J. Phys. Chem. 1082,86,3729-3733
3729
Interchange Perturbation Theory and Phosphorescence: ,Application to CH,O Phlllp Phillipst and Erneot R. Davldson' Department of Chemistry, Unlverslty of Washlngton, Seattle, Washlngton 98 195 (Received: February 2, 1982; In Flnal Form: May 24, 1982)
The equations for applying interchange perturbation theory to the calculation of radiative lifetimes are derived and evaluated for CH20. The individual terms appearing in the expression for the transition amplitude between the ground and 3(n?r*)state are evaluated using either diagrammatic techniques or coupled Hartree-Fock. A comparison is made of the lifetime results obtained from this method with those calculated from the MBPT formalism of Phillips and Davidson. The radiative lifetime of CHzO is predicted to be about 0.02 s.
Introduction Essential to the calculation of a radiative lifetime is the evaluation of the transition amplitude between a singlet ('J,) state and the q = x , y, z sublevels of a triplet (3J,,) state. To first order in the spin-orbit operator, the well-known equation for M(1J,,3J,q)is M(lJ,?J,,) =
cS ( 'J,lPI'J,s) (1J,slhS13J,q)/ (3E- Es) +
In this expression, S runs over all singlet states including lJ, and T,r run over all components of all triplet states including 3J,q. Also, P denotes the dipole operator, hS the spin-orbit operator, and J, the exact eigenfunction of the Bom-Oppenheimer spin-free Hamiltonian for the molecule with eigenvalue E. The radiative lifetime, 7 , is related to M through the Einstein A factors by T = g(A, + A, AJ' (2)
+
where
A, = 4/3( AE)3~-71~2M( 'J,,3J,,)12
(3)
in atomic units. In eq 2, g is the degeneracy of the phosphorescent state and A, is the Einstein A coefficient for emission to or from the triplet sublevel. The primary hurdle in the computation of M(1J,,3J,,) is the evaluation of the sums over the intermediate states in (1). As demonstrated by Langhoff and Davidson,' some of the most important states involve excitations to highlying antibonding orbitals. Hence, the sums in (1)converge slowly if the states are ordered on increasing E. Phillips and Davidson2 recently developed a formalism for evaluating M(lJ,,V,) which retained all terms in (1)but replaced the exact eigenstates of H by approximate states correct to fmt order in the fluctuation potential. Wick's theorem3 was then used to reduce M to a sum over Feynman diagrams. As illustrated with methylene,2 a list of the key intermediate states can be readily determined by examining the individual terms in the calculation of those Feynman diagrams that contribute appreciably to M. However, aside from this success and the retention of all the terms in (l), the MBPT formalism of Phillips and Davidson2is plagued by a problem inherent in all MBPT schemes. The problem stems from the restriction that configurations expanded in the same set of molecular orbitals be used to describe all the states.3 Hence, at most one state in (1)is described Danforth-Compton Predoctoral Fellow.
accurately at the Hartree-Fock level. This problem was clearly exemplified by the methylene lifetime resulh2 Although they were the best choice of orbitals, the 3B1RHF orbitals nonetheless did a poor job of describing the 'A, state and thus the 'A1-3B1 energy gap was two orders of magnitude too large. Since the primary contribution to M for CH2 came from the direct mixing of the 'A, and 3B1states, it was crucial to use an accurate value for this AE, and so an adjusted denominator corrected for higher order correlation (ADHOC approximation)2was introduced. It would be impossible to extend the ADHOC treatment to modify all denominators, however, because the most important configurations often do not represent real physical states. In this paper, we investigate a formalism that is not restricted to the use of one set of SCF molecular orbitals and almost entirely bypasses the problem of accurately describing each intermediate state in the sums in (1). A general theorem is presented for the application of interchange perturbation theory (IPT) to transition matrix elements. Use of this theorem transforms (1) to an equation that can be evaluated accurately from coupled Hartree-Fock wave functions and four simple Feynman diagrams. For simplicity, we have used only the spinown-orbit interaction, although the two-electron spinother-orbit term would be needed for quantitative results. The effects of the other-orbit terms are estimated through atomic shielding factors.
Derivation of the Equations Interchange Perturbation Theory. Dalgarno's interchange theorems have been shown to simplify the calculation of numerous atomic and molecular pr~perties.~ Use of the theorems in the calculation of first-order correlation corrections to polarizabilities and magnetic susceptibilities eliminates the need for computing correlated wave funct i o n ~ In . ~ fact, most double perturbation expansions can be simplified through the use of Dalgarno's theorem^.^ In the following derivation, we show how Dalgarno's ideas can be extended to the evaluation of a transition moment induced by a perturbation. In terms of an unperturbed ground state and an unperturbed excited state the first-order contribution (1)S. R. Langhoff and E. R. Davidson, J. Chem. Phys., 64, 4699 (1976). (2)P. Phillips and E. R. Davidson, J. Chem. Phys., 76, 516 (1982). (3)G.C. Wick, Phys. Reu., 80, 268 (1950). (4)A. Dalgarno and A. L. Stewart,h o c . R. SOC.London, Ser. A , 247, 245 (1958). (5)B.Kirtman, J. Chem. Phys., 49, 3895 (1968).
0022-3654/82/2086-3729$Ql.25/00 1982 American Chemical Society
3730
The Journal of phvsical Chemistry, Vol. 86, No. 19, 1982
of a perturbation A to the transition matrix element of an operator B is given by BG,E(A)=
+
( $GolAI$? ) (EGO - E?)-' ( $?IBI$Eo )
c ($GOIBI$P)(EE0 - E O)-'($?IAI$EO) (4) I
I#E
where HO$? = E?$? and I runs over all states except those explicitly excluded. Using (EGO- E I 0 ) -1 - ( EE0 - E I 0 ) -1 = (EEo- EGo)(EGo- E?)-'(EE0 - E?)-' eq 4 can be rewritten as
In terms of the first-order correction, $Je),to state J from a perturbation 0
Phillips and Davidson
MO's permits an accurate calculation of AE and A (1.1) in this important contribution. The mixing matrix element ('$lhs13$,)can be found from the RHF wave functions for these two states to fair accuracy. Equation 1could have been rewritten as the s u m of two seemingly simple terms
M(1$,3$,) =
('$(hs)lP13$,)
= 2($olPul$(fiJ)
(9)
with u,u = x , y, z. The diagonal components of the tensor define the average polarizability a as
we can rewrite (5) as
=
= ($GOIAI$E'~') + ($G'~'IAI$EO) +
AE[($G'A'I$E(B)) - ($C'B'I$E'A')] + AE-' ( $GoIAI$Eo)[ ( B) G - ( B)E1 AE-'(+c~IBI$E~)[(A)G - ( A ) , ] (6) with AE = EEo- Eoo. For the case that $Eo = 3$q, $Go = '$, B = 1.1, and A = hS, the last term in (6) vanishes giving rise to the final equation for the IPT spin-induced dipole transition amplitude
M(1$,3$,) = MDM+ M1io+ MoJ + M'J
(7a)
MDM= -AE-'( 1$lhS13$,)A(p)
(7b)
with
M'po = ( 1$@)lhS13$s) MLl =
(8)
However, evaluation of the matrix elements of (8) would require the generation of coupled Hartree-Fock wave functions by the addition of a spin-dependent imaginary operator, hS,to the unperturbed Fock operator. To do this calculation, a program would have to be written for truly unrestricted Hartree-Fock calculations for excited states in which (1) the spins were allowed to mix and (2) the orbitals were allowed to become complex. Such a program would be difficult and of limited utility. Further, (8) does not separate into large (MDM)and small terms as does (7). Hence, much of the information concerning the important contributors to M would be lost. Thus, (8) is not a desirable means of evaluating (1). An additional property that can be calculated once first-order (in 1.1) wave functions are available is the dipole polarizability tensor, a. The well-known expression for a from first-order perturbation theory is au"
MOP1
+ (1$lI.113$,(hs))
=
(7c)
( $11 hS I3$ ,( P ) )
a[( l$(hs)13$,(~))
(74
- (1$(~)13$ W ) 9
]
(74
f/3(axx
+ a y y + a,,)
(10)
and the optical anisotropy is given by
p
=
2-'/2[(a,,
- ayJ2 + (a,,- a,,)2
+ (ayy- a,,)2 + 6(aXy2 + aXz2+ a Y ~ ) ] ' (11) /'
in the inertial axis system. Diagrammatic Expansion. Wick's theorem3 applies to expectation values with a single determinant reference function. To put the M'J term of (7) in that form, let us define the one-body operator, X,, so that (3+q9 =
X,l'+O)
(12)
where I1$O) is a single determinant approximation to 11$) and I3+,O) is a single configuration approximation to 13$,) in the same MO basis. The exact expression for X, and perturbation equations that express the transition amplitude (1)to first order in the fluctuation potential in terms of the X, have been given elsewhere.2 In terms of the singlet and triplet projection operators the zeroth-order approximation M'JvO to M'J can be rewritten as
and = (3$143+) - ( ~ w I ~ J / ) (70 If the first-order perturbed wave functions, 3$qb)and '$(J'),are estimated by coupled-Hartree-Fock, the matrix elements M'so and Mop' can be readily evaluated by a computer program for nonorthogonal transition moments. The last term in (7a), M'J, is expeded to be small because it contains a linear AE factor. Hence, this term will be evaluated only to zeroth order in the fluctuation potential using diagrammatic techniques. A large contribution to M is expected to come from the direct mixing, MDM,for molecules of low symmetry whose vertical AE is less than 2.5 eV (measured at the triplet geometry). Configuration interaction expansions of the ground state and excited state in their respective RHF
where
and the E: are as defined in ref 2. Twelve diagrams are produced from the contractions resulting from the application of Wick's theorem3 to M'J,O.
Interchange Perturbation Theory
The Journal of phvsical Chemistry, Vol. 86, No. 19, 1982 3731
TABLE I: Estimates of Energy Differencesa
‘A’ MO’sC E(’A”) - E ( ’ A ’ ) E(’nn*) - E ( ’ A ‘ ) E(’nn*) - E(”’’)
0.128 0.262 0.007
IVO’s for
‘A’
best estimateb
0.105
0.087
0.306 0.001
0.257 0.076
a All numbers in atomic units. POL-CI, ref 7, unpubExcited states calculated with single conlished details. figurations made from indicated MO’s.
Figure 1. The nonzero diagrams for M’*’qo.The symbols are as fdbws: M, dipole “toperator; SO, spin-orbii operator; S, singlet proJector; 1,triplet projector. Up-lines are summed over all virtual orbitals, and down-lines over all occupied orbitals of the singlet reference function. 0.09 c m - ’ 0.03 c m - l
-0.13 c m - l
r i
-2.4 eV
Figwe 2. The three triplet sublevel energies and the lifetimes of each triplet sublevel. The Ilfetimes were computed from IPT using effective charges.
Four of these remain after the identically zero diagrams and canoelling diagrams are eliminated. The four diagrams are pictured in Figure 1 and are equivalent to the expression
In (16) @ runs over spin orbitals occupied in ‘$O and s over those not occupied in ‘$O. The e’s are diagonal elements of appropriate closed shell Fock operators and the Js and ICs are Coulomb and exchange integrals. The operator X, generates the excitation a r.
-
Results for Formaldehyde Initial Considerations. The Cartesian atomic coordinates (in ao) in which all calculations were performed are as follows: C(O., O., O.), O(O., 1.4376, 2.0081), H(f1.7756, O., -1.0669). The coordinates correspond to the JonesCoonss experimental geometry (Rco = 1.306 A, RCH= 1.0962 A, LHCH = 118’) of the 3A” state with a CH2tilt angle of 35.6’. The x axis is perpendicular to the symmetry (6)V. T. Jones and J. B. Coons, J. Mol. Spectroec., 31, 137 (1969).
plane of the molecule and the z axis is 35.6’ from the CO bond axis (and hence in the CH2plane). This axis system was obtained from the diagonalization of the zero field splitting Hamiltonian by Davidson et aL7 Figure 2 shows the triplet sublevel energies at the 3A” g e ~ m e t r y . ~ In this axis system, the components of the dipole moment transform as A”(x), A’(y), A’(z) while the spatial components of the sph-orbit operator transform as A’(x), A”@), A”(z). From (1) it follows that the transitions from the 3$2 and Vysublevels have nonzero y and z components and proceed via ‘A’ and 3A” intermediate states. Similarly, the x-polarized emission from the 3$x sublevel proceeds through 3A’ and ‘A’‘ excited states. All calculations were performed using the Dunning8 [4s, 2p] contraction of the Huzinagag (Ss, 5p) primitive basis for carbon and oxygen, a [3s] contractionloof the Huzinaga (5s) set for hydrogen, and one set of polarization functions (adC= 0.75, ado = 0.85, and apH= 1.2) on each atom. With this basis and geometry, the 3A” SCF energy was -113.81728 au and the ASCF excitation energy was 1.0 eV. Even for the triplet geometry, this is too low as an estimation of the vertical 3A”-1A’ excitation energy. If correlated wave functions are used in the computation of AE, a value of 2.4 eV is obtained.’l This improved estimate of AE was used in the ADHOC and IPT calculations. Shown in Table I are estimates of AE evaluated as expectation values of H with single configuration wave functions. Compared with the best estimate of AE (2.4 eV), the low-order estimate of AE (2.7 eV) using IVO ‘A’ orbitals12 is in fair agreement. However, 3A” MO’s produced a AE of the wrong sign. Canonical ‘A’ orbitals gave a AE (3.6 eV) which was 50% in error. Consequently, only ‘A’ and IVO ‘A’ orbital sets were used in further calculations and the IVO ‘A‘ orbitals were expected to be the better choice. Table I also enumerates the excitation energy to the l m * and 3 7 r ~ *states. The ‘mr* state is expected to contribute strongly1J4to M(’$,3$2) and M(’$, 3$,,). Consequently, we have chosen to use the POL-CI results’ for this energy in M’JvO. Similarly, the 3 ~ 7 r *energy has been chosen to be the POL-CI number because of the large relative error in this excitation energy with either set of MO’s. The ‘nH* energy is even worse but, fortunately, this state does not enter MIJpo. The other excited states have sufficiently large denominators that errors of this size produce only a small relative change in the sum. Calculation of M1pO,Mol’ and the polarizabilities was accomplished by solving the coupled-Hartree-Fock equations using the Fock operator with a finite field E. The (7) E. R. Davidson, J. C. Ellenbogen, and S. R. Langhoff, J. Chem. Phys., 73,865 (1980). (8)T.H. Dunninn. Jr.. J. Chem. Phvs., 53. 2823 (1970). (9)S. Huzinaga, 2: Chem. Phys., 42,- 1293 (1965). (10)L. Stenkamp, Ph.D. Dissertation, University of Washington, 1975. (11)Unpublished results connected with ref 7. (12)The electron was removed from the 2a” orbital and the virtual orbitals defined according to the scheme of W. J. Hunt and W. A. Godd u d , Chem. Phys. Lett., 3,414 (1969).
3732
The Journal of phvsical Chemistry, Vol. 86,No. 19, 1982
TABLE 11: Transition Amplitude Results from the MBPT Formalism with IVO 3A' Orbitals" sublevel polarity X
X
Y Y
z
Y z Y
z
z
X
X
Y
Y z Y z
Phillips and Davidson
TABLE 111: IPT Results"
resummed
ADHOC
- iMo -0.0002 -0.0020 -0.0018 0.0041 0.0028
-0.0002 -0.0024 -0.0020 0.0047 0.0031
X
X
Y Y z z
Y z Y z
0.0002 0.0015 0.0010 -0.0031 -0.0021
X
X
Y Y
Y
z
2
sublevel polarity
-
Y z
z
z
z Y
Y z
av
38.0 0.037 0.018 0.037
a M o and M' are the zeroth-order and first-order contributions to the transition amplitudes defined in ref 2. All values are reported in Hartree atomic units. The ADOHC results used the POL-CI value for the 3A" - 'A' energy difference from Table I. T is the mean lifetime in seconds.
X
X
Y Y z z
Y
X
X
0.0001 -0.0004 -0.0005 0.0007 0.0005 -0.0002 -0.0001 -0.0001 0.0002 0.0000
Y Y z
Y z Y z
X
X
Y Y
Y z Y z
The value of E for which (17) is nearly constant yet has the most significant figures left after the subtraction is the optimal E. We found it necessary to converge the SCF energy to W4au to facilitate an accurate evaluation of (17) and the transition amplitudes. The optimal field strength was found to be 0.0015 au. Lifetime Results. Table I1 presents the values for the transition amplitude computed using NO 'A' orbitals with the resummed MBPT formalism of ref 2. Close inspection of Table I1 reveals that the absolute magnitude of the first-order corrections to the transition amplitudes is comparable to the zeroth order and opposite in sign. Consequently, the zeroth-order estimate of the average lifetime, 0.005 s, is much shorter than the first-order estimate. As discussed in ref 2, MBPT is slowly convergent. Unlike the zeroth-order amplitudes to which the direct mixing term makes the largest contribution, the first-order amplitudes are not affected by this terma2The important excitations in first order involve high-lying antibonding u orbitals as well as the lmr* low-lying state. Specifically, the important intermediate states are formed from the excitations: 5a' 12a', 2a" 3a", 6a' 12a', and 5a' 9a'. The 12a' orbital is primarily a radial correlation orbital for the CO bond, the 3a" a CH antibond, and 9a' an antibonding CO u orbital. Table I11 shows the lifetime and transition amplitude calculations from the IPT equations. The average lifetime predicted by this method is 0.007 s. I t was verified that the dependence of the lifetime on the MO choice for the computation of MIJ-Ois indeed small when POL-CI denominators are used for the states listed in Table I. Order of magnitude differences were found when the other energies from Table I were used for the 1,31r1r*states. Fortunately, M'J*O contributes only about 10% of any component of M, so small errors and higher-order correlation effects in this quantity are not important. The contribution from MDMto My and M, ranged between 60 and 75%. As is well known,2the dominant effect of the spin-other orbit term can be accounted for by replacing the nuclear charge by an effective charge. The ratio of the computed
- -
-
z Y
2
0.0000 -0.0012 -0.0012 0.0021 0.0021 -
z
optimal value of the field strength was found from a sequence of calculations of
-
iM'8 0.0000 -0.0003 0.0004 0.0004 0.0004 -0.0001 -0.0001 -0.0001 0.0001 0.0000
-~ M D M
T
22.3 0.028 0.015 0.030
X
effective chargesb
- jMQ>1
-iM' 0.0002 0.0013 0.0009 -0.0027 -0.0018
unshielded results
z
z
0.0000 -0.0008 -0.0008 0.0014 0.0014
iM',',o
-0.0000 -0.0001 -0.0001 0.0003 0.0002
-0.0000 -0.0001 -0.000 1 0.0002 0.0001
- iM -0.0002 -0.0017 -0.0020 0.0032 0.0029
-0.0002 -0.0012 -0.0013 0.0022 0.0020
T
X
Y z av
1.6 0.010 0.0037 0.0081
3.2 0.021 0.0079 0.017
" POL-CI energi es used for 'nn*, % A * , and 3nn* states. 'A' IVO's used for M'.'.o. T in seconds, M in atomic units. 20 = 5.45, Zc = 2.8. All results are reported in atomic units. spin-own-orbit parameters to the experimental ones requires an effective charge of 2.8 for carbon and 5.45 for oxygen.13 If the spin-orbit integrals are modified with the effective charges, the IPT formalism predicts a phosphorescence lifetime of 0.016 s for CH20. As the oxygen spin-orbit term dominates, the shielded results are nearly 5.4518 times the unshielded ones for M and the inverse squares of this ratio for 7. Langhoff and Davidson' calculated the average phosphorescence lifetime of CH20 at the planar ground state geometry including the spin-other orbit effect. Their value of 0.063 s is in excellent agreement with the Bendazzali and PalmieriI4 value of 0.061 but is somewhat different from the IPT value of 0.016 s. At the planar geometry of CH20,the excitation energies to the 3A" and l*a* states were found by Langhoff and Davidson to be 2.75 and 11.5 eV, respectively, whereas the 3A" and lira* are placed at 2.4 and 9.4 eV above the lA' state at the Jones-Coons geometry in the present work. It appears then that the discrepancy between the Langhoff and Davidson lifetimes and the IPT results is partly due to the difference in geometry, although the use of a different basis set, a different treatment of the spin-other-orbit effect, and a different (13) S. Fraga and G. Malli, 'Many Electron Systems: Properties and Interactions",W. B. Saunders, Philadelphia, 1968. (14) G. L. Bendazzoli and P. Palmieri, Znt. J. Quantum Chem., 8 941 (1974).
J. Phys. Chem. 1902, 86, 3733-3737
TABLE IV: Theoretical CH,O Polarizabilitiesa I A’ 3A” “xx “YY
“
YZ
a22 -
cy
P
12.8 9.2 0.4 21.9 14.6 11.4
13.5 9.8 -0.6 15.7 13.0 5.3
(I In the inertial axis system at the 3A“ geometry, the z axis is very nearly the CO bond axis. The lack of diffuse basis functions should make these results somewhat too small.
computational method make it difficult to isolate the source of the discrepancy. McGlynnlS has used a semiempirical approach to calculate the radiative lifetime of CH20by using an experimental value of 4.5 eV for the l ~ a * - ~ Aenergy ” difference and 3.25 eV for the 3A”-1A‘ excitation energy and considering only the mixing of the lmr* and 3na* states at the ground-state geometry. His calculations produce a lifetime of 0.005 s for 7, in remarkable agreement with the corresponding IPT number of 0.0079. Our transition moment and spin-orbit integrals agree quite well with McGlynn’s empirical estimates, but our energies are quite different. McGlynn neglected to include the important direct mixing term, however. If this term were dropped from the IPT transition amplitude, we would obtain a lifetime of 0.08 s for 7,. Further, it is now generally believed that the strong vertical excitation at 7.8 eV above the ground state is due to a Rydberg state and not to laa* at Hence, any agreement with McGlynn must be regarded as purely fortuitous as his hE and transition moment are derived from an incorrect assignment of the spectrum. (15)S.P. McGlynn, “Introductionto Applied Quantum Chemistry”, Holt, Rinehart, and Winston, Inc., New York, 1972. (16)S. D. Peyerimhoff, R. J. Buenker, W. E. Kammer, and H. Hsu, Chem. Phys. Lett., 8, 129 (1971).
3733
Table IV contains the polarizabilities for the 3A‘and 3A” states of CH20 at the 3A” geometry. Average polarizabilities (2) of 14.6 and 13.0 au were computed for the ‘A’ and 3A” states, respectively, in the inertial axis system. In this axis system the z axis is nearly coincident with the CO bond. Because of the small basis set used, these polarizabilities are likely to be somewhat in error, but the trends are probably significant. The low-lying laa* state is the principal state contributing to the azzpolarizability of the lA’ state. A a ?r* excitation from the excited state should contribute only half as much to the excited state polarizability; hence, the ground-state polarizability is larger than that for the excited state. Also reported in Table IV are the anisotropies of the polarizabilities. As is evident from the table, the ground-state a tensor is strongly asymmetric because of the laa* polarization effect. This asymmetry is probably larger at this geometry than it would be at the ground-state geometry where “r*should contribute less.
-
Conclusion Equation 6 is a general equation for the application of IPT to transition moments. As illustrated with the CH20 calculations, the IPT lifetime results were relatively insensitive to the choice of the molecular orbitals used in the computation of M1.’vo. Further, the problem of describing a series of excited states by one set of MO’s has been confined to one term, MIJiO,whose contribution to M was verified to be small. It is in these two respects that the IPT approach improves upon the MBPT formalism of Phillips and Davidson2 As of yet, there have been no gas-phase studies done on the phosphorescence of monomeric CH20. However, experimental lifetime data on a* n phosphorescence of various carbonyls in rigid glasses at 77 K range from 0.08 to 0.0006 s. Hence, our estimate of 16 ms seems reasonable. Acknowledgment. We thank the National Science Foundation for partially funding this research.
-
Conservative Hamiltonian for the Theory of Multiple-Photon Absorption J. R. Stlne’ Los Alamos National Laboratory, Los Alamos. New Mexlco 87545
and D. W. Nold’ Chemistry Division, Oak Rhlge National Laboratory, Oak RMge, Tennessee 37830, and Depsrtment of Chemistry, University of Tennessee, Knoxvllte, Tennessee 37918 (Recelved:February 5, 1982; In Final Form: May 7, 1982)
A conservative Hamiltonian is developed for the classical dynamical theory of the multiple-photon absorption process and is the analogue of the time-dependent Hamiltonian. Applications to semiclwical theory are discussed for the interaction of a diatomic molecule with one and two lasers. It is demonstrated that the two-laser system can significantly lower the threshold to chaotic motion.
I. Introduction In the past, the Hamiltonian-based methods used to model the multiple-photon process assumed a time-dependent Hamiltonian that consists of a molecular Hamiltonian plus a term that describes the interaction of the molecule with the This interaction term contains (1)R. B.Walker and R. K. Preston, J. Chem. Phys., 67,2017(1977). 0022-365418212086-3733$01.25/0
as factors the dipole moment of the molecule and the time-dependent electric field of the laser. These methods have been applied to a number of multiple-photon processes including the vibrational ex(2)D. W.Noid, M. L. Koszykowski, R. A. Marcus, and J. D. McDonald, Chem. Phys. Lett., 51, 540 (1977). (3)K. D. Hansel, Chem. Phys. Lett., 57, 619 (1978). (4)D. W. Noid, C. Bottcher, and M. L. Koszykowski, Chem. Phys. Lett., 72,397 (1980).
@ 1982 American Chemical Society
