J . Phys. Chem. 1985, 89, 964-969
964
Semiclassical Selection of Basis Sets for the Calculation of Vibrational Wave Functlons Steven M. Adler-Golden Spectral Sciences, Inc., Burlington, Massachusetts 01803 (Received: August 3, 1984)
A semiclassical analysis is shown to be highly useful for the variational calculation of vibrational energy levels and wave functions using a basis set of zeroth-order, separable eigenfunctions. The convergence properties of the variational calculation may be qualitatively described via the concept of semiclassical convergence. An eigenstate is said to be semiclassically converged when the phase space covered by the corresponding classical trajectory is fully spanned by the basis set. The criterion of semiclassical convergence yields an energetic limit to the range of eigenstates which may be converged with a given basis set. It also leads to methods for generating efficient basis sets, which may be geared to converging either many eigenstates or a single one. Applications of the semiclassical analysis are made to Morse, Henon-Heiles, and triatomic molecule model Hamiltonians.
I. Introduction A number of techniques exist for calculating quantum-mechanical vibrational wave functions and energies of small molecules. These include the traditional perturbation theory approach’J as well as more recent SCF*5 and variational“1° methods. When applicable, the variational method is probably the best for accurate work. All of these methods face severe difficulties as the energy and dimensionality are increased, especially when high accuracy is also desired; thus, a major focus has concerned optimizing the efficiency of the calculation. In particular, different choices of variational basis sets have been explored, and several systematic studies of convergence as a function of basis set size and type have been p e r f ~ r m e d . ~ - lMuch ~ of the recent y o r k has concerned finding the Ubest”separable Hamiltonian Howhose eigenstates form the basis set (14)). “Local mode” and “normal mode” models have been studied to determine which is the best zeroth-order description for a given system and energy rar~ge.’~-~O An alternative way of choosing a basis set relies on functions which are not tied to a zeroth-order Hamiltonian but instead can be positioned at will in coordinate space or phase space.”-’3 This approach is exemplified by the “grid” and “orbit” methods of Davis and Heller,13 who utilize Gaussian coherent state basis functions centered at discrete phase space points. In the grid method the points form a lattice within classically allowed phase space. In the orbit method, which is geared to converging a single or a few eigenstates, the Gaussians are placed along a classical trajectory. An extension of the orbit approach led to semiclassical methods
(1) (2) (3) (4) (5) 2376. (6) (7)
Nielson, H. H.Rev. Mod. Phys. 1951, 23, 90. Darling, B. T.; Dennison, D. M. Phys. Reu. 1940, 57, 128. Bowman, J. M. J . Chem. Phys. 1978, 68, 608. Thompson, T. C.; Truhlar, D. G. J. Chem. Phys. 1982, 77, 3031. Roth, R. M.; Gerber, R. B.; Ratner, M. A. J . Phys. Chem. 1983,87,
Suzuki, I. Bull. Chem. SOC.Jpn. 1971, 44, 3277. Foord, A.; Smith, J. G.; Whiffen, D. H. Mol. Phys. 1975, 29, 1685. (8) Whitehead, R. J.; Handy, N . C. J . Mol. Spectrosc. 1975, 55, 356. (9) Carney, G. D.; Langhoff, S. R.; Curtiss, L. A. J . Chem. Phys. 1977, 66, 3724. (10) Carney, G. D.; Sprandel, L. L.; Kern, C. W. Adu. Chem. Phys. 1978, 37, 305. (1 1) Chesick, J. P. J . Chem. Phys. 1968, 49, 3772. (12) Shore, B. W. J . Chem. Phys. 1973, 59, 6450. (13) Davis, M. J.; Heller, E. J. J. Chem. Phys. 1979, 71, 3383. (14) Burton, P. G.; Von Nagy-Felsobuki, E.; Doherty, G. Chem. Phys. Lett. 1984, 104, 323. (15) Henry, B. R.; Siebrand, W. J. Chem. Phys. 1968, 49, 5369. (16) Wallace, R. Chem. Phys. 1975, 1I, 189. (17) Swofford, R. L.; Long, M. E.; Albrecht, A. C. J . Chem. Phys. 1976, 65, 179. (18) Lawton, R. T.; Child, M. S. Mol. Phys. 1979, 37, 1799. (19) Sage, M. L.; Jortner, J. Ado. Chem. Phys. 1981, 47, 293. (20) Stannard, P. R.; Elert, M. L.; Gelbart, W. M. J . Chem. Phys. 1981, 74, 6050.
0022-3654/S5/2089-0964$01.50/0
by which approximate wave functions are built from Gaussians using classical trajectories alone, without the need for a variational matrix d i a g o n a l i z a t i ~ n . ~ ’ - ~ ~ The key questions are, what type of basis functions (Le., local mode, normal mode, Gaussian, etc.) will be most efficient for a given problem, and how can the best parameters (such as position, frequency, range of quantum numbers, etc.) of the basis set be chosen? This paper shows that a semiclassical analysis is highly useful for answering these questions. We find that the same phase space considerations used in the construction of efficient Gaussian c_oherent state basis setsI3 are equally applicable to basis sets of Hoeigenfunctions. In particular, the phase space picture enables us to define a “semiclassical convergence” which gives proper behavior in the classical limit and which even in the quantum domain qualitatively predicts the convergence properties of a variational calculation. The criterion of semiclassical convergence also enables us to develop methods for choosing efficient basis sets. Furthermore, Gnce the semiclassical approach is helpful for chming the “best”Ho, it may also find application in perturbation or other nonvariational treatments which rely on a zeroth-order Hamiltonian. The organization of this paper is as follows. Section I1 presents the basis semiclassical theory and introduces semiclassical convergence. Semiclassically predicted convergence properties are then compared with the results of actual variational calculations. In section I11 the semiclassical convergence criterion is used to generate basis set optimization strategies. Several of these strategies are applied by using model Hamiltonians, and they are found to be quite successful. Finally, an overall summary and discussion are provided in section IV. 11. Semiclassical Model of the Variational Method The literature on semiclassical methods for molecular vibrations is enormous24and will not be summarized here. Of greatest relevance to the current work are the papers by Heller25and Davis and HellerI3 which present the basic concepts of a semiclassical approach to the variational method. The emphasis here is slightly different, since we focus on the use of basis functions of a zeroth-order Hamiltonian rather than Gaussian coherent states. The semiclassical model relevant to this type of calculation is quite simple and is presented in section 1I.A. Further developments and applications are presented in subsequent sections. A . Theory. The convergence of a variational calculation relies on the inclusion of the proper functions in the basis set. _More precisely, a basis set {I4))will converge eigenstate I$) of H to a (21) Davis, M. J.; Heller, E.J. J . Chem. Phys. 1981, 75, 3916. (22) DeLeon, N.; Heller, E. J. J . Chem. Phys. 1983, 78, 4005. (23) Heller, E. J. Faraday Discuss. Chem. SOC.1983, 75, 141. (24) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. Annu. Reu. Phys. Chem. 1981, 32, 267. (25) Heller, E. J. J. Chem. Phys. 1977, 67, 3339.
0 1985 American Chemical Society
Semiclassical Selection of Basis Sets specified accuracy provided that 114)) contains all states whose overlap integral squared with I$), 1($14)12, is greater than a sufficiently small value. The strategy employed here, and previously by Heller,13*25 is to utilize classical mechanics to predict which basis functions significantly overlap I$), and hence must be included in the basis set. The use of classical or semiclassical approximations for overlap integrals is rather new in the context of the variational calculation of vibrational energy levels, but it is a well-established approach for Franck-Condon spectra. These Franck-Condon methods include semiclassical wave packet propagation the semiclassical WKB stationary phase appro xi ma ti or^,^^ and fully classical treatments.2s Let us begin by assuming that the eigenstate I$) has a corresponding classical trajectory, which we deflote $. Since 14) is the eigenstate of a separable Hamiltonian, H,, 14) also has a corresponding trajectory, 4. Now, we know that in the classical limit overlap between I$) and 14) corresponds to an intersection of $ and 4 in phase space. Thus, we can form a definition of convergence which in the classical limit is consistent with the convergence of a variational calculation: A basis set (14))converges I$) semiclassically if the set contains all states whose classical trajectory 4 intersects $. The above definition of semiclassical convergence provides a necessary condition for the adequacy of a variational basis set. It is a necessary condition because if 4 and $ intersect, then the overlap I(41$)12will be substantial, except if by chance it is zeroed by totally destructive quantum interference. This is simply a statement of the Franck-Condon principle. However, since in the quantum domain position and momentum are uncertain, the converse is not necessarily true-that is, 14) and I$) can overlap somewhat even if the classical trajectories do not meet. Therefore, a semiclassically converged state may not in reality be accurately variationally converged. Given this difference between classical and quantum behavior, how useful is the criterion of semiclassical convergence? This question is probably best answered by studying examples representative of typical variational problems. We_do this in sections 1I.B and 111.4 using a Morse oscillator for H and a harmonic oscillator for Ho. It should be noted that, in an important limiting case, semiclassical convergence is_suffi$ient for quantum convergence. That case is the limit H = H,, i.e., I$) = 14) and $ = 4, Thus, we can expect that even in the quantum domain the semiclassical convergence criterion will be useful-as a nearly sufficient condition for quantum convergence when H is very well approximated by Hoe ”Nearly sufficient” means that only a relatively small number of extra basis functions will be needed as a cushion to ensure reasonable quantum convergence. Of course, what we mean by “reasonable” depends to an extent on the error we are willing to tolerate. We shall see in the examples which follow that although semiclassical convergence does not imply quantum convergence to the high accuracy demanded for most spectroscopy, it still provides a qualitative description of the variational calculation. This predictive power of the semiclassical picture leads straightforwardly to methods for basis set optimization, as will be seen in section 111. To complete our theoretical discussion, we should also consider cases in which a unique corresponding trajectory $ may not exist, such as when the classical dynamics are ergodic. We can retain our definition of semiclassical convergence by appropriately extending the notion of “corresponding trajectory”, regarding it simply as that portion of phase space to which 14) is assigned in the classical limit. In practice, it may be impossible to specify the correspondence any more explicitly than this, but that need not be a problem. It is safe to say that the “corresponding trajectory”, whatever it may be, will cover a subspace of the full phase space of the classical Hamiltonian H at energy E . ( 2 6 ) Heller, E. J. Acc. Chem. Res. 1981, 14, 368. (27) Mies, F. H. J . Chem. Phys. 1968, 48, 482. (28) Foth, H.-J.; Polanyi, J. C.; Telle, H. H. J . Phys. Chem. 1982, 86, 5027.
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985 965
a
-2.5
-5.0
2.5
0.0
5.0
0 Figure 1. Classical trajectories of Morse and harmonic oscillator Hamiltonians, eq 1 and 3, respectively. The heavy curve is the Morse u = 5 state, and the concentric circles are the u = 0-12 harmonic states. The
solid circles are those which intersect the Morse state.
-
m 8 -
& -A
EXRCT
0
VRRIRTIONRL
- SEWICLRSSICRL
DIVERGENCE
? -. m 3
I
I
I
I
I
4
5
6
7
8
9
V Figure 2. Energy levels of the Morse oscillator Hamiltonian, eq 1. Semiclassical energy E(u) is derived in Appendix A.
Therefore, if the basis set (14))contains all states whose trajectory 4 at some point (p,q‘) acquires the value H e , ; ) = E , then we are assured that I$) is semiclassically converged. This idea forms the basis of methods discussed in sections 1I.C and 1II.B. B. Example: the Morse Oscillator. To illustrate the above semiclassical picture, we consider the example of a Morse oscillator diagonalized in a harmonic oscillator basis set. The Morse oscillator Hamiltonian is chosen as 2H = p 2
+ [ I - exp(-0.17q)12 (0.17)2
where the exact energy levels are given by E = (V
+ 1/2) - 0.01445(~+ 1/2)2
(2)
The basis set is chosen as eigenstates of the harmonic oscillator 2Ho = p 2 + q2
(3)
Figure 1 shows the classical trajectories of the u = 5 Morse state (egg-shaped heavy curve) and the u = 0-12 harmonic states (concentric circles). Since the u = 3-10 harmonic state trajectories cross the Morse trajectory, it is those harmonic states which must be included in a basis set capable of semiclassically converging the Morse state. It is also evident that by including u = 0-10 harmonic states the u = 0-5 Morse states are semiclassically converged. The corresponding variational calculation was performed, and the result is depicted in Figure 2. Below u = 5 , the variational eigenvalues (squares) agree quite well with the exact
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The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
Adler-Golden
energy levels (triangles) on the scale of Figure 2. Above u = 5, the variational error is very large. The error is also substantial for u = 5, the highest level which is semiclassically converged. All of these results are fully consistent with the previous discussion. C. Semiclassical Threshold for the Onset of Divergence, Ed, In the previous example, all Morse states below a certain energy were semiclassically converged. Let us denote this energy Ed,the threshold for semiclassical divergence. To compute Ed explicitly, we note that Edis the energy such that for E I Ed the phase space of H lies entirely within the basis set’s phase space boundary. The equation of this boundary is Emax = Ho(P,q)
(4)
where E,,, is the maximum energy of the functions included in the bask set. From Figure 1 it is apparent that Ed is the minimum value of the exact Hamiltonian H(p,q) on the curve given by eq
-
w
8
5000
10800
15000
20800
25088
4. To illustrate the behavior of the variational eigenvalues in the vicinity of Ed, we have plotted Ed as the dotted line in Figure 2. As predicted by the earlier discussion, it is found that Ed lies somewhat above the energy where convergence is acceptable for spectroscopic purposes, but it does serve to indicate the onset of severe divergence. The above prescription for Ed, Le., minimizing H(p,q) on the curve defined by eq 4,will not be valid as a divergence threshold in all cases, however. For example, if outside of the phase space boundary defined by eq 4 the potential function declines instead of monotonically increases, values of E less than Ed would be found both outside and inside of this boundary; only eigenstates associated with the phase space within the boundary would be semiclassically converged. In this case, Ed would not monotonically increase with E,, as it does for the Morse oscillator, but it would reach a maximum and then decline. If a corresponding variational calculation is performed, this decline will signify the appearance of spurious variational eigenstates located outside of the potential well. Edcan be calculated for multidimensional systems in essentially the same way as for one-dimensional systems. For example, if the bask functions are selected by the cutoff E,,, Ed is calculated from the multidimensional analogue of eq 4,which now describes a (2n - 1)-dimensional hypersurface. In some cases, Ed can be obtained analytically. Otherwise, it may be calculated by Monte Carlo sampling phase space points on the hypersurface to find the minimum of H . The procedure can also be reversed: given an energy Ed which represents the range of eigenstates one desires to converge, one can compute E,,,, which in turn specifies the size of the basis set required. There are, of course, an almost limitless number of ways to choose members of a basis set, especially in a multidimensional case. For instance, rather than using an energy limit E,,, another choice which has been used in the calculation of eigenstates of triatomic molecules9J4is to include all basis functions having up to a specified number of total quanta, u,,,. For a harmonic oscillator basis set the classical phase space boundary is given by n
CHoi(pi,qi)/ha, = umax
+ n/2
(Sa)
I=]
where the Hot are the components of Ho Ho = C H o i i= 1
(5b)
Equation 5a replaces eq 4. From this point on, Ed may be computed in the same way as before. As an example of the calculation of Edin a multidimensional system, we consider the Hamiltonian
Figure 3. Semiclassical threshold for divergence using a model Hamiltonian for ozone and harmonic oscillator basis functions having energy 5
Enlax.
normal-coordinate potential energy surface for the ground electronic state of ozone due to Barbe et al.29 Ho is taken to be the quadratic part of eq 6 3
HO= !hCai@?+ 4;)
(7)
I=]
The basis set is chosen as all Ho eigenstates having energy _
