J . Phys. Chem. 1986, 90, 1599-1603
1599
A Generalized Semiclassical Self-Consistent-Field Procedure for Nonseparable Vibrationally Bound States David Farrelly* Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024
and Andrew D. Smitht Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1, and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742 (Received: September 18, 1985)
A completely uniform semiclassical self-consistent-field (SCF) procedure is developed which is applicable to nonseparable potentials containing arbitrary configurations of wells and barriers. The method is a generalization of the primitive semiclassical self-consistent-field method developed by Ratner, Gerber, and co-workers and is in principle extendible to the calculation of complex resonance energies. A detailed numerical investigation is presented with reference to a coupled double-well system which models a two-dimensional isomerization reaction. Our calculations suggest that for many bound problems a partially uniform approach is sufficient in which a uniform semiclassical quantization is combined with a primitive semiclassical procedure for evaluating the corrections to the SCF equations. In addition, we find that very good accuracy may be obtained by evaluating the single-modeSCF corrections using a primitive semiclassical approximation within the context of a quantum SCF calculation, with a substantial reduction in computational effort as compared to fully quantum SCF calculations.
1. Introduction The majority of semiclassical methods which have been developed for the quantization of multidimensional bound states take as their starting point the exact, possibly chaotic, classical dynamics.lb In the most well-known approach, pioneered largely by Marcus and co-workers: quantization is effected by searching for classical trajectories satisfying the EBK quantization conditions.' However, trajectory quantization rapidly becomes computationally expensive as the number of degrees of freedom increases. In addition, the method is not readily applicable to problems involving tunneling, or when the dynamics is ~ h a o t i c . ~ - ~ A number of alternative semiclassical schemes have thus been developed to avoid some of the problems and limitations associated with quantization of classical trajectories. For example, methods based on canonical perturbation theory*-l I have proved highly successful both in quantizing chaotic regions of phase space as well as in describing quantum tunneling.12 It is worth noting, however, that many semiclassical approaches are currently limited to model problems (e.g., coupled oscillators), and that even in these cases their practical application is not straightforward, often requiring the development of fairly sophisticated computational techniques. It is therefore important that simpler alternatives be investigated which at the expense of making (perhaps severe) global approximations to the exact dynamics may be applied fairly straightforwardly to realistic models of polyatomic molecules. Two examples of these more approximate approaches are provided by the semiclassical self-consistent-field (SCF)13-16and adiabatic approximations (AA).17-20 In this paper we will focus our attention on the S C F method. Originally developed for vibrational problems by Carney et al.,I3 and Bowman and c o - ~ o r k e r s , ' ~ the - ' ~ S C F method provides a simple and accurate pmcedure for the calculation of eigenvalues corresponding to bound or resonance states in arbitrary nonseparable systems. The S C F approach as applied to vibrational dynamics is closely related to the more well-known Hartree-Fock method for many electron problems. Like the Hartree-Fock method it is assumed that the wave function is separable, which in combination with a variational principle reduces the multidimensional problem to a set of coupled one-dimensional Schriidinger equations, one for each mode. The effective potential in each mode contains a contribution obtained by averaging over all other modes. 'Present address: British Aerospace Dynamics Group, Space and Communication Division, Argyle Way, Stevenage, Herts, SG1 2AS U.K.
0022-3654/86/2090-1599$01.50/0
Extraction of the energy levels is straightforward since it involves the solution of a set of one-dimensional problems. Thus the procedure is readily applicable to systems of fairly high dimensionality and usually provides very good agreement with exact results. By means of complex coordinate rotation, S C F has been extended to the calculation of complex energy eigenvalues by Christoffel and Bowman.2' A clear computational drawback to the quantum mechanical implementation of the S C F approach is, however, the need to evaluate highly oscillatory integrals over the numerically generated wave functions in order to evaluate the successive corrections to the single-mode potentials in the iterative procedure.22 An additional problem in the S C F calculation of
(1) Percival, I. C. A h . Chem. Phys. 1977, 36, 1. (2) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. Annu. Reu. Phys. Chem. 1981, 32, 267. (3) Rice, S. A. A h . Chem. Phys. 1981, 47, 117. (4) Tabor, M. Adu. Chem. Phys. 1981, 46, 73. (5) Stechel, E. B.; Heller, E. J. Annu. Reu. Phys. Chem. 1984, 35, 563. (6) Reinhardt, W. P. In "The Mathematical Analysis of Physical Systems"; Mickens, R., Ed.; Van Nostrand-Reinhold: New York, 1985; pp 169-245. (7) Einstein, A. Verh. Dtsch. Phys. Ges. (Berlin)1917, 19, 82. Brillouin, L. J . Phys. Radium 1926, 7,353. Keller, J. B. Ann. Phys. (New Yo&) 1958, 4, 180. (8) Nayfeh, A. H. "Perturbation Methods"; Wiley: New York, 1973. (9) Goldstein, H. "Classical Mechanics", 2nd ed.; Addison-Wesley: Reading, MA, 1980. (10) Swimm, R. T.; Delos, J. B. J . Chem. Phys. 1979, 71, 1906. (1 1) Birkhoff, G. D. "Dynamical Systems"; American Mathematical Society: Providence, RI, 1966. (12) Jaffc, C.; Reinhardt, W. P. J . Chem. Phys. 1979, 71, 1862. (13) Carney, G.D.; Sprandel, L. I.; Kern, C. W. Adu. Chem. Phys. 1978, 37, 305. (14) Bowman, J. M.; Christoffel, K. M.; Tobin, F. J . Phys. Chem. 1979, 83, 905. (15) Christoffel, K. M.; Bowman, J. M. J . Chem. Phys. 1981, 74, 5057. (16) Bowman, J. M. J . Chem. Phys. 1978, 68, 608. (17) Ezra, G. S. Cfiem. Phys. Lerr. 1983, 101, 259. (18) Saini, S . ; Hose, G.; Stefanski, K.; Taylor, H. S. Chem. Phys. Lett. 1985, 116, 35. (19) Stefanski, K.; Taylor, H . S. Phys. Reu. A 1985, 31, 2810. (20) Romanowski, H.; Bowman, J. M. Chem. Pfiys. Lett. 1984,110,235. (21) Christoffel, K. M.; Bowman, J . M. J . Cfiem. Pfiys. 1982, 76, 5370. (22) Farrelly, D.; Hedges, Jr., R. M.; Reinhardt, W. P. Chem. Phys. Lett 1983, 96, 599.
0 1986 American Chemical Societv
1600 The Journal of Physical Chemistry, Vol. 90, No. 8,1986
complex eigenvalues is that since a finite basis is employed the energies are not independent of the angle by which the coordinates are rotated into the complex plane and consequently the behavior of the complex eigenvalue must be examined as a function of rotation angle in order to obtain a converged result. The numerical problems associated with the quantum SCF procedure have been successfully avoided by Gerber and R a t r ~ e r ? ~ and by Ratner et al.24who developed a primitive semiclassical version of SCF equally suitable for the calculation of bound-state energies or resonance positions (energies). In this approach the single-mode energies are obtained by use of a primitive BohrSommerfeld quantization and the quantum wave functions are replaced by a primitive semiclassical probability density in computing the corrections (essentially expectation values of the coupling) to the single-mode effective potentials. This approach has been applied to a number of systems, giving excellent agreement with the corresponding quantum S C F calculations. Except for the case of very weak coupling, the error induced by using a semiclassical approximation is usually less than that introduced by using the S C F approach in the first place. As pointed out by Garrett and Truhlar,2s however, use of a primitive approximation is a major restriction since problems involving nonclassical paths cannot be handled. For example, the primitive semiclassical S C F cannot be used to calculate level splittings due to barrier penetration in bound problems, or resonance widths in quasibound systems. Farrelly et aLz2have extended the semiclassical S C F procedure to the calculation of resonance widths by a partial uniformization in which use of a primitive probability density was retained but the Bohr-Sommerfeld quantization formula was replaced by a uniform semiclassical expression appropriate to quasi-bound states. Excellent agreement was obtained for states below the barrier top for the double-barrier problem considered. In this paper we develop a completely uniform semiclassical S C F procedure, which, in principle, can be applied to bound problems involving arbitrary configurations of potential wells and barriers. (Note, however. that the SCF approach is not directly suitable for treating dynamic tunneling26in which no physical barrier in the potential occurs.) The fully uniform version of S C F involves little additional computational effor over the primitive S C F method, and, like the primitive version, completely avoids the evaluation of any wave functions. Application is made to the calculation of bound states in a coupled double-well problem. In section 2 we give a brief description of the primitive S C F procedure and in particular point out where a uniform approximation is required. In section 3 we describe how a fully uniform S C F procedure may be developed. Application is made in section 4 to a two-dimensional problem involving a double-well potential in one of the modes. Conclusions are in section 6.
2. The Semiclassical SCF Method For simplicity we will restrict our discussion to two-dimensional Hamiltonians of the form
+ h, + V,(x,y)
7f = h,
(2.1)
where
Farrelly and Smith has been considered in a number of recent paper^.^^^^'^^* In the SCF method, a Hartree product wave function of the form $flrn,(wI
(2.3)
= dh,(X)&,(Y)
is assumed in the Schrodinger equation w f l r n J ( x ?= Y )Eflxfl!$flxflJ(x,Y)
(2.4)
The condition that the energy be stationary subject to the variational principle
wn>fl,lm4fxfl,) =0
(2.5a)
and the orthonormality condition
(4tlIdJdl) =
(2.5b)
6flfl.
leads to the S C F equations [h,y(x)+ Wx(il(~)]$+,:i)(x) = enA(i)(ny)+n>i)(x) (2.6) [h,.Oi) + W,("CV)I4fl,(i)Oi) =
c~,(')(~.J+~~(~)CV) (2.7)
where the superscripts denote the iteration. The corrections to the single-mode Schrodinger equations, W,(x), and W y ( j )are given by W , W =
( ~flJ(~ilv,(.~,Y)l~fl,(~)) (2.8)
W;i)Oi) =
(4Jn,(Y Vc(x,Y)l@"r(i))
(2.9)
and are seen to be simply expectation values of the nonseparable coupling potential with respect to the single-mode wave functions. The effective potential in each mode thus contains a contribution from the average of the coupling potential over the other mode. The eigenvalues are given by the formula EflLfl,(il = en,())
+ en (i) - Wn,n, (i) J
(2.10)
together with
w r p = ( ic,l,,,~"I~,(x~Y)licfl~n,'~')
(2.1 1 )
where e,,>) and e,,y(l) are the single-mode eigenvalues obtained from a solution of eq 2.6 and 2.7. To evaluate eq 2.8, 2.9, and 2.1 1 the wave functions &x and &, are required. In the primitive semiclassical S C F explicit evaluation of the wave functions is avoided by introducing the following a p p r ~ x i m a t i o n ~ ~ . ' ~ Io(q)lz=
1-;
1 Pq(9)
qa
4
qb
(2.12)
otherwise
where q is a coordinate and qa and qb are the two classical turning points bounding the potential well. This approximation reduces the expressions for the expectation values in eq 2.8 and 2.9 to easily evaluable integrals of the form
leqbdq
(2.2a) h, =
h2
--
2m
a*
aY2
+ V,b)
(2.13)
(2.2b)
and V,(x,y) is a nonseparable coupling potential (extension to higher dimensions is straightforward). Although we have indicated Cartesian coordinates, in practice, improved results can sometimes be obtained by searching for the coordinate system which most nearly separates the problem. The optimum choice of coordinates (23) Gerber, R. B.; Ratner, M . A. Chem. Phys. Lett. 1979, 68, 195. (24) Ratner, M . A.: Buch, V.; Gerber, R . B. Chem. Phys. Lett. 1980, 53.
At this point it is clear that if the effective potentials contain multiple wells, the approximation of eq 2.12 will break down. One possible solution would be to assume a contribution of similar form to eq 2.13 from each well (with the appropriate normalization). In the next section we show how a more general uniform approximation for W,(x) and W , b ) may be obtained. The remaining problem is to obtain the single-mode eigenvalues, cnt and en,. In the primitive semiclassical version of S C F they are
345.
( 2 5 ) Garrett, B. C.; Truhlar, D. G. Chem. Phys. Lett. 1982, 92, 64. (26) Heller, E. J.: Davis. M. J. J . Phys. Chem. 1981, 85, 307.
(27) Thompson, T C.; Truhlar, D. G. J . Chem. Phys. 1982, 77, 3031 (28) Lefebvre, R. Int. J . Quantum Chem. 1983, 23, 543
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986 1601
Quantization of Multidimensional Bound States TABLE I: Eigenvalues for the Coupled Double-Well System of Eq 3.1 with wY = 0.7, a = 1.0, c = 1.0, p = 1.5, V , = 9.0, X = 0.1, and h = m = 1.0 nr*, nu QSCF" PSCF" USCF" O+, 0 0-, 0 I+, 0 1-, 0 2+, 0 2-9 0 3+, 0 3-, 0 4+, 0 4-,O 5+, 0 5-, 0
3.424 84488 3.428000 16 5.486 561 15 5.51304830 7.316978 75 7.446 10046 8.91803905 9.314 805 63 10.4645320 11.1708231 12.1744474 13.042 1958
3.434 207 61 3.43689498 5.497 277 56 5.52141233 7.336 285 22 7.456 987 10 8.94550634 9.326 384 14 10.4868103 11.1813635 12.1882573 13.051 2466
3.434 207 61 3.43689498 5.497 27745 5.52141231 7.336 28273 7.456 985 97 8.94550283 9.326 268 67 10.4868007 11.1813607 12.1882548 13.051 2465
I. In zero order, the x-mode potential is a symmetric double well, illustrated in Figure 1 , while that in t h e y mode is a harmonic oscillator. (i) Uniform Quantization of the Double Well. The uniform semiclassical quantization of double-well problems has been the subject of a number of articles, so it is sufficient simply to quote the quantization rule which is used to replace the primitive quantization of eq 2.14. A number of equivalent formulas exist in the literature. Since it explicitly shows the splitting of the levels due to tunneling, the most useful formula is that of F r O n ~ a nand ~~ PauIsson et aL30 where (3.4a)
"QSCF = quantum SCF; PSCF = partially uniform SCF (uniform quantization, primitive correction);USCF = uniform SCF. l5
VX(X) -
' a=
1 h
- Re l : p , ( x ) dx; E > Vo i m = - -hi ; p , ( x )
dx
(3.4b) (3.5)
and the quantum correction function, $(E), is given by
10
$(E) =
c
+ arg r(y2+ it) - e In It1
(3.6)
where x,, xb, and x, are the classical turning points shown in Figure 1. The branch of p x is chosen so that t < 0 for energies below the barrier top. The levels are seen to occur in pairs, E*, with the splitting being given' approximately by the expression3'
A = tan-' (e"')/(da/dE); 0 1
-5
I
-3
,
I
I
-1
1
3
5
X Figure 1. The uncoupled x-mode potential. The turning points at an arbitrary energy are labeled x,, xb. x,, and xd. The barrier maximum occurs at E = 9.0.
obtained by imposing a Bohr-Sommerfeld quantization in each mode
+ Y2)r
-1x : p J i ) ( x ) d x = (n, h
(2.14) (2.15)
where
a = a - t/z4(c)
(3.7)
These formulas are valid for the turning points pairwise propinquous. For problems involving different or more complicated distributions of turning points the correct quantization formula may be obtained fairly easily by a piecewise application of semiclassical connection formulas, provided that three or more turning points never coalesce simultaneously. It is important to note that the effort involved in solving the uniform formula (3.3) is only slightly greater than that in solving by using a primitive quantization. An additional quadrature must be performed to obtain the barrier phase integral and the arg r function must be evaluated. Both are straightforward. We now turn our attention to the uniform evaluation of WJx) and W y b ) . (ii) Uniform Evaluation of the Single-Mode Corrections W,(')(q).We first examine in more detail the derivation of the formula for the corrections, W,c"(q),eq 2.13. For a single-well potential the semiclassical wave function is
px(l) = (2m[tnX(') - Vx(x) - W X ( ' ) ( x ) ] ) ' / * (2.16) (2.17) p,(l) = ( 2 m [ ~ , , (-~ )V,b) - Wy(i)(y)])'/2 The S C F procedure is to solve eq 2.14, 2.15, 2.10, and 2.1 1 until self-consistency is achieved.
3. Uniform SCF In order to develop a uniform S C F procedure applicable to arbitrary configurations of single-mode potential wells and barriers, two refinements of the primitive S C F are needed; (i) the primitive quantization conditions (2.14) and (2.15) must be replaced by the appropriate unijwm quantizatipn rule, and (ii) a uniform procedure must be used to evaluate the corrections to the effective potentials, W,(x) and W,b). For clarity we will present our discussion with specific reference to a potential similar to that considered by Christoffel and Bowmani5 and designed to model an isomerization reaction
+
+ Vo exp(-cx2)
V(x,y) = '/zmwy2(x)y2 y2ax2
(3.1)
where w y ( x ) = wo[I - X e ~ p ( - p x ) ~ ]
(3.2)
and the parameter values are summarized in the caption to Table
The expectation value of some multiplicative operatorflq) in terms of this wave function is given by Mq)) =
(3.9) assuming that contributions from outside the well are negligible. Replacement of the square of the cosine by its average value of leads to the expression
which is equivalent to eq 2.13. Although a rather rough app r ~ x i m a t i o n ,eq ~ ~ 3.10 gives surprisingly accurate results in (29) Froman, N. Ark. Fiz. 1966, 32, 79. (30) Paulsson, R.; Karlsson, F.; LeRoy, R. J. J . Chem. Phys. 1983, 79, 4346. (31) Connor, J . N. L.; Uzer, T.; Marcus, R. A,; Smith, A . D.J . Chem. Phys. 1984, 80, 5095.
1602
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986
practice, even for the ground state. The most obvious extension to general potentials containing multiple wells is to replace the wave function in eq 3.8 by a uniform semiclassical wave function and evaluate the integral 3.9 by direct quadrature. However, this approach would considerably detract from the simplicity of the semiclassical S C F method. In addition, since semiclassical wave functions are in general less accurate than semiclassical quantization formulas it is not clear now valuable this approach would be. Froman3*x33and Froman and fro ma^^^^ have developed a procedure for calculating semiclassical expectation values which gives results of comparable accuracy to the uniform quantization formula (which the method starts from) and, in addition, explains the unexpected accuracy of the primitive formula (2.13) for single-well problems. We now briefly outline the Froman’s method. 32-34 Assume that the uniform quantization formula for an arbitrary potential, V ( q ) , is given implicitly by (3.1 1 ) N(E,n) = 0: n = 0,I , 2, ... In order to evaluate the expectation value of an operatorf(q). the auxillary potential3s (3.12) V(q,K) = V ( q ) - Kf(q) is introduced where K is 3n arbitrary small parameter. Provided that the auxillary potential does not differ drastically from the true potential, we may assume that the quantization formula appropriate to the auxillary potential becomes N(E,ti,n) = 0 ; n = 0, I , 2, ... (3.13) According to the Hellmann-Feynman theorem,36 (dV(q,K)/ a K ) = aEn(K)/ a K
(3.14)
Inserting eq 3.12 for v(q,K) we obtain c f ( q ) ) n = ---dEn(K)/dK
(3.15)
-
Differentiation of eq 3.13 with respect to then letting K 0 gives the formula’*
K
x
(3.17) dx
i - -
x
with px(x) defined by eq 2.16, and $(e)
= -%@(e)
f
’/* tan-] (e*‘)
3.18)
and 3.19) where
0’. 3 0-, 3 I+, 3 I-, 3 2+, 3 2-, 3 31, 3 3-, 3 4+, 3 4-> 3 5+, 3 5-, 3
5.52492601 5,528 509 80 7.57584201 7.604 983 86 9.389 423 86 9.532 11076 10.9679587 11.396 177 4 12.5182897 13.2503452 14.2429536 15.122 1123
5.531 90097 5.534 820 51 7.58761240 7.614 21201 9.409 745 69 9.54353605 10.995 1536 11.408033 0 12.5390599 13.2607449 14.2557986 15.1309032
5.531 90095 5.534 820 51 7.58761 1 4 4 7.614211 80 9.409724 19 9.543 524 99 10.995 145 5 11.407 768 5 12.538 993 2 13.260 729 8 14.255 783 8 15.130 903 8
“ Q S C F = quantum SCF; PSCF = partially uniform S C F (uniform quantization. primitive correction); U S C F = uniform S C F .
It is easy to show that eq 3.17 reduces to the expected primitive result in the two limits: 0, giving (i) E
V,,. In this case
t
>>
dx
0 and d g t / d t
-
0, giving
or, since the double well is symmetric,
c
-
TABLE II: Eigenvalues for the Coupled Double-Well System of Eq 3.1 with the Same Parameters as Table I QSCF“ PSCF“ USCF“ n,*, ny
(for fixed n ) and
Since this formula has been derived from the uniform quantization formula, eq 3.13 without explicit use of the wavefunction it is expected to be of comparable accuracy to the quantization formula. It is straightforward to show that the primitive formula, eq 3.10 is in fact, identical with the uniform result for a single well: hence the “unexpected” accuracy of eq 3.10. For the symmetric double-well problem we obtain (wheref(x) is the x-dependent part of V J X , J J ) ) dx
Farrelly and Smith
+ is the digamma f ~ n c t i o n . ~ ’
(32) Froman, N. In “Proceedings of the NATO Advanced Study Institute, Cambridge, England, September 1979”; Child, M. S., Ed.: Reidel: Dordrecht, 1979: Chapter 1 . (33) Froman, N . Phys. Lerr. 1974, 48A, 137. (34) Froman, N.; Froman, P. 0. J . Math. Phys. 1977, 18, 903. (35) Delves, L. M . Nucl. Phys. 1963, 41, 497. (36) Levine, I . N. “Quantum Chemistry”, 3rd ed.; Allyn and Bacon: Boston, MA, 1983; Chapter 14.
(3.22)
4. Calculations for the Coupled Double-Well Oscillator In Tables I and I1 we compare quantum S C F results with partially uniform semiclassical (uniform quantization, primitive correction) and fully uniform semiclassical S C F calculations for the sequence of states, (nx, nJ = (n,, 0) and (nx, nv) = ( n x , 3). The quantum results were obtained by direct numerical integration of the x-mode S C F equation, with the correction to the y-mode potential being evaluated by a Bode’s method integrator with an h7 error ( h is the step size).37 In both the quantum and semiclassical calculations, the y-mode equation was solved analytically as was the correction to the x-mode potential. The first point to note is that overall both sets of semiclassical calculations are in very g o d agreement with the quantum results. The semiclassical splittings, portrayed in Figure 2 , are also in good accord with the quantum S C F results. It is apparent from examination of Tables I and I1 that there is very little improvement obtained by using the uniform formula for the single-mode corrections as compared to the primitive formula. As expected, the,two sets of semiclassical calculations are almost identical for states far below or above the barrier, with any differences being most pronounced for states very close to the barrier top. In particular, the slight improvement obtained by using the uniform formula is less than the error introduced by using a first-order semiclassical quantization formula. These results thus suggest that the partially uniform (37) Abramowitz, M.: Stegun, 1. A. “Handbook of Mathematical Functions”; National Bureau of Standards: Washington, DC, 1964.
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986
Quantization of Multidimensional Bound States 1. 0
1603
TABLE 111: Eigenvalues for the Coupled Double-Well System of Eq 3.1 for the Same Parameters as Table I"
0 quantum
n semiclassical
ny*. n,,
QSCF (a)
OSCF (b)
o+, 3
5.52492601 5.528 509 80 7.575 84201 7.604983 86 9.389 423 86 9.532 110 76 10.967 958 7 11.3961774 12.5182897 13.250 345 2 14.242953 6 15.1221123
5.524926 32 5.528 508 37 7.575 8 4 4 4 9 7.604 983 86 9.389451 42 9.532 11967 10.967 969 5 1 1.396 449 2 12.5 18 342 2 13.250361 2 14.242 964 7 15.122 114 1
0-, 3 I+, 3 1-, 3 2+, 3 2-, 3 3+, 3 3-, 3 4+, 3 4-, 3 5+, 3 5-, 3 "X
Figure 2. Exact quantum SCF (0) and uniform SCF (A) splittings for the sequence of states corresponding to Table I.
"Results labeled (a) are the full quantum SCF calculation while those labeled (b) are from the hybrid quantum SCF where the singlemode corrections are evaluated by using the primitive semiclassical formula.
0.
quantum S C F results in which the uniform formula is used to calculate the single-mode corrections are identical with the fully quantum results to at least seven figures. Since the evaluation of the single-mode corrections is the most time consuming part of the fully quantum SCF, the hybrid procedure is especially valuable. As a final note, we point out that replacement of the first-order quantization formula by a higher order phase integral q ~ a n t i z a t i o nin~ conjunction ~,~~ with a uniform formula for the expectation values would be expected to provide extremely close agreement with fully quantum S C F calculations, with some savings in computational effort, particularly for systems of high dimensionality.
Wy(Y) -0.
-0.
1
-0.5
4
1I -0.8
i 2
4
6
8
T ,
10
12
14
converged SCF energy Figure 3. Quantum ( O ) , primitive semiclassical (D), and uniform semiclassical ( A ) corrections for the sequence of states corresponding to Table I. The top of the barrier is indicated by an arrow. The curves appear spiky since both n,+ and n; states have been included.
semiclassical S C F procedure can be expected to provide acceptable results for problems containing barriers, even for states close to the tops of the barriers. In Figure 3 we plot the quantum, primitive semiclassical, and uniform semiclassical corrections for the sequence of states, (nx, 0) from the corresponding converged S C F calculation. The uniform formula clearly provides much better agreement with the quantum results than does the primitive formula; however, because of the way the single-mode corrections enter into the S C F equations, evidently this improvement has a fairly minimal effect on the final eigenvalue. Even for more strongly coupled systems with narrower and taller barriers (more tunneling), this picture persists since it is the error introduced in the quantization itself which dominates. Although the original aim of this study was to improve the semiclassical S C F by using a uniform formula for the expectation values, the partially uniform semiclassical S C F has emerged as a flexible and robust procedure in that it is fairly insensitive to the accuracy of the single-mode corrections. Note that the primitive semiklassical S C F of Ratner, Gerber, and c o - w o r k e r ~would ~ ~ , ~be ~ unable to provide any information about the splittings. These results immediately suggest a hybrid quantum S C F approach in which the eigenvalues are computed exactly, while the corrections to the S C F equations are obtained by a primitive or uniform semiclassical calculation. For the set of parameters corresponding to Table I, we present the results of the hybrid quantum S C F procedure in Table 111, obtaining excellent agreement with the corresponding exact quantum S C F calculations. Evidently the error introduced by evaluating the single-mode corrections semiclassically is very much less than the intrinsic error of the S C F approximation itself. Hybrid
5. Conclusions We have developed a fully uniform semiclassicalSCF procedure which is applicable to nonseparable problems containing arbitrary configurations of wells and barriers. Our calculations for a coupled double-well problem have demonstrated the accuracy of the semiclassical S C F procedure and, in particular, have established the uniform semiclassical S C F method as a viable alternative to the more time-consuming quantum procedure. The semiclassical method has the further advantage that the states are uniquely assigned quantum numbers which is especially useful if the semiclassical results are to be used as initial guesses in the quantum S C F procedure. Since the S C F method is not very sensitive to the precise accuracy of the single-mode corrections, a useful alternative to the fully quantum S C F is a hybrid procedure in which the single-mode eigenvalue problems are solved exactly, with the corrections being provided by a primitive semiclassical method. The uniform semiclassical S C F procedure is also extendible to the calculation of complex eigenvalues corresponding to quasi-bound states.38 Note Added in Proof: After this article was accepted, an excellent review of the semiclassical self-consistent field approach by Ratner and Gerber has appeared.39
Acknowledgment. We thank Dr. J. M. Bowman for providing his unpublished quantum results from ref 15, and Mandy Ceccarelli for her help in preparing the figures. Partial support of this research by the donors of the Petroleum Research Fund, administered by the American Chemical Society, the UCLA Committee on Research, and the UCLA Office of Academic Computing is gratefully acknowledged. A.D.S. thanks Dr. W-K. Liu for his hospitality and for partial support through the NSERC of Canada. (38) Farrelly, D. J . Chem. Phys., in press. (39) Ratner, M . A,; Gerber, R. B. J . Phys. Chem. 1986, 90, 20.
