Electron-phonon optical resonance and vibrational chaos in molecules

Physica A 197 (1993) 260-274 North-Holland SDZ: 0378-4371(93)E0033-B

Electron-phonon optical resonance vibrational chaos in molecules

and

V. S . Yarunin Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 1OlaK) Moscow, Russian Federation

Received 14 September

H.P. Office Box 79,

1992

The electron-nucleon interaction of different kinds in various molecules is considered. Two types of optical resonance, according to the cases of weak and strong electron-phonon interaction in comparison with the electron-photon one, are found for the two-atomic molecule. The strong nonlinear electron-phonon interaction via the soft mode of the deformational phonon for a multi-atomic molecule is considered. The optical electron absorption is found to produce the vibrational chaos in the molecule in this case.

1. Introduction

The electron-phonon interaction is a fundamental part of condensed matter physics. Various types of it are represented in molecules. A special kind of phonons, deformational and librational phonons, exists in multi-atomic molecules. The frequencies of these phonons are much smaller than those of ordinary “valence bond” movements of nuclei. This smallness of the phonon energy quanta leads to intensive interphonon energy exchange in the molecule. The increase of the influence of phonon nonlinearity is the result of the presence of such a “soft mode”. The experiments on the absorption and luminescence give important information about the electron-phonon interaction. The theory of this interaction in molecules, which leads to the phonon chaos, is presented in this paper. The next section of this paper considers an electron-phonon optical response of a two-atomic molecule in the case of weak and strong electron-phonon coupling as compared with the electron-photon coupling. The critical behavior of a multi-atomic molecule via the infrared multi-phonon absorption is given in the third section. The fourth section considers the randomization of the phonon system of such a molecule during the optical electron-phonon transition in the presence of the deformational “soft phonon mode”. The functional 03784371/93/$06.00

(CJ 1993 - Elsevier Science Publishers B.V. All rights reserved

V.S.

Yarunin

I Electron-phonon

integral method for Bose- and Fermi-variables paper.

2. Electron-nucleon

optical resonance

optical resonance

261

is applied in all sections of this

in two-atomic

molecules

Let a molecule have two electronic levels E1,z, the electron transition between these happening due to the dipole interaction A with the external field 9 = t&(t) exp(-iw,t), Im & = 0, E, +

w(c++ A)(c + A) wJ*

[c,c’] = 1.

The frequency of nuclear oscillations is w and A is a shift of the nuclear potential curve in the upper electronic state. Molecular states are taken in the form

IA& 4 = AllO, z2) + B101, z,)

,

IAl2 + 1111~ = 1

in the basis of electron occupation numbers ln2n1), nl,* = 0,l and coherent states Iz) of nuclear oscillations. The Rabi-type oscillations of polarization and level population are a wellknown example of the two-level atomic electron response to the resonant optical field. Oscillations of the same kind for an electron in a molecule are in general destroyed by the electron-vibration interaction. However, some conditions can happen when the Rabi-type electron oscillations for a molecule in the external optical field are expected. These conditions are quite different in the cases when the electron-field interaction is much larger or much smaller than the electron-vibration interaction [ 1,2]. The matrix transition element between the initial (t = 0) and final (t) states for the evolution operator is considered. The coherent representation for nuclear vibrations is taken so that the zero-point of the z1 plane is placed at the minimum of the vibrational curve in the ground electronic state. In order to simplify the calculations we choose the initial and final states of a molecule such that A, = B,= 0,B, = A, = 1, zi = z. and z2 = z. The matrix transition element can be written in the form [2] 9 = (10, zI exp(-iM,t)lOl, =

I

zo)

9c* 9c (Tm)12 exp [‘[c*(iz-w)cdr+z*c(t)], 1 0

262

V.S. Yarunin

40) = 20

c*(t) =

7

T,=Texp

z*

I Electron-phonon

)

I4 =

optical resonance

so & IW

( -1‘[Mdr)=exp(-itM,)

9

cfclk) = klk)

)

Texp(-ijm(r)dr), 0

-=exp(ijMoir)

M, exp(-ijM,di),

The TM symbol in this formula in the case of an exact resonance, w,=E,-E,+A’q can be reduced to the form TM = exp(-itlll,)

FM, d7

0

1

I

= Texp(-i/m,dr). 0

It is clear that the TM matrix, and therefore the function 9, cannot be calculated analytically without some additive approximation. In order to choose such an approximation let us notice that the diagonal part of m, describes the interaction between an electron and vibrations of nuclei, while the nondiagonal part of no describes the interaction of an electron with the optical field. This means that the two approximations are possible depending on whether the electron-field interaction is large or small in comparison with the interaction of an electron with nuclear vibrations. The first of these cases is described by the inequality h,bo>wA. After expressing the ?,,,, matrix in the form

(2)

V.S. Yarunin

I Electron-phonon

263

optical resonance

f

FM = RT exp(-i)

R=

cos A isin A

wA(c + c*)q(T) dr,

i sin A cos A > ’

(3) 4=

-icosAsinA sin’ll

COS’A icosAsinA

(where R is the Rabi matrix), we use the first approximation tion theory ht,/~~(oA)-~)l and obtain the following formula:

in the perturba-

t 9 = [exp(e -iO*z*zo eiMo’)R(f)]1,2(1 + p) ,

A=A

I

$odr,

0

f

ei”c’-t)q dr + z.

A singularity of function p can be noticed in the case of constant amplitude JI, and equality o=2Aeo.

(4)

It means that when the condition (4) is satisfied, the resonance between the field and nuclear system of a molecule due to the electron-vibration interaction takes place despite the smallness (2) of this interaction. If the electron-vibration interaction is large,

(5)

Wo
the resonant response of molecule is also possible under some conditions for o, A and the frequency oo. In order to find these conditions, the new representation for the function FM and 9 can be taken as

FM=exp[(ii

8)9(t)]

To,

To=Texp(-ih~~oWdr), 0 f

exdW)l 0

’

9=aA

I

(c+c*)dr.

0

The approximation (5) makes it suitable to use the t-ordering decomposition To, the ( To)1,2 component of which has the form

of

264

V.S.

(T,),,,

=

t, = t

)

I 0

/ Electron-phonon

/ ei’(f~) . . .

7eie

0

0

0

(c + c*)i?(t, - 7) dr tk ’ tk+l

optical resonance

1

e-ip(tl)

(-iA)“‘+’

_;.

tp(tk) = WA

Yarunin

T

dT1

. . .

dT2n+l

,

,

k = 1,3,.

. . , (2n + 1).

Each n-term (To)1,2 @) of this decomposition produces a Gaussian functional integral over the functions c,c* so that it can be calculated exactly, while the extremals c, and c: are found from the Lagrange-Euler equation

and the canonically conjugated equation for c:. In order to simplify these formulae let us notice that the vibrational coordinate does not change if the electron-vibrational excitation of the molecule occurs. This fact can happen for molecules of the type

(6)

C.O
In this case, the matrix T, has the form

TM = exp(-Mot)

T exp(-ih)

I r,ko(0

OK)d7,

K = (-A’

- 2rriA*).

0

If we require the surplus of field quantum energy WA* over the energy of the electron transition E, - E, to contain the integer number of vibrational quanta w, the transition matrix element becomes 9 = exp[e-iWfz*zo - it(E, - E, + wA’)](-i)

sin(e-A2tA$o) .

This formula shows that the optical transition in a heavy molecule ( A+0> w) is accompanied by Rabi-type oscillations of an electron if the electron-vibration interaction is strong (WA > At,bo) and the excitation energy of the vibrational subsystem A’o is a multiple of the vibrational quantum energy o. Thus, in the case of a strong electron-field interaction (2), a small perturbation of electron Rabi-type oscillations by nuclei is found. The exception occurs in the case of the resonance w = 2AJI, between nuclear vibrations and the

V.S. Yarunin I Electron-phonon optical resonance

265

external field, so that a large part of the field energy will be transported to nuclei. The opposite case of a large electron-nucleon interaction (5), (6) in big two-atomic molecules (WA > A+,, > o) can happen if the shift A is sufficiently big (A + 1) and A* is an integer. The frequency of the electron oscillations (exp(-A*) M,) is small in this case. Concerning the value of the shift A, the two cases can be imaged. In the case of a valence bond we have A < 1 (in units of the atomic radius). In the case of a deformational degree of freedom we have A > 1 because a large number A2 of soft quanta o are excited by the optical field. A situation like that is impossible for the two-atomic molecules, but can be expected in many-atomic molecules. In the next sections of the paper we will consider such a situation and we will show the importance of inter-mode phonon anharmonism for an electronvibron optical transition.

3. The multi-phonon

infrared

excitation of a large molecule

We are going to generalize the model (1) for the case of a multi-nuclear molecule with the nonlinear interaction between phonon modes. The first step is to consider the infra-red excitation of the molecular nuclear sub-system in the ground electron state. Let us take a system of 2N + 1 phonon modes with the Hamiltonian

h, =

oJc+c +

2 [O&z,

g=l

+ o,bt,b, + .(ut,b’gc+ C.C.)]

)

which describes the resonant w = wn + wb interaction of the o oscillator with N pairs of Bose-oscillators with frequencies w~,~. Such a resonance interaction takes place in the phonon subsystem of a molecule between an optically active mode w and a number of other phonon modes w, b. We suppose the Hibbs distribution to be present for the whole system and an external field pulse j with the frequency o to be so strong that a thermal number of optical-active phonons is less than the number of them, excited by the field rz, = (em’ -

1))’ < 1jl*/o* .

If it is possible to approximate distribution

the dynamics for this system with the help of the

266

V.S. Yarunin

p=e

eiht

-i”‘Po

p.

9

=

I Electron-phonon

e-@ho)

optical resonance

p = T-’

,

wc+c +jc++j*c + 2 (Waa;a,+ w,b;b,)

h, =

(8) )

g=l

and to write the generative functional F, for Green’s functions as the functional integral [3]

F,=

I I[c* 2 +5 (a;f$ S2QeS,

I

SE-

9’Q

+bi

d$)]drijXdr,

g=l

0

= 919

9c

ff 9ag g=l

[O,t>

0

9ag 96;

9bg

,

1=2t+p,

7

9 It + P, 4 .

[4 t + P)

Here the action S describes the dynamics of the system on the sectors of time [0, t) and [t + p, I] and the initial Hibbs distribution on the sector of temperature [t, t + /3). The functional integral over the a,b variables can be calculated exactly: Fl =

I

Seff = -

9c* 9c exp(S,,,) ,

I

0

dc + i ~(1 -f’) ( dr

c* -

>

c d7 + (1 -f’)(jc*

+ i(1 -f

+j*c) - Nln(Det

M),

2)wa,b.

With the help of a scale transformation c+ fl c, c* + fl c, one can prepare Serf for the asymptotical (if N --, m) calculation of Fl and the following variational equation appears:

V.S. Yarunin

I Electron-phonon

--

iT + i[fo + i(1 -f’)w])c p+

(vR)-lp

= &

267

optical resonance

ln(Det M) - (1

-f”)j ,

.

(9)

This equation is asymptotically exact in the sense of the quasiclassical limit N-t 03 and it describes the evolution of the oscillator with frequency w which was disturbed by the field j, j* in the initial moment. Some other approximation in (9) may be done if the intermode phonon interaction p is not big (p 4 wa,b ). In this case &ln(DetM)=&spln

(

LLifCtL-i

P

6 L,‘cL,‘c* = 6c* sP 0

Gfcc)* .

(10)

The decomposition of the In into matrix series over j.~/w,,, has been used here. So we receive the linear equation instead of (9) for the trajectory c and it is suitable to introduce the new variables c_(r) = C(T), C+(T) = c(l - T) for the purpose of dealing with the sector [0, t] only,

(-&+im)y-Fij

Y(+) exp]i(w, +

Wb)(t

-

T)l

0

A

(11) 1.2 = y=c+

l/L1 -

exp(ob,ap>l

=

B,,2=11[exp(w,,bP)-11

?

-c_.

The Laplace transformation function y” are &1,2

)

$(w +

Wb

+

0.) f

y + y”may be applied to (11) and the poles of the

[$d2- &.b2(n,+ rzb)]“’

)

(12)

where d = o - o, - ob and n, b = [exp( PO, b) - 11-l. The second term in ,sl 2 looks like the Rabi frequency and depends on the detuning of the resonance ‘d in the phonon system, its nonlinear interaction p and the equilibrium occupation numbers of thermostat oscillators. The condition for e1,2 to be complex numbers,

268

V.S.

Yarunin

I Electron-phonon

optical resonance

shows that the non-stability of the optical-active mode appears if the intermode interaction and the temperature are large and the detuning is small. The oscillator’s non-stability of this kind was observed in infra-red molecular spectroscopy [4] and is called a “thermal explosion”. In the next section of this paper we get rid of the approximation of linearisation (10) and we see that the “complex Rabi frequency” disappears. The exact nonlinear motion of an oscillator has a tendency toward the semiclassical chaotisation as far as the trajectory y comes near the separatrix of the phase plane. This situation happens for electron-oscillator optical transitions, which are the subject of our following research.

4. The optical electron transition and vibrational chaos In this part of the paper we combine the formulae (1) and (7) in order to construct the model of a multi-phonon molecule with two electron states. The presence of a large nuclear subsystem and its participation in the electron optical transition makes it necessary to distinguish the phonon behavior in the various stages of this transition. A good possibility for this purpose is given by the theory of irreversibility. A phenomenological scheme of irreversibility was proposed by Schwinger [5] and Keldish [6]. It was based on the assumption that the evolution of a quantum system “forward” and “backward” on the axis of time, is described by different Hamiltonians so that the motion of a system in the closed time contour is nonunitary. A situation like this takes place in a molecule in processes of two-level electron absorption and luminescence. A microscopic analysis of this process for large molecule shows that the nonregularity of nonlinear nuclear dynamics can be induced by the irreversibility because of the electron-phonon and interphonon interactions [7]. This mechanism of chaotisation can be associated with some experiments in the molecular spectroscopy [8]. We wish to discuss the most important point: the connection between irreversibility of the electron optical transition and the chaotisation of nuclear degrees of freedom in large molecules. The main idea is that the optical electron transition in a molecule produces a sudden and nonperturbative excitation of a nuclear system. Then this excitation transforms into the nuclear irregular motion because of the nonperturbative nonlinear interaction between phonons (intermode oscillator anharmonism). So the Hamiltonian of the molecule is composed with the help of a 2 X 2 matrix of an electron system (1) and 2N + 1 bose-operators a,b,c of nuclear oscillations of type (7) in the ground (h) and excited (h,) electron states,

V.S. Yarunin

I Electron-phonon

H=(i ;)hA+(;l$h-EL

E

269

optical resonance

=E2 - E, ,

h=oc’c+g~l(a’b’)g(;yt ;I)(;),, wa,*+ww The Hamiltonian h optical-active mode interaction constant. the upper electronic

describes the resonance interaction o, - wb = o of an o with N pairs of Frank-Condon modes o, b; p is the There is a shift A of the optical-active mode coordinate in state, so that

h, = h( c-w+A,ct-tc’+A

.

The electron-phonon excitations in the weak external optical field due to the interaction of an electron with light, are described by the correlation function

K(t) = (sp /U(t)

p = eeBH ,

7

The average over the electron variables is calculated in the last term of this formula, so that the non-unitary of the evolution is present because the Hamiltonians h and h, are not equal. To avoid the perturbation theory in the Stokes shift A and the anharmonical interaction p between modes, we use the functional-integral representation for K, Z(t) =

1d2z d2zr d2,?’ (z

=

,-JZz d2=l

d2zf1

I

I

) ei*(h~-E)I

CiB2$_

z’)

!3’$

(z’

iB2*+

1 emphI

2”)

(z”

I emihtJ

2)

edwiEr,

(13)

4 = iS+(t, 0) + S(p, 0) - iS,(t, 0) , d2z = d2z, fi (d2z, d2z,),

,

d2*,

=

e-bd2

dKi;K

,

K =

a,

b, c ,

g=1

9’$,

=

5J2c,

fi

(B2ar

B2b,),

g=l

The actions S’ and S, correspond

,

CB2$ = 53’~ fi

(9’2a S’b),

.

g=1

to the motion of an oscillating system of a

270

V. S. Yarunin

I Electron-phonon

optical resonance

molecule from the moment 0 to the moment t with the Hamiltonian h, and from the moment t to the moment 0 with the Hamiltonian h,. The action S corresponds to the thermal equilibrium state with temperature p-l. The boundary conditions for the trajectories are

u*(t) = 2; a_(O) =

a*(p) =

)

z;

a(0) =

)

z:*

z;

a;(t) =

)

z;*

)

a+(O)= z, >

)

and the same for b, b,, c, c,. An exact calculation of the integral Z(t) over the variables c, c,, z,, z:, 21 can be done:

n = (ewa - 1))’ ,

S.~~=-ijrdrirYdriij%drif, 0

0

~*=f

0

[ur(i&-Wll)u~+bf(i~-~~)b=]

y g

g=l

P

F

=

dr dr’ f(r) f(r’) [n + S(T - T’)] + in eior

I

0

P

0

I

f d7

0

-ine-‘.‘\f*dr/(f, 0

(f*-

-

f*+) dr’

*

-f_)dT’ *

+fT(T) (f- -f+Lfi(T’ P

-PA

I

- T)l I

(Q+Q*)dT-ipA

r (Q_+Q!-2wA)dr,

0 Nr

z= 2

I

[a*(-

g=l

The terms bilinear in f, fk of the phase F coincide with the same terms of Schwinger [5], but the difference is that instead of Schwinger’s phenomeno-

V.S. Yarunin I Electron-phonon

optical resonance

271

logical forces P, P, we have here a closed nonlinear system with trajectories f* =

eioTQ* ,

f* =eewTQ*,

f=eO'Q,

Q, = i: g; >

Q_=wd+?

w+_bZ ,

q = pub*.

g=l

q* =

g=l

g,,

Q=f

g=l

qg,

The terms, proportional to A and A’, describe the influence of an electron transition on the nuclear amplitudes. Now the quasiclassical approximation must be used. The first step of it consists of the thermodynamical limit N+ m and the variational principle

w+ =o,

The corresponding closed and a:, b z, a*, b* can be a, and a:, b, and bz ; a, the system of equations conditions

qs+>* =0

ss, =o.

w=o,

)

system of the variational equations for a,, b, , a, b written and analysed. It can be noticed that the pairs b and a*, b* are not canonically conjugated because discussed here and a system of equations for the

tG*=0,

f&s,)* = 0

do not coincide. This result of nonunitarity of evolution disappears when A+- 0. The second step in the quasiclassical approximation is to suppose that many phonons of the optical-active nuclear degree of freedom appear through the electron transition. This condition may be expressed by the inequality A* s=-1 and can be satisfied in the case of a small frequency w that corresponds to the deformational motion of nuclei. This situation takes place in molecules of experiments [8] where the values w - 200 cm-’ and o, b - 1000 cm-’ were observed. An additional simplification must be made to analyse the system of variational trajectories. This consists of the following two (alternative) inequalities: A* > 1 > rt =

(ews- I)-’ ,

A*>n>l.

The left-hand side of both of them gives a possibility to get rid of the phonon interaction on the temperature part of the evolution and this means that the temperature should not be far from 300 K = To. A closed equation,

272

V.S. Yarunin

d*u 2 - x1,2&L(2~u,~2 dr2

&: = L,

U, =

)

.q*u: ,

u_(t) = -u+(t)

/ Electron-phonon

optical resonance

+ .$Z) = 0 7

(14)

&; = L, - n21iq2

E; = L, - A*,

)

u_(O) = -u+(O) e-@ + id, + id,

may be obtained for the generalized

nuclear amplitudes,

which are written as

u, = - lI,dr’+(n+l)j(f,-f_)dr’+A(n+l), 0

u_

=

0

If-dr’-nj(f,-f_)dr--nd+iA, 0

d”= iA(e’“’ - 1)

.

0

The same equations can be found for the amplitudes u,. The definition of x1,* is a very important point because it means the determination of the conjugation condition. This definition is very close to the solution of the Hamiltonian and non-Hamiltonian nonlinear behaviour of three bose-oscillators 191. It must be noticed that the definition of x1,* substitute the definitions of (a:)* and (br)*. In a particular case of high temperatures (n > l), one can see that an ordinary conjugation condition (a:)* = a,, (bT)* = b, is satisfied. This may be explained by the well-known spectroscopical fact that the difference between absorption and luminescence disappears when the temperature is high, as the shift A becomes very small and we have an almost unitary evolution over the closed time contour. So, the sign ‘+’ in eq. (14) means absorption and the sign ‘-’ means luminescence. In the case of low temperatures (X = -1, T < To), the aperiodiis possible, when the energy of noncal solution u, = IE;I$ ch-‘(r/_+;(+) linear motion is close to zero and ET < 0. This solution describes the “solitonlike” excitation of the phonon system with the resonant exchange o * w, - o,, of frequencies, and this happens when Ap > 0. For the case of large temperatures (X = 1, T > To), the aperiodical solution ]&,‘I1 that) i) is possible, when the energy of nonlinear motion is close to zero and E: < 0. This solution

V.S. Yarunin I Electron-phonon optical resonance

273

describes the “kink-like” excitation of the phonon system with the resonance conditions w --, on - wb, w + w, - wb and takes place when Apn > 0. This solution is infinite in the phase plane and the decay part of its trajectory corresponds to the dissociation. The aperiodical solutions have an infinite period, and irregular motion of the nonlinear oscillator takes place in its neighbourhood. This motion may explain the absence of an ordinary phonon structure of an electron molecular spectrum, observed in [8]. The conditions of such a situation are the inequalities Ap > f2 or Apn > 0. These conditions are satisfied in [8] due to the shift detuning A2 = 8, the anharmonism p = 10 cm-‘, and the Fermi-resonance L2=5cm-‘. Thus, the nonperturbative analysis of the molecular intermode interaction p shows that a “soft” phonon mode w can be brought to the irregular motion due to the influence of the electron transition shift A in the potential curve. The molecular spectroscopy chaos may result from this process.

5. Conclusion

Thus, the role of the electron-phonon interaction in a molecule depends on its value, which is determined by the value of the Stokes shift A. For small shifts of valence vibrations, the motion of nuclei is almost harmonical. It gives the Rabi-type response of an electron, which strongly interacts with the external field. The exception happens in the special case of a resonance in an external field (4), when the frequency of the field is close to the nuclei frequency and their moving is considerably disturbed. A large shift A in a two-atomic molecule indicates the strong interaction of an electron with vibrations of nuclei. The resonant response of a molecule is possible in the case of heavy atoms and low field frequency. In the multi-atomic molecules we have the following situation. For a large shift (deformational vibration) the electron transition produces a lot of phonons and the intermode anharmonicity p between them disturbs the vibrational spectra considerably. The large magnitude of A is connected with the smallness of the frequency of the deformational vibration o. When the detuning of the Fermi resonance Ll is small compared with the product pA, the electron transition leads to chaos in the phonon system, which makes the situation similar to the infra-red catastrophy in electrodynamics. Since the constant p is always small in comparison with the frequency w, the approximation of weak anharmonicity in the absence of electronic transition [4] is justified. Still, for an electron-phonon excitation with a large shift A [7] the

V.S.

274

Yarunin

I Electron-phonon

optical resonance

nonlinear dynamics of the nuclei must be treated outside the framework of the theory of perturbation, in spite of the smallness of p(o)-‘: w

a,b

F=-w 4 p > Old

,

A>l.

The motion of the nuclei in this case is described by an aperiodic solution of eq. (14). The Landay-Singer theory is valid when A < 1; it implies that the electronphonon interaction is small and valid for valence vibrations, and it is always valid for diatomic molecules. The large value of A implies a substantial change in the “landscape” for the point of the vibrational phase space during the electron transition. Just this circumstance leads to the participation of a large number of phonon degrees of freedom, and the result is the manifestation [8] of the randomization of the vibrations during the electron-phonon transition in a weak external field.

Acknowledgements

The author has the best recollections about scientific collaboration on the subject of this paper with Professors B.S. Neporent and V.N. Popov in St. Petersburg.

References [l] V.N. Popov and V.S. Yarunin, Collective Effects in Quantum Statistics of Radiation and Matter (Kluwer, Dordrecht, 1988). [2] V.N. Popov and VS. Yarunin, Functional methods in quantum optics, in: Series on Condensed Matter Physics, vol. 7 (World Scientific, Singapore, 1990) pp. 49-102. [3] S.A. Fedotov and M.B. Schirschov, Theor. Math. Phys. (USSR) 69 (1986) 279. [4] N.V. Kusmin, V.S. Letokhov and A.A. Stuchebryukhov, J. Exp. Theor. Phys. (USSR) 90 (1986) 458. [5] J. Schwinger, J. Math. Phys. 2 (1961) 407. [6] L.V. Keldish, J. Exp. Theor. Phys. (USSR) 47 (1964) 15. [7] B.S. Neporent, S.A. Fedotov and VS. Yarunin, J. Exp. Theor. Phys. (USSR) 99 (1991) 447. [S] B.S. Neporent, S.V. Kulia and A.G. Spiro, Exp. Theor. Phys. Lett. (USSR) 48 (1988) 367. [9] V.N. Popov and VS. Yarunin, Theor. Math. Phys. (USSR) 57 (1983) 115.