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On the theory of intramolecular vibrational relaxation in highly excited polyatomic molecules: Quasi-harmonic versus random-phase approximation
Volume 145, number 5
CHEMICAL PHYSICS LETTERS
15 April 1988
ON THE THEORY OF INTRAMOLECULAR VIBRATIONAL RELAXATION IN HIGHLY EXCITED POLYATOMIC MOLECULES: QUASI-HARMONIC VERSUS RANDOM-PHASE APPROXIMATION
A.A. STUCHEBRUKHOV Research Center on Technological hers,
Academy of Sciences of the USSR. Troitzk. Moscow District 142092, USSR
Received 3 November 1986; in final form 19 February 1988
Two alternative approaches, the quasi-harmonic approximation (QHA) and the random-phase approximation (RPA), are compared and their applicability to describe the kinetics of non-equilibrium vibrational states in highly excited polyatomic molecules analyzed in the framework of the pumped-mode-heat-bath (PMHB ) model. It is shown that RPA, which is used by several authors, is invalid in this model, since it neglects essential coherent effects in the damping of off-diagonal elements of the density matrix describing the pumped mode. The resulting error in the estimated width of the IR transition exceeds an order of magnitude. The applicability of the Markovian approximation is discussed.
1. Introduction The intramolecular dynamics of polyatomic molecules in highly excited vibrational levels (with energies E x D, where D is the dissociation energy) are adequately described by statistical theories like RRKM [ 11. One reason why such descriptions are possible is because of the fairly rapid intramolecular vibrational energy redistribution [ 21 between different modes, proceeding via Fermi-resonance chains [ 3-51. The presence of the subsequent resonances causes a stochastization of vibrations [ 6-81, which takes place at energies of the order of E,x (3-10) x lo3 cm-’ in molecules like CF,I, SF6, etc. [9-l I]. The stochastic behaviour of molecules with energies above Es implies a uniform average energy distribution over the molecular modes, for time intervals t ZBo -I, where o is a specific molecular frequency. If a mode is displaced from equilibrium, i.e. from the equidistribution of mode energies, anharmonical interactions induce a relaxation to a new equilibrium state. Such non-equilibrium states can arise, for example, as a result of multi-photon excitation of molecules [ 121, if a particular mode is pumped continuously by laser radiation, or in experiments on selective excitation of specific highly excited states by picosecond laser pulses [ 13,141.
Let us consider a situation where a mode of a strongly excited polyatomic molecule is “knocked out” of the equilibrium state by, for instance, absorption of an IR quantum, while other molecular modes remain in equilibrium. The relaxation of such a non-equilibrium state is described most naturally in the framework of the pumped-mode-heat-bath (PMHB) model [ 15-261. The bath represents the other modes which are inactive as far as the absorption of the IR radiation is concerned. In this approach the molecular state is given by a density matrix u for the excited mode and a vibrational temperature, T, for the bath. The relaxation to the equilibrium state is described, in general, by a closed set of evolution equations for a and T [ 15,24,25 1. (The variables are coupled because of anharmonic interactions between the pumped mode and the heat bath.) In a weak non-equilibrium situation, where the final bath temperature after relaxation does not differ markedly from the initial temperature, the mode relaxation can be treated as proceeding at constant temperature. When considering the linear absorption spectrum for the vibrational mode, one can also assume that the bath temperature is constant; this is a standard approach in linear response theory [ 271. The equation for the bath temperature is rather trivial [ 25 1, so we will not consider it further and 387
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henceforth assume that the bath temperature is constant. The present communication seeks to answer two important questions relevant to the PMHB model. (i) Is the PMHB model applicable to polyatomic molecules and, if so, in which energy region? (ii) What is the equation describing the evolution of the density matrix u for the pumped mode in the bath? In spite of the extensive use of the PMHB model in polyatomic excitation theory [ 15-251, no accurate investigation of its applicability has been reported in the literature. An equation for o has been proposed on phenomenological grounds [ 18,19 ] and derived on a number of occasions [ 15-l 7 1. Interaction between the pumped mode and the bath, due to anharmonical terms in the molecule Hamiltonian, induces dephasing processes (and consequent energy relaxation from the pumped mode) which may be represented by incorporating a relaxation operator ZRin the kinetic equation for 0. In addition, the anharmonica1 interaction results in a change in the dynamical part of the equation for a, which can sometimes be described by a modification of the Hamiltonian for the pumped mode. The equation is au/at= -i[Z&,
a] +ZR ,
(1)
The problem is to find flefland ZRfor the pumped mode. In the absence of any interaction with the bath one has &=Z!Z, and ZR= 0, where HP is the pumped mode Hamiltonian. We now consider the operator ZR, as it determines both the relaxation and the broadening of the mode absorption spectrum. Two alternative approximations have been employed for ZR: the random-phase approximation [ 15-23,261 and the quasi-harmonic approximation [ 5,24,25,28,29]. The correspondence between these two approaches has not yet been fully analysed; however, the former approach is not at all applicable to a highly excited molecule. There is a large error, which arises as discussed below. A particularly simple case [ 301 for which a kinetic equation can be derived occurs when: (i) the Markovian approximation is valid, i.e. the system relaxation time r is much larger than the correlation time r, for the bath, (ii) the system frequencies are not degenerate, i.e. for any different level pairs (m, n) 388
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and (m’, n’) one has lo,,-co,,,,,,1 %r-‘.Thefirst condition is necessary in order to obtain a differential kinetic equation (not an integro-differential equation such as those describing systems with memory [ 3 11). The second condition provides an independent damping of the off-diagonal elements of the density matrix, so the off-diagonal part of the kinetic operator is 6,,=-
(l/T,
)mn~mn,
where T2 is the phase relaxation time, which is specific for any pair of levels (m, n). These two conditions in fact form the random phase approximation, see section 3. If the latter condition is satisfied, the system relaxation proceeds in the same manner as for a set of non-coupled two-level subsystems. If the system frequencies are degenerate, as is the case for the standard harmonic oscillator, the relaxation of the off-diagonal elements is much more complicated. There is a coupling between different matrix elements, and the general equation takes the form
where Rm,,kl are coefficients, see section 3. Thus a coherence appears in the damping of the off-diagonal elements, and relaxation involves the whole system. As relaxation of the off-diagonal elements determines the absorption spectrum, the character of the relaxation is of fundamental importance. It was noted [ 321 that by taking account of the abovementioned coherent effects a substantial narrowing of the absorption spectrum may result. We shall assume that the Markovian approximation is adequate in some energy region, that is to say, t> z,. We still have to determine if the second condition is satisfied in the excited molecule. It is not difficult to deduce that it is not. The quantity T-’ determines a specific width for the absorption spectrum. As is known [ 121, multiphoton excitation of polyatomic molecules in the quasi-continuum region is possible solely because the broadening of the spectrum exceeds the anharmonic level shifts 6, so r-r s S. The quantity 6 characterizes the deviation of the vibrational spectrum from an equidistant pattern; it amounts to about l-4 cm-‘. Line widths in the spectra of excited molecules are typically 1O-30 cm-’ [12],sooneindeedhast-‘*&AsSdepends
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on the difference (w,, - o,,,,,, 1, the second wndition is clearly invalid. One can consider the opposite situation, where 1corn,- o,,~. I *: 7-‘, so that the nonequidistant structure of the spectrum is not essential for the relaxation process, and the vibrational mode can be represented by a sequence of equidistant levels. This is the quasi-harmonic model [ $25 1, where the mode frequencies depend on the molecular energy, but the mode levels remain equidistant. It has been implied by some authors [ 15-23,261 that condition (ii) is fulfilled, which is not the case as we have argued, and consequently the width of the IR absorption spectra of excited molecules has been substantially overestimated. For example, if one calculates the transition spectrum between levels of the harmonic oscillator with the (incorrect) assumption that the second condition holds, the well known formula [ 15-l 91 is obtained
where k f%,k is the rate of vibrational relaxation from level n to level k under the action of the heat bath (the purely phase relaxation is neglected for the moment, see section 2). For an oscillator with linear friction [ 33 1, the spectral width, calculated in this way, is and
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CHEMICAL PHYSICSLETTERS
PMHB, is applicable for polyatomic molecules. Under the assumptions adopted here, the anharmanic interaction coupling the pumped mode to heat bath is of order m + 1, where m 2 2. The essential part of the anharmonic interaction can be represented [25] by a term q&(ql, .... qs), where qP is the coordinate of the pumped mode, and ql, .... qs are the coordinates of the bath. This form is chosen because the number of terms in the anharmonic potential linear in the mode coordinate is WS~, s>> 1, the number of quadratic terms is z:s”‘- ‘, etc. The bath function B,,, can be written as (m+l) &(q,...q,)=
5
.
i
a, -am= I
qo1,...qa, 9
(3)
where $(“‘+I) are specific constants of the (m + 1 )th order anharmonicity. The bath correlation time, z,, is defined in terms of quantities like ( B, ( t )B, (0 ) ) , where B,(t) is an operator in the Heisenberg representation, with averaging over the bath states implied. Correlators of this type characterize the bath in its equilibrium state. If we assume [ 51 that (qi(t)qi(O)) Pexp( -yt), where y=y(E) is a specific relaxation rate for the molecule, depending on its energy E, then the correlator (B(t) B(O)} is a sum of oscillating exponentials with composite frequencies Oil + Wizf ... &wi, and with a damping rate of my, so (B(t)B(O))ccexp(-myt)
C exp(i&t) Lx
.
(4)
=y1[(2n+1)(2n+1)+2A],
where ff is the mean occupation number for the oscillator, and yI determines the rate of energy relaxation towards equilibrium Ai = - 2y,At, see eq. (8) below. It is known, in fact, that the transition spectrum simply has width yI, see eq. (8) below. The factor [(2~+1)(2n+1)+2ff] amounts to 10 and higher for excited molecules (AS l-2), so the spectral width is typically overestimated by more than an order of magnitude.
2. Markovian approximation We now investigate conditions under which the Markovian approximation, which is fundamental for
The range of the composite frequencies in a polyatomic molecule, d, is of the order of a typical molecular frequency, AZ t3. The spacing between neighbouring composite frequencies depends on the number of frequencies involved in the composite oscillation; it is determined by the inverse density of the corresponding Fermi resonances [ 25 1, (~2)~‘. If one has (pr)-‘Amy, then the correlator (B(t)B(O)) is damped at time intervals &WA- *wc7.1-‘,because of dephasing of the oscillating exponents, and recurrence phenomena, which are possible at times 7, %pF, are suppressed by the exponential exp( - myt), since myk (pF ) -I. Thus, under the condition myk(pE)-‘, the bath correlation time, z,, is of the same order as a typical inverted molecular frequency. Clearly, in this case 389
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TZ y-’ >>T,, which is necessary for the Markovian approximation. Typical values of the inverse density of resonances in molecules like CFSI, SF6 and (CF,) &I lie within the interval l-l 0 cm-’ and depend on the number of degrees of freedom and the resonance type (see ref. [ 25 ] ) , For molecules excited to energies close to the dissociation energy, the width of the absorption spectrum is usually 2 10 cm- 1 [ 121. The condition myk (pz ) - ’is already fulfilled for third-order resonances, having a minimal densityp”“. On the other hand, the width depends on the molecular energy [ 5 1, so the Markovian approximation is valid, in general, only for high energies and sufficiently large molecules. We have considered the case of a definite order of anharmonicity, m + 1. If at certain energies the Markovian approximation is valid for third-order resonances, m= 2, then it is surely adequate for resonances of higher orders, since the resonance density p F increases rapidly with m [ 25 1. In order to be sure that the Markovian approximation is valid, one needs to estimate the order of anharmonicity, m* + 1, dominating in the intramolecular vibrational energy redistribution. This problem has been considered [ 34 1. At low energies m* = 2, and at higher energies, ER D,m* may be very large, e.g. > 10. The energy dependence of m* is determined by the molecular frequencies, the number of degrees of freedom, and the rate of the decrease of the anharmonicity constants with increase of the order m + 1. If m* is known [ 34 1, the necessary condition, under which the approximation is reasonable, is m*y> (pz:)-‘.
3. RPA and QHA for excited polyatomic molecules The condition necessary for the random phase approximation is obtained as follows. A general matrix form of the kinetic equation for a system density matrix cr is [ 30 ] (5)
where wl,, are the system transition frequencies, renormalized because of the interaction with the heat bath, &,,kl are kinetic coefficients. These coefficients determine the relaxation rate y, which has an 390
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where R is a typical value order of magnitude yx IR 1, of&,H. Let us assume that the system levels are not degenerate and consider the relaxation of the diagonal elements, m=n. The rhs of eq. (5) oscillates rapidly in all terms of the sum, except for k= 1. Therefore the diagonal terms are relaxing independently of the off-diagonal terms. An estimate for the off-diagonal terms, to first order in 1R I, is
(6) If the system frequencies are not degenerate, i.e. IeLn -wL, Ix 6 3 yz R for any different pairs of levels (n, m) and (k, I), then such pairs (k, 1) can be discarded from (5). Only the terms with (k, I) = (m, n) are retained, and the relaxation term is reduced to - ( ~/T,),,,,u~,,, where (l/T,),,= This is the random phase approximation. -L,,,. The criterion of applicability of this approximation is 6> y, that is to say, anharmonic shifts of the system levels must substantially exceed their width. In the opposite situation, 6 5 y, the anharmonicity can be neglected, and the levels can be considered as equidistant. This determines the applicability of the quasi-harmonic approximation, while the random phase approximation is invalid. Note that ref. [ 301 gives a considerably overestimated condition for the random phase approximation, namely, 6* T; , where T, is the damping time for correlations in the heat bath. As Y-XT; I, one has 6 B y provided that 6 % T; ’, while the inverse statement is false. A typical situation for a vibrational mode of a highly excited polyatomic molecule is 6 4z y * 7; ’, corresponding to the Markovian quasi-harmonic approximation. We now consider the kinetic equation for the pumped mode in the quasi-harmonic approximation. The most general situation, taking account of all orders in the anharmonicity and of the phase relaxation, has been considered previously [ 25 1. In a particularly important case, where the coupling of the pumped mode to the heat bath is linear in the mode coordinate, the equation for the density matrix is [25]
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CHEMICAL PHYSICSLETTERS
+Y,{(2aaa+- atao--au+a)(fif +~(2a+au-aa+a-aua+)},
1)
(7)
where a, a+ are creation and annihilation operators for the pumped oscillator, A is the mean occupation number for the mode when it is in equilibrium with the heat bath, and the relaxation constant y1 determines the relaxation rate of the mode energy E to its value in the equilibrium state, 5, k= -2y, (E-F)
(8)
as well as the absorption spectrum of the excited molecule [ 25 1,
15 April 1988
approximation and the random phase approximation. It is shown that the latter, which has been widely utilized [ 15-26 1, is invalid for a highly excited molecule. It neglects an essential coherence in the damping of off-diagonal matrix elements of the density operator describing the pumped mode; as a consequence, the width of the IR transition spectrum is overestimated by at least an order of magnitude. An adequate approach to highly excited molecules is the quasi-harmonic approximation, which admits an exhaustive analytical investigation in terms of the quasiprobability function and coherent states. References [ 11P.D.Robinson and K.A. Holbrook, Theory of monomolecular reactions (Academic Press, New York, 1974).
[ 2 ] S.A. Rice, in: Advances in laser chemistry (Springer, Ber-
where 4 is the quasi-harmonic frequency of the mode [ 5,25 1. The off-diagonal matrix elements are in fact coupled in the quasi-harmonic approximation. The matrix form of eq. (7) is
t-Y,{(flfl)(2J(n+l)(m+l)c,+,,~+, -Jmfl,,
-&a,,)
+A(2JZzrr&,_, -Jn+lo,,)).
-JmTick, (10)
The relaxation operator on the rhs contains a num ber of terms, besides a,,. This is the essential di! fcrencc between the quasi-harmonic approximation and the random phase approximation. The matrix form of the quasi-harmonic approximation is much more complicated than that for the random phase approximation. If one starts from the operator equation (7), however, and employs the coherent state basis and the quasiprobability representation [28,29,33], the quasi-harmonic approximation admits an exhaustive analytical investigation.
4. Conclusion Two alternative approximations to the intramolecular vibrational relaxation in polyatomic molecules have been considered here, the quasi-harmonic
lin, 1978). [3] V.S. Letokhov and A.A. Makarov, Usp. Fiz. Nauk 134 (1981) 45. [4] J.S. Hutchinson, J.T. Hynes and W.P. Reinhardt, Chem. Phys. Letters 108 (1984)353. [ 51A.A. Stuchebrukhov, M.V. Kuzmin, V.N. Bagratashvili and V.S. Letokhov, Chem. Phys. 107 (1986) 429. [ 6 ] G.M. Zaslavsky, Stochasticity ofdynamical systems (Nauka, Moscow, 1983) [in Russian]. [ 7 ] D.W. Oxtoby and S.A. Rice, J. Chem. Phys. 65 ( 1976) 1676. [ 81 F.M. Izrailev, B.V. Chirikov and D.L. Shepelyansky, preprint 80-210, Institute of Nuclear Physics, Novosibirsk (1980). [9] V.N. Bagratashvili, Yu.G. Vayner, V.S. Doljikov, S.F. Koljakov, VS. Letokhov, L.P. Malkin, E.A. Rybov, E.G. Silkis and V.D. Titov, Zh. Exp. Teor. Fis. 80 (1981) 1008. [lo] V.M. Akulin, V.D. Vurdov, G.G. Esadse, N.V. Karlov, A.M. Prokhorov and E.M. Khohlov, Pisma v Zh. Exp. Teor. Fis. 41 (1985) 239. [ 111 V.S. Doljikov, Yu.S. Doljikov, VS. Letokhov, A.A. Makarov, A.L. Malinovsky and E.A. Ryabov, Chem. Phys. 102 (1986) 155. [ 121 V.S. Letokhov, Nonlinear selective photoprocesses in atoms and molecules (Nauka, Moscow, 1983) [in Russian 1, [ 131 P.M. Felker and A.H. Zewail, Chem. Phys. Letters 108 (1984) 303. [ 141 N. Bloembergen and A.H. Zcwail, J. Phys. Chem. 88 (1984) 5459. [ 151 D.P. Hodgkinson and J. B&g, Phys. Letters A 43 (1976) 451. [ 161 C.D. Cantrell, H.W. Galbraith and J. Ackerhalt, in: Multiphoton processes, ed. J.H. Eberly (Springer, Berlin, 1978). [ 171 C.D. Cantrell, in: Laser spectroscopy, Vol. 3, eds. J.L. Hall and J.L. Carlsten (Springer, Berlin, 1977) p. 109. [ 18 1H. Friedman and V. Ahiman, Opt. Commun. 33 (1980) 163.
391
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[ 191J.C. Stephenson, D.S. King, M.F. Goodman and J. Stone, J. Chem. Phys. 70 ( 1979) 4496. [20] J. Black, E. Yablonovich, N. Bloembergen and S. Mukamel, Phys. Rev. Letters 38 (1977) 1131. [21] J.L. Lyman, J. Chem. Phys. 67 (1978) 1868. [22]E.R. Grant, M.J. Coggiola, Y.T. Lee, P.A. Schulz, AaS. Sudbo and Y.R. Shen, Chem. Phys. Letters 52 ( 1977) 595. [23] M.J. Shultz and E. Yablonovitch, J. Chem. Phys. 68 (1978) 3007. [24] L.M. Narducci, S.S. Mitra, R.A. Shatas and C.A. Coulter, Phys. Rev. A 16 (1977) 247. [ 251 V.N. Bagratashvili, M.V. Kuz’min, V.S. Letokhov and A.A. Stuchebrukhov, Chem. Phys. 97 (1985) 13. [26] S. Rashev, Chem. Phys. Letters 100 (1986) 578.
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[27] L.D. Landau and E.M. Lifshits, Statistical physics (Nauka, Moscow, 1977). [ 281 V.N. Sazonov and A.A. Stuchebrukhov, Chem. Phys. 56 (1981) 391. [29] V.N. Sazonov, A.A. Stuchebrukhov and S.V. Zatsepin, Chem. Phys. 69 (1982) 459. [30] V.M. Fain, Photons and nonlinear media (Soviet Radio, Moscow, 1972). [ 311 S. Mukamel, in: Photoselective chemistry, eds. J. Jortner, R.D. Levine and S.A. Rice (Wiley, New York, 198 1) [32] K. Kay, J. Chem. Phys. 75 (1981) 1690. [33] B.Ya. Zeldovitch, A.M. Perelomov and V.S. Popov, Zh. Eksp. Teor..Fiz. 57 (1968) 196. [ 341 A.A. Stuchebrukhov, Soviet Phys. JETP 64 (1986) 1195.
CHEMICAL PHYSICS LETTERS
15 April 1988
ON THE THEORY OF INTRAMOLECULAR VIBRATIONAL RELAXATION IN HIGHLY EXCITED POLYATOMIC MOLECULES: QUASI-HARMONIC VERSUS RANDOM-PHASE APPROXIMATION
A.A. STUCHEBRUKHOV Research Center on Technological hers,
Academy of Sciences of the USSR. Troitzk. Moscow District 142092, USSR
Received 3 November 1986; in final form 19 February 1988
Two alternative approaches, the quasi-harmonic approximation (QHA) and the random-phase approximation (RPA), are compared and their applicability to describe the kinetics of non-equilibrium vibrational states in highly excited polyatomic molecules analyzed in the framework of the pumped-mode-heat-bath (PMHB ) model. It is shown that RPA, which is used by several authors, is invalid in this model, since it neglects essential coherent effects in the damping of off-diagonal elements of the density matrix describing the pumped mode. The resulting error in the estimated width of the IR transition exceeds an order of magnitude. The applicability of the Markovian approximation is discussed.
1. Introduction The intramolecular dynamics of polyatomic molecules in highly excited vibrational levels (with energies E x D, where D is the dissociation energy) are adequately described by statistical theories like RRKM [ 11. One reason why such descriptions are possible is because of the fairly rapid intramolecular vibrational energy redistribution [ 21 between different modes, proceeding via Fermi-resonance chains [ 3-51. The presence of the subsequent resonances causes a stochastization of vibrations [ 6-81, which takes place at energies of the order of E,x (3-10) x lo3 cm-’ in molecules like CF,I, SF6, etc. [9-l I]. The stochastic behaviour of molecules with energies above Es implies a uniform average energy distribution over the molecular modes, for time intervals t ZBo -I, where o is a specific molecular frequency. If a mode is displaced from equilibrium, i.e. from the equidistribution of mode energies, anharmonical interactions induce a relaxation to a new equilibrium state. Such non-equilibrium states can arise, for example, as a result of multi-photon excitation of molecules [ 121, if a particular mode is pumped continuously by laser radiation, or in experiments on selective excitation of specific highly excited states by picosecond laser pulses [ 13,141.
Let us consider a situation where a mode of a strongly excited polyatomic molecule is “knocked out” of the equilibrium state by, for instance, absorption of an IR quantum, while other molecular modes remain in equilibrium. The relaxation of such a non-equilibrium state is described most naturally in the framework of the pumped-mode-heat-bath (PMHB) model [ 15-261. The bath represents the other modes which are inactive as far as the absorption of the IR radiation is concerned. In this approach the molecular state is given by a density matrix u for the excited mode and a vibrational temperature, T, for the bath. The relaxation to the equilibrium state is described, in general, by a closed set of evolution equations for a and T [ 15,24,25 1. (The variables are coupled because of anharmonic interactions between the pumped mode and the heat bath.) In a weak non-equilibrium situation, where the final bath temperature after relaxation does not differ markedly from the initial temperature, the mode relaxation can be treated as proceeding at constant temperature. When considering the linear absorption spectrum for the vibrational mode, one can also assume that the bath temperature is constant; this is a standard approach in linear response theory [ 271. The equation for the bath temperature is rather trivial [ 25 1, so we will not consider it further and 387
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henceforth assume that the bath temperature is constant. The present communication seeks to answer two important questions relevant to the PMHB model. (i) Is the PMHB model applicable to polyatomic molecules and, if so, in which energy region? (ii) What is the equation describing the evolution of the density matrix u for the pumped mode in the bath? In spite of the extensive use of the PMHB model in polyatomic excitation theory [ 15-251, no accurate investigation of its applicability has been reported in the literature. An equation for o has been proposed on phenomenological grounds [ 18,19 ] and derived on a number of occasions [ 15-l 7 1. Interaction between the pumped mode and the bath, due to anharmonical terms in the molecule Hamiltonian, induces dephasing processes (and consequent energy relaxation from the pumped mode) which may be represented by incorporating a relaxation operator ZRin the kinetic equation for 0. In addition, the anharmonica1 interaction results in a change in the dynamical part of the equation for a, which can sometimes be described by a modification of the Hamiltonian for the pumped mode. The equation is au/at= -i[Z&,
a] +ZR ,
(1)
The problem is to find flefland ZRfor the pumped mode. In the absence of any interaction with the bath one has &=Z!Z, and ZR= 0, where HP is the pumped mode Hamiltonian. We now consider the operator ZR, as it determines both the relaxation and the broadening of the mode absorption spectrum. Two alternative approximations have been employed for ZR: the random-phase approximation [ 15-23,261 and the quasi-harmonic approximation [ 5,24,25,28,29]. The correspondence between these two approaches has not yet been fully analysed; however, the former approach is not at all applicable to a highly excited molecule. There is a large error, which arises as discussed below. A particularly simple case [ 301 for which a kinetic equation can be derived occurs when: (i) the Markovian approximation is valid, i.e. the system relaxation time r is much larger than the correlation time r, for the bath, (ii) the system frequencies are not degenerate, i.e. for any different level pairs (m, n) 388
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and (m’, n’) one has lo,,-co,,,,,,1 %r-‘.Thefirst condition is necessary in order to obtain a differential kinetic equation (not an integro-differential equation such as those describing systems with memory [ 3 11). The second condition provides an independent damping of the off-diagonal elements of the density matrix, so the off-diagonal part of the kinetic operator is 6,,=-
(l/T,
)mn~mn,
where T2 is the phase relaxation time, which is specific for any pair of levels (m, n). These two conditions in fact form the random phase approximation, see section 3. If the latter condition is satisfied, the system relaxation proceeds in the same manner as for a set of non-coupled two-level subsystems. If the system frequencies are degenerate, as is the case for the standard harmonic oscillator, the relaxation of the off-diagonal elements is much more complicated. There is a coupling between different matrix elements, and the general equation takes the form
where Rm,,kl are coefficients, see section 3. Thus a coherence appears in the damping of the off-diagonal elements, and relaxation involves the whole system. As relaxation of the off-diagonal elements determines the absorption spectrum, the character of the relaxation is of fundamental importance. It was noted [ 321 that by taking account of the abovementioned coherent effects a substantial narrowing of the absorption spectrum may result. We shall assume that the Markovian approximation is adequate in some energy region, that is to say, t> z,. We still have to determine if the second condition is satisfied in the excited molecule. It is not difficult to deduce that it is not. The quantity T-’ determines a specific width for the absorption spectrum. As is known [ 121, multiphoton excitation of polyatomic molecules in the quasi-continuum region is possible solely because the broadening of the spectrum exceeds the anharmonic level shifts 6, so r-r s S. The quantity 6 characterizes the deviation of the vibrational spectrum from an equidistant pattern; it amounts to about l-4 cm-‘. Line widths in the spectra of excited molecules are typically 1O-30 cm-’ [12],sooneindeedhast-‘*&AsSdepends
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on the difference (w,, - o,,,,,, 1, the second wndition is clearly invalid. One can consider the opposite situation, where 1corn,- o,,~. I *: 7-‘, so that the nonequidistant structure of the spectrum is not essential for the relaxation process, and the vibrational mode can be represented by a sequence of equidistant levels. This is the quasi-harmonic model [ $25 1, where the mode frequencies depend on the molecular energy, but the mode levels remain equidistant. It has been implied by some authors [ 15-23,261 that condition (ii) is fulfilled, which is not the case as we have argued, and consequently the width of the IR absorption spectra of excited molecules has been substantially overestimated. For example, if one calculates the transition spectrum between levels of the harmonic oscillator with the (incorrect) assumption that the second condition holds, the well known formula [ 15-l 91 is obtained
where k f%,k is the rate of vibrational relaxation from level n to level k under the action of the heat bath (the purely phase relaxation is neglected for the moment, see section 2). For an oscillator with linear friction [ 33 1, the spectral width, calculated in this way, is and
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PMHB, is applicable for polyatomic molecules. Under the assumptions adopted here, the anharmanic interaction coupling the pumped mode to heat bath is of order m + 1, where m 2 2. The essential part of the anharmonic interaction can be represented [25] by a term q&(ql, .... qs), where qP is the coordinate of the pumped mode, and ql, .... qs are the coordinates of the bath. This form is chosen because the number of terms in the anharmonic potential linear in the mode coordinate is WS~, s>> 1, the number of quadratic terms is z:s”‘- ‘, etc. The bath function B,,, can be written as (m+l) &(q,...q,)=
5
.
i
a, -am= I
qo1,...qa, 9
(3)
where $(“‘+I) are specific constants of the (m + 1 )th order anharmonicity. The bath correlation time, z,, is defined in terms of quantities like ( B, ( t )B, (0 ) ) , where B,(t) is an operator in the Heisenberg representation, with averaging over the bath states implied. Correlators of this type characterize the bath in its equilibrium state. If we assume [ 51 that (qi(t)qi(O)) Pexp( -yt), where y=y(E) is a specific relaxation rate for the molecule, depending on its energy E, then the correlator (B(t) B(O)} is a sum of oscillating exponentials with composite frequencies Oil + Wizf ... &wi, and with a damping rate of my, so (B(t)B(O))ccexp(-myt)
C exp(i&t) Lx
.
(4)
=y1[(2n+1)(2n+1)+2A],
where ff is the mean occupation number for the oscillator, and yI determines the rate of energy relaxation towards equilibrium Ai = - 2y,At, see eq. (8) below. It is known, in fact, that the transition spectrum simply has width yI, see eq. (8) below. The factor [(2~+1)(2n+1)+2ff] amounts to 10 and higher for excited molecules (AS l-2), so the spectral width is typically overestimated by more than an order of magnitude.
2. Markovian approximation We now investigate conditions under which the Markovian approximation, which is fundamental for
The range of the composite frequencies in a polyatomic molecule, d, is of the order of a typical molecular frequency, AZ t3. The spacing between neighbouring composite frequencies depends on the number of frequencies involved in the composite oscillation; it is determined by the inverse density of the corresponding Fermi resonances [ 25 1, (~2)~‘. If one has (pr)-‘Amy, then the correlator (B(t)B(O)) is damped at time intervals &WA- *wc7.1-‘,because of dephasing of the oscillating exponents, and recurrence phenomena, which are possible at times 7, %pF, are suppressed by the exponential exp( - myt), since myk (pF ) -I. Thus, under the condition myk(pE)-‘, the bath correlation time, z,, is of the same order as a typical inverted molecular frequency. Clearly, in this case 389
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TZ y-’ >>T,, which is necessary for the Markovian approximation. Typical values of the inverse density of resonances in molecules like CFSI, SF6 and (CF,) &I lie within the interval l-l 0 cm-’ and depend on the number of degrees of freedom and the resonance type (see ref. [ 25 ] ) , For molecules excited to energies close to the dissociation energy, the width of the absorption spectrum is usually 2 10 cm- 1 [ 121. The condition myk (pz ) - ’is already fulfilled for third-order resonances, having a minimal densityp”“. On the other hand, the width depends on the molecular energy [ 5 1, so the Markovian approximation is valid, in general, only for high energies and sufficiently large molecules. We have considered the case of a definite order of anharmonicity, m + 1. If at certain energies the Markovian approximation is valid for third-order resonances, m= 2, then it is surely adequate for resonances of higher orders, since the resonance density p F increases rapidly with m [ 25 1. In order to be sure that the Markovian approximation is valid, one needs to estimate the order of anharmonicity, m* + 1, dominating in the intramolecular vibrational energy redistribution. This problem has been considered [ 34 1. At low energies m* = 2, and at higher energies, ER D,m* may be very large, e.g. > 10. The energy dependence of m* is determined by the molecular frequencies, the number of degrees of freedom, and the rate of the decrease of the anharmonicity constants with increase of the order m + 1. If m* is known [ 34 1, the necessary condition, under which the approximation is reasonable, is m*y> (pz:)-‘.
3. RPA and QHA for excited polyatomic molecules The condition necessary for the random phase approximation is obtained as follows. A general matrix form of the kinetic equation for a system density matrix cr is [ 30 ] (5)
where wl,, are the system transition frequencies, renormalized because of the interaction with the heat bath, &,,kl are kinetic coefficients. These coefficients determine the relaxation rate y, which has an 390
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where R is a typical value order of magnitude yx IR 1, of&,H. Let us assume that the system levels are not degenerate and consider the relaxation of the diagonal elements, m=n. The rhs of eq. (5) oscillates rapidly in all terms of the sum, except for k= 1. Therefore the diagonal terms are relaxing independently of the off-diagonal terms. An estimate for the off-diagonal terms, to first order in 1R I, is
(6) If the system frequencies are not degenerate, i.e. IeLn -wL, Ix 6 3 yz R for any different pairs of levels (n, m) and (k, I), then such pairs (k, 1) can be discarded from (5). Only the terms with (k, I) = (m, n) are retained, and the relaxation term is reduced to - ( ~/T,),,,,u~,,, where (l/T,),,= This is the random phase approximation. -L,,,. The criterion of applicability of this approximation is 6> y, that is to say, anharmonic shifts of the system levels must substantially exceed their width. In the opposite situation, 6 5 y, the anharmonicity can be neglected, and the levels can be considered as equidistant. This determines the applicability of the quasi-harmonic approximation, while the random phase approximation is invalid. Note that ref. [ 301 gives a considerably overestimated condition for the random phase approximation, namely, 6* T; , where T, is the damping time for correlations in the heat bath. As Y-XT; I, one has 6 B y provided that 6 % T; ’, while the inverse statement is false. A typical situation for a vibrational mode of a highly excited polyatomic molecule is 6 4z y * 7; ’, corresponding to the Markovian quasi-harmonic approximation. We now consider the kinetic equation for the pumped mode in the quasi-harmonic approximation. The most general situation, taking account of all orders in the anharmonicity and of the phase relaxation, has been considered previously [ 25 1. In a particularly important case, where the coupling of the pumped mode to the heat bath is linear in the mode coordinate, the equation for the density matrix is [25]
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+Y,{(2aaa+- atao--au+a)(fif +~(2a+au-aa+a-aua+)},
1)
(7)
where a, a+ are creation and annihilation operators for the pumped oscillator, A is the mean occupation number for the mode when it is in equilibrium with the heat bath, and the relaxation constant y1 determines the relaxation rate of the mode energy E to its value in the equilibrium state, 5, k= -2y, (E-F)
(8)
as well as the absorption spectrum of the excited molecule [ 25 1,
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approximation and the random phase approximation. It is shown that the latter, which has been widely utilized [ 15-26 1, is invalid for a highly excited molecule. It neglects an essential coherence in the damping of off-diagonal matrix elements of the density operator describing the pumped mode; as a consequence, the width of the IR transition spectrum is overestimated by at least an order of magnitude. An adequate approach to highly excited molecules is the quasi-harmonic approximation, which admits an exhaustive analytical investigation in terms of the quasiprobability function and coherent states. References [ 11P.D.Robinson and K.A. Holbrook, Theory of monomolecular reactions (Academic Press, New York, 1974).
[ 2 ] S.A. Rice, in: Advances in laser chemistry (Springer, Ber-
where 4 is the quasi-harmonic frequency of the mode [ 5,25 1. The off-diagonal matrix elements are in fact coupled in the quasi-harmonic approximation. The matrix form of eq. (7) is
t-Y,{(flfl)(2J(n+l)(m+l)c,+,,~+, -Jmfl,,
-&a,,)
+A(2JZzrr&,_, -Jn+lo,,)).
-JmTick, (10)
The relaxation operator on the rhs contains a num ber of terms, besides a,,. This is the essential di! fcrencc between the quasi-harmonic approximation and the random phase approximation. The matrix form of the quasi-harmonic approximation is much more complicated than that for the random phase approximation. If one starts from the operator equation (7), however, and employs the coherent state basis and the quasiprobability representation [28,29,33], the quasi-harmonic approximation admits an exhaustive analytical investigation.
4. Conclusion Two alternative approximations to the intramolecular vibrational relaxation in polyatomic molecules have been considered here, the quasi-harmonic
lin, 1978). [3] V.S. Letokhov and A.A. Makarov, Usp. Fiz. Nauk 134 (1981) 45. [4] J.S. Hutchinson, J.T. Hynes and W.P. Reinhardt, Chem. Phys. Letters 108 (1984)353. [ 51A.A. Stuchebrukhov, M.V. Kuzmin, V.N. Bagratashvili and V.S. Letokhov, Chem. Phys. 107 (1986) 429. [ 6 ] G.M. Zaslavsky, Stochasticity ofdynamical systems (Nauka, Moscow, 1983) [in Russian]. [ 7 ] D.W. Oxtoby and S.A. Rice, J. Chem. Phys. 65 ( 1976) 1676. [ 81 F.M. Izrailev, B.V. Chirikov and D.L. Shepelyansky, preprint 80-210, Institute of Nuclear Physics, Novosibirsk (1980). [9] V.N. Bagratashvili, Yu.G. Vayner, V.S. Doljikov, S.F. Koljakov, VS. Letokhov, L.P. Malkin, E.A. Rybov, E.G. Silkis and V.D. Titov, Zh. Exp. Teor. Fis. 80 (1981) 1008. [lo] V.M. Akulin, V.D. Vurdov, G.G. Esadse, N.V. Karlov, A.M. Prokhorov and E.M. Khohlov, Pisma v Zh. Exp. Teor. Fis. 41 (1985) 239. [ 111 V.S. Doljikov, Yu.S. Doljikov, VS. Letokhov, A.A. Makarov, A.L. Malinovsky and E.A. Ryabov, Chem. Phys. 102 (1986) 155. [ 121 V.S. Letokhov, Nonlinear selective photoprocesses in atoms and molecules (Nauka, Moscow, 1983) [in Russian 1, [ 131 P.M. Felker and A.H. Zewail, Chem. Phys. Letters 108 (1984) 303. [ 141 N. Bloembergen and A.H. Zcwail, J. Phys. Chem. 88 (1984) 5459. [ 151 D.P. Hodgkinson and J. B&g, Phys. Letters A 43 (1976) 451. [ 161 C.D. Cantrell, H.W. Galbraith and J. Ackerhalt, in: Multiphoton processes, ed. J.H. Eberly (Springer, Berlin, 1978). [ 171 C.D. Cantrell, in: Laser spectroscopy, Vol. 3, eds. J.L. Hall and J.L. Carlsten (Springer, Berlin, 1977) p. 109. [ 18 1H. Friedman and V. Ahiman, Opt. Commun. 33 (1980) 163.
391
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[ 191J.C. Stephenson, D.S. King, M.F. Goodman and J. Stone, J. Chem. Phys. 70 ( 1979) 4496. [20] J. Black, E. Yablonovich, N. Bloembergen and S. Mukamel, Phys. Rev. Letters 38 (1977) 1131. [21] J.L. Lyman, J. Chem. Phys. 67 (1978) 1868. [22]E.R. Grant, M.J. Coggiola, Y.T. Lee, P.A. Schulz, AaS. Sudbo and Y.R. Shen, Chem. Phys. Letters 52 ( 1977) 595. [23] M.J. Shultz and E. Yablonovitch, J. Chem. Phys. 68 (1978) 3007. [24] L.M. Narducci, S.S. Mitra, R.A. Shatas and C.A. Coulter, Phys. Rev. A 16 (1977) 247. [ 251 V.N. Bagratashvili, M.V. Kuz’min, V.S. Letokhov and A.A. Stuchebrukhov, Chem. Phys. 97 (1985) 13. [26] S. Rashev, Chem. Phys. Letters 100 (1986) 578.
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[27] L.D. Landau and E.M. Lifshits, Statistical physics (Nauka, Moscow, 1977). [ 281 V.N. Sazonov and A.A. Stuchebrukhov, Chem. Phys. 56 (1981) 391. [29] V.N. Sazonov, A.A. Stuchebrukhov and S.V. Zatsepin, Chem. Phys. 69 (1982) 459. [30] V.M. Fain, Photons and nonlinear media (Soviet Radio, Moscow, 1972). [ 311 S. Mukamel, in: Photoselective chemistry, eds. J. Jortner, R.D. Levine and S.A. Rice (Wiley, New York, 198 1) [32] K. Kay, J. Chem. Phys. 75 (1981) 1690. [33] B.Ya. Zeldovitch, A.M. Perelomov and V.S. Popov, Zh. Eksp. Teor..Fiz. 57 (1968) 196. [ 341 A.A. Stuchebrukhov, Soviet Phys. JETP 64 (1986) 1195.