- Email: [email protected]
Tunnelling in low-temperature atom-transfer processes
Vohme 63, number 2
TUNNELLING
CHEMICAL PHYSICS LETTERS
IN LOW-TEiW’ERATURE
ATOM-TRANSFER
PROCESSES
Joshua JORTNER
Received 6 Ianuary 1979
We demonstrate that the i-ran&-Condon vibrational overhp factors, which determine the rates of molewku group nuclear tunnellini processes at IONtemperatures, can be expressed in terms of the Camov formula_
EJementsry reactions involving atom or molecular group transfer (AT) between donor and acceptor centers occur in ;1 variety of processes in solid-state physics, soIution chemistry and bioiogy. Typical solid-state processes invohre rotational rela-xrtion of anionic dopants [I ,2), diffusion of ions in insuhting cr+als [3J, diffusion of hydrogen in metals [LtJ and low-tempr,‘rature tunnelling in amorphous mnteriais [S J _ Notable exampIes of AT in chemical systems invohe proton-transfer reactions [6], ligand substitution 171, inner-sphere electron transfer in transitionnietd compIexes IS]. the intmmolecuirtr inversion of ammonia and other ffusional molecuLs [9j and some po@nerization reactions [IO]_ Internsting biophysical processes are the reversible uptake of osygen, carbon monoxide and other small iigand molecules by hemoglobin and myogJobin [I I ] and proton translontion in rJrodopsin [JzJ_ At higJr temperatures these reactions invoIve a transition of 3 moIecuIar group from the donor to the acceptor by thermaJJy-activated passage over a potential energy barrier, separating the cquihbrium positicns of the molecular group in the initial and final states? while at low temperatures the proccss is expected to proceed via temperatureindependent nuclear tunneJJing of the molecular group- Such a pattern has been reported for the radi256
ation-induced low-temperature polymeriation of formaldeiryde [J Of ~ the radiation-induced Jow-temperature formation of prehrmirhodopsin from rhodopsin [I 21) and the recombination of CO with hemogIobin subunits [I 11. The theory of nonadiabatic outer-sphere electron transfer (ET) processes in condensed media is ROW we11 deveioped rJ3---151, the non-adiabatic ET rate being expressed commonly as tlte product of a square of a two-center one-eJectron exchange integraf and a therrnahy-averaged Fmnck-Condon nuclear overlap integd, Tke quantum theory of AT processes is still underdeveloped_ Some studies considered AT in 1erm.i of a double adiabatic approsimation with the electrons and the high-frequency motion of the Jight atom being viewed together as the fast subsystem in ;i modified Born-Oppenheimer scheme [J 6,J 7]_ The anaJysJs of experimental data on Iow-temperature AT processes II 1.12-181 often rests on the Garnov tunneling formula II 91. This approach requires further justification, as the Gamov formula was derived for the decay of a bound state into a continuum 119) and is therefore not directly applicable to AT between manifolds of bound states_ In the present note we 2alI demonstrate that the nuclear Franck-Condon factors which appear in the quantum-mechrmical rate
CHEMICAL
Volume 63, number 2
PHYSICS LETTERS
expressions for AT processes can be recast in a form akin to the Gamov formula. Consider nonadiabatic AT in 3 simple model system which is represented by two zero-order electronic states, a and 6, coupled to a single nuclear coordinate X, and which represents the motion of the molecular group to be transferred. We ignore reorganizntion of other intramolecular modes and only view the role of the medium (iattice) modes as ensuring vibrationJ reixsation [IS] (irreversrbility) subsequent to the
AT step. The nonadiabatic mte expression for the AT process is then IV = (~T$QZ-~
I Vab j” CC~X~(--&!~) ”
15 May 1979
vibrational reorganization energy Er = .S&. The temperature dependence of W is reflected in the Bose occupation number ii = [exp(fic&nT) - 11-l, and the single-mode rate expression becomes [20,21] W=A exp[-S(28+
1)]1, {ZS[G($+ I)] ‘j7)
X [(c-i- I)i/r,-]pP/3, where A = 2 1v& j2/li2u. p = f &!?ilfia is die nomidized energy gap. and IP {=I stands for the modified BtsseI function of order p. Eq. (4) e.xhibits a continuous transition from 3 low-temperature nuclear tunnehing expression W= A ekp(-S)p/p!
1”
0 where _‘c,, ’ and Xblv are the nuclear wavefunctions corresponding to the electronic states a and 6 3nd the nuclear quantum numbers u and IV, respectively. 15:~ and Ebqv the corresponding energies. v& the electronic matris elerucnts which couple the zero-order states LZand 6, and ( ) denotes the integration with respect to the nuclear coordinates; AE is the energy rgSp_z= 4, e\o(-flEz)_ __ the nuclex pxtition function in the initial state. and /I= (knT)-l, where kS
is Boltzmann’s constsnt and T the absolute temperature. This norurdidb.rtic multrphonon formalism has been spphfd previously to the analysis of the lowtemperature recombination of CO and hemoglobin [20] If we assume that the potenthl surfaces of the initial and final states, U, and r/,, respectively, are displaced harmonic curves Ui = $uw’,u” )
(2)
vr = p&loZ(LY - ci)? + -ilE:
(3)
we can characterize the nuclear configurational chano,e in the AT process by the reduced displacement fi = fY(UW/fi>~. where d is the coordinate dismnce between the minima of the potential surf3ces, w the frequency of the vibrational motion which, for the s3ke of simplicity, is assumed to be identical in the initial and final states, and ~_r is the reduced mass 3ssociated with the nuclear motion. The coupling strength of the nuclear mode is then S= 4’/2 and the
(4)
(kBT
G)
determined by the low-temperature harmonic poissonian overlap, to a I@h-temperature activated rate espression w= A(2irs~*[email protected])-‘!’
exp(+EA)
(kgT%-?U). (6)
with a %3aussian-typeactivation energy EA = (SRo f .U)7-l4SKLL The low-temperature rate expression. eq. (5). has the form Iv= AFh(O_p), where the Franck-Condon factor Fh(O,p)= ((~&,]xbP)]2 represents the overlap between the ground vibrationai wavefunction, xc0 (u = 0) of the initisl electronic state and the pth vibrational wavefunction in the final electronic state_ For the symmetrical reaction. corresponding to p = 0, Fh = esp(-S). The notion of a zero-temperature rate of 3 symmetrical process is fraught with some conceptual difficulties. 3s only essentiahy non-diagonal transitions contribute to the reaction rste { 131. An esplicit expression for the finite-temperature reaction rdte of a symmetrical process c3r-rbe derived by the application of the genersting function method [13.15] to eqs. (l)-(3) followed by a saddle point integration, which results in 1133 rv = .a [(2/Ti)S
cosech @JW2)]
X exp [-S tanh(jP%&)]
.
-1/2 (7)
We note that in the hi&-temperature limit eq. (7) reduces to eq. (6) with rlE = O_ Eqs. (4) and (7) constitute the central results of nonadiabatic AT rate theory_ These rate espressions 237
are expressed in terms of a product of an electronic term and a nucJear Franck-Condon vibrationa overlap term_ We shah now demonstrate tliat the nuciear contribution to these rate equations at low temperatures cm be recast in an alternative form which is pmcticaJJy equivalent to tire celebrated Ga_mov’s formuIa_ AnaIysis of low-temperature AT processes commonIy apply 16,11,12.18] the Gamov formula 119) for one_dimensionaJ nucJear motion in the form = v.
IV,
esp[-~dW~)Uz/fi]
7
(S)
where EA isthe barrier height, d is its width and p is the effective mass of the tunnelling particle. y is a constant which depends on the barrier sirape. For a rectanguIar barrier y is 3’13-, and for a paraboiic barrier y = ?r/+_ v. is a preexponentia1 factor which is identified with the effective frequency with which the particle of mass fl hits the barrier. We shall now demonstrate that the exponential term in eq. (8) is practicaliy equivalent to the nucIear vibrationa overlap factor_ Consider fust the symmetrical reaction wltere M= 0 and E4 = S&/4_ From tJle Iatter relation, together with S = (d2/2) r_co/fi, we get S= (2”*/fi)X fjZA)%f_ Thus, eq. (7) can be recast in tire form W = A [(Z/ir)S
cosech @fk&)J
tanh (prrw/l)]
_
(9)
Eq_ (9) bears a close family resemblance to eq_ (S), as in tJre Iimit kTefiw the exponenrhl Octor appearing in this muJtiphonon rate is practically identical with tJrat of the Gamov formuh with T = 2=, which is quite cJose to the vaJue of T for a paraboJic barrier_ Next, we consider a low-temperature exoergic process (M < 0). For hge values of p, which is appropriate for such exoergic processes, we can utiJize Stirling’s approximation o! = esp [p(Inp - l)] in eq. (5) wI?ich results in m = In (p/S) -
exp (-uzp),
I_
(JO)
This can be written in tfre form iV=A
exp[-(Sfiw
X exp(-(I Introducing 238
- IxI)/fiw]
(JJ)
+ m)IMIlfi~]. the relations E,
and A2 = d2(,x@) ZU =
we can express eq. (11)
as
A exp [-23/2d~A)U’fi]
X exp [-( I + m)I 4fWio],
(12)
which bears a cfose formal reJationship to the Gamov formuIa, eq. (8) The second exponent in eq. (12) is of minor importance, since (I + m) is a small (II + rn I = O-3) negative quantity (asp
-w
X exp ~-21~2~(&A)%-’
W = A exp (-S)
15 hfay 1979
CHEMICAL PHYSICS LETI-ERS
Volume 63. number 2
= (.SZc,l -
;UDZ/4!iTzw
References (I J V_ Narayaaamurti and R-0. PohI, Rev_ Mod. Phys. 42 (1970) 201[2] F-K_ Fang and DJ_ DiesUer, J_ Chem_ Phys- 57 (1972) 4953. [3] F-K. Fang, Theory of molecular relaxation (Wiley. New York, 1975). [4I C_P_ Flynn and A.hI. Stoneham. Phys- Rev- Bl (1970) 3966_ [SJ P-W_ And&on, B.I. Halperin and C.hI. Varma, Phil_ hfag_ 25 (1971) 1; J_ JHckIe.2. Physik 2.57 (1972) 212; W-A. PhiIIips, J. Low Temp- Phys- 7 (1972) 331. [6J R-P. Bell. The proton in chemistry, 2nd ed- (Chapman and HalI, London, 1973). 171 F_ BasoIo and R-G. Pearson, hfechanisms of inorganic reactions. 2nd ed. (Wiley. New York, 1967). [SJ H. Taube, Electron transfer reactions in solution (Audemic Press. New York, 1970)_ 191 hI.D_ Harmony, Chem. Sot. Rev- l(1972) 211. [lo) V.I. Goldanslcii, M.D. Frank-Kamenetskij and I.M. Barkalov. Science 189 (1973) 1344.
Volume 63, number 2
CHEMICAL
1.5 May 1979
PHYSICS LEL-TERS
1111 RX Austin, K-W. Beeson, L. Eisenstein, H. Frauenfelder and 1-C. Gunsalus, Biochemistry 14 (1975) 5355; N. AIberdinrJ, R-H. Austin, K_N’_ Beeson, S-S. than, L. Eisenstein, H_ FrauenfcIder and TM. Nordhrnd, Science 192 (1976) 1OOZ [12] K. Peters, M.L. Applebury and P.M. Rentzepis, Proc. NatL Acad Sci. US 74 (1977) 3119. [13J T. Holstein, Ann. Phys. 8 (1959) 343; Phil. Ma& B37 (1978) 49,4991141 R-R_ Dogonadze and A%_ Kuzuetsov, Prog. Surface sci- 5 (1975) 1. 1151 N.R. Kestner, J. Logan and J. Jortner, 3. Phys. Chem. 78 (1974) 2148-
1161 V-G. Levich, R-R_ Dogonadze, E.D. German, A.hI_ Kuznetsov and Yu.L Kharkats, EIectrochim. Acta 15 (1970) 353: [ 171 R-R- Dosonadze, A.M. Kuznetsov and %A. Vorotyntscv, DOW. Akad_ Nauk SSSR Ser_ Fii- IChim_ 209 (1973) 1135. 118) V.I. GoldanskB. Ann- Rev. Phys. Chem. 27 (1976) 85. 1191 G. Gamov, 2. Physik 51(1928) 204. 1201 J. Jortner and J. UIstrup, J. Am. Chem. Sot, submitted for publication. [Zl] J. Jortner, J_ Chem. Phys. 64 (1976) 4860. 1221 N. BrZiniche-Olsen and J. UIstrup, J. Chem. Sot. Faraday Trans. II 74 (1978) 1690. 1231 L-D. Landau and EM- Lifshitz, Quantum mechanics, 2nd ed. (Perpmon
Press, Oxford,
1965).
239
TUNNELLING
CHEMICAL PHYSICS LETTERS
IN LOW-TEiW’ERATURE
ATOM-TRANSFER
PROCESSES
Joshua JORTNER
Received 6 Ianuary 1979
We demonstrate that the i-ran&-Condon vibrational overhp factors, which determine the rates of molewku group nuclear tunnellini processes at IONtemperatures, can be expressed in terms of the Camov formula_
EJementsry reactions involving atom or molecular group transfer (AT) between donor and acceptor centers occur in ;1 variety of processes in solid-state physics, soIution chemistry and bioiogy. Typical solid-state processes invohre rotational rela-xrtion of anionic dopants [I ,2), diffusion of ions in insuhting cr+als [3J, diffusion of hydrogen in metals [LtJ and low-tempr,‘rature tunnelling in amorphous mnteriais [S J _ Notable exampIes of AT in chemical systems invohe proton-transfer reactions [6], ligand substitution 171, inner-sphere electron transfer in transitionnietd compIexes IS]. the intmmolecuirtr inversion of ammonia and other ffusional molecuLs [9j and some po@nerization reactions [IO]_ Internsting biophysical processes are the reversible uptake of osygen, carbon monoxide and other small iigand molecules by hemoglobin and myogJobin [I I ] and proton translontion in rJrodopsin [JzJ_ At higJr temperatures these reactions invoIve a transition of 3 moIecuIar group from the donor to the acceptor by thermaJJy-activated passage over a potential energy barrier, separating the cquihbrium positicns of the molecular group in the initial and final states? while at low temperatures the proccss is expected to proceed via temperatureindependent nuclear tunneJJing of the molecular group- Such a pattern has been reported for the radi256
ation-induced low-temperature polymeriation of formaldeiryde [J Of ~ the radiation-induced Jow-temperature formation of prehrmirhodopsin from rhodopsin [I 21) and the recombination of CO with hemogIobin subunits [I 11. The theory of nonadiabatic outer-sphere electron transfer (ET) processes in condensed media is ROW we11 deveioped rJ3---151, the non-adiabatic ET rate being expressed commonly as tlte product of a square of a two-center one-eJectron exchange integraf and a therrnahy-averaged Fmnck-Condon nuclear overlap integd, Tke quantum theory of AT processes is still underdeveloped_ Some studies considered AT in 1erm.i of a double adiabatic approsimation with the electrons and the high-frequency motion of the Jight atom being viewed together as the fast subsystem in ;i modified Born-Oppenheimer scheme [J 6,J 7]_ The anaJysJs of experimental data on Iow-temperature AT processes II 1.12-181 often rests on the Garnov tunneling formula II 91. This approach requires further justification, as the Gamov formula was derived for the decay of a bound state into a continuum 119) and is therefore not directly applicable to AT between manifolds of bound states_ In the present note we 2alI demonstrate that the nuclear Franck-Condon factors which appear in the quantum-mechrmical rate
CHEMICAL
Volume 63, number 2
PHYSICS LETTERS
expressions for AT processes can be recast in a form akin to the Gamov formula. Consider nonadiabatic AT in 3 simple model system which is represented by two zero-order electronic states, a and 6, coupled to a single nuclear coordinate X, and which represents the motion of the molecular group to be transferred. We ignore reorganizntion of other intramolecular modes and only view the role of the medium (iattice) modes as ensuring vibrationJ reixsation [IS] (irreversrbility) subsequent to the
AT step. The nonadiabatic mte expression for the AT process is then IV = (~T$QZ-~
I Vab j” CC~X~(--&!~) ”
15 May 1979
vibrational reorganization energy Er = .S&. The temperature dependence of W is reflected in the Bose occupation number ii = [exp(fic&nT) - 11-l, and the single-mode rate expression becomes [20,21] W=A exp[-S(28+
1)]1, {ZS[G($+ I)] ‘j7)
X [(c-i- I)i/r,-]pP/3, where A = 2 1v& j2/li2u. p = f &!?ilfia is die nomidized energy gap. and IP {=I stands for the modified BtsseI function of order p. Eq. (4) e.xhibits a continuous transition from 3 low-temperature nuclear tunnehing expression W= A ekp(-S)p/p!
1”
0 where _‘c,, ’ and Xblv are the nuclear wavefunctions corresponding to the electronic states a and 6 3nd the nuclear quantum numbers u and IV, respectively. 15:~ and Ebqv the corresponding energies. v& the electronic matris elerucnts which couple the zero-order states LZand 6, and ( ) denotes the integration with respect to the nuclear coordinates; AE is the energy rgSp_z= 4, e\o(-flEz)_ __ the nuclex pxtition function in the initial state. and /I= (knT)-l, where kS
is Boltzmann’s constsnt and T the absolute temperature. This norurdidb.rtic multrphonon formalism has been spphfd previously to the analysis of the lowtemperature recombination of CO and hemoglobin [20] If we assume that the potenthl surfaces of the initial and final states, U, and r/,, respectively, are displaced harmonic curves Ui = $uw’,u” )
(2)
vr = p&loZ(LY - ci)? + -ilE:
(3)
we can characterize the nuclear configurational chano,e in the AT process by the reduced displacement fi = fY(UW/fi>~. where d is the coordinate dismnce between the minima of the potential surf3ces, w the frequency of the vibrational motion which, for the s3ke of simplicity, is assumed to be identical in the initial and final states, and ~_r is the reduced mass 3ssociated with the nuclear motion. The coupling strength of the nuclear mode is then S= 4’/2 and the
(4)
(kBT
G)
determined by the low-temperature harmonic poissonian overlap, to a I@h-temperature activated rate espression w= A(2irs~*[email protected])-‘!’
exp(+EA)
(kgT%-?U). (6)
with a %3aussian-typeactivation energy EA = (SRo f .U)7-l4SKLL The low-temperature rate expression. eq. (5). has the form Iv= AFh(O_p), where the Franck-Condon factor Fh(O,p)= ((~&,]xbP)]2 represents the overlap between the ground vibrationai wavefunction, xc0 (u = 0) of the initisl electronic state and the pth vibrational wavefunction in the final electronic state_ For the symmetrical reaction. corresponding to p = 0, Fh = esp(-S). The notion of a zero-temperature rate of 3 symmetrical process is fraught with some conceptual difficulties. 3s only essentiahy non-diagonal transitions contribute to the reaction rste { 131. An esplicit expression for the finite-temperature reaction rdte of a symmetrical process c3r-rbe derived by the application of the genersting function method [13.15] to eqs. (l)-(3) followed by a saddle point integration, which results in 1133 rv = .a [(2/Ti)S
cosech @JW2)]
X exp [-S tanh(jP%&)]
.
-1/2 (7)
We note that in the hi&-temperature limit eq. (7) reduces to eq. (6) with rlE = O_ Eqs. (4) and (7) constitute the central results of nonadiabatic AT rate theory_ These rate espressions 237
are expressed in terms of a product of an electronic term and a nucJear Franck-Condon vibrationa overlap term_ We shah now demonstrate tliat the nuciear contribution to these rate equations at low temperatures cm be recast in an alternative form which is pmcticaJJy equivalent to tire celebrated Ga_mov’s formuIa_ AnaIysis of low-temperature AT processes commonIy apply 16,11,12.18] the Gamov formula 119) for one_dimensionaJ nucJear motion in the form = v.
IV,
esp[-~dW~)Uz/fi]
7
(S)
where EA isthe barrier height, d is its width and p is the effective mass of the tunnelling particle. y is a constant which depends on the barrier sirape. For a rectanguIar barrier y is 3’13-, and for a paraboiic barrier y = ?r/+_ v. is a preexponentia1 factor which is identified with the effective frequency with which the particle of mass fl hits the barrier. We shall now demonstrate that the exponential term in eq. (8) is practicaliy equivalent to the nucIear vibrationa overlap factor_ Consider fust the symmetrical reaction wltere M= 0 and E4 = S&/4_ From tJle Iatter relation, together with S = (d2/2) r_co/fi, we get S= (2”*/fi)X fjZA)%f_ Thus, eq. (7) can be recast in tire form W = A [(Z/ir)S
cosech @fk&)J
tanh (prrw/l)]
_
(9)
Eq_ (9) bears a close family resemblance to eq_ (S), as in tJre Iimit kTefiw the exponenrhl Octor appearing in this muJtiphonon rate is practically identical with tJrat of the Gamov formuh with T = 2=, which is quite cJose to the vaJue of T for a paraboJic barrier_ Next, we consider a low-temperature exoergic process (M < 0). For hge values of p, which is appropriate for such exoergic processes, we can utiJize Stirling’s approximation o! = esp [p(Inp - l)] in eq. (5) wI?ich results in m = In (p/S) -
exp (-uzp),
I_
(JO)
This can be written in tfre form iV=A
exp[-(Sfiw
X exp(-(I Introducing 238
- IxI)/fiw]
(JJ)
+ m)IMIlfi~]. the relations E,
and A2 = d2(,x@) ZU =
we can express eq. (11)
as
A exp [-23/2d~A)U’fi]
X exp [-( I + m)I 4fWio],
(12)
which bears a cfose formal reJationship to the Gamov formuIa, eq. (8) The second exponent in eq. (12) is of minor importance, since (I + m) is a small (II + rn I = O-3) negative quantity (asp
-w
X exp ~-21~2~(&A)%-’
W = A exp (-S)
15 hfay 1979
CHEMICAL PHYSICS LETI-ERS
Volume 63. number 2
= (.SZc,l -
;UDZ/4!iTzw
References (I J V_ Narayaaamurti and R-0. PohI, Rev_ Mod. Phys. 42 (1970) 201[2] F-K_ Fang and DJ_ DiesUer, J_ Chem_ Phys- 57 (1972) 4953. [3] F-K. Fang, Theory of molecular relaxation (Wiley. New York, 1975). [4I C_P_ Flynn and A.hI. Stoneham. Phys- Rev- Bl (1970) 3966_ [SJ P-W_ And&on, B.I. Halperin and C.hI. Varma, Phil_ hfag_ 25 (1971) 1; J_ JHckIe.2. Physik 2.57 (1972) 212; W-A. PhiIIips, J. Low Temp- Phys- 7 (1972) 331. [6J R-P. Bell. The proton in chemistry, 2nd ed- (Chapman and HalI, London, 1973). 171 F_ BasoIo and R-G. Pearson, hfechanisms of inorganic reactions. 2nd ed. (Wiley. New York, 1967). [SJ H. Taube, Electron transfer reactions in solution (Audemic Press. New York, 1970)_ 191 hI.D_ Harmony, Chem. Sot. Rev- l(1972) 211. [lo) V.I. Goldanslcii, M.D. Frank-Kamenetskij and I.M. Barkalov. Science 189 (1973) 1344.
Volume 63, number 2
CHEMICAL
1.5 May 1979
PHYSICS LEL-TERS
1111 RX Austin, K-W. Beeson, L. Eisenstein, H. Frauenfelder and 1-C. Gunsalus, Biochemistry 14 (1975) 5355; N. AIberdinrJ, R-H. Austin, K_N’_ Beeson, S-S. than, L. Eisenstein, H_ FrauenfcIder and TM. Nordhrnd, Science 192 (1976) 1OOZ [12] K. Peters, M.L. Applebury and P.M. Rentzepis, Proc. NatL Acad Sci. US 74 (1977) 3119. [13J T. Holstein, Ann. Phys. 8 (1959) 343; Phil. Ma& B37 (1978) 49,4991141 R-R_ Dogonadze and A%_ Kuzuetsov, Prog. Surface sci- 5 (1975) 1. 1151 N.R. Kestner, J. Logan and J. Jortner, 3. Phys. Chem. 78 (1974) 2148-
1161 V-G. Levich, R-R_ Dogonadze, E.D. German, A.hI_ Kuznetsov and Yu.L Kharkats, EIectrochim. Acta 15 (1970) 353: [ 171 R-R- Dosonadze, A.M. Kuznetsov and %A. Vorotyntscv, DOW. Akad_ Nauk SSSR Ser_ Fii- IChim_ 209 (1973) 1135. 118) V.I. GoldanskB. Ann- Rev. Phys. Chem. 27 (1976) 85. 1191 G. Gamov, 2. Physik 51(1928) 204. 1201 J. Jortner and J. UIstrup, J. Am. Chem. Sot, submitted for publication. [Zl] J. Jortner, J_ Chem. Phys. 64 (1976) 4860. 1221 N. BrZiniche-Olsen and J. UIstrup, J. Chem. Sot. Faraday Trans. II 74 (1978) 1690. 1231 L-D. Landau and EM- Lifshitz, Quantum mechanics, 2nd ed. (Perpmon
Press, Oxford,
1965).
239