J . Phys. Chem. 1985, 89, 4460-4464
4460
An ab Initio Random Phase Approximation Study of the Excited-State Intramolecular Proton Transfer In 3-Hydroxychromone Thomas D. Bouman,* Michael A. Knobeloch,+and Steve Bohant Department of Chemistry, Southern Illinois University, Edwardsville, Illinois 62026 (Received: March 25, 1985; I n Final Form: May 13, 1985) Ab initio SCF calculations are reported for geometry-optimized structures of the normal and tautomeric forms of 3hydroxychromone. The K,A* excited state believed to be responsible for the intramolecular proton transfer is modeled in the random phase approximation, and potential energy curves for the process are constructed. The charge rearrangements occurring in the four steps of the Sengupta-Kasha mechanism and the nature of the bonding in the states involved are discussed and illustrated with contour plots.
Introduction Excited-state intramolecular proton transfer (ESIPT) has recently attracted a great deal of interest from experimentalists.’ Although ESIPT is observed in a number of systems, much of the initial and continuing effort has been directed at the molecules 3-hydroxyflavone (3HF, and its skeletal precursor 3-
:m 8
1
6---H I1
II(N)
E(T)
hydroxychromone (3HC, II).12-14 In 3HF, the primary UV absorption at ca. 350 nm results in an intense yellow-green fluorescence at ca. 520 nm in the absence of hydrogen-bonding perturbations; if the latter are present, only the “normal” mirror-image (violet) fluorescence band appears just to the longwavelength side of the UV a b s o r p t i ~ n . ~ - ’The ~ yellow-green emission has been ascribed by Sengupta and Kasha (SK)z to deexcitation of a tautomeric species (T*) resulting from an FSIPT from the normal form (N*). The steps of the mechanism are the primary absorption ( N N*), excited-state tautomerization (N* T*), emission (T* T), and finally, radiationless relaxation N).2 to the ground state of the normal form (T This basic four-level mechanism has been amply supported by experiment for both 3HC and 3HF2-I4and has acquired additional importance of late because it afforded a prediction, later confirmed, of amplified spontaneous emission in 3HF.’0.’1 The transitions involved in the first and third steps of the mechanism are assigned as a a*,based on observed absorption intensities and radiative lifetimes. The N* T* step in 3 H F is fast enough (apart from solvation effects) to imply no barrier separating the excited-state tautomer^,^-^ while Itoh and Fujiwara14 find that the reverse T N process is slowed by a nonnegligible barrier that is higher in 3HF than in 3HC. The tautomeric forms have been postulated to have a resonance-stabilized pyrylium structure (111) in the
--
-
-
where a and p refer to orbitals occupied in the ground state, m and n refer to virtual orbitals, and Bam,Bn are Hamiltonian m and p matrix elements involving primitive excitations a n, and the X s and the Ys are the amplitudes of these primitive excitations in the qth excited state.”J8 The effects of electron correlation are brought in through the B matrix elements.
-
12
-
excited state.* The purpose of this article is to study the electronic structure of 3HC in the four states involved in the mechanism with particular reference to the magnitude of the observed red shift and to estimate the potential energy curves for proton transfer in the ground and excited states. We base our calculations on ab initio S C F molecular orbitals, with first-order correlation effects on low-lying singlet excitations computed in the random phase approximation (RPA).15-’7 In the RPA, excitation energies ow are obtained from the coupled sets of linear equations
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In American Chemical Society/Petroleum Research Fund Scholar. *Present address: Department of Chemistry, University of California, Berkeley, CA.
0022-3654/85/2089-4460$01.50/0
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Geometry Optimization Initial geometries for 3HC(N) and 3HC(T) were obtained from Dreiding models and were optimized by using Allinger’s 1973 force
(1) For references, see H. Shizuka, M. Machii, Y. Higaki, M. Tanaka, and I. Tanaka, J . Phys. Chem., 89, 320-326 (1985). (2) P. K. Sengupta and M. Kasha, Chem. Phys. Lett., 68, 382-385 (1979). (3) G. J. Woolfe and P. J. Thistlethwaite, J . Am. Chem. SOC., 103, 6916-6923 (1981). (4) A. J. G. Strandjord, S. H . Courtney, D. M. Friedrich, and P. F. Barbara, J . Phys. Chem., 87, 1125-1133 (1983). ( 5 ) A. J. G. Strandjord and P. F. Barbara, Chem. Phys. Lett. 98, 21-26 (1983). (6) D. McMorrow and M. Kasha, J . Am. Chem. SOC.,105, 5133-5134 (1983). (7) D. McMorrow and M. Kasha, J . Phys. Chem. 88,2235-2243 (1984).
(8) D. McMorrow, T. P. Dzugan, and T. J. Aartsma, Chem. Phys. Lett., 103, 492-496 (1984). (9) D. McMorrow and M. Kasha, Proc. Natl. Acad. Sci. U.S.A.,81, 3375-3378 (1984). (10) A. U. Khan and M. Kasha, Proc. Natl. Acud. Soc. U.S.A.,80, 1767-1770 (1983). (1 1) P. Chou, D. McMorrow, T. J. Aartsma, and M. Kasha, J . Phys. Chem., 88, 4596-4599 (1984). (12) M. Itoh, K. Tokumura, Y. Tanimoto, Y. Okada, H. Takeuchi, K. Obi, and I . Tanaka, J . Am. Chem. SOC.,104, 4146-4150 (1982). (13) M. Itoh, Y.Tanimoto, and K. Tokumura, J . Am. Chem. SOC.,105, 3339-3340 (1983). (14) M. Itoh and Y. Fujiwara, J . Phys. Chem., 87, 4558-4560 (1983). (1 5 ) C. W. McCurdy, Jr., R. N. Rescigno, D. L. Yeager, and V. McKoy in ‘Methods of Electronic Structure Theory”,H. F. Schaefer 111, Ed. Plenum, New York, 1977. (16) J. Oddershede, Adu. Quantum Chem., 11, 275-352 (1978). (17) T. D. Bouman, Aa. E. Hansen, B. Voigt, and S . Rettrup, Inr. J . Quantum Chem., 23, 595-61 1 (1983), and references therein. (18) Aa. E. Hansen and T. D. Bouman, Mol. Phys., 37, 1713-1724 (1979).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 21, 1985 4461
Proton Transfer in 3-Hydroxychromone
TABLE I: Major Contributions‘ to the Lowest T 3HC (N and T Forms)
x+ Y
a-m A7
*5
*6 *6
*s
T7
0 (u (u,
---
-
-
K*
x- Y
N
Normal Form 0.860 0.866 -0.296 -0.241 0.413 0.25 1 -0.138 -0.085 0.144 0.156
=s TS
T9 TIO
*IO
Excitation in
0.745 0.07 1 0.104 0.012
0.023
0.954 Tautomeric Form 0.927 1.036
*II
0.961
H
OAt least 1% of the excitation normalization Figure 1. Optimized geometry of the normal (N) form of IV. Lengths are in angstroms and angles are in degrees. The chromone is obtained by building a standard aromatic ring onto the left-hand carbons.
800.2 0.998 Figure 2. Optimized geometry of the tautomerized portion (T) of 3hydroxy-4-pyrone, as in Figure 1.
field (MMPI).I9 The effects of the hydrogen-bonding interactions between the carbonyl and hydroxyl groups on the geometry were further treated in a partial gradient optimization, at the SCF level, of 3-hydroxy-4-pyrone (IV) and its tautomeric form corresponding
T
-
Figure 3. Contour plots of the effective occupied (IGO)and virtual ( N O )molecular orbitals involved in the lowest T T* excitation of 3HC in N and T forms, at 1.0 au above the molecular plane.
Calculations of Steps in Mechanism The final geometries of IV, above, were combined with the MMPI-optimized aromatic ring (A-ring) geometry to give the atomic coordinates for II(N) and II(T) used in the calculations. For the ground states of these forms of 3HC, we were limited by the size of the system to a minimal basis (STO-5G) S C F calculation. The 66-orbital basis set yielded spin-restricted HartreeFcck ground-state energies of -566.755 au for II(N) and -566.652 au for II(T). Low-lying singlet excitations out of these ground states were modeled in the normal RPA, using program R P A C . ~ ~
The chromone basis set supports a total of 720 valence particlehole excitations, all of which were included in the calculation. The lowest computed a a* excitation is at 6.4 eV for II(N) and at 3.1 eV for II(T). Table I shows the composition of this excitation in terms of contributions Xom,g2- Yam,:of primitive excitations a m to the normalization of the excitation eigenvector. In both N and T forms, the principal contribution to the excitation is a7 but other primitive excitations mix in nontrivially. Table I also shows that a better single-promotion representation of the excitations may be obtained by constructing effective improved ground orbitals (IGO) or improved virtual orbitals Thus excitation from the IGO obtained from a, and a5into accounts for 8 1.6% of the normalization of this excitation. In the T form, the excitation is 98.1% out of a, into an IVO consisting mostly of agwith a small admixture of aI0.These effective “H0MO”s a* excitation are plotted in and “LUM0”s for the lowest a Figure 3 for II(N) and II(T), in a plane 1 au above the molecular plane, using the PLOT 76 graphics package.22 The lowering of the a a* excitation energy from N to T forms is already apparent at the orbital level. The orbital energies T), 0.1881 (in au) of a7and a8are -0,2310 (a7,N), -0.1502 (a7, (ag, N), and 0.1329 (as,T), respectively. Qualitatively, it appears from Figure 3 that the destabilization of the occupied a orbital in the proton transfer process is due to a weakening of the C2-C3 and Cg-C9-C,, bonds, and to the development of antibonding character in this orbital on the C=O group. The a* orbital, on the other hand, is qualitatively similar in the two forms, so that
(19) N. L. Allinger, Adu. Phys. Org. Chem., 13, 1-82 (1976). (20) P. N. van Kampen, F. A. A. M. de Leeuw, G. F. Smits, and C. Altona, QCPE, No. 437 (1982); J. A. Pople et al., QCPE No. 406 (1981).
(21) Aa. E. Hansen and T. D. Bouman, J . Am. Chem. Soc., 107, 4828-4839 (1985). (22) N. H. F. Beebe, private communication.
-
Ip
to 3HC(T). In these calculations, using the GAUSSIAN so program system,20 the carbonyl oxygen (Oil) and the hydroxyl group (O12-H) were permitted to move, while the remaining atoms were constrained to the MMPI geometry of 3HC. A split-valence (5-31G) basis set, augmented by a set of single Gaussian polarization functions (“d(C) = 0.97, q ( 0 ) = 0.97, ap(H) = 0.12) was used on the movable atoms and carbons C3 and C4, and a minimal (STO-5G) basis was placed on the remainder of the molecule. The optimized geometries of IV, shown in Figures 1 and 2, yielded S C F energies of -413.791 20 and -413.744 14 au, respectively.
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4462 The Journal of Physical Chemistry, Vol. 89, No. 21, 1985
Bouman et al.
P result inconsistent with the observed photophysical processes in 3HC. Believing that this result was an artifact of the basis set, we carried out extended-basis calculations on the N form of two smaller systems, namely 3-hydroxy-4-pyrone (IV) and 1,2-dihydroxyacrolein (VI). In IV, we used both the STO-5G basis
Figure 4. *-Electron ground-state and excited-state charge density contour plots for 3HC structures, at 1.0 au above the molecular plane. Contours are not displayed above 0.02
the source of the energy lowering of this orbital is not evident. Since the excitations in steps I and I11 of the SK mechanism involve primarily the a orbital set, some information about the process may be gleaned from an inspection of the a-electron part of the electron density in the four states involved. Figure 4 shows contour plots of these densities in the plane defined above. The ground-state density is obtained in the usual way; in the RPA, however, excited states are not computed explicitly. The excited-state density is obtained indirectly from that of the ground state through the RPA prescription for expectation value differences of one-electron operator^:^'^^^
A
In this context, is the density _operator;later we shall also present results obtained from using A as the electric dipole moment operator. In the plots in Figure 4 the aromatic character of the A ring in N is readily apparent from the equal x bonding shown around the ring; the isolation of the C=O group and the near isolation of the C=C double bond from the adjacent oxygen lone pairs are also evident. Moving through the steps in the mechanism, in N* the aromaticity of the A ring is destroyed, and charge density increases on C4 and C7 while decreasing on C3 and C,. In T*, the A ring is again stabilized by regaining its aromatic character, bonding increases between C3 and O,,while charge is also transferred from O,,to OI2. In T, the charge flow is again reversed, the A ring is somewhat destabilized, and there is substantial conjugation along 01-C2-C3-O12. From a Mulliken population analysis of the ground-state wave functions for II(N) and II(T), together with a similar analysis of the charge rearrangement density described above, we obtained the net atomic charges for N, N*, T*, and T shown in Figure 5. The computed ground-state dipole moments and those derived from eq 3 for the excited states are also depicted in Figure 5. The net charges generally provide quantitative confirmation of the trends noted above; of particular interest is the complete loss of excess charge on O,,in T*. These charges, together with the direction of the dipole moment in T*, suggest a zwitterionic structure for this species of the form V while in the tautomer ground state, the structure II(T) appears to dominate the description. In the present minimal-basis calculations for II(N), an n a * excitation is computed to lie below the a x* excitation, a
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-
(23) D. Lynch, M. F. Herman, and D. L. Yeager, Chem. Phys., 64,69-81 (1982).
and Dunning's [3s2p/2s] split-valence set,24while in VI we used both these bases plus two additional sets including, respectively, diffuse and polarization functions. The results of these calculations are as follows: the expansion of the basis set, as evinced in calculations on dihydroxyacrolein, narrows the gap between the a * energies from 2.79 to 1.08 eV but leaves n n a * and a a* lowest. The effect of increasing the size of the conjugated system, going from VI to IV to 11, decreases the gap from 2.79 to 2.18 eV. If we assume that the two effects are additive, the a * below n A* in 3HC. Having extrapolation places A established that an expanded basis set suffices to restore the correct ordering of states, we will not treat the n a * excitation further.
--
-
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-
Potential Energy Curves In order to examine further the proton-transfer process in 3HC, we constructed potential energy curves for both ground and excited states, connecting N and T forms. As starting points we used the optimized coordinates for II(N and T) discussed above. The reaction coordinate for the proton transfer was taken to be the concerted, linear motion of the proton and oxygen atoms 0,,and OI2between their respective equilibrium positions in N and T forms. Calculations were also carried out at points approximately 25%, 50%, 65%, 75%, and 82% of the distance along the path, with S C F energies of -566.739, -566.698, -566.672, -566.660, and -566.658 au, respectively. Since the RPA description of the excitation spectrum includes a balance of electron correlation effects between ground and excited states,'* we chose to present the ground-state curve as one that has been corrected by second-order Mdler-Plesset (MP2) perturbation theory.25 Starting from the N form, the MP2 corrections to each point on the curve in au are -0.664, -0.673, -0.689, -0.698, -0.698, -0.692, and -0.68 1 (T form), respectively. The resulting curve is shown as the bottom curve in Figure 6. We note that the MP2 approach has no first-order consequences for the charge distribution, so that Figures 3 and 4 are still good approximations to the charge densities. The fact that the T form is no longer at a stationary point shows that the MP2 equilibrium geometry is not quite identical with the geometry at the S C F level. We are left with a computed T form 54 kcal/mol above N, with no barrier opposing N step. the T The potential energy curve for the excited state was constructed by adding the RPA excitation energy to the ground-state energy at each point;26the result is the upper curve in Figure 6. In the excited state, we calculate N* 25 kcal/mol higher than T*, with a barrier of 6 kcal/mol to the excited-state proton transfer. The a * excitation decreases along the intensity of the allowed a
-
-+
(24) T. H. Dunning and P. J. Hay in H. F. Schaefer, Ed., "Methods in Electronic Structure Theory", Plenum Press, New York, 1977. (25) J. S. Binkley and J. A. Pople, In?. J. Quunfum Chem., 9, 229-236 (1975). (26) W. Coughran, J. Rose, T.-I. Shibuya, and V. McKoy, J. Chem. Phys., 58, 2699-2709 (1973).
The Journal of Physical Chemistry, Vol. 89, No. 21, 1985 4463
Proton Transfer in 3-Hydroxychromone
I
I
-.l 1
-.22
.o 0 -. 3 9
N*
T*
N
T
+.32
Figure 5. Net ground-state and excited-state atomic charges in the four forms of 3HC. Charges on ring hydrogens (not shown) are within 0.01 electron of +0.08 in all cases. Dipole moment directions and relative magnitudes are indicated as well. E(a.u)
-567.1
- 567.2
Sl(7VT*)
the normal ground-state form. Second, the excited-state potential curves show that in the T,T*state the tautomeric form lies well below the normal form, and that the barrier separating them is small. The barrier to proton transfer in the T,T*state appears to be characterized by a small decrease in electron density on C3 coupled with an increase on the labile hydrogen. The results are qualitatively reasonable, but the numerical agreement with experiment is only fair. In particular, the N* T* barrier is rather larger than the experiments indicate, while ~ this may the computed T N barrier is too ~ m a l l . ~ - ~Inq 'part, be due to our choice of reaction coordinate; the motion of the proton and the two oxygens may be a stepwise rather than a concerted process, for example. An equally serious source of error is the minimal atomic orbital basis set to which we have been limited. Further refinement of this point must await enhanced computational resources. In any event, errors of 6 kcal/mol in the potential energy curves are not unexpected within the present computational scheme. The actual energy separation between the N and T forms is not known, but an upper limit may be estimated from the observed absorption and fluorescence. If we assume no energy difference between N * and T* in 3HC, the observed N N* energy of ca. 3.9 eV and T* T fluorescence at ca. 2.5 eV imply an N-T energy separation of no more than ca. 1.4 eV, or about 32 kcal/mol. Our minimal basis set calculations on 3HC give an energy difference of about 54 kcal/mol; however, the calculated difference of 1.28 eV for the tautomeric forms of IV in the extended basis set described above is in reasonably good agreement with the estimate given above. It is clear that adjustment of the ground-state potential energy curve by this amount, scaled across the reaction coordinate, would lead to an improvment. Our qualitative discussion would not be affected, however, and we will not pursue it further. Finally, we see that the qualitative features of the SK mechanism are reproduced by the straightforward calculations, and that adjustment of these numbers by reasonable enhancements leads to improved quantitative agreement a s well. Analysis of the charge rearrangements in the various mechanistic steps lends insight into the nature of the proton transfer process.
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-
- 567.3
-567.4
-567.5
r; RE ACT1 ON COORDINATE
Figure 6. Calculated ground-state (So) and excited-singlet-state (S,) potential energy curves for motion along the presumed reaction coordinate in 3-hydroxychromone.
reaction coordinate, from f = 0.10 (mixed form) at N to f = 0.05 at T. In the first case, the transition moment is directed nearly along the long axis of the fused ring system, whereas the moment in the T form is nearly parallel to the C3-C4 bond.
Discussion This straightforward set of RPA calculations exhibits a number of the qualitative features of the SK mechanism. As stated above, it shows the existence of an energetically accessible protontransferred tautomer with little or no barrier separating it from
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4464
J . Phys. Chem. 1985, 89, 4464-4412
Acknowledgment. This work was supported in part by grants from the Scientific Affairs Division of NATO (RG. 138.81), the donors of the Petroleum Research Fund, administered by the American Chemical Society, and the National Science Foundation (CHE-8218216). We are grateful to Professor Michael Kasha
for bringing this problem to our attention, and to Dr. Aage E, Hansen for stimulating and helpful discussions. Registry No. I, 577-85-5; II(N), 13400-26-5; 111, 97877-63-9; 1V. 496-63-9; VI, 636-38-4.
Natural Orbital Analysis of Vibration-Rotation Wave Functions Bruce K. Holmert and Phillip R. Certain* Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706 (Received: April 8, 1985)
Natural orbital analysis has been applied to vibration-rotation wave functions of a model triatomic van der Waals system. Model potentials for both linear and T-shaped molecular geometries, along with various degrees of anisotropy, were considered. Calculations were not restricted to the ground state but included states with nonzero total angular momentum and excited bending vibrations. The vibration-rotation wave function was determined variationally by using a body-fixed Hamiltonian and then transformed and expressed in terms of natural orbitals. The natural orbital expansion of the wave function converged more quickly than the original expansion and thus facilitated the interpretation of the wave function. For the model potentials used, the natural orbital expansion consisted of one dominant term plus several correction terms. Through the use of the harmonic limit of the body-fixed wave equation, the nodal structure and spatial distributions of the correction terms were attributed to either vibration-rotation coupling or potential anharmonicity.
1. Introduction Advances in the spectroscopy of small and medium-sized have spurred the development of a variety of computational methods2v3which are used to calculate vibration-rotation spectra. One of the computational methods used for triatomic molecules employs the variational principle and assumes that the vibration-rotation wave function can be expressed as a sum of products of single variable vibrational functions and symmetric top rotational functions.* Alternatively, close-coupling methods have been applied to triatomic van der Waals molecules in which the set of close-coupled equations are solved by either explicit integration: perturbation t h e ~ r yor , ~basis set expansion.6 In all of these approaches, emphasis has been placed on the calculation of energies and other expectation values, with only minimal consideration of methods for analyzing the wave function. In this paper, the emphasis will be placed on the natural orbital analysis of variationally obtained wave functions in order to recover as simple a picture as possible of rotation-vibration states which are far from the harmonic oscillator-rigid rotor limit. An abundance of tools exist for the analysis of electronic wave functions: natural natural bond orbitals,I0 localized molecular orbitals,” etc. Similar tools for analyzing vibrationrotation wave functions have been introduced to build a greater understanding of molecular vibrations; however, application of the ideas of natural orbital analysis to nuclear motion has been limited. A model problem of two coupled harmonic oscillators was solved in closed form for its natural orbitals.12 Bishop and Cheung performed accurate nonadiabatic computations for H2+ and its isotopesi3 and then submitted the resulting wave functions, which involved both nuclear and electronic coordinates and consisted of over 300 terms, to natural orbital a n a l y ~ i s . ’ ~Stratt, Handy, and Miller used natural orbital analysis to examine the question of quantum chaos in a coupled oscillator system.15 Sage and Williams have used the analysis to study local modes in a coupled Morse oscillator system.16 None of these applications considered the rotational degrees of freedom. The purpose of the present contribution is to demonstrate the utility of natural orbital analysis when applied to vibration-rotation wave functions. Present address: Miller Fellow, Department of Chemistry, University of California-Berkeley, Berkeley, CA 94720.
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To illustrate the application of natural orbital analysis to vibrational problems we studied a model triatomic van der Waals system and considered both linear and T-shaped molecular geometries with various degrees of anisotropy. We choose this example because it models a weakly bound, strongly anisotropic molecule that exhibits large amplitude motion. Thus, there is no obvious “best” zero-order approximation to describe its states. The vibrational-rotational motion was solved variationally by using a body-fixed Hamiltonian. Rather than using harmonic or Morse oscillator basis functions for the van der Waals stretching coordinate, we used basis functions constructed from products of polynomials and the ground-state wave function of a LennardJones potential. The total wave function was then transformed and expressed in terms of natural orbitals. As expected, the natural orbital expansion of the wave function converged much more quickly than the original expansion, thus allowing for easier interpretation of the wave function. In order to better understand the results of the natural orbital analysis we also made use of the harmonic limit of the wave equation by relating the nodal behavior in this limit to that of the natural basis function expansion.
(1) D. H. Levy, Annu. Reu. Phys. Chem., 31, 197 (1980). (2) G. D. Carney, L. L. Sprandel, and C. W. Kern, Adu. Chem. Phys.. 37, 305 (1978). (3) R. J. LeRoy and J. S . Carley, Ado. Chem. Phys., 42, 353 (1980); see
also the original work: C. F. Curtiss, J. 0. Hirschfelder, and F. T. Adler, J . Chem. Phys., 18, 1638 (1950). (4) A. M. Dunker and R. G. Gordon, J . Chem. Phys., 64, 354 (1976). (5) J. M. Hutson and B. J. Howard, Mol. Phys., 41, 1123 (1980). (6) R. J. LeRoy and J. Van Kranendonk, J . Chem. Phys., 61,4750 (1974). (7) P. 0. Lowdin, Phys. Reo., 97, 1474 (1955). (8) E. R. Davidson, Reo. Mod. Phys., 44, 451 (1972). (9) E. R. Davidson, “Reduced Density Matrices in Quantum Chemistry”, Academic Press, New York, 1976. (IO) A. E. Reed and F. Weinhold, J . Chem. Phys., 78, 4066 (1983). (1 1) C. Edmiston and K. Ruedenberg, Reo. Mod. Phys., 35, 457 (1963). (12) P. D. Robinson, J . Chem. Phys., 66, 3307 (1977). (13) D. M. Bishop and L. M. Cheung, Phys. Rec. A, 16, 640 (1977). (14) D. M. Bishop and L. M. Cheung, In?.J . Quanrum Chem., 15, 517 (1979). (15) R. M. Stratt, N. C. Handy, and W. H. Miller, J . Chem. Phys.. 71, 3311 (1979). (16) M. L. Sage and J. A. Williams 111, J . Chem. Phys.. 78. 1348 (1983).
0 1985 American Chemical Society
