J . Phys. Chem. 1988, 92, 3252-3260
3252
can be rather large. We were surprised by this since our experience in using FFT to solve Schrodinger equation was that very often the phase of the wave function causes rapid temporal and spatial oscillations. In other words if we write ~ ( s , t = ) A(s,t) exp(-iF(s,t)) with A and F real, then whenever F(s,t) changes by 2 a the wave function oscillates. One can easily see how this could cause difficulties. Our success in using large As and time step suggested that this situation does not occur in the present case. To make sure we calculated the real and the imaginary parts of (SI$,) and found them to be quite tame. Two examples of C,(t) = (SI$[) are shown in Figure 7a,b. IV.3;f. The Thermal Wave Function (SI&). The thermal wave function (si&) satisfies eq IV.14 but the initial condition is 86(s-$/83 rather than 6(s-3). Arguments similar to those made in section IV.3.d. suggest that (SI&) will have a minimum at s < 0, a maximum a t s > 0 and is zero at s = 0. Moreover, the minimum and the maximum get broader as p is increased and the values of s at which the extreme occur will move away from
8s at s = S (for a symmetric barrier the second term in eq 11.18
s=s=o.
is zero). For this reason we studied the behavior of dA(s,S;t)/&, whose derivative with s is needed. The results are illustrated in Figure 8a,b which show the absolute value of 8A(s,s;t)/ds.Since this function has the property f(-s) = -f(s) (for symmetric barriers) the absolute value has a cusp. The derivative d2A(s,s;t)/8s83 for s = S is the derivative 0. of the absolute value at s E, for E In Figure 9 we show the decay of the complex function C z ( t ) 8(fl$t)/8f (see eq 11.21). Again we notice that we do not have any rapid temporal oscillations of the phase of the functions that are propagated. The decay of the flux-flux correlation function for a symmetric potential barrier is determined by the decay of C , ( t ) and C z ( t )(eq 11.19). Comparing Figures 7 and 9 we find that in case I, the high-temperature case, it is mainly C2(t)that determines the decay rate, but at low temperatures, case 11, the decay of C,(t) seems to be more important. In the case of a parabolic barrier, valid at high temperatures, Ic,(t)l exp(-obt/2) and Ic2(t)l exp(-3wbt/2) when wbt >> 1.
The examples shown in Figure 8a,b illustrate this-behavior SI$^) is also shown in the same figure. IV.3.g. The Behavior of (SI$,). A glance at eq 11.18 shows that the calculation of Cdt) requires that we compute 82A(s,s)/8s
Acknowledgment. This work was supported by the National Science Foundation (NSF-CHE82-06130) and in part by the Office of Naval Research. We are grateful to Benny Carmeli, Bill Miller, John Tromp, and Rob Heather for useful discussions.
((SI&) corresponds to t = 0). The time evolution of
+
-
-
-
Mean-FieM Models for Molecular States and Dynamics: New Developments R. B. Gerber Department of Physical Chemistry and The Fritz Haber Institute for Molecular Dynamics, Hebrew University of Jerusalem, Jerusalem, Israel
and Mark A. Ratner* Department of Chemistry, Northwestern University, Evanston, Illinois 60208 (Received: July 24, 1987; In Final Form: October 27, 1987)
For accurate description of vibration/rotation eigenstates of molecules, and for discussion of reactions and relaxation processes, simple independent-modepictures are generally inadequate. Mean-field methods, in which each mode acts subject to a mean potential (static or dynamic) that is just the exact potential averaged over all other modes of the system, are attractive for treating such problems for several reasons: they are conceptually simple, numerically tractable, quantitatively quite accurate, and generally applicable to a wide variety of molecular species, energies, and coupling conditions. For these reasons, such mean-field, or self-consistent-field,techniques have been applied to molecular problems quite extensively within the last decade. We discuss several aspects of recent and current work on mean-field applications to molecular problems. In the class of static mean-field, or self-consistent-field, methods such situations include inversion of vibration/rotation spectra to obtain potential energy surfaces, distorted-wave Born approximation work on vibrational predissociation lifetimes of long-lived van der Waals complexes, and an extension of the Slater theory for unimolecular decay rates in the weak-coupling regime. Applications of time-dependent SCF, or TDSCF, include a linearized approximation for investigation of long-time processes and study of the Fourier representation to derive random-phase approximations for direct calculation both of excitation energies and of instabilities and lifetimes. Several intriguing problems remain in the general area of mean-field methods for molecular systems: these include optimal choice of coordinate systems, selection of initial state in dynamical problems, and methods for tractable extension of mean-field approximations.
I. Introduction Nearly all of chemistry is concerned with the structure or dynamics of many-particle systems. Problems as diverse as the second-order optical properties of m-nitroaniline, solvent effects on isomerization reactions, rates of ligand exchange, or relative stabilities of isomeric species are determined by the interactions of many particles or modes. For qualitative purposes, these interactions can sometimes be neglected; the ideal gas law, the Huckel model for ?r-electron systems, and the normal-mode scheme for molecular vibrations are examples of situations in which a fully independent-particle, or independent-mode, scheme is extremely useful for interpreting the behavior of a chemical system. For 0022-3654/88/2092-3252$01.50/0
many situations, however, and particularly when quantitative accuracy is required, the interactions among modes or among particles cannot be neglected, and a description is required in which these interactions are taken into account. The simplest, and most common, procedure for including interaction effects is the mean-field approximation. In mean-field schemes for describing a given single mode or particle, the exact potential providing interactions among modes or particles is approximated by its average (or mean) value over all the other modes or particles of the system. Generally these mean potentials and single-mode, or single-particle, states must be solved for selfconsistently, so that methods of this type are often called SCF 0 1988 American Chemical Society
Models for Molecular States and Dynamics (self-consistent field) methods. Important examples include the Weiss mean-field approximation for the study of magnetism and of order/disorder phenomena,’ the Hartree and Hartree-Fock methods for electronic structure problems: and the time-dependent Hartree-Fock and random-phase approximations for both electronic and nuclear excitations and dynamic^.^-^ In the past dozen years, such mean-field, or SCF, methods have been discussed, and fairly extensively applied, in the context of the vibrational/rotational dynamicsbZS and e n e r g e t i ~ s * ~ of -~l
(1) Cf. e.g. Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena; Oxford: New York, 1971. (2) Hartree, D. R. The Calculation of Atomic Structures; Wiley: New York, 1957. Parr, R. G. Quantum Theory of Molecular Electronic Structure; Benjamin: New York, 1964. (3) Linderberg, J.; Ohm, Y . Propagators in Quantum Chemistry; Academic: London, 1973. Olsen, J.; Yeager, D. L.; Jmgensen, P. Adu. Chem. Phys. 1983, 54, 1. (4) Thouless, D. J. The Quantum Mechanics of Many-Body Systems; Academic: New York, 1972. Towner, J. S. A Shell-Model Description of Light Nuclei; Clarendon: Oxford, England, 1977. (5) E.g. Negele, J. Phys. Today 1985, 38. Alhassid, Y.; Koonin, S. Y. Phys. Reu. C 1981, 23, 1890. (6) Harris, R. A. J . Chem. Phys. 1980, 72, 1776. (7) Heller, E. J. J . Chem. Phys. 1976, 64, 83. (8) Halcomb, C. R.; Diestler, D. J. J . Chem. Phys. 1986, 84, 3130. (9) Sawada, S.; Heather, R.; Jackson, B.; Metiu, H. J. Chem. Phys. 1985, 83, 3009. Sawada, S.; Nitzan, A,; Metiu, H. Phys. Rev. E Condens. Matter 1985, 32, 85. (IO) Coalson, R. D.; Karplus, M. Chem. Phys. Lett. 1982, 90, 301. (11) Clary, D. C.; DePristo, A. E. J . Chem. Phys. 1984, 81, 5167. (12) Skodje, R. T.; Truhlar, D. H. J . Chem. Phys. 1984, 80, 3123. (13) Kosloff, R.; Cerjan, C. J. Chem. Phys. 1984, 81, 3722. (14) Kotler, Z.; Nitzan, A.; Kosloff, R. Proceedings of the Nineteenth Jerusalem Symposium; Pullman, B., Jortner, J., Eds.; Reidel: Dordrecht, Netherlands, 1986. (15) Mukamel, S.; Yan, Y . J.; Grad, J. In Stochasticity and Intramolecualr Redistribution of Energy; Lefebvre, R., Mukamel, S., Eds.; Reidel: Dordrecht, Netherlands, 1987. (16) Miller, W. H. In Stocasticity and Intramolecular Redistribution of Energy; Lefebvre, R., Mukamel, S., Eds.; Reidel: Dordrecht, Netherlands, 1987. (17) Gerber, R. B.; Buch, V.;Ratner, M. A. J . Chem. Phys. 1982, 77, 3022. (18) Schatz, G. C.; Buch, V.;Ratner, M. A.; Gerber, R. B. J . Chem. Phys. 1983, 79, 1808. (19) Buch, V.;Gerber, R. B.; Ratner, M. A. Chem. Phys. Lett. 1983,101, 44. (20) Kirson, Z.; Gerber, R. B.; Nitzan, A.; Ratner, M. A. Surf. Sci. 1984, 137, 527. (21) Eslava, L. A.; Gerber, R. B.; Ratner, M. A. Mol. Phys. 1985, 56, 47. (22) Gerber, R. B.; Bacic, Z.; Ratner, M. A. In Proceedings of the Nineteenth Jerusalem Symposium; Pullman, B., Jortner, J., Eds.; Reidel: Dordrecht, Netherlands, 1986. (23) Ratner, M. A,; Gerber, R. B. J . Phys. Chem. 1986, 90, 20. (24) Gerber, R. B.; Ratner, M. A. Adu. Chem. Phys., in press. (25) Ratner, M. A.; Gerber, R. B.; Buch, V. In Stochasticity and Inrramolecular Redistribution of Energy; Lefebvre, R., Mukamel, S., EMS.; Reidel Dordrecht, Netherlands, 1987. (26) Bowman, J. M. J . Chem. Phys. 1978,68,608. Bowman, J. M. Acc. Chem. Res. 1986, 19, 202. (27) Carney, G. D.; Sprandel, L. I.; Kern, C. W. Adu. Chem. Phys. 1978, 37, 305. (28) Cohen, M.; Greita, S.; McEachran, R. P. Chem. Phys. Lett. 1979,60, 445. (29) Bowman, J. M.; Christoffel, K.; Tobin, F. J . Phys. Chem. 1979, 83, 905. (30) Romanowski, H.; Bowman, J. M.; Harding, L. B. J . Chem. Phys. 1985, 82, 4155. (31) Gerber, R. B.; Ratner, M. A. Chem. Phys. Lett. 1979, 68, 195. Farrelly, D. H.; Smith, A. D. J . Phys. Chem. 1986, 90, 1599. (32) Ratner, M. A.; Buch, V.;Gerber, R. B. Chem. Phys. 1980,53, 345. (33) Roth, R. M.; Ratner, M. A,; Gerber, R. B. J . Phys. Chem. 1983, 87, 2376. (34) Gerber, R. B.; Roth, R. M.; Ratner, M. A. Mol. Phys. 1981,44, 1335.
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3253 molecules. Static S C F has been used to calculate vibrational eigen~tates,~~-~O to d i m s chaotic behavior and molecular stab i l i t ~ to , ~analyze ~ vibration/rotation spectra,30 to compute lifetimes of relaxation, is~merization,~’ and dissociation problems,42 and to invert spectroscopic data to obtain an accurate molecular potential energy Dynamic S C F has been used to examine unimolecular decay rates both in strong-coupling and in weak-coupling regimes,l7-lg to study many-mode systems in which some modes can be treated classically while others require quantal description,” and to calculate relaxation,2’ reorientation,22 energy-exchange, and deactivation rates.20 Mean-field methods have many attractive features, which account for their extensive use. Probably the most important of these features is conceptual: it is far simpler to visualize the behavior of a complicated system in terms of single-particle or single-mode behavior (since this article will be devoted to the dynamics of molecules and the important features for such problems involve nuclear motions, we shall from now on speak only of modes, rather than particles). Thus, for example, most discussion of molecular electronic structure is in terms of mean-field concepts such as orbital energies and one-electron molecular orbital wavefunctions, while mean-field-type “order parameters” are very widely used to discuss phase transitions in magnetic, ferroelectric, and lattice statistics problem.’ A second important advantage of mean-field methods is their broad applicability: thus Hartree-Fock descriptions are nearly always used as the initial electronic structure description of any molecule (even those for which the H F structure is unbound), and time-dependent S C F has been shown to describe well both the weak-coupling and strong-coupling regimes of unimolecular decay.”-19 Any multimode system for which a Hamiltonian can be simply written in terms of a single mode part and a sum of n-body interactions is susceptible to mean-field description. A third feature of mean-field approximations involves formal advantages: thus time-dependent S C F methods conserve both energy and norm.I7 A very important practical advantage of mean-field techniques is their great computational efficiency: generally, an n-mode problem requiring effort proportional to cln for exact description requires effort proportional to nc2 for a mean-field calculation (c, and c2 are constants); therefore, mean-field methods become more useful as the physical problems being studied increase in size. Finally, mean-field methods can often be easily extended, either by use of perturbation theories (which are now very widely used in electronic structure probl e m ~ )by , ~use ~ of simple multiconfiguration methods, or by optimization of a parameter of the physical description itself (such as a coordinate t r a n s f o r m a t i ~ n ) . ~ ~ The chief impetus for use of more accurate theoretical descriptions, including mean-field methods, to multimode problems in dynamics of molecules has been the very rapid progress in the measurement of time-resolved and frequency-resolved molecular behavior. The use of laser, beam, jet, and matrix techniques has permitted experimental probing of molecular behavior at a level of precision and detail that permits investigation of chemical processes at the time scale of elementary events such as molecular
(35) Roth, R. M.; Ratner, M. A.; Gerber, R. B. Phys. Rev. Lett. 1984, 52, 1288. (36) Barboy, B.; Schatz, G. C.; Gerber, R. B.; Ratner, M. A. Mol. Phys. 1983. 50. 353. (37) Gibson, L. L.; Roth, R. M.; Ratner, M. A.; Gerber, R. B. J . Chem. Phys. 1986,85, 3425. ( 3 8 ) Smith, A. D.; Liu, W. K.; Noid, D. W. Chem. Phys. 1984.89, 345. (39) Schatzberger, 0.; Halevi, E. A,; Moiseyev, N. J . Phys. Chem. 1985, 89. 4691. (40) Thompson, T. C.; Truhlar, D. G. J . Chem. Phys. 1982, 77, 3031. Garrett, B. C.; Truhlar, D. G. Chem. Phys. Lett. 1982, 92, 64. (41) Bacic, Z.; Gerber, R. B.; Ratner, M. A. J . Phys. Chem. 1986, 90, 3606. (42) Schatz, G. C.; Gerber, R. B.; Ratner, M. A,, to be submitted for
publication in J . Chem. Phys. (43) Romanowski, H.; Gerber, R. B.; Ratner, M. A,, to be submitted for publication in J . Chem. Phys. (44) E.g., Jsrgensen, P.; Simons, J. Second-Quantization-BasedMethods in Quantum Chemistry; Academic: New York, 1981. (45) Lefebvre, R. Znt. J . Quantum Chem. 1983, 23, 543.
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collisions or even (in some favorable cases) vibrational periods. Indeed, entirely new areas of chemistry, such as structure and dynamics of van der Waals have been developed by use of these elegant experimental probes. The detailed and beautiful experimental observations now available for a wide variety of molecular species and processes demand far more accurate theoretical description than can be provided by independent-mode approximation, and indeed constitute the chief motivation for development of mean-field (and other) methods for more accurate description of structure and dynamics of these molecules. In this paper we indicate some of the recent and current work from our own laboratories involving development and use of mean-field methods. Section I1 is devoted to some new work on static S C F problems. These include inversion of spectroscopic data to obtain a very accurate three-dimensional potential surface for the COz molecule, use of perturbation-theory techniques based on S C F states to calculate vibrational predissociation rates, and extension of standard, normal-mode based methods to calculate unimolecular decay rates. Section I11 discusses the time-dependent S C F (TDSCF) area. It includes extension of TDSCF ideas both in the time-dependent (linearized TDSCF) and in the frequency-dependent (random phase approximation) regimes. Applications to calculation of excitation energies and instabilities (lifetimes) are outlined. Finally, in section IV we mention some theoretical problems in the general S C F area that are still not fully resolved. These include choice of coordinate systems, initial-state choice for TDSCF calculations, special problems for specific applications, extension to multiconfiguration or perturbation schemes, and melding of S C F to statistical ideas. 11. Static Mean-Field Studies
Static mean-field, or SCF, approximations to a multimode problem can be discussed in either quantum-mechanical or classical situations; we limit ourselves here to the quantum situation. The problem, then, is to derive static properties of the solution to the Schrodinger equation H+(ql...qn) = E+(qi...qn)
where ql...qnare a set of suitable chosen coordinates. We denote the Hamiltonian for convenience of illustration as H = Ch,(qJ + CW,(q,,q,) I
(1)
lj
that is, we do not write explicitly coupling terms involving three, four, ... modes. Such coupling terms can be easily included within S C F studies (indeed they are crucial for accurate inversion of potential surfaces),43but we can present the S C F ideas without them. Note that the interaction W,, can include both potential coupling of the usual type and kinetic energy coupling, which will appear whenever certain coordinate systems, such as local modes, are chosen in H . The S C F approximation, called the Hartree approximation for the electronic structure situation,2is then simply to approximate the wave function by the product form n
+(414”) =
mJlcsl)
1=1
(2)
with the single-mode wave functions 4!(qZ) to be determined. For notational ease, we consider here only a two-mode system. One then obtains, from the variational principle with the ansatz of ( 2 ) , the S C F equations in the form26%27 hIcff(n)4m(1)(ql) = cm(1)4m(1)(q1)
(46) Levy, D. H. Annu. Rev. Phys. Chem. 1980, 31, 197. (47) Janda, K. C. Adv. Chem. Phys. 1985, 60, 701. (48) Miller, R. E. J . Phys. Chem. 1986, 90, 3301.
(3a)
Gerber and Ratner The effective potential for motion along a given mode (say q l ) then includes the bare q1 potential in h,, plus the average of the interaction potential Wi2 over $n(2). Note that the single-mode wave functions qdi) and energies di) in fact depend on all the quantum numbers for all modes of the system, since they are obtained by using an effective potential obtained by averaging over all other modes. The equations (3c) and (3d) for the effective Hamiltonians and (3a) and (3b) for the eigenvalues must be solved self-consistently. The single-mode eigenvalue equations (3a) and (3b) can be solved either quantum mechanicallyz6~27~42 or s e m i c l a s s i ~ a l l y ;the ~~~~~ semiclassical formulation involves only negligible errors and, since it defines turning points, is very useful (indeed necessary) for application to such problems as unimolecular decay or inversion of spectroscopic data34,35,43 to obtain potential surfaces. The simplest (Bohr-Sommerfeld) semiclassical solution is obtained from (3a) as31
with ml the effective mass for motion along q l , q2] and qirthe right the potential in the S C F and left turning points, and pcF(ql) Hamiltonian hiscF. Solution of (4) yields both the energies t m ( l ) and the turning points q I rand qI1for mode 1. The SCF equations (3) and (4) have been used widely in studies of vibrational eigenstates of molecules. Specific examples include many energy level determinations, examinations of isomerization reaction barriers and rates,29s36~41 structural in~tabilities,~~ definition of optimal coordinate^,^^^^^ and criteria for onset of chaotic beprogram for performing such calculations is h a v i ~ r .A~ general ~ available.50 We discuss here three very particular and novel applications of static S C F methods, two of which require the semiclassical version of (4). All three of them focus not so much on the calculation of eigenfunctions or eigenvalues per se as on the use of the S C F scheme to calculate other observable molecular properties. Molecular Potential Energy Surfaces Obtained from Inversion of Spectroscopic Datu. Given the superb accuracy and precision of such current spectroscopic techniques as Fourier transform i r ~ f r a r e d ~or’ .stimulated ~~ emission pumping,53or intracavity laser a b s ~ r p t i o n we , ~ ~can confidently expect to have available in the near future extensive, highly accurate measurements of molecular rotation/vibration spectroscopic energy levels for small molecules such as NzO, 03,H2C0, HzO, CH30H, NH3, and C 0 2 . Indeed, for several of these data are already a ~ a i l a b l e . ~ *These - ~ ~ spectroscopic observations correspond uniquely to the molecular potential energy surface and therefore provide, in principle, a direct experimental way to obtain such surfaces. In the case of diatomic molecules, the semiclassical Rydberg-Klein-Rees (RKR) methodS5indeed permits just such a direct inversion. The RKR method is based on the use of Bohr-Sommerfeld quantization, as in (4). Intuitively, it is clear that knowledge of the energies e m ( ’ ) uniquely determines the turning points qlr and qI1of (4); the RKR method essentially uses this to find the diatomic potential surface as the locus of all turning points. Extension of the RKR method to polyatomics is not straightforward, since semiclassical quantization for multimode systems is itself a complicated problem. If the S C F idea is used, however, (49) Gerber, R. B.; Schatz, G. C.; Ratner, M . A,, work in progress. (50) Romanowski, H.; Bowman, J. M. QCPE No 496 (POLYMODE). (5 1) Bailly, D.; Farrenq, R.; Guelachvili, G.;Rosetti, C. J. Mol. Specfrosc. 1981, 90, 74. (52) Chedin, A. J . Mol. Spectrosc. 1979, 76, 430. (53) Abramson, E.; Field, R. W.; Imre, D.; Innes, K. K.; Kinsey, J. J . Chem. Phys. 1985, 83, 453. Hamilton, C. E.; Kinsey, J. L.; Field, R. W. Annu. Rev. Phys. Chem. 1986, 37, 493. Reisner, D. E.; Field, R. W.; Kinsey, J . L.; Dai, H. L. J . Chem. Phys. 1983, 78, 2817. (54) Lehmann, K. K.; Scherer, G.;Klemperer, W. A. J . Chem. Phys. 1982, 77, 2853. 1983, 78, 2817. (55) Rees, A. L. G. Proc. Phys. SOC.,London 1947,59,998. Schutte, C. J. Theory of Molecular Specrroscopy: North-Holland: Amsterdam, Netherlands, 1976.
Models for Molecular States and Dynamics
The JournaI of Physical Chemistry, Vol. 92, No. 11, 1988 3255
TABLE I: Sample Calculated and Observed Energy Levels for the CO, Molecule, Obtained Experimentally and from Different Approximate Potential Surfaces’
vibr energies for CO, molecule ~ ~
~
_
_
vibr state
exptl
RKRscf
RKRsc-cor
Chedin
Carter/ Murrell
10 0 10 I 30 0 14 0 06 0 04 1 24 0 10 2 02 2
1285.4 3612.8 3792.7 4064.1 4225.0 5099.7 5329.8 5915.2 6016.7
1301.6 3617.8 3859.2 4041.9 4166.1 5068.0 5319.9 5893.5 5980.7
1284.2 3609.3 3792.2 4060.7 4223.6 5098.1 5327.7 5917.5 6018.3
1287.6 3610.9 3802.4 4057.0 4212.0 5090.6 5324.6 5907.1 6005.7
1286.0 3611.0 3795.0 4095.0 4229.0 5100.0 5315.0 5912.0 6013.0
_
_
cm-’ for the perturbatively corrected SCF-based inverted potential, compared to 6.67 and 5.1 cm-I for the two fitted potentials. Figure 1 shows cuts along the (Ql,Qz) coordinate (symmetric/asy”etric stretch); the four potentials (SCF, perturbatively corrected SCF, Chedin, and Carter/Murrell) are all qualitatively very similar. The differences among these potentials occur, not surprisingly, at larger values of the displacements of Q1 and Q2. An alternative formulation of the perturbative correction to the SCF-inverted potential might be called the SCF-directed, or locally inverted, method.57 Here one writes, in the spirit of Frost’s local-energy idea (O) V(Qi
Qz Q3) = [(EJn,,,n, - T)Vn,n,n,(Qi Q2Q3) 1 /+Jn,n,n,(Q~Q2Q3)
(7)
where Tis the kinetic energy operator. Unlike the scheme of (5) and (6), in which the corrected form of the potential is obtained by least-squares fit to all observations, the correction implied by the mode separability inherent in the simple product wave function (7) is a local one: the potential surface in a given region of the form of (2) means that semiclassical self-consistent quantization coordinate space (Q1,Q2,Q3)is corrected by using the energy levels of each mode, using (4),can be completed. This in turn means closest to having their turning point in that region of space, as that, within the S C F approximation, the RKR inversion scheme determined by the S C F wave function, whose turning points are can be generalized. Such a generalization has been p r e ~ e n t e d ~ ~ ? fixed ~ ~ by (3) and (4). Just as (5) and (6) can be iterated to obtain and used to determine a potential surface (two-dimensional) for convergence, so can (7). nonbending COz.3SVery recently, we have extended this This perturbatively corrected S C F inversion scheme has, until to include, using iterative perturbation analysis, corrections beyond now, been used only to produce a potential surface for COz, but the S C F level and have derived what we believe to be the most this surface is, we feel, a very good one (indeed, it appears to be accurate local potential surface currently available for any triathe most accurate surface currently available as judged by ability tomic. to reproduce spectroscopic energy levels). The remaining small The SCF-level potential energy surface, which we denote as error averaging 1.96 cm-’ for 60 low levels of COz arises from po)(QI,Qz,Q3)for a three-mode molecule such as COzor HzO, two features: the first is a truncated expansion for the moment is obtained as an (approximate) solution to a multimode of inertia and the second is the fairly small number of terms chosen Schriidinger equation. We can, then write the fust-order correction for the correction potential of (6). Neither of these errors is to the energy as inherent in the method, and it therefore appears that the perturbatively corrected SCF-inversion scheme can be used to obtain highly accurate potentials for any small (three or four atoms) where AE(’) is the first-order energy correction, (o)+J(QlQ2Q3) stable molecule for which the required spectroscopic data are is the numerically exact (configuration interaction) wave function available. with angular momentum J, obtained from the SCF-inverted poPerturbatiue Calculation of Lifetimes f o r van der Waals tential Vfo), and AV(QlQzQ3) is the correction, representing the Dissociation. The study of weakly bound van der Waals species The difference difference between the correct potential and po). is one of the newest and most active research areas of chemical A@’)is then taken to be simply the observed (exact) energy minus physics.These species are of great inherent interest as limiting that of the inexact potential: cases of very weak chemical bonds and are also important for the insights that they provide into clustering, nucleation, and dynamics in the weak-bonding and weak-coupling limits. Some of the very first important measurements made on van der Waals species were with n,, nz, and n3 being the quantum numbers in Ql, Qz, and Q3 the observations by Levy’s of vibrational predissociation and (exptl)Eand (O)Ebeing respectively the experimental energy lifetimes of the triatomic species Iz*(u)X, where X is a rare gas and the energy associated with the wave function (O)+. Equations and Iz*(v)denotes the uth vibrational state of electronically excited Sa and 5b can be solved as a linear integral equation for AV12(3Bo). The most surprising finding was that these van der Waals (QlQ2Q3),the correction potential. Alternatively, and more simply, clusters live for quite long times (from several picoseconds down one can choose a simple power-series form to longer than a microsecond for some other van der Waals clusters); these times correspond to many vibrational periods, and one important theoretical issue raised by these studies has been understanding the dynamical processes of energy flow and disand then determine the coefficients Axlrvby least-squares fitting sociation in these highly anharmonic, weakly bound species. to (5). Static S C F methods are, in principle, inappropriate for deTo obtain very precise surfaces, one may (depending on how scription of dynamical processes, since no energy transfer among close OVis to the correct potential) have to iterate the perturbation modes is allowed in static S C F (the potentials in hiCffare timecalculation based on (S), simply using (l)$ and ( , ) E ,the values independent, and the eigenvalues &) are effectively constants of computed by using AVof (6) and defining a (2)AVby an equation the motion). Nevertheless, since static S C F yields rather good (z)AV,and (I)+ replacing identical with (S), but with (l)E, descriptions both of initial and of final states for, say, a vibrational (O)E,A@, AV, and (O)+, respectively. predissociation processes, the static S C F might be used for calPrecisely such an iterative, perturbatively corrected SCF inculation of rates within a time-independent rate expression such version scheme has been used by Romanowski et to invert as those found in static scattering or perturbation theories. In the experimental data on the COz molecule, originally measured particular, for most van der Waals predissociation reactions, the with FTIR methodsS and initially analyzed by Chedin.S2 Table coupling is weak (that is, extensive intramolecular energy pooling I presents some energy levels calculated with this SCF-based or energy sharing does not occur prior to the predissociation step) inverted potential, compared to fitted potentials obtained by and the probability of dissociation per period is very small, so ChedinSzand by Carter and M ~ r r e l l . For ~ ~ 60 observed levels, perturbation theories might work well. the root-mean-square energy differences with experiment are 1.96 “Energies are in cm-’. From ref 43.
(56) Carter, S . ; Murrell, J. N. Croat. Chem. Acta 1984, 57, 355
(57) Gerber, R. B.; Ratner, M. A,, work in progress.
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Gerber and Ratner
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
A Potential f r m RK&scf
'otentlal
from RKR-scf + PT corr
1.
1.
1.3-
a
a
.l.
.1.2-
(Y (L
(Y (L
1.1
-
1.01
1.
'
1.1I
1
.
1.2
1
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R1, R R1. A Figure 1. (A) Calculated potential contours to the (Ql,Q2) surface of the C 0 2 molecule. Results from ref 43, 52, and 56. Note that qualitatively all potentials are comparable. Contour intervals are 1000 cm-I. (B) Differences in calculated potentials, along surface, for C 0 2 . Contour
(el,&)
intervals are 200 cm-'.
Accordingly, Schatz et al.42 haver recently studied the predissociation rates for some linear triatomic van der Waals sDecies using the distorted-wave Born approximation with static SCF states. The idea behind this calculation is that the residual effects of mode-mode correlation, which are not included in SCF, result in energy transfer and dissociation. Accordingly, the residual coupling potential is defined as - h,SCFb) (8) VmUp H - hISCF(X) where the Jacobi coordinates x and y are respectively the 1-1 distance and the distance from the I2 center of mass to the rare
gas. The distorted-wave Born approximation (DWBA) for the dissociation rate R is then used. in the form R,,.,
= 2 * / h l ( ~ " " ( ~ , y ) l V ~ ' , , , ~ l ~)12S(E, ' ( ~ , y )- E , ) (9)
where and u , denote the iodine vibrational quantum numbers in the initial complex and the dissociated product, respectively. The initial state Il"""(x,y)is simply a bound SCF product state of (2). The final state corresponds to a scattering, rather than a quasibound, wave function and is written as $fi""(x,y) = l&'o'(x)p"'(j)
(10)
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3257
Models for Molecular States and Dynamics
c 4
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-
73
3
47
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-3.0
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i n i t i a l 12 s t a t e Figure 2. Calculated lifetimes (in the form of line width for dissociation) for 12.-He. The boxes are from ref 58. The solid line represents calcu-
lations using the distorted-wave Born approximation with SCF states; the dotted line uses simple first Born approximation. From ref 42. where $&O)(X) is just a (Morse) eigenstate of the separated I2 in the v'vibrational state. The scattering state xfml(y) is determined from the motion of the rare gas in the potential
Generalized Slater Theory for Unimolecular Reaction Rates. Unimolecular reactions generally can be classified in terms of whether or not extensive intramolecular mode-mode energy transfer occurs before reaction. In the most common situation, such transfer is extensive, and reaction occurs from an ensemble in which the intramolecular distribution of energy is quasithermal. This is the strong-coupling limit, in which statistically based theories, in particular RRKM theory, are accurate and The other limiting situation, referred to as the weak-coupling case, is found when extensive energy sharing does not occur prior to reaction. In this limit, the reacting species may be far from equilibrium. Slater long ago developed60an intuitive and elegant theory for dealing with weak-coupling unimolecular decay reactions; the Slater theory is based on calculating the rate a t which the reactant species obtains a critical configuration; once that configuration is attained, reaction is assumed to occur uniformly. Slater theory is limit-ed in two important ways: first, it assumes that all molecular vibrations are harmonic, with frequencies to be obtained from normal-coordinate analysis. Second, it is expressed in terms of classical oscillators and has not been properly extended to the fully quantum case. The S C F scheme clearly offers and approximation to correct the harmonic oscillator limitation of Slater theory. An extension to the quantum regime can also be obtained by using the Wigner distribution. Although that discussion lies outside our scope here, it is important to point out that the quantum extended Slater theory49can deal both with barrier tunneling and with zero-point effects, which can, in principle, wreak havoc with classical rate expressions, for example, by unphysically pooling all of the zero-point energy to dissociate classically, even when correct (quantum) dissociation is forbidden. The critical configuration, in the Slater model, is generally taken as a critical extension R1* of some bond distance R , . Slater assumes that the molecule carries out harmonic vibrations and that no energy exchange occurs prior to dissociation. The extension R 1 can be written as a sum of normal-mode displacements Qi
Schatz et al. completed DWBA calculations for dissociation of linear 1,He and IzNe complexes. The potential function was taken, as in several previous theoretical studies, as a sum of two Morse potentials, one between the two I atoms and one between the rare gas and the neighboring I. Over a broad range of v values, the calculated values for the predissociation rates are in excellent with coefficients ai describing the projection of the displacement agreement with highly accurate quantum-mechanical and classical R , onto the normal modes. Now the displacements Qican, by calculations. The calculations focused on the final state u' = u using SCF, be extended beyond the normal-mode (harmonic) - 1, which is found both experimentally and theoretically to be limit.49 An approximation to the behavior in the anharmonic case strongly favored (propensity rule). Since DWBA is based on can be written as first-order perturbation theory, it is expected to be most accurate for v'= u - 1 . Qi(t) X ( Q P ) - Q;')) COS (a,(')? 4j) (13) These DWBA calculations differ from simple Born approxiHere Q,(I) and Q,(l) are the turning points obtained from the mation in that the initial and final states take into account the semiclassical SCF of eq (3) and (4), while a>? is a local frequency interactions among modes-in particular, the initial state is chosen defined by as an S C F eigenfunction. Figure 2 shows the results obtained by using DWBA with S C F states for the 12--He decay. The dashed line shows results for ordinary (uncoupled, no S C F corrections) Born approximation. Note that the DWBA results are and diis a phase. The approximate solution (1 3) amends the in excellent agreement with the results of Beswick and J ~ r t n e r , ~ ~ displacement QP) - Qi(l)and the local frequency ~ $ 3to deal with while the uncoupled results show much larger errors. This is as S C F corrections. A more exact formulation, including overtone expected, since the bound species live long enough (very many contributions, is also available.49 vibrational periods) that the two modes vibrate not in their bare Once the criterion for R 1* is determined, the SCF-corrected Morse potentials but in the full potential including interactions. Slater theory49can indeed provide important corrections to the Schatz et al.42also calculated eigenvalues (energy levels) of original, harmonic-limit Slater prescription. The actual calcuthe triatomic complexes using static SCF and compared the anlations of the rate, involving use of Kac's result for the frequency swers with the exact (configuration interaction) results in the same with which the critical extension R1* is reached, follows potential. The S C F values for the energies are remarkably acstraightforwardly. There is some ambiguity involved in the choice curate: even though the interactions are very weak (binding of threshold for reaction. The simple choice of R , * follows Slater's energies on the order of 13 and 96 cm-I for 1.-He and 1.-Ne), ideas and seems intuitively attractive. Other possibilities also exist, compared to the strong 1-1 binding, nevertheless the S C F results however. For example, one might select a velocity criterion to obtain essentially perfect agreement for the binding energy and define the transition state. Such a notion is suggested by classical for the excited states both of the I2 and of the van der Waals trajectory studies of, for example, the dissociation of the 1,He van modes. This high numerical accuracy even for a weakly bound der Waals c o m p l e ~ . 'One ~ might thus define a velocity u ~ equal , and very anharmonic system constitutes further proof of the broad to (2D1/pl)1/z,where D , and p , are the dissociation energy and applicability of static S C F methods for bound-state eigenvalue analysis. (59) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley:
+
( 5 8 ) Beswick, J. A.; Jortner, J. J. Chem. Phys. 1918, 68,2277; 1978, 69, 512.
New York, 1972. (60) Slater, N. B. Theory of Unimolecular Reactions; Cornell University Press: Ithaca, NY, 1959.
3258 The Journal of Physical Chemistry, Vol. 92, No. 11 1988
Gerber and Ratner
I
effective mass of a given dissociating bond. The criterion for reaction is then
Other possibilities to define the reaction criterion, or transition state, include a condition on the inner turning point or, intriguingly, a phase-space criterion involving simultaneous requirements on velocity and coordinates. Phase-space criteria of this type have been employed in recent studies of intramolecular energy transfer and p r e d i ~ s o c i a t i o n . ~ ~ - ~ ~ Preliminary calculations indicate49 that, once a reasonable threshold criterion is defined, the SCF-extended Slater scheme does quite well in calculating lifetimes of small clusters. In particular, the corrections, arising from the SCF treatment, of the vibrations to deal with anharmonicities are important in the dissociation dynamics of weakly bound van der Waals complexes. This use of static S C F to correct an oversimplified harmonic treatment should prove useful not only in Slater-type theories, but also in any other situation where anharmonic effects can be important. For example, in dissociation dynamics of weakly bound homogeneous species, such as rare gas or COz clusters, the dynamics is of RRKM (strong-coupling) type, but the oscillations are highly anharmonic. Recalculation of the statistical, transition-state theory (RRKM) rate constant with SCF-derived frequencies and partition functions might well improve accuracy of the calculated rates. Several important fundamental aspects of static S C F theory are under active scrutiny. Perhaps the most important of these issues involves the choice of coordinates in which to solve any given problem. For example, it has been found that normal coordinates are better than local coordinates for most small-molecule vibration problems solved in an S C F context,33that the more collective the coordinates are the better the dynamical screening is, and the more accurate S C F will be,23 that hyperspherical (highly collective) coordinates provide quite accurate eigenval~es,~~ and that, in some situations, coordinate transformations can be chosen variationally, leading to a different optimal coordinate description for each e i g e n ~ a l u e . ~Perhaps ~ . ~ ~ the most instructive case studied so far has been the HCN molecule. Here Back et aL4' selected elliptical coordinates, chosen for the good physical reason that the minimum energy path for isomerization is nearly elliptical. SCF calculations in these coordinates gave excellent results for eigenvalues, as well as interesting predictions about isomerization rates and modespecificity effects. We feet that, whenever possible, S C F (or any other) calculations should be carried out in coordinates one of whose nodal surfaces follows the minimum energy path on the molecule's potential energy surface. The static S C F method is simple, easily applied, and accurate. It also suffers from several drawbacks; the most important of these is, as already mentioned, that the single-mode energies are constants of motion, so that no intermode energy transfer can occur in static SCF. To deal with problems in which one wishes to describe energy flow dynamically, a time-dependent treatment is needed. Such a dynamical mean-field scheme is the TDSCF. to which we now turn. 111. Time-Dependent Mean-Field Studies
In static mean-field, or SCF, theories, one computes the energy and wave function for a single mode acting in the mean static potential of the other modes. In dynamic mean-field, or timedependent self-consistent field (TDSCF), models one considers the time evolution of a single mode subject to a dynamical average potential due to the other mode^.'^.^^ Because these mean-field potentials are time-dependent, the single-mode energies are not conserved, so that energy flow among modes is observed in TDSCF studies. Like static SCF, TDSCF can be formulated in classical, semiclassical, or quantum limit^.'^**^ We limit our formal dis(61) Davis, M. J . Chem. Phys. 1985, 83, 1016. (62) Gibson, L.; Schatz, G. C.; Ratner, M. A,; Davis, M. J. J . Chem. Phys. 1987, 86, 3263. (63) Gray, S . K.; Rice, S. A. J . Chem. Phys. 1987, 86, 2020.
cussion to the quantum case but observe that, because TDSCF does provide independent treatment of each mode, it offers a fully consistent way to treat parts of a system quantum mechanically and other parts classically; this can be very useful in problems such as vibrational relaxation in liquids or hydrogen-bond situations, in which strong quantum effects are expected only in some of the modes. The derivation of the TDSCF equations is s t r a i g h t f ~ r w a r d : ' ~ , ~ ~ to solve the time-dependent Schrodinger equation ih a / & ' k ( ql...q,;t) = H 9 ( q,...q,;t ) (16) one chooses an approximate product wave function n
'k(q,...q,;t) = n $ k ( q k J ) k=l
(17)
Substitution of (17) in (16), followed by calculation of the scalar product and some algebra, produces the desired single-mode TDSCF equations ih
a/at
4 k ( q k J ) = hkSCF(qkJ)4k(qkJ)
(18)
where eXP(i%)$k(qkJ) (19) with the unimportant phase qk. The time-dependent single-mode TDSCF Hamiltonian is 4k(qkrt)
hkSCF(9kJ) = hk(qk) + V k ( q k J ) and the time-dependent potential is Vk(qkJ) = (
n4i(qiJ)lCwL,ln 4/(4iJ))
I#k
ri
I#&
(20) (21)
Once again, the simultaneous solution of (21) and (18) is generally obtained by SCF-type iteration. TDSCF methods were first developed by Dirac more than a half-century ago.64 Though chemists have used them fairly extensively in study of electronic system^,^,^^ application to chemical reactions is much newer; within the past decade, TDSCF has been used, among other things,'z5 for study of reactions and processes at s ~ r f a c e sproton , ~ ~ ~transfer ~ in H-bonded systems,66 vibrational relaxation,*' tunneling in solid^,^*^*^ and unimolecular decay reactions in both strong-coupling and weak-coupling limi t ~ . ~In ~some - ~ of~ these situations the single-product form of (17) is clearly too restrictive, at least with ordinary choice of but in general the results have been very satisfactory. The TDSCF methods enjoy many of the same advantages already outlined for static SCF, including broad applicability (for example, both strong-coupling and weak-coupling limits of unimolecular decay can be described a c c ~ r a t e l y ' ~ - 'great ~ ) , economy of effort (scaling like n for an n-mode problem), and simple physical interpretation; TDSCF also conserves both norm and total energy,17 while allowing for extensive energy transfer among modes. Recent TDSCF work in our own laboratories has centered on comparison to exact dynamics67@using quasi-spectral methods,69 on extension to multimode systems, on linearization and Fourier analysis to deal with slow processes and stability analysis, and on extensions to different coordinate systems and multiconfiguration schemes. We consider here only the newest issues, linearization, stability analysis, and vibrational excitation spectra. Linearized TDSCF, the Random-Phase Approximation, Excitation Energies, and Stability Calculations. O n e of the interesting features of the TDSCF approximation is that if an initial state is chosen as an eigenstate of the static S C F problem, it will not evolve in time within the TDSCF scheme. This implies that states close to being static S C F eigenstates will evolve slowly in (64) Dirac, P.A. M . Proc. Cambridge Philos. SOC.1930, 26, 367. (65) McLachlan, A. D.; Ball, M. A. Rev. Mod. Phys. 1964, 36, 844. (66) Makri, N.; Miller, '8.H. J . Chem. Phys., in press. (67) Bisseling, R. H.; Kosloff, R.; Gerber, R. B.; Gibson, L. C.; Ratner, M . A.; Cerjan, C. J . Chem. Phys. 1987,87, 2760. (68) Kosloff, R.; Hammerich, A.; Ratner, M. A. In Proceedings ofrhe 21 Jerusalem Conference; Jortner, J., Pullman, B., Eds.; Reidel: Dordrecht, Netherlands, in press. (69) Kosloff, R.; Kosloff, D. J . Chem. Phys. 1983, 7 9 , 1823. J . Compur. Phys. 1983, 52, 35.
Models for Molecular States and Dynamics
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3259
time, within the TDSCF scheme. Now if we are interested in calculating relatively slow processes, such as the predissociation rate of a very weakly coupled van der Waals complex like Ar. OCS48970 or overtone line shapes in small, "nonstatistical" polyatomics like C2H2or H202, then the TDSCF evolution will be very slow, and indeed, direct study of such rates by time-dependent means is a difficult task.70 In section I1 we examined situations in which static methods, based either on perturbation theory or on a transition-state idea, could be used to calculate slow rate processes in special cases. In this section, we wish to examine the behavior of a system in which the initial state is chosen as very nearly, but not precisely, an S C F eigenstate. We will then examine the evolution of that initial state in time, linearizing the TDSCF equations (valid because the non-SCF component of the initial state is small) to obtain linearized TDSCF (LTDSCF) equations. By examining the Fourier transform of these LTDSCF equations, we can obtain expressions for the characteristic excitation frequencies relevant to the initial, near-SCF state. Examination of these equations for complex values of the Fourier frequency variable supplies a scheme for the calculation both of excitation energies and of lifetimes, from the real and imaginary parts of the excitation frequency, respectively. A stability criterion can be obtained similar to that first suggested by Thouless for the analysis of the nuclear and electronic many-body problem^.^,^' Physically, any initial state that is not an exact eigenstate of the system Hamiltonian will change with time. Most kinetic experiments in chemistry are performed on initial states (prepared, say, by optical excitation or molecular collision) that are not eigenstates of the total Hamiltonian. Moreover, since such initial states are generally prepared by interaction with an external perturbation (electromagnetic or particle collision), they will not usually be eigenstates of the SCF Hamiltonian, either. Therefore, such initial states will evolve in time, and indeed, it is solving for the dynamical evolution of their behavior by using the TDSCF equations, for systems like unimolecular dissociation, surface collision, or vibrational relaxation, that has constituted the chief application thus far of TDSCF theories in dynamics of molecules. But if the initial state is very close to being an S C F eigenstate (for instance, the first vibrationally excited state of HCN), its evolution will be slow. By using Fourier analysis, we can examine the characteristic excitation and decay frequencies that near-SCF initial state-that is precisely what the random-phase approximation (the Fourier space analogue of LTDSCF) does. 1. Linearized TDSCF Equations. Consider for illustrative purposes a two-mode system, denoting the two coordinates x1 and x2. The TDSCF equations follow as eq (18) and (20) above. Suppose now that the initial state in the ith mode is nearly an S C F eigenstate, so that we write 4i(xj,t) = Xj(xi,t)
+ 6xi(xitt)
(22)
where Xi(xi,t) = g i ( x i ) exp(-icit/h) (23) is the static SCF part (solution to the static S C F (3) with energy ti and static eigenfunction gi). The TDSCF equation for + j then becomes (Ti + (gjijlVlgj))6xj(xi,t)
+ ( 6 x j l ~ g i ) g ~ ( x i ) e - i [ ~ ~++ € j l ' / h
(X j J ~ G X i ) g j ( x i ) e - i [ f ~ - = f ~ ih ] ' ~ hd(6xi)/dt (24)
On the left-hand side, we have dropped terms proportional to 6xi 6xj and to ( 6 ~ ~in) ~that ; sense, (24) is a linearized TDSCF equation, in that only terms linear in the small admixture 6xi are retained. The solution to the LTDSCF (24) has both advantages and disadvantages over the TDSCF form (18), to which it is an approximation. The most serious disadvantage is simply that the smallness assumption may not hold: for some physical situations, 6xi may not be so small that the linearization is satisfactory, and in this case the LTDSCF results will be in error. There are two (70) Gibson, L. L.; Schatz, G . C. J . Chem. Phys. 1985, 83, 3433. (71) Thouless, D. J. Nucl. Phys. 1961, 22, 78.
obvious and substantial advantages: the first is that linearized equation sets like (24) (of course, there are n such equations for an n-mode system) are in general easier to integrate than the nonlinear TDSCF set (18). We have begun testing LTDSCF, as opposed to TDSCF and exact dynamics, on some model problems; it promises to be very useful for slow processes. 2. Random-Phase Approximation ( R P A ) and Stability Analysis. A second, very important advantage of LTDSCF compared to TDSCF equations of motion is that they are susceptible to Fourier transformation and analysis. Suppose we seek a solution of (18) in the f 0 r m ~ 9 ~ 6xi = x [ Xmi.e-'w' + y mi.*&'I 2me-"'t/h (25) m
with the Fourier frequency w being allowed to assume complex values. Then the RPA equation is obtained as
hwXni =
(E,
-hwY,, =
(en
- ei)Xni+ Cm ( i j l n m ) ( Y m ,+ Xm,)
(26)
- ci)Yni+ Cm ( i j l n m ) ( Y m+j Xm,)
(27)
with ( iilmn )
E
)lij(2)1 V(XI3x2) Ixm( 1) ~ n ( 2))
(28)
These equations represent a non-Hermitian set, to be solved for the excitation frequency w and the amplitudes X and Y . They are nearly identical (except for the lack of exchange terms) with the RPA equations c?f electronic or nuclear t h e ~ r y . ~The , ~ frequency w = A i r contains a real part A corresponding to an excitation frequency and an imaginary part r describing a lifetime. Equations 26 and 27, the RPA equations for excitation of the molecule, have several very interesting features. First, suppose that r = 0, so that the additional part 6xi is multiply periodic'rather than decaying. Then the initial single-mode state 4i(xi,t) will oscillate in time but will not damp: it is a stable state, with well-defined characteristic oscillation frequency A. The RPA set of (26) and (27) can then be solved directly for the excitation energies, hA, Essentially, the small perturbation 6x acts as a test probe to evaluate the characteristic excitation behavior of the system-this is very much in the spirit of the original development of RPA ideas for excitations in solids.72 The RPA equations can be used to calculate excitation energies directly (by solving (26) and (27)) just as they are in electronic structure t h e ~ r y .The ~ choice of initial state is fairly clear: it must be chosen such that w is purely real (that is, r = 0). Only in such a case will the excitation be stable. The excitation energies calculated from the RPA equations are not simply static excitation energies: they correspond to excitations from a correlated ground state to a correlated excited state and should generally be more accurate than simple S C F answers. If the amplitude yi, is ignored in (26) and (27), one obtains the Tamm-Dancoff, or monoexcited CI, result. To date, we are not aware of any calculations in which RPA excitaion energies have been calculated for molecular vibrations, but clearly such calculations are feasible and should be straightforward. Now consider the imaginary part r. Note that if r # 0, the original supposed single-mode state 4i of (22) is unstable, in the sense that its norm is not conserved. Physically, this is easily interpreted from the LTDSCF form of (24): as energy is transferred among modes, the single-mode states 4ideviate more and more strongly in time from the simple static S C F state xi. Thouless pointed that the existence of nonvanishing r corresponds to an instability in the original SCF state; that is, one of the characteristic excitation modes is complex, corresponding to an unstable motion. Thus the lifetime = r-1
+
should correspond to the characteristic time before the assumed initial state has decayed considerably. Thus the RPA equations not only permit calculations of the excitation frequencies A but (72) Ehrenreich, H.; Cohen, M. H. Phys. Rev. 1959, 115, 786
3260 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 also allow time-dependent calculations of decay lifetimes +. Once again, we are not aware of any calculations on actual systems using RPA to estimate lifetimes, but such calculations would be of real interest.
IV. Remarks The use of mean-field methods has already strongly influenced theoretical approaches to many problems in chemical dynamics. Indeed, in some cases the very failure, at a quantitative level, of the mean-field description has led to important physical insights into the mechanism of a particular process.66 While the use of mean-field techniques in molecular dynamics is still far less extensive than is its application in statistical mechanics, nuclear theory, or electronic structure, nevertheless a good deal has been learned both about the nature of the static and dynamic mean-field approximations themselves and about their applicability to vibration and dissociation problems. Several important issues concerning these mean-field techniques still await final answers. Probably the most important and pervasive of these difficulties involves choice of coordinates. A number of studies using static SCF in different coordinate systems have pointed out the great sensitivity of the result to the coordinate choice a d ~ p t e d . Several ~ ~ , ~ ~general principles can be used to select coordinates: for static SCF, the best coordinates are collective, conform to our physical intuition about what notions are important (elliptical for H C N isomerization,'" hyperspherical for excitations of small homogeneous clusters),37can be chosen variationally, and will be different for different states. For TDSCF, there has to our knowledge been no explicit study of the relative merits of different coordinate systems, though it is clear that, once again, the results obtained will depend strongly on coordinate choice.73 This sensitivity to coordinate is not unusual in molecular dynamics; the concept of reaction coordinates and of reaction path Hamiltonian, because of its intuitive appeal and its close link to transition-state theory, has been an important fixture of most quantitative descriptions of chemical kinetics for the past halfcent~ry.'~-~~ Other important aspects of these mean-field methods are still not fully understood. One example involves initial state choice for TDSCF problems. Here the difficulty involves theoretical modeling of an experiment: for instance, if one wishes to study the energy flow dynamics following excitation of CH3CCH to the u = 6 state of the methyl group, just what state is to be chosen to represent the laser-excited species? Again this is not a new ~~
(73) As a trivial example, if the coupling W . in (1) is chosen as the momentum coupling PiPj/pijbetween local modes and if the initial state is chosen as an eigenstate of H - xWij, then no evolution will occur. In normal modes for the same Hamiltonian, evolution will occur. (74) Marcus, R. A. J . Chem. Phys. 1968, 49, 2610. 1968, 45, 4500. Hofacker, G. L. 2.Naturforsch A: Phys., Phys. Chem., Kosmophys. 1963, 18A,607. Hofacker, G. L.; Levine, R. D. Chem. Phys. Lett. 1971,9,617. (75) Light, J. C. Adv. Chem. Phys. 1971,19, 1. (76) Fischer, S. F.; Ratner, M. A. J . Chem. Phys. 1972,51, 2769. (77) Miller, W. H.; Hardy, N. C.; Adams, J. E. J . Chem. Phys. 1982,86, 1136. Truhlar, D. G.: Kilpatrick, N. J.; Garrett, B. C. Ibid. 1983,78,2438.
bJ),
Gerber and Ratner problem (it was a very intensively debated topic in the treatment of nonradiative decay problems), but is understanding will be important for kinetics or line shape applications of TDSCF. A third issue involves improvements to mean-field methods: it is clearly very desirable to know both when the simple product function ansatze ((2), (17)) are inadequate and how to improve them. Use has been made of simple multiconfiguration wave f ~ n c t i o n sor~ of ~ .perturbative ~~ corrections to SCF schemes,78and these ideas clearly merit further development (for example, in the context of branching or bifurcating reactions, where single-configuration TDSCF cannot be used without special adjustmentI8). This issue is clearly related to the problem of coordinate choice since several problems, such as isomerizations, might well require a multiconfiguration approach in one set of coordinates yet be treatable with simple mean-field methods if a different coordinate scheme is chosen. Special problems might be encountered in applying mean-field schemes to any given situation. For example, in the generalized Slater theory42one has the difficulty of choosing a criterion for dissociation. A final important and intriguing problem is the possibility of melding S C F ideas with statistical concepts. For example, one can show on very general bases that, in the chaotic regime, most eigenfunctions, if expanded in an uncoupled basis set, will have coefficients that drop off as a gap law in the energy difference between this eigenstate and the basis function.79 At the same time, some S C F type states are fairly unmixed (and occur as sharp lines e ~ p e r i m e n t a l l y even ) ~ ~ quite far above the threshold for chaos.80 A unified description that uses S C F ideas for strong states or resonances coupled with statistical results for the weakly coupled remaining manifold of states would be intriguing and perhaps very useful. While some progress has been made on this problem,*' a great deal remains to be done. It seems that mean-field methods will continue to grow in importance and applicability for the understanding of problems in molecular energetics and dynamics. Acknowledgment. We thank the Chemistry Division of the National Science Foundation and the U S . DOE for partial support of this research. The Fritz Haber Institute is supported by the Minerva Gesellschaft fur die Forschung, Munchen, FRG. We are very grateful to R. Kosloff and G. C. Schatz for enjoyable and stimulating collaboration. Finally, it is a pleasure to thank Z . Bacic, B. Barboy, V. Buch, L. Eslava, L. Gibson, A. Hammerich, V. Keller, R. Roth, and L. Shen for their incisive and critical contributions to this research. (78) Shen, L.; Gerber, R. B.; Ratner, M. A. Mol. Phys., to be submitted for publication. (79) Buch, V.; Gerber, R. B.; Ratner, M. A. J . Chem. Phys. 1982, 76, 5397. Mol. Phvs. 1982,46, 1129. Chem. Phvs. Lett. 1982. 89, 171. (80) Taylor,.H. S. In Stochasticity and Intramolecular Redistribution of Energy; Lefebvre, R., Mukamel, S., Eds.; Reidel: Dordrecht, Netherlands, 1987. (81) Buch, V.; Gerber, R. B.; Ratner, M. A. J . Chem. Phys. 1984,81, 3393.
