but complicates the derivations of formulas. The last term of (6) is proportional to the dipole- dipole operator T(r) = ®(T —

p f c )VV|r|

, where 0 is the step function and r^ is the hard core diameter of

a molecule. We also divide the electron action Sc into two parts, the first one Sj^c is independent of the electron shell displacements, while the second Sf^e is due to the interaction between the excess electron and closed electron shells. In general, the latter can be intricately connected with the dipole momentum of the fluid particle, but we assume that it is proportional to the dipole momentum

155

The action Sc of the electron in the medium can be calculated if we know the electron- medium potential. We present the potential as a sum of the pairwise electron-isolated site interactions. In Eq. (7) m is the electron mass, and u is the pair- wise potential, which can be estimated by scattering at an isolated atom. In some cases, approximations of the potential, such as the hard sphere potential can be used. In fact, u in Eq. (7) can depend on the additional indices denoting the sort of molecules or the type of sites for the same molecule. To simplify the formulas, we drop these indices. All the expressions can be easily generalized. We will be interested in the case when there is a gap between the energy of the excess electron and bound electrons. In other words, the state of the solvated electron is different from the states of closed electronic shells (the relation with the band structure in liquids will be discussed in Sec. 7.2). Thus, our problem is transformed to the calculation of the density matrix for the solvated electron. On the whole, this problem consists in the evaluation of the integral (5) having N3 classical and (JV + 1)3 quantum degrees of freedom, but the solvated electron state differs from the bound electrons. The dimension of (5) is too huge, thus the theoretical problem is to reduce the dimensionality of the integral, retaining all the interesting physical properties of the system. In some cases, it can be done by extraction of the degrees of freedom, which strongly interact with the solvated electron. In other cases, the system reduction is due to the choice of action form allowing to integrate (5) analytically for some degrees of freedom (for quadratic form of the action ). Sometimes, it can be possible due to low dimension of space (see 8 ). From the theoretical viewpoint, the successful employment of the path integrals method in statistical physics is due to the fact that the production functional of the system can be obtained in closed form by the path integrals37. By this, its functional derivatives determine a set of equations for the correlation functions of the system in the compact form. This treatment allows one to make a theoretical analysis of the statistical behavior of the system. 156

3.S

Description of medium by density correlations

The form of the electron action (6) is quadratic over dipole momentum. Therefore these degrees of freedom can be analytically excluded, and action (6) can be reduced by the effective potential (see Appendix 1)

-j

where «^, teraction

is the nonlocal effective potential due to the electron - dipole in-

Here A±j are the Fourier elements of matrix

Thus, the problem of the solvated electron in the quantum polarizable fluid can be treated as the problem of the electron in some effective fluid with potential (9). It should be noted that effective action (9) is similar to the electron- phonon interaction. (If the medium interaction potential Ug is harmonica!, phonon coordinates can be introduced. Assuming the linear

dependence of the electron- particle potential on the phonon coordinates, we can obtain the Pekar-Frohlich hamiltonian 4'38 from (9).) The next stage of the reduction is a short-cut description of the medium by a set of correlation functions. Let us introduce the equilibrium correlation functions of the density medium Xn defined as 39

Here Xi (R.) >8 tne average medium density, while Xz(H-i R-') denotes the densitydensity correlation function connected with the structure factor. The correlation function can be obtained from the analytical statistical theory of liquids or 157

by molecular modeling . Some information about this function can be found experimentally for simple liquids. We can determine Xs(K-i H/> R.") in a similar way. Figure 2 plots schematically the distance dependence of density-density

correlation functions and potential u(r~). This potential includes short-range repulsive part u, and long-range attractive part u\. We introduce the generalized Mayer functions / of the first order and / of the second order for the electron,

Then, the partition function can be transformed (see Appendix 2) by effective action S e , which only depends on the electron path and the density correlation functions of all the orders. In general, we may consider such higher correlation functions over medium density as we need. But in many practical cases, one restricts the molecular structure treatment by the second order cor-

relation functions, and assumes that xs = • • • = 0. This is common practice in statistical physics 39>4° to reduce the dimension of the studied system. As a result, we obtain

In expression (14), the symbol * denotes the convolution integration

158

Figure 2: Distance dependence of density-density correlation functions (a) (1 for gas, 2 for liquid, and 3 for solid state) and potential u(r) (b).

Formulas (9)- (14) determine the effective interaction between an excess electron and the medium by the correlation functions of the medium density, which are the statistical quantities while the electron- particle potential

is determined by the quantum-chemical calculations.

In some problems of

disordered systems 8 , the electron - medium potential is the statistical quantity, and its different approximation are used . From our viewpoint, relation

(14) is more convenient, since it uses independent characteristics of the medium and electron-particle interaction. Therefore, they can be defined within independent approaches. The density correlation functions can be obtained

by statistical theory, while the pair- wise electron- particle potential may be gained from quantum- chemical calculations. Thus, if we ignore nongaussian correlations, we may reduce the dimension of the configurational integral to the two-dimensional integrals. As a whole, our problem is to calculate the path integral (14) where effective action is 159

nonlocal in time and space.

4 4-1

The effective potential for the solvated electron Formalism of the reference action

In fact, the integral of the type (14) is analytically calculated only for a special form of the action . There are a few mathematical methods allowing one to calculate the path integrals. These are the perturbation methods and the saddle point method for functional integrals. Both the methods are based on the formalism of the reference action 34 . Using a reference action So, we can estimate the partition function as

where the symbols denote

If the partition function Zg(Tn) is known, we can include the high order corrections using the perturbation methods and thereby ascertain the inequality. The variational estimation of (15) is another approach analogies to the saddle point method. Relation (15) is valid for any real 5 o (r n ), however, the estimation of the partition function by (15) will be optimal, if we fit the parameters Fn of So to maximize the right side of (15). Hence, we can replace the order of integration by an electron paths and medium coordinates and reduce the problem to calculation of the averaged Mayer function

We can define < /(R)/(R') > and < f(R,R') > in a similar way. Usually, the quadratic form of the reference action is used 34 > 36 ) it enables ones to calculate the integrals (16) analytically. In the case when the electron strongly interacts with the medium and forms a localized state, the electron state can be calculated in the mean- field approximation by the reference action which is determined in self-consistent manner, (see Fig.3).

160

Figure 3: Instantaneous configuration of an excess electron in polar liquid obtained by the Monte-Carlo method: delocalized (a) and localized (b) states. Dark circles denote electron distribution, light circles are the positions of surrounding molecules.

4-2

Mean- field approximation

It is possible to obtain the effective Shrodinger equation determining the electron state in the molecular medium by the reference action method 13 ' 18 . Let us introduce the reference action So, which is defined by the onepartition Green function G(r,r') as

where n denotes a set of the electron wave functions <j>n with energy En. In the mean-field approximation, the averaged values can be expressed by the 161

one-partition Green function (see Appendix 3). Consequently

where G =< G(r, r) >= / G(i, r)dr, and u,(r), UI(T) are the short- and longrange parts of the electron-particle potential. For the Coulomb polarizable systems, where the term proportional to the first power of uj(r) vanishes, we

should include into considerations the terms proportional to U|(r)wj(r ). If the ground electron state with ^o is not degenerate, that is /3(E0 — Ei) » 1, it follows from (19) that

To compute the averaged Mayer function / for nonlocal time-dependent potential, it is necessary to know the dynamic behavior of the system, and the frequency-dependent correlation functions of the medium. There are a few

approaches to avoid these difficulties. They are self-consistent (SCA) and adiabatic approximations (AA). The adiabatic treatment yields the low-frequency limit, when uo/E —> 0, and the electron coordinate can be regarded as an adiabatic variable. The other limit <*Jo/E ~* °° corresponds to the self-consistent approximation, when the environment can be treated as a classical polarizable fluid. As a result, ones can express the second order Mayer function of the electron as

In the self-consistent treatment we get UQ = oo.

The obtained relations are the consequence of the mean- field approximation and the definition of the Mayer function. Thus, the free energy functional can be expressed by a set of the electron wave functions determined by the

reference action S0. solvated electron

We may also estimate the entropy change r,.nt for the

162

The optimal choice of So giving the best estimation of (14) is obtained from the extremum of the functional ^j = 0, witch yields the effective Shrodinger equation for the ground and excited electron wave functions

We may obtain the continual limit of this equation, if we postulate that Xz — 0The Shrodinger equation has a simple physical nature. The effective electron- medium potential is denned by the sum of the electron interaction

with all the particles Vfj (r) ~ / dRp(R, )u(r — R), however, the effective medium density results from the mean field induced by the electron. From this viewpoint, our model is one of self- consistent field models. Similar Shrodinger

equations were obtained both in solid-state and liquid disordered systems (see Sec.7). It should be noted, that although in these cases the steady electron states are described by the same equations, the dynamical behavior of the electron localization may be different. In the solids there is a distribution of

localized electron states and the main contribution to the free energy functional is due to the electron state described by (23), moreover, the medium density changes insufficiently. While in liquids the electron induces large fluctuations in the medium density, and it is self-localized on these local fluctuations. If the interaction between electron and medium particle is week and

ft < u >«

1, Eq.( 23) has the instanton form 4 1

This form is well known in the field theory, for ordered systems it corre-

sponds to the polaron problem 4'5 (the excited states of polaron were investigated i n 4 2 ) , while in disordered medium it is obtained in the problem of the electron localization in gaussian random field (see Sec. 7.1).

Thus, we have reduced the calculation of path integral (9) to the investigation of the Shrodinger equation. The solution to the problem can be obtained by direct numerical calculations or by variational estimations. Both 163

the methods have conventional mathematical supplement and do not require huge computing resources. As a result, it can be found how the electron state

depends on the environment. The examples of the estimations will be given in Sec.6. 4-3

Semicontinual treatment of the effective potential

Eq.(23) determines the self- consistent effective potential of the electron being in the equilibrium in the molecular medium. The long-ranged and shortranged parts of the potential coincide with the results of the Semicontinual theories2. Particularly, it can be found that the shot- ranged part has a cavity form V,j (k —» oo) = Vo where Vo is the cavity depth. On the other hand, we can find for Ve/ (k -* 0) that

where u,(k) denotes the Fourier transform of the Coulomb part of the electronparticle interaction. The expression in square brackets is connected with static dielectric constant e of fluids 40 .

while the second term of (24) can be expressed by the high-frequency dielectric constant too for our model

In the self-consistent limit, we should use e^ = 1. Therefore, the effective potential of the solvated electron in the polar disordered medium has the following asymptotics

Considering the Debye screening for an electrolyte, we can obtain the result similar to 2O. The difference between the self-consistent field and adiabatic treatments of the solvated electron in a polar fluid was discussed in 43. In the simplest case, this difference can be taken into account by replacing c = 1 — - with the Pekar factor cc} = — - -. J

«oo

€

164

Simple joining of asymptotics results in the semicontinual effective potential 2 , which includes cavity and polaron tails

In essence, the semicontinual treatment is the approximation of our potential with additional phenomelogical parameter, i.e. cavity radius (see Fig.4). Using electron distribution, we can calculate also the electron-solvent correlation function ge,(r), which mean the probability to find a solvent molecule at distance T from the center of electron localization. Figure 5 presents the comparison of the simulation data with the mean-field calculation for ^.(r).

In our statistical formulation, this parameter is not necessary and the form of the potential can be calculated from the microscopic consideration of the medi-

um, i.e., from the electron- particle interaction u(r — R) and the correlation functions of the medium density XiiXl-

Figure 4: Effective electron-medium potential. Solid line corresponds to the mean-field model, the dashed one to the semicontinual treatment, triangles present data of quantum molecular dynamic simulations.

165

Figure 5: Comparison of the simulation data (line 1) with the mean-field

calculation (line 2) for electrron-positive ion correlation function ge+(r) in molten salts.

5

Mean-field treatment of the dielectron states

The dielectron state in liquids has also been the subject of experimental and theoretical study (see references to earlier works in 4 4 ). Although there are experimental data indicating that spin pairing occurs in metal- alkali halide and metal-ammonia solutions 45'46, the theory is not yet as comprehensive as experiment. Recently, the success combination of the path simulation tech-

nique and the density functional approach 2r has stimulated the interest in the problem. We sketch here the employment of our model for the dielectron problem. All the above consideration can be extended to the dielectron problem. In this case we should include into consideration the electron-electron interaction.

166

Then, the action of the dielectron problem can be written as

Here the second term represents the electron-electron interaction due to the

Coulomb repulsion. Repeating the above derivation, we express the effective action of the dielectron as

where fdi, fdi are respectively the generalized first order and second order

Mayer functions for the dielectron

To reveal the physical nature, we consider only the ground state of the localized dielectron. Then using the mean-field approximation, we can present the averaged Mayer functions of the dielectron as

where $(r, r') is the dielectron wave function of the ground state. Here we take into account the physical equivalence of both electrons. As in the one-electron case, the complete physical nature of the dielectron is determined by the frequency-dependent correlation functions. The second

167

order Mayer function fn of the dielectron can be evaluated by the wave function $(r, r') only in self-consistent and adiabatic limits. As a result, we can obtain

Here the first term in the exponent of (35) represents the one-electron correlation, while the second is due to the two-electron correlations. The free energy for the dielectron formation can be expressed as

As with the electron problem, we can reduce the functional to the biinstanton form for the week electron-particle interaction

where r?(r, r') is the potential due to the charge of the bi-instanton, while Wj corresponds to the nonlocal self-interaction

The continual limit of the functional yields the standard bipolaron

functional 48

168

In the self-consistent limit, (.&> is equal to unit. In principle, the investigation of the functional (36) and the related Shrodinger-like equation for bipolaron state requires the relation between oneelectron and dielectron wave functions. Different approaches can be used for this purpose. Direct variational method uses ia special variational form of the wave functions 48 . Another method considers the dielectron correlations by the Kohn-Sham formalism and to use the density functional model for the spin pairing problem. The semicontinual form of the functional was studied in 44 , where the shortrange cavity form and bipolaronic tails of the effective potential were assumed. In our model it is a natural consequence of the microscopical consideration. Recently, the analogous form of the effective potential for the bipolarons has been obtained by path simulations 27 , where the peanut form of the dielectron distribution was found.

The adiabatic limit of functional (39) is well known 4 and is investigated for different types of the bipolaron wave function by the variational method. The problem of the bipolaron stability in the presence of short-range and longrange interactions was studied in 4 9 ' 5 0 . Without considering the problem as a whole, we point out that the presence of the sharp cavity-like short-range potential should increase the stability of the bipolaron formation.

6 6.1

Application of the model The shape of the absorption spectrum

In general, the absorption of light is defined by the dynamics of the electron behavior as well as the medium state. However, we suppose that the time of the electron relaxation exceeds greatly the relaxation time of the environment, and the medium adiabatically follows the electron. The probability of the light absorption W(ui) can be presented as integral over complex variable 9 = it/h

Here u is the frequency of light, while dij is the matrix electron-photon element for the initial -final states transition, and F{, F] are the initial and final timedependent free energy functionals, respectively.

Note that the electron and medium states are strongly correlated, they are connected by common relations determining the free energy surfaces of the 169

initial and final states. The problem of the medium influence on the absorption spectrum reduces to computation of integral (40), with taking into account these correlations between the electron and environment states. The difference between the light absorption problem and the effective potential potential is the complexity of the variable 9. All consideration of Sec. 4 can be repeated, but the analytical continuation is needed for Ff(0), Fj(6) 31. As a result, we can obtain

We will show below, that 2?, are connected with the reorganization energy. Although the general expressions are are easily calculated, we present here the continual limit (x? = 0) °f these expressions to save room

where e = 1 + /, e = 1 + /, and subscripts in averaging brackets denote the averaging over initial and final states, respectively. For a weak electron-particle interaction, the above equations are transformed to the conventional expression of the reorganization energy

where E° and E* is the short-range and long-range parts of the reorganization energy. There are two limiting cases, determined by the value S = ET(3 1. S » 1 (Strong coupling)

170

Integral (40) can be estimated by the saddle point method The main contribution is due to the vicinity of the saddle point and the spectrum has a gaussian form

Here Wf:max,Aj are the maximum frequency and half-width of /-th line.

2. 5 «

1 (Weak coupling).

In this case it is necessary to consider the range of the integration where |7m0| is large. But therein, 8F(9) can be presented as a linear over 0 and the

spectrum has the Lorentz form

6.S

The environment effect on the maximum and width of the absorption line

In general, the absorption spectrum should be calculated numerically by the Shrodinger equation (23). To demonstrate the possibilities of our method, we estimate how the medium state (its temperature, pressure and concentration of added salts) affects the maximum and with of the gaussian absorption spectrum of the solvated electron 18. Note that for the Coulomb potential

Here Uei is the potential electron energy. The latter follows from the virial theorem. For the polaron potential one can obtain that h<jjmax = |(7|f/e||, where

C ~ 1 is the numerical coefficient, which is of the order of unity. The shortranged interaction can slightly reduce the coefficient, C ~ 0.5-=-l. Despite this fact we can put ( 7 = 1 for the estimation. a.) The temperature effect. It should be pointed out that the dielectric constant weakly depends on the temperature. Neglecting this dependence, we can get

171

In this estimation, we also neglect the dependencies of the cavity radius and density on the temperature. Substituting Vo ~ 1 -=- 2 ev, we obtain

which agrees well with the experimental data 1 . For the half-width of the absorption line, we can find the known squared dependence on the temperature A ~ V T. b.) The pressure influence. From these reasons, we can find the pressure dependence of the absorption maximum.

For polar liquids with £ > 10 the last term is less then 10 % and pressure dependence is due to the density effect. The correlation between the absorption maximum and density dependencies was experimentally found in c.) The added salts concentration influence. The presence of added ions causes the Debye screening of the polarization potential. For the dilute electrolyte it can be obtained that

where TV = (e2nf))~1'2 is the Debye radius, and n is the concentration of added salts. It follows

The estimation obtained in 2 0 , correctly determines the sign of the maximum shift due to the added salts concentration. This shift is widely experimentally observed, (see for example 52 ). For the width of the absorption line, it is necessary to consider the next terms of the expansion, which is proportional to n. As a result, we have

here r2p is the size of the first excited electron state. This dependence explains the asymmetry of broadening absorption band. Since the broadening is proportional to the size of the excited state, the broadening is larger for the high-

172

energy transition la —> 3p, which broadens the spectrum in the short-length range. This asymmetry of the broadening was experimentally observed in 5 2 . For dipolar liquids Ufi ~
6.3

Nonadiabatic electron transfer

The reasons of the previous Sec. is valid for the nonadiabatic electron transfer. We should treat an integral similar to (40) and add donor and accepter - electron potential for the electron states localized in donor and accepter respectively. Considering the ground electron states only, we can find in twolevel approximation that the transfer probability is described by the following relation

and saddle point 6* is determined by the extremum of the functional SF(0"). In general, it yields a nonlinear algebraic equation for 6*. It differs from the Marcus electron transfer theory 55 where SF(6*) is a quadratic function over 0. The solution of this algebraic equation should be found numerically. However, we present some interesting considerations following from (52). First, the temperature dependence of the electron transfer rate constant is not generally presented by Arrehnius law. The rate constant does not exponentially increase with temperature but it can decrease for some parameters of the problem. This possibility was pointed out in 5 6 . Second, the derived relations of the electron transfer parameters may be useful when the calculation of the reorganization energy via the conventional electron transfer theory 55 is rather difficult. This is the case for the substances with essential contributions of short- range interactions, when the calculation or measurement of the dielectric constants is very difficult. Third, various crude approximations of the results obtained can estimate the dependence of the transfer probability on the medium parameters. For example, for molecular nonpolar liquids the electron pseudopotential can be approximated as

173

where d is the scattering length. Then, in classic approximation we may get

here re denotes the electron radius, S(k) is the structure factor and L ~

/ dr<£? n (r)<£? (r) is the exchange integral. This relation can be used for the estimation of the temperature, density and scattering length effects on the activation barrier for the nonpolar molecular medium. Fourth, it should be noted that all the results are also valid for inhomogeneous fluids. Particularly, they can be used for the electron transfer near the phase boundary or near electrodes or in the presence of an external field. There are several models of inhomogeneous fluids for which the correlation functions were calculated ; these data can be used to estimate the effect of

the medium heterogeneity on the electron transfer. 7

7.1

Comparison •with other models of the electron state in disordered medium Solid-state systems

The problem of an excess electron in disordered lattice is the simplest example of the electron problem in disordered systems 57. On the other hand, these studies have been initiated by the development of the electron band theory in ordered lattices. However, in the disordered systems the band structure has more complicated character related to new physical phenomena. A more illustrative example is the Anderson localization problem 58. A wide set of other illustrations can be found in 5 9 . The key question studied intensely in this field (see reviews ' ' 8 ' 8 ), is the

calculation of the density of the electron states (DOS) D(E) related to the oneelectron partition Green function averaged over all the impurities configuration

Thus, the problem is reduced to the calculation of the imaginary part of the Green function, and its dependence on the concentration and type of the impurities and on the electron energy. In this sense, this problem for the so-called deep tails region is analogous to our treatment. Leaving aside huge volume of investigations, reviewed i n 6 ' ' 8 ' 9 , we will compare our model with basic theoretical approaches in this field. a.)The optimal- fluctuation method

174

If the inter-impurity distance is less than the electron radius, the (DOS) can be evaluated by the averaged concentration of the impurities p(r). cording to 8

Ac-

Here F(/J) is the entropy of the medium, depending on the impurity concentration p. The variational estimations of this equation yield the extremum of the concentration Po(r)

where 7 is the Lagrange multiplier. It is determined by the fact that E, (r) are the energy and electron wave function of the ground state such as

If the entropy is proportional to the logarithm of the impurities concentration F oc lnp(r), then from (57-58) we can get equation (23) in the continual limit. This case corresponds to the Poisson range of spectrum. When the deviations from the averaged concentration po = const are rather small, the

gaussian statistics can be used and F oc p(r) — p0. For this case, the electron wave function is determined by the Shrodinger equation of the instanton type (24), which coincides with the equation obtained by Halperin and Lax 80 ' 81 . Note that the results of the approach differ from our model in continual limit only in the Lagrange multiplier 7 (in our model 7 = /3). b.) Path-integral approach. It seems that the path-integral approach was first used in e2 to calculate of the DOS in disordered systems. The method was actively studied in Sayakanit's works 63'84. It is based on the Feynman idea 65 to estimate the action by the reference quadratic action. The statement of the problem is the following. The one-particle Green function is determined by the continual integral

175

Then, the effective action is evaluated according to (15) by the quadratic

reference action

The parameter 7 is fitted to get the optimal estimation of (15). As a result, the nonlinear algebraic equation relating the electron energy E to the parameter 7(xi,ti(r)) is obtained. The analytical calculations were made in (60) where the Mayer function was expanded over the interaction potential till second order. Sa-yakanit used the method to evaluate the D(E) for the screened Coulomb potential. The effective action depending on the two parameters TI , 72 was investigated in 63 . The estimation of D(E) corrected by the higher cumulants was obtained in 6 8 . On the whole for the deep tails region, the method gets the estimations of the DOS close to hose obtained by the optimal-fluctuation theory. The only difference is in the higher energy region where the DOS is overestimated by the path-integral model. It should be pointed out that equation (60) is the continual limit of effective action (14), when Xi = const, Xi = 0> i > 1 and /3 is replaced by it/h. Thus, all evaluations obtained by the path-integral method are similar to our consideration and only differ in the reference action form. The quadratic form of the reference action is chosen in the path-integral model to make analytical evaluation. c.) Replica method. Another approach connected with our treatment is the replica method 6 7 . This approach was first introduced by Edwards and Anderson .In ' the method was used to evaluate the DOS in heavily doped semiconductors. The starting point of the method is the representation of the Green function by the gaussian integrals

here H is the matrix determining the Hamiltonian of the system. The problem is too complicated, since the Hamiltonian depending on the impurities configuration is involved both in the numerator and denominator. However, using the replica trick identity, this equation can be expressed as a multi-dimensional integral. For any functions g(x) and f(x) we can write

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where x = (z • • • zn) is the replica vector. As a result, the order of averaging and integration over inner degrees of freedom can be changed, and the Green function can be expressed as

To average the Green function, the data on impurity distribution are re-

quired.

These data can be obtained from the density correlation functions

formalism as in our approach. The final result is the expression of the Green function by the density correlation functions and distribution function over replica. For the gaussian statistics, the averaging and evaluation of the multidimensional integral can be made analytically to obtain the instanton type (24) of the equation for the electron wave function. The numerical calculation of (24) was obtained in 71 for different distributions of the correlations. The polaronic effect is neglected in these methods, and the medium density is assumed to be unchanged. The influence of both the polaron effect and the medium disorder was first examined in 2. and bipolaron effect was investigated in 73 The polaron and bipolaron were shown to be localized and more stable due to disorder. The polaron state in disordered medium was investigated 74 by the path-integral method. The polaron effect was discussed in detail in 75 where it was shown to result in an additional peak of the DOS in the deep

tails region. In principle, the consideration of 74 ' 75 coincide with our model. The evaluation of the absorption spectrum presented there for the acoustic polaron is similar to our estimation (40), the difference is only in the quadratic form of the reference action, and simple approximation of potential wells by gaussian-like distribution.

7.2

Liquid systems

a.)Semicontinual models. It has long been known that the disorder of liquid can affect the solvated electron. The earlier attempts to include the disorder into the consideration were based on the semicontinual models, where several statistical quantities of the effective electron- medium interaction were treated as a parameter of the averaging. For example, the number of molecules in the first coordination shell was used as a fluctuating value in 7 6 , while in 77 this role was played by the polarization of the medium, the depth of the cavity was used as the fluctuating quantity in 78 . The modern version of these models includes different types of

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disorder: the radius of the cavity and the energy of hydrogen bounds in 79, the interaction potential between electron and point charges in 80. All these approaches are the limiting cases of the general formula (23) for the effective potential. b.) Density functional models The continual limit of our model also coincides with the self-trapped electron theory for dense gases and liquids10. It is shown in13 that both the models have an identical starting formulation. The key of the model is that the free energy includes the kinetic energy of the electron, the term proportional to

the chemical potential of the medium, and the term determined by the change in energy of the environment particles due to the excess field induced by the electron

Here fm is the Helmholtz energy of the medium. The variational estimations of the equation yield the extremum of the density po(r) according to (57) and the electron wave function determined by (58).

In these models the pair correlations between the medium molecules are ignored or treated as small corrections to the free energy functional F and to the effective potential Vej , while the pair-wise interaction is represented by

pseudopotential (53), which is generalized to the case of Van der Waals fluid in l0'81. Another generalization includes surface interaction effects. c.) RISM and other related models It was pointed out that the RISM- polaron theory was a starting point of the mean-field models for the solvated electron. This theory uses the direct correlation function cc(r) of the electron instead of the Mayer function. These functions have the same long-range asymptotics, but differ significantly in the short-range region. As was emphasized in , the difference weakly affects the solvated electron state, if the state is rather diffusive. The direct correlation function is connected with the electron-isolated site potential u by some additional relations. This brings up a necessity to compute the additional RISMlike integral equation

where he = gf — 1, gc is the pair electron-solvent distribution function, and w is the response function of the electron

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To solve the equation, the additional closing approximation of type c(r,<) = exp[—j0u(r)]/(/i s (r) is required. The RISM-polaron model uses the meanspherical approximation. The resulting effective action of the electron is evaluated by reference action of type (60) with a set of fitting parameters 7i, • • • , 7 n The variational estimation leads to the set of the nonlinear algebraic equations connecting fn and the electron state parameters. The problem of the electron state is reduced to the calculation of a nonlinear algebraic set of equations for the parameters -yn (force constants). These equations are calculated by some iterative procedure. Note that the use of the quadratic reference action allows one to estimate the localized electron state as well as quasi- free electron states. The theory of the RISM-polaron model was considered in 28 . Numerical calculations for hard sphere liquids were presented in . The theory results were compared with the simulations in 11 . I n 2 9 , the model was generalized to polarizable fluids, and in 31 to the real time electron behavior. The method was used in 12 to calculate the electron mobility. The RISM- polaron theory was used for the hydrated electron in 17 , for calculation of the electron state 15 and its absorption spectrum 18 in molten salts. The calculation of the effective electron mass was obtained in 14 for nonpolar fluids. The connection between the mean-field treatment and the RISM-polaron theory was investigated in 13 for classical fluids. Another treatment, which interlies between our and the RISM-polaron consideration is the work of Zhu and Cukier 32 ' 33 . The Shrodinger-like equation was also obtained for electron state in nonpolar 32 and polar fluids 33 in the mean-field treatment. In contrast to our model, the effective potential in this equation was expressed by the electron-solvent distribution function. Thus, an additional relation connecting this function and pair-wise potential is required. The Ornstein-Zernike-like integral equation was used as this connection. The calculations based on this model were compared with simulation. d.) Theories of band structure in liquid In contrast to our case, the problem is reduced to the investigation of collective excitations (band structure) for topologically disordered systems, when the excess electron energy is rather close to the energy of bound electrons. This problem was repeatedly considered (see reviews82 and references therein). The result is the calculation of the self-consistent equation for the one-particle Green function connecting the distribution function