- Email: [email protected]
A derivation of the master equation by restricted quantum exchange
Volume 82, number 3
CHEMICAL PHYSICS LETTERS
A DERIVATION
OF THE MASTER
EQUATION
BY RESTRICTED
15 September 1981
QUANTUM
EXCHANGE
Kenneth G. KAY Departnlent
of Chemistry,
Kansas State Ukiversity,
Manhattan,
Kansas 66506,
USA
and James STONE,
Everett THIELE
and Myron F. GOODMAN
Department of Blologzcal Scrences and Center for Laser Studzes, Los Angeles, California 90007, i-K4
Universrty of Southern
CahfOrnla.
Recewed 27 March 1981; m final form 1 June 1981
The Pauli master equation for intramolecular viiratmnal relaxation and the heat bath feedback Bloch equations for radiative pumping of polyatomlc molecules can be derived by replacing the standard assumption of random matrw element coupling between zero-order vlbratlonal states by an assumption that relaxation is governed by restricted quantum exchange
The Pauli master equation has frequently been applied to describe the dynamics of vibrational energy transfer in isolated polyatomic molecules. It has been useful for interpreting kinetic and spectroscopic evidence concerning intramolecular vibrational relaxation and for addressing the issue of statistical equilibration in collisionless systems [I -6]_ A related equation known as the heat bath feedback Bloch equation has recently been introduced to describe the competition between intramolecular vibrational energy transfer and radiative transitions in molecules exposed to intense electromagnetic fields [7,8] _This equation leads to explicit formulas for radiative transition rate coefficients appearing in the master equation for multiphoton excitation and to expressions for optical lineshapes of molecules undergoing transitions between excited vibrational levels [7,9] _ Fust-prmcrples derivations of these two equations which identify the conditions for their validity have been presented [ 1O-l 21 T _ The purpose of tms paper is to describe the basis for alternative derivations that may, under certain conditions, be more appropriate than the existing ones. One result is that the range of vahdity of the Pauli and Bloch equations can be extended to cover a significant class of real molecules previously excluded. We are concerned exclusively with first-principles derivations; the only assumptions we allow are those that describe properties of the hamlltonian in a basis of zero-order states. One critrcal assumption needed to derive both the master equation and the heat bath feedback equations is that the matrix elements V,, = bz IV Inz) of the vibrational interaction v between zero-order states In>, ]nz) obey a version of the diagonal singularity condition introduced origmally by van Hove [ 141. To state this condition, replace V with hV where h is a dimensionless scale factor, and let Aci) be operators dragonal in the zero-order representation, such that AZ) = (n ]A(‘)]n) vary slowly with 12. Then appropriate perturbation techniques express occupation probabilities for zero-order states in terms of certain quantities
c c -..c
F,$$=NJ’~
ml “1
#n,rz’
)
m2
V,tm1A~~Vmlm2 AC21 m2
___A@)V
mp
m2 fiz,ml,n',
mp f n,
ml,
m2,
mp
---
mp’
mp_l,
0)
3
n’,
p=l
, 3- _._ _
T Ar. alternative derivation of the heat bath feedback equations has been provided by Friedmann [ 131. 539
Volume 8 2, number 3
CHEhlICAL PHYSICS LETTERS
15 September 1981
The dldgonal singularity condltlon 1s that, for suftkently small X, the ratio of off-diagonal elements I$$)# (II’ # 12) for the transition probto diagonal elements F!Ej must be small enougfi to allow neglect of F,,,@‘)pin the expressions abiflties In previous work [ 1 l-l 31. the dmgonaf singularity property was presumed to arise from randomness of the agns of the V,,,,, _ Yet, there is no direct evidence to support the idea that the signs of the I&?, are indeed random. Tfds exposes A major weakness in current derlvatlons of Bloc11 and master equations. We argue flere that, for large enough molecules, the Paul1 and Bloch equarions can be derived more convincing-
Iy by repfacmg this random matrix hypothesis with an alternative assumption that has previously been used in interpreting the energy dependence of optical fmewldths - the restricted quantum exchange hypothesis (RQE) [ 15.161. Like tfle random matrix assumption. the RQE assumption leads to a dlagonaf smgufarity property and, thus. to tfle validity of tfle desired dynamical equations. Unlike the random matrix assumption, the RQE assumptlon 1s supported by empincaf evidence. The restricted quantum exchange hypothesis is the assumption that the only pairs of zero-order states on a c@en energy shell whose mutual couplmg through intramolecular interaction is significant are those that differ by a feu quanta m a few vlbrationaf degrees of freedom. Evidence for tile validity of this hypothesis comes from anaf1 sls of the expressions from absorption rate constants and lmeshape functions resultmg from the heat bath feedback formahsm. In the absence of RQE, tflese expressions predict that radiative transition rate coefficients decrease exponentially to zero and that absorption linewidths increase exponentially to infinity as the energy in the moIecule increases [ I5,I 6]_ Such behavior IS Inconsistent with the occurrence of detectable infrared multiphoton dissociation for molecules such ds SF6 and the experimentally observed tendency of absorption lmewidths, especlafil overtone fnlewdtf~s, to level off or even decrease with increasing energy [ 17.181. By applying the RQE ansatz to rhe rate coefficient expressions a successful detailed comparison between theory dnd experiment was obtained for the ldser-mduced decomposition of CF, CFCf [ 19]_ Reasonable rates and hnewldths which level off in a reasonable manner were also obtained for CFzHCf-[30] and SF6 [2 1 ] . We note tflat a form of restrlcted quantum exchange operates in certain infinite systems such as gases and anharmanic crystals [22] and that this property [14] is the ultimate source of the diagonal singularity in these cases. However. etisting demonstrations of the singularity of these systems are confined to very IOW perturbation orders p_ Validity of a master equation describing dynamics of finite molecules for a chemically relevant time scale requlres proof of the singuIarity for all a ranging from I to values much greater than rhe number of degrees of freedom _ In the foilowing. we show how restricted quantum exchange leads to a diagonal singularity condition for vibratlonaf relaxation in finite molecules and examine the nature of the singularity. A full derivation of the master equatlon based on the restricted quantum exchange assumption (and, thus, a demonstration that the singularity described here is mdeed strong enough for these purposes) will be presented elsewhere_ We consider a large, highly excited molecule. The number of vlbrationaf modes, s, is much greater than 10 * and the number of these modes with at feast one quantum of excitation is on the order of s. We assume that the form of the interaction I/leads to restricted quantum exchange. states that are directly coupled differ by the gain of t?z
540
Volume 82, number 3
CHEMICAL PHYSICS LETTERS
15 September 1581
ence in quantum numbers for all the various modes. We now count the number of states m eq. (1) reached after each of the p + 1 transitions and keep track of their distance D. The states reached in the first transition differ from the Initial state by 2mz quanta (nz quanta gained plus nz quanta lost); hence, D = 2m. Since each of these quanta can go into e:s modes. the number of states reached is
as2m
f
., p, the states that are reached differ from the previous states by 2F?Z At each subsequent transition i = 1,2, quanta which can be used either to increase or decrease D. A change in D by the amount 2ri, where ‘i- = m, nz - 1, -.-) -CV, is achieved by using 17z+ ‘E quanta to Increase and )zz - ‘f quanta to decrease D. Each of the IPZ+ ri quanta can go into on the order of s modes; these contribute a fclctor of --“srntri to the number of states reached. Each of the z7z- z-i quanta can go only into those modes which are already excited or de-excited m a particular way relatzve to the initial state; these contribute a factor& (not calculated here) to the number of states reached. Thusf,~“‘+‘~ is the contribution to the number of states in eq (1) from transition i which increases D by 2z-i. The zzumber of states with D = 2fif z 2rzz + XiE1 2 ‘i that are reached in a tota of p f 1 transitions is now calculated as
where Cp,nr = Xr fif2 ___ f, and the sum IS over all ‘;- conszstent with a given Z!M_ Smce this count mcludes transitions to al1 the =s2111 states with D = 2M, where F $> , is proportional to the number of states involved In transitions to a particular final state iz’, we dlvzde the above number by sZAf to obtain
(2) Neglecting the M-dependence of CP,nf for the moment, we find that F,,,, (f’), ISat least a factor of $ larger for dzagozlal (M= 0) than for non-diagonal (Ma 1) transitions. Elsewhere, we show that thzs is preczsely the strength of the diagonal smgularity needed to derive master and heat bath feedback equations_ In fact, the M-dependence of CP,hf cannot generally be neglected and can be shown to reduce the ratio fi$z)/eH? by a factor on the order of min(p, s). However, a complete derivation of the master equation shows that F$? contains additional factors which compensate for this contnbution. These new factors arise when: (a) it is recognized that the functions A,W in eq. (1) that are relevant to the derivation can be expressed as [E - E,l - G,,(E)]-’ where G is a complex functzon of energy E; and (b) the summations in eq. (1) are replaced by integrations over continous energies E,, _ As a result, the diagonal singularity is mdeed of order s, as described earher. These additional factors are also dependent on restricted quantum exchange and thus the restricted exchange ansatz enters a complete derivation in two separate but related ways. We emphasize the essential role of restricted quantum exchange in producing the diagonal singularity- If an arbitrary number of quanta are exchanged with each interaction, the number of terms in eq. (1) is independent of Mand no diagonal singularity IS predicted by our analysis. Accurate formulae for counting the terms in eq. (1) have been derived [34] for low values of z?z.Analytical and numerical investigation of these expresslons confirm the conclusions reported above. It is interesting to recall that evidence for the valrdity of restricted quantum exchange originally arose from an examination of the heat bath feedback expressions for absorption rates and lmewidths. However, there remained a possible question of logical consistency_ The arguments presented above show that the RQE assumption is not only consistent with the heat bath feedback equation but may be responsible for Its very validity. The correctness of the derivation, whether by restricted quantum exchange or by the random sign assumption [1 l-13] is ultimately subject to another consistency requirement restricting the magnitude of the linewidths r to be much less than the range AE over which average values of the matrix elements are constant [l 11. Siznple ana’ In these estimates we neglect combinatorial factors which do not alter the order of magnitude
of our final results. 541
Volume
82, number 3
CHEMICAL PHYSICS LETTERS
15 September1981
14hc expressions for both T, and T2 contributions to the Iinewidth, which depend only on the vibrational degeneracy and the number of quanta in the heat bath have been derived when only a few quanta are exchanged [15,16]. Thus. we not only obtain a consistency check, but also a complete set of T, and T2 relaxation rates required to fully specify the final equations. We note, with regard to the consistency question. that the leveling off of the llncwi&h. obtauled with the restrlcted quantum exchange ansatz, may be essential. Linewidths which increase xcordmg to the full density of states would exceed the al!owable range AE at unacceptably low energies. This work was supported in part by NSF Grants CHE 73-10020 and CHE 79-10385. Much of the work was carried out durmg d workshop sponsored by NRCC and CECAM at the UniversitC Paris-Sud, France; we thank Dr. Cxl &loser and Dr. William A. Lester for support. We also wish to thank Ms.Sarah Wright for skillful typing of the m.muscript.
References
[ 1 1 J.W. [I] W.Zl.
Brsuner and D.J. W&on, J. Phqs Chem. 67 (1963) 1134. Gelbxt, S.A. RW and K I‘. Treed, _I.Chem. Phys 52 (1970) 5718. [3\ J.D Rynbrandt and B S Rabmo\~tch, J Phys. Chem. 75 (1971) 2164. [4] S.H. Lin, K.11 Lau, W. Richardson, L. Volk and H. Eyrmg, Proc. Nntl. Aud SCI. US 69 (1972) 2778. (5 1 1.. Ttuele. MI. Goodman and J. Stone, Opt. Eng. 19 (1980) 10; Cbem. Phys Letters 69 (1980) 18. [6] S \luhamel and R.E. Smalley, J. Chem. Phls. 73 (1980) 4156. 17) J Stone .~nd XI-‘. Goodman, J. Chem. Phys. 71 (1979) 408. [8 1 Ii. rrwdmann. III L.aser-Induced processes in molecules, eds. K.L. Kompa and S D. Smith (Springer, Berhn, [9 ] II. i‘rledmsnn and V. Ahlman. Opt. Commun. 33 (1980) 163. [ 1 O] B. Cameli and A. Nirznn, J. Chem. Phys. 72 (1980) 2070. Ill] KG.l+.J.Ch=zm Phys.61 (1974)52OS_ [ 121 K.G. IQy, I. Chem. Phys. (1981), to be published. [ 131 H. rrxdmann, private communiutlon. [ 141 L van Hove, Phys~ca 21 (19.55) 527,23 (1957) 441. [ 1.5 I J. Stone, C. Thlele and h1.T. Goodman, Chem. Phys Letters 7 1 (1980) 171. [ 161 J. Stone. XI-. Goodman and E. Thiele, J. Chem. Phys. (1981), to be published. [ 171 SN. Beck. D.E. Powers. J.B. Hopkins and R.E. Smalley, J. Chem. Phys_ 73 (1980) 2019; D D Smith and A.H. Zewad. J. Chem. Phys. 71 (1979) 540. [ 18 1 R.G. Bray and M.J. Berry, J. Chem. Phys. 71 (1979) 4909. [ 191 J. Stone, E. Thele, h1.F. Goodman, J. Stephenson and D. S. Kin g, J. Chem. Phys. 73 (1980) 2259. [70] J Stephenson, D. King, M-F. Goodman and J. Stone, J. Chem. Phys. 70 (1979) 4496. 121 ) J A. Ilorsley, J. Stone, h1.F. Goodman and D.A. Dows, Chem. Phys. Letters 66 (1979) 461. [22] [23] [24]
542
1. Prlgogme, Noncquxlibrmm statlsticai mechamcs R. Brout and 1. Prlgogine, Physiu 22 (1956) 621. L. Verboven, PhysIca 26 (1976) 1091. J. Ftone, unpublished results.
(InterscIence, New York, 1962);
1979).
CHEMICAL PHYSICS LETTERS
A DERIVATION
OF THE MASTER
EQUATION
BY RESTRICTED
15 September 1981
QUANTUM
EXCHANGE
Kenneth G. KAY Departnlent
of Chemistry,
Kansas State Ukiversity,
Manhattan,
Kansas 66506,
USA
and James STONE,
Everett THIELE
and Myron F. GOODMAN
Department of Blologzcal Scrences and Center for Laser Studzes, Los Angeles, California 90007, i-K4
Universrty of Southern
CahfOrnla.
Recewed 27 March 1981; m final form 1 June 1981
The Pauli master equation for intramolecular viiratmnal relaxation and the heat bath feedback Bloch equations for radiative pumping of polyatomlc molecules can be derived by replacing the standard assumption of random matrw element coupling between zero-order vlbratlonal states by an assumption that relaxation is governed by restricted quantum exchange
The Pauli master equation has frequently been applied to describe the dynamics of vibrational energy transfer in isolated polyatomic molecules. It has been useful for interpreting kinetic and spectroscopic evidence concerning intramolecular vibrational relaxation and for addressing the issue of statistical equilibration in collisionless systems [I -6]_ A related equation known as the heat bath feedback Bloch equation has recently been introduced to describe the competition between intramolecular vibrational energy transfer and radiative transitions in molecules exposed to intense electromagnetic fields [7,8] _This equation leads to explicit formulas for radiative transition rate coefficients appearing in the master equation for multiphoton excitation and to expressions for optical lineshapes of molecules undergoing transitions between excited vibrational levels [7,9] _ Fust-prmcrples derivations of these two equations which identify the conditions for their validity have been presented [ 1O-l 21 T _ The purpose of tms paper is to describe the basis for alternative derivations that may, under certain conditions, be more appropriate than the existing ones. One result is that the range of vahdity of the Pauli and Bloch equations can be extended to cover a significant class of real molecules previously excluded. We are concerned exclusively with first-principles derivations; the only assumptions we allow are those that describe properties of the hamlltonian in a basis of zero-order states. One critrcal assumption needed to derive both the master equation and the heat bath feedback equations is that the matrix elements V,, = bz IV Inz) of the vibrational interaction v between zero-order states In>, ]nz) obey a version of the diagonal singularity condition introduced origmally by van Hove [ 141. To state this condition, replace V with hV where h is a dimensionless scale factor, and let Aci) be operators dragonal in the zero-order representation, such that AZ) = (n ]A(‘)]n) vary slowly with 12. Then appropriate perturbation techniques express occupation probabilities for zero-order states in terms of certain quantities
c c -..c
F,$$=NJ’~
ml “1
#n,rz’
)
m2
V,tm1A~~Vmlm2 AC21 m2
___A@)V
mp
m2 fiz,ml,n',
mp f n,
ml,
m2,
mp
---
mp’
mp_l,
0)
3
n’,
p=l
, 3- _._ _
T Ar. alternative derivation of the heat bath feedback equations has been provided by Friedmann [ 131. 539
Volume 8 2, number 3
CHEhlICAL PHYSICS LETTERS
15 September 1981
The dldgonal singularity condltlon 1s that, for suftkently small X, the ratio of off-diagonal elements I$$)# (II’ # 12) for the transition probto diagonal elements F!Ej must be small enougfi to allow neglect of F,,,@‘)pin the expressions abiflties In previous work [ 1 l-l 31. the dmgonaf singularity property was presumed to arise from randomness of the agns of the V,,,,, _ Yet, there is no direct evidence to support the idea that the signs of the I&?, are indeed random. Tfds exposes A major weakness in current derlvatlons of Bloc11 and master equations. We argue flere that, for large enough molecules, the Paul1 and Bloch equarions can be derived more convincing-
Iy by repfacmg this random matrix hypothesis with an alternative assumption that has previously been used in interpreting the energy dependence of optical fmewldths - the restricted quantum exchange hypothesis (RQE) [ 15.161. Like tfle random matrix assumption. the RQE assumption leads to a dlagonaf smgufarity property and, thus. to tfle validity of tfle desired dynamical equations. Unlike the random matrix assumption, the RQE assumptlon 1s supported by empincaf evidence. The restricted quantum exchange hypothesis is the assumption that the only pairs of zero-order states on a c@en energy shell whose mutual couplmg through intramolecular interaction is significant are those that differ by a feu quanta m a few vlbrationaf degrees of freedom. Evidence for tile validity of this hypothesis comes from anaf1 sls of the expressions from absorption rate constants and lmeshape functions resultmg from the heat bath feedback formahsm. In the absence of RQE, tflese expressions predict that radiative transition rate coefficients decrease exponentially to zero and that absorption linewidths increase exponentially to infinity as the energy in the moIecule increases [ I5,I 6]_ Such behavior IS Inconsistent with the occurrence of detectable infrared multiphoton dissociation for molecules such ds SF6 and the experimentally observed tendency of absorption lmewidths, especlafil overtone fnlewdtf~s, to level off or even decrease with increasing energy [ 17.181. By applying the RQE ansatz to rhe rate coefficient expressions a successful detailed comparison between theory dnd experiment was obtained for the ldser-mduced decomposition of CF, CFCf [ 19]_ Reasonable rates and hnewldths which level off in a reasonable manner were also obtained for CFzHCf-[30] and SF6 [2 1 ] . We note tflat a form of restrlcted quantum exchange operates in certain infinite systems such as gases and anharmanic crystals [22] and that this property [14] is the ultimate source of the diagonal singularity in these cases. However. etisting demonstrations of the singularity of these systems are confined to very IOW perturbation orders p_ Validity of a master equation describing dynamics of finite molecules for a chemically relevant time scale requlres proof of the singuIarity for all a ranging from I to values much greater than rhe number of degrees of freedom _ In the foilowing. we show how restricted quantum exchange leads to a diagonal singularity condition for vibratlonaf relaxation in finite molecules and examine the nature of the singularity. A full derivation of the master equatlon based on the restricted quantum exchange assumption (and, thus, a demonstration that the singularity described here is mdeed strong enough for these purposes) will be presented elsewhere_ We consider a large, highly excited molecule. The number of vlbrationaf modes, s, is much greater than 10 * and the number of these modes with at feast one quantum of excitation is on the order of s. We assume that the form of the interaction I/leads to restricted quantum exchange. states that are directly coupled differ by the gain of t?z
540
Volume 82, number 3
CHEMICAL PHYSICS LETTERS
15 September 1581
ence in quantum numbers for all the various modes. We now count the number of states m eq. (1) reached after each of the p + 1 transitions and keep track of their distance D. The states reached in the first transition differ from the Initial state by 2mz quanta (nz quanta gained plus nz quanta lost); hence, D = 2m. Since each of these quanta can go into e:s modes. the number of states reached is
as2m
f
., p, the states that are reached differ from the previous states by 2F?Z At each subsequent transition i = 1,2, quanta which can be used either to increase or decrease D. A change in D by the amount 2ri, where ‘i- = m, nz - 1, -.-) -CV, is achieved by using 17z+ ‘E quanta to Increase and )zz - ‘f quanta to decrease D. Each of the IPZ+ ri quanta can go into on the order of s modes; these contribute a fclctor of --“srntri to the number of states reached. Each of the z7z- z-i quanta can go only into those modes which are already excited or de-excited m a particular way relatzve to the initial state; these contribute a factor& (not calculated here) to the number of states reached. Thusf,~“‘+‘~ is the contribution to the number of states in eq (1) from transition i which increases D by 2z-i. The zzumber of states with D = 2fif z 2rzz + XiE1 2 ‘i that are reached in a tota of p f 1 transitions is now calculated as
where Cp,nr = Xr fif2 ___ f, and the sum IS over all ‘;- conszstent with a given Z!M_ Smce this count mcludes transitions to al1 the =s2111 states with D = 2M, where F $> , is proportional to the number of states involved In transitions to a particular final state iz’, we dlvzde the above number by sZAf to obtain
(2) Neglecting the M-dependence of CP,nf for the moment, we find that F,,,, (f’), ISat least a factor of $ larger for dzagozlal (M= 0) than for non-diagonal (Ma 1) transitions. Elsewhere, we show that thzs is preczsely the strength of the diagonal smgularity needed to derive master and heat bath feedback equations_ In fact, the M-dependence of CP,hf cannot generally be neglected and can be shown to reduce the ratio fi$z)/eH? by a factor on the order of min(p, s). However, a complete derivation of the master equation shows that F$? contains additional factors which compensate for this contnbution. These new factors arise when: (a) it is recognized that the functions A,W in eq. (1) that are relevant to the derivation can be expressed as [E - E,l - G,,(E)]-’ where G is a complex functzon of energy E; and (b) the summations in eq. (1) are replaced by integrations over continous energies E,, _ As a result, the diagonal singularity is mdeed of order s, as described earher. These additional factors are also dependent on restricted quantum exchange and thus the restricted exchange ansatz enters a complete derivation in two separate but related ways. We emphasize the essential role of restricted quantum exchange in producing the diagonal singularity- If an arbitrary number of quanta are exchanged with each interaction, the number of terms in eq. (1) is independent of Mand no diagonal singularity IS predicted by our analysis. Accurate formulae for counting the terms in eq. (1) have been derived [34] for low values of z?z.Analytical and numerical investigation of these expresslons confirm the conclusions reported above. It is interesting to recall that evidence for the valrdity of restricted quantum exchange originally arose from an examination of the heat bath feedback expressions for absorption rates and lmewidths. However, there remained a possible question of logical consistency_ The arguments presented above show that the RQE assumption is not only consistent with the heat bath feedback equation but may be responsible for Its very validity. The correctness of the derivation, whether by restricted quantum exchange or by the random sign assumption [1 l-13] is ultimately subject to another consistency requirement restricting the magnitude of the linewidths r to be much less than the range AE over which average values of the matrix elements are constant [l 11. Siznple ana’ In these estimates we neglect combinatorial factors which do not alter the order of magnitude
of our final results. 541
Volume
82, number 3
CHEMICAL PHYSICS LETTERS
15 September1981
14hc expressions for both T, and T2 contributions to the Iinewidth, which depend only on the vibrational degeneracy and the number of quanta in the heat bath have been derived when only a few quanta are exchanged [15,16]. Thus. we not only obtain a consistency check, but also a complete set of T, and T2 relaxation rates required to fully specify the final equations. We note, with regard to the consistency question. that the leveling off of the llncwi&h. obtauled with the restrlcted quantum exchange ansatz, may be essential. Linewidths which increase xcordmg to the full density of states would exceed the al!owable range AE at unacceptably low energies. This work was supported in part by NSF Grants CHE 73-10020 and CHE 79-10385. Much of the work was carried out durmg d workshop sponsored by NRCC and CECAM at the UniversitC Paris-Sud, France; we thank Dr. Cxl &loser and Dr. William A. Lester for support. We also wish to thank Ms.Sarah Wright for skillful typing of the m.muscript.
References
[ 1 1 J.W. [I] W.Zl.
Brsuner and D.J. W&on, J. Phqs Chem. 67 (1963) 1134. Gelbxt, S.A. RW and K I‘. Treed, _I.Chem. Phys 52 (1970) 5718. [3\ J.D Rynbrandt and B S Rabmo\~tch, J Phys. Chem. 75 (1971) 2164. [4] S.H. Lin, K.11 Lau, W. Richardson, L. Volk and H. Eyrmg, Proc. Nntl. Aud SCI. US 69 (1972) 2778. (5 1 1.. Ttuele. MI. Goodman and J. Stone, Opt. Eng. 19 (1980) 10; Cbem. Phys Letters 69 (1980) 18. [6] S \luhamel and R.E. Smalley, J. Chem. Phls. 73 (1980) 4156. 17) J Stone .~nd XI-‘. Goodman, J. Chem. Phys. 71 (1979) 408. [8 1 Ii. rrwdmann. III L.aser-Induced processes in molecules, eds. K.L. Kompa and S D. Smith (Springer, Berhn, [9 ] II. i‘rledmsnn and V. Ahlman. Opt. Commun. 33 (1980) 163. [ 1 O] B. Cameli and A. Nirznn, J. Chem. Phys. 72 (1980) 2070. Ill] KG.l+.J.Ch=zm Phys.61 (1974)52OS_ [ 121 K.G. IQy, I. Chem. Phys. (1981), to be published. [ 131 H. rrxdmann, private communiutlon. [ 141 L van Hove, Phys~ca 21 (19.55) 527,23 (1957) 441. [ 1.5 I J. Stone, C. Thlele and h1.T. Goodman, Chem. Phys Letters 7 1 (1980) 171. [ 161 J. Stone. XI-. Goodman and E. Thiele, J. Chem. Phys. (1981), to be published. [ 171 SN. Beck. D.E. Powers. J.B. Hopkins and R.E. Smalley, J. Chem. Phys_ 73 (1980) 2019; D D Smith and A.H. Zewad. J. Chem. Phys. 71 (1979) 540. [ 18 1 R.G. Bray and M.J. Berry, J. Chem. Phys. 71 (1979) 4909. [ 191 J. Stone, E. Thele, h1.F. Goodman, J. Stephenson and D. S. Kin g, J. Chem. Phys. 73 (1980) 2259. [70] J Stephenson, D. King, M-F. Goodman and J. Stone, J. Chem. Phys. 70 (1979) 4496. 121 ) J A. Ilorsley, J. Stone, h1.F. Goodman and D.A. Dows, Chem. Phys. Letters 66 (1979) 461. [22] [23] [24]
542
1. Prlgogme, Noncquxlibrmm statlsticai mechamcs R. Brout and 1. Prlgogine, Physiu 22 (1956) 621. L. Verboven, PhysIca 26 (1976) 1091. J. Ftone, unpublished results.
(InterscIence, New York, 1962);
1979).