J . Phys. Chem. 1986, 90, 20-30
20
FEATURE ARTICLE Excited Vibrational States of Polyatomic Molecules: The Semiclassical Self-Consistent Field Approach Mark A. Ratner* Department of Chemistry and Materials Research Center, Northwestern University, Evanston, Illinois 60201
and R. B. Gerber Department of Physical Chemistry and Fritz Haber Institute for Chemical Dynamics, Hebrew University, Jerusalem, Israel (Received: July 25, 1985)
An outline is given of self-consistent field methods for treating the coupled vibrationsof polyatomic systems and some applications to energy level structure and dissociation dynamics of molecules and clusters are described. The SCF approximation describes each mode as moving in an average field of all other modes; the mean fields for the single modes are determined by a self-consistency condition. The method is computationally simple, applicable to relatively large systems, and can be formulated for static or for time-dependent problems, in classical, semiclassical,or quantum representations. We discuss several aspects of static SCF, including vibrational energy level and eigenstate determination, with applications to spectroscopy of highly excited states, inversion methods for obtaining polyatomic potential surfaces, metastable states, and finally the validity range of SCF and its extensions. For time-dependent SCF, we discuss dissociation of van der Waals molecules both in the strong coupling (Ar3, RRKM-like) and weak coupling (12Ne,Slater-like) regimes, and vibrational relaxation in polyatomic clusters (12NeN).In each of these cases, SCF techniques are fairly accurate, and offer convenient physical interpretation. We believe that these advantages will result in an important place for these SCF methods in the theoretical discussion of vibrational energy states, flow, and dynamics.
I. Introduction The recent advent of laser techniques, in the visible, the Raman, and the infrared, has enormously enlarged both the frequency range over which vibrational spectra have been observed and the accuracy of the spectral determination. In addition, the combination of precise time-resolution and frequency-resolution techniques has led to substantial interest in the lifetimes and decay pathways of the excited states of polyatomic molecules reached by multiphoton excitation and other methods. Such recent techniques as intracavity photoacoustic spectroscopy,’ secondary emission pumping,2 and high-resolution Fourier transform infrared spectroscopy3 provide detailed information on highly excited vibrational states of many molecules. These excited states involve significant vibrational excursions away from the equilibrium geometry, and therefore the ordinary normal-mode descriptions, which are valid in the limit of small oscillation^,^ become inadequate. More specifically, there are a number of important questions which have been raised by recent experimental work and whose answers remain unclear. These include (a) In molecular dissociation dynamics, the traditional RRKM-type behavior, in which intramolecular vibrational equilibration is essentially complete before dissociation begins and the unimolecular decay pathway is determined by statistical considerations, characterizes very well most unimolecular reactions of covalent species. Several special cases have been observed, however, in which this strong-coupling behavior in the vibrational manifold is not found but, rather, the dissociation is dominated (1) Lehmann, K. K.; Scherer, G.; Klemperer, W. A. J . Chem. Phys. 1982,
77, 2853. ( 2 ) Reisner, D. E.; Vaccaro, P. H.; Kittrell, C.; Field, R . W.; Kinsey, J. L.; Dai, H. L. J . Chem. Phys. 1982, 77, 575. (3) Bailly, D.; Farrenq, R.; Guelachvili, G.; Rossetti, C. J . Mol. Spectrosc. 1981, 90, 74. (4) Wilson, Jr., E. B.; Decius, J. C.; Cross, P.C. ‘Molecular Vibrations”; McGraw-Hill: New York, 1955.
0022-3654/86/2090-0020$01.50/0
by dynamical effects, the coupling in the vibrational manifold is weak, the Slater-type description is more appropriate than is RRKM. Examples include weakly bonded species such as clusters and van der Waals molecules, and some high energy organic intermediates. Can a unified theoretical description of these two behaviors (Slater and RRKM) be formulated? (b) Even for a molecule like ozone which seems qualitatively to decay by a strong coupling route, spectroscopic evidence indicates the existence of “good” modes up rather high in vibrational energy, close to dissociation. Is there a unified theoretical picture which can comprehend both strong coupling dissociation dynamics and the existence of these strong, rather sharp spectroscopic states? (c) Some polyatomics of intermediate size, such as (CH3)4X (X = C, Si, Ge), vinyl isocyanate, (CzH4)2,(HF),, p-difluorobenzene, and so forth, have been studied in detail, either spectroscopically or kinetically or both. Which theoretical techniques can be used conveniently to describe these large systems as well as triatomics such as H 2 0 or O3or H C N ? These experimental advances and questions have engendered a very large theoretical effort in the general area of highly excited vibrational state^.^ Attention has been paid primarily to conceptual questions, such as chaotic motion in both classical and quantum cases.6 There seems to have been much less work on developing dynamical approximations applicable to cases of many degrees of freedom, into which most real molecules fall. The main exception to this are the celebrated statistical theories of dissociation dynamics, recent statistical treatments of time-independent properties,’-’ and a few dynamical approaches and models.l0>’
’
( 5 ) A convenient review is provided by: Noid, D. W.; Kosykowski, M. L.; Marcus, R. A. Annu. Rev. Phys Chem. 1981, 32, 267. (6) Cf. e.g. J . Phys. Chem. 1982, 86, No. 12. (7) Buch, V.; Gerber, R. B.; Ratner, M. A. J . Chem. Phys. 1982, 76,5377; Mol. Phys. 1982, 46, 1129. Gerber, R . B.; Buch, V.; Ratner, M. A. Chem. Phys. Lett. 1982, 89, 121. (8) Pechukas, P. Phys. Rev. Len. 1983, 51, 943. Haller, E.; Koppel, H.; Cederbaum, L. S. Chem. Phys. Lett. 1983, 101, 215. Stechel, E. B.; Heller, E. J . Annu. Rec. Phys. Chem. 1984, 3 S , 563.
0 1986 American Chemical Society
Feature Article
The Journal of Physical Chemistry, Vol. 90, No. 1. 1986 21
The focus of the present article is a method aimed at treating level structure and dynamics in cases where “exact” quantal or classical calculations may be unfeasible due to computational complexity. A simple and attractive approach to the problems presented by the description of highly excited vibrational states is offered by the self-consistent field (SCF) picture. In this approach, each vibrational mode, in any given vibrational state of the molecule, is described by the averaged potential due to all of the other modes, the different modes being treated self-consistently. This picture is the vibrational analogue of the familiar Hartree-Fock method for the many-electron problem, and like Hartree-Fock it offers several signal advantages: it is easily interpreted in terms of single-mode potentials, energies, and eigenstates; it can be used for essentially any form of the molecular Hamiltonian: it can be developed either in static form to study eigenstates or in timedependent form to examine dynamics, and it permits study of quite large systems. Thus, the S C F technique appears of real promise for answering the sorts of questions outlined above. In this article we sketch the nature of the S C F treatments both for the static (eigenvalue) and the dynamic (energy transfer and dissociation) problems, and review some of the recent applications of these techniques which have already been completed and a number of outstanding problems for which the SCF method seems ideal. Section I1 considers excited energy levels and static SCF, section 111 deals with energy transfer and time-dependent SCF, and section IV sketches some future directions.
Before proceding to outline formally the S C F and SC-SCF methods, it is useful to describe their qualitative physical content. These S C F approximations define the best single-mode wave functions $i(qi),fixed by an effective potential WSCF(qj) which is the sum of the bare potential along coordinate qi and the average of the interaction potential over all of the other vibrations. The wave function qi(qi)along qi may then be found by solving the one-dimensional Schrodinger equation by using either quantal (SCF) or semiclassical (SC-SCF) techniques; the SC-SCF is usually simpler. The wave function in the harmonic limit ( Wanh 0) can be written exactly as
11. The
-
*(qi...qn)
=
(4)
fi$i(qi) I=I
where $i(qi) is in this case a harmonic-oscillator state in the coordinate qi. The S C F method assume^'^,^^ that the simple product form (4) is approximately valid even for the full potential of ( 2 ) . It then uses the variational principle
4(*17fI*)/(*l*)l to define an equation for the modes equations are (heff(i)- ei)l/ii = 0
(5)
=0 $i.
These single-mode
i = I...n
(6)
SCF and SC-SCF Methods
The problem of coupled vibrations arises, physically, because vibrationally excited states of polyatomic molecules exhibit displacements so large that anharmonicities become important, causing the harmonic modes to mix. Formally, we can express this by starting with the vibrational Hamiltonian corresponding to a single Born-Oppenheimer potential energy surface, expressed in the normal-mode representation as4 n
7f = CP,2/2k + V(q1...qfl) I=
1
(1)
Here the momentum p i is conjugate to the normal coordinate qi, with reduced mass pi; there are n vibrations in the molecule. The potential is written
where ki is the force constant in normal mode i and Wanhis the anharmonic potential which couples the normal modes; in the usual Taylor’s series expansion, Wanhcontains cubic and higher powers of the coordinates. (In taking the simple form (l), we have neglected Coriolis and!vibronic couplings.) If the displacements are so small that cubic and higher order terms may be neglected compared to quadratic ones, then only the first term remains in (2), and (1) is fully separable as a simple sum of harmonic oscillators. The resulting frequencies wj = (ki/pi)1/2
(3)
are quite accurate descriptions for vibrational spectroscopic fundamentals. For vibrationally excited states, Wanh generally cannot be neglected. It is the mixing due to these anharmonic terms which determines the character of the vibrationally excited states, and which the S C F methods are designed to treat. (9) Abramson, E.; Field, R. W.; Imre, D.; Innes, K. K.; Kinsey, J. L. J . Chem. Phys. 1984,80,2298. Mukamel, S.; Sue, J.; Pandey, A. Chem. Phys. Lett. 1984, 105, 134. Dai, H.-L.; Field, R. W.; Kinsey, J. L. J. Chem. Phys. 1985,82,2161. (10) Cf. e&: Sibrand, W. J . Chem. Phys. 1967, 46, 440. Henry, B. R.; Siebrand, W. Ibid. 1968,49, 5369. Swofford, R. L.;Long, M. E.; Albrecht, A. C. Ibid. 1976,65, 179. Wallace, R. Chem. Phys. 1971.11, 189. Halonen, L.; Child, M. S. J . Chem. Phys. 1983, 79, 559, 4355. (1 1) A quantum-mechanical algebraic approach to energy levels of poly-
atomics was proposed by: van Roosmalen, 0. s.;Iachello, F.; Levine, R. D.; Dieperink, A. E. L. J . Chem. Phys. 1983, 79, 2515.
The self-consistent field potential WscF(qi) acting on the ith mode is simply the average of the total potential over all of the other modes of the molecule. The states $i of ( 6 ) are the best (variationally) possible independent-mode states for the molecule. They are coupled by the S C F potentials WSCF. The SC-SCF equations14 involve use of a simple Bohr-Sommerfeld quantization to solve (6). To simplify the notation, we will consider throughout the rest of this section only the two-mode case, though extension to several modes is quite Using m,n as the quantum numbers in modes 1,2, respectively, we write 7f = h ( q J + h2(qz) + W12(q142)
(8)
hi(qi) = P?/2pi + V,(qi)
(9)
(12) Miller, W. H. J . Chem. Phys. 1976, 64, 2880. (13) (a) Carney, G. D.; Sprandel, L. I.; Kern, C. W. Adu. Chem. Phys. 1978, 37, 305. (b) Bowman, J. M. J . Chem. Phys. 1978, 68, 608. (14) Gerber, R. B.; Ratner, M. A. Chem. Phys. Lett. 1979, 68, 195. (15) Thompson, T. C.; Truhlar, D. G. J . Chem. Phys. 1982, 77, 3031. (16) Garrett, B. C.; Truhlar, D. G. Chem. Phys. Lett. 1982, 92, 64. (17) Lefebvre, R. In?. J . Quuntum Chem. 1983, 23, 543. (18) Roth, R. M.; Gerber, R. B.; Ratner, M. A. J . Phys. Chem. 1983,87, 2376. (19) Thompson, T. C.; Truhlar, D. G. J . Chem. Phys. 1982, 76, 1790. (20) Farrelly, D.; Hedges, R. M.; Reinhardt, W. P. Chem. Phys. Letr. 1983, 96, 599. (21) Barboy, B.; Schatz, G. C.; Gerber, R. B.; Ratner, M. A. Mol. Phys. 1983, 50, 353. (22) Gerber, R. B.; Roth, R. M.; Ratner, M. A. Mol. Phys. 1981,44, 1335. (23) Roth, R. M.; Ratner, M. A,; Gerber, R. B. Phys. Rev. Left. 1984, 52, 1288. (24) Schatzberger, P.; Halevi, E. A,; Moiseyev, N. J . Phys. Chem. 1985, 89, 4691. (25) Sellers, H. J. Mol. Sfruct. 1983, 92, 361. Lesseski, D.; Reed, W. E.;
Pavlovich, L. J.; Carney, G. D. Physical Chemistry Research Report, Allegheny College, June 1985. (26) Bowman, J. M.; Christoffel, K.; Tobin, F. J . Phys. Chem. 1979, 83, 905. (27) Romanowski, H.; Bowman, J. M.; Harding, L. B. J . Chem. Phys. 1985,82,4155. (28) Ratner, M. A.; Buch, V.;Gerber, R. B. Chem. Phys. 1980, 53, 345.
22
The Journal of Physical Chemistry, Vol. 90, No. 1, 1986
Ratner and Gerber .04
where is a potential acting along normal coordinate i and W12 is a coupling potential. The the S C F equations areI3,l4
Linear HDO
(m,o) Overtones
-%
with the average potentials defined by
.03
/
v
0,
0
9L
Thus the potential acting on coordinate 1 is the sum of the bare potential V,(ql) and the average of the coupling potential over mode 2. Note that, to determine $m(ql),we require not the wave function +,(q2),but an expectation value of that wave function. Evaluating that expectation value (12) in the semiclassical limit yieldsI4
2
3 .02
H.0
h M Ll
2
w
.01
with the normalization and momentum defined by [C2'")l2= p 2 ( q 2 )=
s,p' dq2/p2(42) - V2(q2)- @2(m)(q2)11''2
0.00
(14)
(15)
the q?, q: being turning points. Finally, the energy ti of the turning points are fixed by Bohr-Sommerfeld quantization:
- -
Equations for mode 1 are the same as (13-16), with the proper 1, n m). substitutions (2 Thus an SC-SCF solution for two modes means a knowledge of six numbers (two tr, four qa) for an S C F state, which is in turn labeled by quantum numbers (m,n). These six values are obtained self-consistently: we solve (16) for the t and turning points in one mode. Then (13, 11) are used to find the potential in the other mode. Using this potential and (16), we find the t and turning points for this second mode, which in turn determines a new potential (via (1 3)) to redetermine the original t and turning points. The entire process is iterated until self-consistency is attained.I4 The SC-SCF technique differs from the SCF in two ways: first, it never computes any wave functions numerically, since the probability determines T (from (12)), and this probability is proportional to p-I; this makes it less demanding computationally. Second, it uses the semiclassical condition (16) rather than a boundary value problem (1 1) to determine the energy eigenvalue t. In an actual SC-SCF calculation, the only mathematical work required is evaluation of one-dimensional integrals such as those in (13, 14, 16). Verbally, the S C F and SC-SCF methods can be characterized similarly to any other self-consistent field: the wave functions determine (via (13)) effective potentials, which are averages of the total potential over the other modes, and which in turn determine the wave function. Note that the SC-SCF never really uses the wave functions per se. It computes classical averages, and quantizes energies by the Bohr-Sommerfeld rule. Formally, the S C F methods replace one differential equation in n variables (H\E = E 9 ( ql...qn))by n equations ( 6 ) each in one variable. Therefore the computational effort is a linear function of n. This is to be contrasted with basis-set expansion methods, where the effort is proportional to the nth power. Thus these static S C F techniques are of real promise for the analysis of large systems; two-mode, t h r e - e - m ~ d e , ~and ~ , ~six-mde ~ . ~ ~ problems (with Coriolis coupling)27have been solved until now. Coordinate Systems and SCF. We have formulated the S C F equations using normal coordinates. This choice was made because the normal modes make the kinetic energy expressions simple (as in (l)), because they transform according to the symmetry rep(29) Christoffel, K. M.; Bowman, J . M . Chem. Phys. Letr. 1982, 85, 220. (30) Thompson, T C.; Truhlar, D. G. Chem. Phys. Lett. 1980. 75, 8 7 .
0
1
2
3
4
5
6
m Figure 1. Energy difference between approximate and exact results for the "asymmetric stretch" overtones of linear HDO; in local coordinates, m is the quantum number in the 0-D bond. The approximations are uncoupled harmonic oscillators,uncoupled local modes, uncoupled normal modes, and normal-mode SCF (from ref 18).
resentations of the molecular point group, and because it is standard in older treatments of vibrational problem^.^ The S C F techniques themselves, however, can be employed in any convenient coordinates. One alternative choice involves coordinates generated by orthogonal transformation of the normal coordinate set (ql...qn). Such coordinates, which we may call (QI...Qn), leave the kinetic energy diagonal, but change the form of the potential energy, and therefore the form of the SCF wave function. Elegant applications of this transformation have been given by Truhlar and co-workers,I5J6by Lefebvre,I7 and by M ~ i s e y e v . They ~ ~ generalized the application of the variational principle (5) to determine not only the single-mode wave functions $ but also the coordinate set. That is, for a two-mode case, they defined new coordinates x via x, = q 1 cos 0 q2 sin 6' (17) x2 = -ql
+ sin 0 + q2 cos 0
(18)
The optimal angle 6' was found by energy minimization, from the variational condition
a ~ / a e= o
(19)
For 6' = 0, the new coordinates xi are just the normal coordinates qi, and this is the best value for the first few states. For high overtones, however ( m > 4, n = 0), the angle starts to increase rapidly, becoming close to 45" for m = 14 (fully localized mode). A second alternative coordinate choice involves a more general transformation of the coordinates, one which introduces momentum coupling into the Hamiltonian. We have considered'* the molecules C 0 2 and H 2 0 and their isotopic variants, treated as two-mode systems with bond angle fixed. We performed SC-SCF calculations both in normal modes and in local modes. The local-mode picture has been applied very fruitfullyI0 to hydrocarbons, hydrides, and other molecules with heavy middles and light outsides, and should work well for H 2 0 . If the mixing terms (plp2/p in local modes, W12in normal modes) are simply ignored, we do indeed findL8that the local-mode picture is better, for H 2 0 and its isotopes, for m,n > 2. When S C F states and energies are used, however, the normal-coordinate choice is preferable for all states for HDO, DTO, etc. the only exception being excited H 2 0 . Indeed, the simple SCF normal-mode values are in very favorable agreement with the exact results, with the largest errors occurring (31) Moiseyev, N . Chem. Phys. Lett. 1983, 98, 233
Feature Article
The Journal of Physical Chemistry, Vol. 90, No. 1, 1986 23
for quite high energy states (- 18 000 cm-l), and even there being only -4%. Figure 1 shows the errors in energies calculated by using a simple harmonic picture, an independent-mode normalcoordinate picture (neglecting all cross-terms in (3)), a noninteracting local-mode picture (neglecting p1p2/p) and the SC-SCF based on (1). As expected, the local mode picture gets better as the excitation numbers increase, but the SCF-corrected normal mode is always to be preferred. The most important conclusion from these results is that a zeroth-order separable approximation can usually be better improved by finding the best possible mode potentials within a given set of coordinates than by changing the coordinates themselves. Within SCF, one expects the best coordinate choice to be that which leads (a) to maximal separation of mode frequencies and (b) to minimal mode-mode correlations. In this sense, we expect that collective-typecoordinates, which avoid the crowding of atoms in the same region of space, should be best for the SCF description. In particular, hyperspherical coordinates are very promising in this regard, and we have begun studying S C F treatments in these coordinates. Vibrational Spectroscopic Energies and SCF. The most obvious application of vibrational S C F techniques is to the calculation of Both vibrational energy levels and wave functions.13~'4~17~18~25-30 S C F and SC-SCF calculations have been reported for a wide variety of two-mode systems, and a few applications to three-mode larger systems have been reported.z5~z9In general, the S C F results for ground states are very good (to far better than 1%); errors become larger as the vibrational energies and amplitudes increase, but even for some fairly high modes (one quantum of symmetric stretch, two of asymmetric) of a very unfavorable case (linear HzO), the error in the S C F energy id8 only -2%. The vibrational transition frequencies for linear H D O differ18 from their exact value by an average of 13 cm-' (-0.2%) for 17 states whose vibrational energy ranges up to 15017 cm-l! For a two-mode model of formaldehyde, even 100 metastable states (containing enough energy to dissociate) exact and SCF energies which differ by no more than 1.5%. The S C F results for wave functions are also quite good. For HzO, Romanowski, Bowman, and Harding were ablez7to fit the vibrational spectrum (positions and intensities) very well using S C F wave functions, and we for the two-mode Henon-Heiles model, that the S C F state was the dominant contributor (92%) to the exact eigenstate even for quite high energies ( m = 4, n = 0, E 6Eo). The errors arising from the semiclassical approximation are even smaller than those of the S C F itself. Thus Garrett and Truhlar16 found, for the resonances in HF H collisions, that the SC-SCF error average 0.8 kcal/mol (for resonance energies up to 13.9 kcal), while the quantum S C F errors for comparable data averaged 1.1 kcal/mol. Farrelly et aLzoobserved, again in calculating resonances, that "the error introduced by S C F itself is much greater than that induced by using a semiclassical approximation within the SCF", while we found, for Henon-Heiles, a mean disparityz8 of
