1212
J. Phys. Chem. 1982, 86, 1212-1213
model spaces are no particular problem for typical CI methods,61and have also recently been incorporated into diagrammatic perturbation theoryF2 Multiconfigurational equations-of-motion theories exist as well.42 The BCSbased method will work well only when the degenerate states are approximately described by a BCS Ansatz. Thus, at best, only special open shells could be considered. The pseudo-HF method can be precisely defined for any BCS wave function, and offers BCS-based orbital equations which are effectively HF equations. In short, the BCS fractional orbital occupancies appear explicitly in the electrostatic energy of a single HF-type Fock operator whose one-body energy is renormalized. Given the success and relative simplicity of quantum chemical HF orbital algorithms, the pseudo-HF method holds promise. This is of practical and aesthetical interest: if the pseudo-HF method proves equivalent to the general BCS orbital equations, then a BCS method is available whose orbital and field theories are isomorphic to HF theories. (56) Oberlechner, G.; Owano-N-Guema; Richert, J. Nuouo Cim. B 1970, 68,23. (57) Johnson, M. B.; Baranger, M. Ann. Phys. (N.Y) 1971, 62, 172. (58) Kuo, T. T. S.; Lee, S. Y.; Ratcliff, K. F. Nucl. Phys. A 1971,176, 65. (59) Lindgren, I. J.Phys. B 1974, 7, 2441. (60) Kvasnicka, V. Ado. Chem. Phys. 1977, 36, 345, and references
therein. . ~.~ . (61) Shavitt,I. “Modern Theoretical Chemistry”;Plenum: New York, 1977; Vol. 111, and references therein. (62) Hose, G.; Kaldor, V. J. Phys. B 1979,12, 3827; Phys. Scr. 1980, ~
~~~
21, 357.
The ab initio HF and GVB methods have been successfully applied to large systems which represent real surfaces. The HF calculations have been largely concerned with photoioni~ation,~~ whereas the GVB studies have pursued catalytic reaction mechanisms.64 The applicability of the rigorous BCS method to such systems should be at least comparable to that of the GVB method. The infinite crystal%and pseudopotential% techniques developed there will also work with BCS wave functions. If the pseudo-HF method proves useful, then the BCS method could be applied to large systems almost as easily as a corresponding HF method, and would, in principle, provide the simplest possible ab initio method for the investigation of reactive processes. Any of the model potential^^^ or p a r t i t i ~ n i n g developed s ~ ~ ~ ~ for single Fock operators could be used. If desired, semiempirical parameterizations could be introduced straightforwardly.
Acknowledgment. Acknowledgment is made to Dr. David Silver for helpful comments. This work was supported by the Graduate School, University of Wisconsin-Milwaukee. (63) Bagus, P. S.; Hermann, K.; Seel, M. J. Vac. Sci. Technol. 1980, 18, 435. (64) Upton, T. H.; Goddard, W. A. ‘Chemistry and Physics of Solid Surfaces”; CRC: Cleveland, 1981; and references therein. (65) Upton, T. H.; Goddard, W. A. Phys. Reo. E 1980,22, 1534.
(66)Melius, C. F.; Olafson, B. D.; Goddard, W. A. Chem. Phys. Lett.
1974,28,457. (67) Suritalski, J. D.; Schwartz, M. E. J. Chem. Phys. 1975,62,1521; 1976,64, 4245.
Why Does the Diatomtcs-in-Molecules Method Appear To Fail for H,? Philip J. Kuntz* and Christian C. Chang Hahn-h<ner-Institut fiir Kernforschung, Berlin QmbM Bereich Strahlenchemie, P 1000 Berlin 39, Federal Republic of Germany (Received: October 27, 1981)
Using extended-basis valence bond calculations, we computed the projections of the X ‘Zs+ and b 3Zu+ states of H2 onto several diatomics-in-molecules(DIM) asymptotic basis sets as a function of internuclear distance. From these it is concluded that 2Pstates of H must be included in the DIM basis in order to represent the triplet state adequately at short distances. This suggests that previous DIM calculationson H4may be inadequate and that new work with a larger basis is necessary.
The method of diatomics-in-molecules (DIM)1-3 is fast becoming a very useful tool for representing potential energy surfaces in chemical dynamics ~ t u d i e s . ~It, ~has enjoyed considerable success with many systems, especially H3+;596however, some molecules present d i f f i ~ u l t i e s . In ~~ this paper, we argue that a possible explanation as to why the DIM energy of the coplanar H4 molecule in its ground electronic state is much too low compared to ab initio is that the previous DIM calculations may not (1) F. 0. Ellison, J.Am. Chern. SOC., 85, 3540 (1963). (2) E. Shiner, P. R. Certain, and P. J. Kuntz, J. Chem. Phys., 59,47 (1973). (3) M. B. Faist and J. T. Muckerman, J. Chem. Phys., 71, 225 (1979). (4) J. C. Tully, Ado. Chem. Phys., 42, 63 (1980). (5) J. C. Tully, Ber. Bunsenges. Phys. Chem., 77, 557 (1973). (6) C. W. Bauschlicher,Jr., S.V. O’Neil, R. K. Preston, H. F. Schaefer, 111, and C. F. Bender, J. Chem. Phys., 59 1286 (1973). (7) M. B. Faiat and J. T. Muckerman, J. Chem. Phys., 71,233 (1979). (8) C. W. Eaker and C. A. Parr, J. Chem. Phys., 65, 5155 (1976). (9) C. W. Eaker and L. R. Allard, J. Chern. Phys., 74, 1821-3 (1981). 0022-3654/82/2086-12 12$0 1.25/0
have used the correct basis. Our analysis serves as an example of how one can go about obtaining self-consistent DIM models for other systems as well. The structure of a DIM model depends on the number and type of polyatomic basis functions contained in it; hence, the numerical results of even an empirical DIM model may be sensitive to the choice of basis. Until recently, there has been no prescription for selecting an appropriate basis set. This is partly because there is as yet no variation principle for the method, so that the errors associated with a particular model are difficult to estimate. Furthermore, the DIM method is often applied semiempirically in the hope that the use of exact diatomic potential curves will compensate for other shortcomings of the method. In other words, the basis set is too often viewed as a convenient framework on which to hang the DIM model but not as something important in itself; afterall, the DIM wave functions are hardly ever used. This viewpoint is, however, fundamentally wrong and 0 1982 American Chemical Society
Dlatomlcs-in-Molecules Method for H4
constitutes an abuse of the method. It is essential that the DIM basis adequately represent the bonding in polyatomic and diatomic molecules; this is especially true of those diatomic states whose potential curves are to be adjusted empirically. For example, semiempirical DIM models for the H4 molecule must have a basis which at the very least represents the bonding in the X '2,. and b 3Xu+ states of H,. Recently,'" a basis function selection criterion for DIM models has been proposed which makes it possible to employ ab initio eigenvectors of diatomic molecules to help define an appropriate basis set. T o apply this criterion, one must first list those diatomic state manifolds which are needed for the DIM model. It is then required that a t least the ground states of each manifold be well-described by the DIM basis. To test for this, good-quality ab initio eigenvectors for these states are projected onto the DIM diatomic basis. If the projections are 20.90,the DIM model can be accepted; if not, the basis must be enlarged and the procedure applied again until a suitable model is found. This procedure was recently applied with success to the BeFH moleculelo and was able to select an optimum DIM basis for that system. The projections required for the analysis can be calculated from the ab initio wave functions alone if an asymptotic bask is used to define the DIM model: Let (IEi)] be a set of ab initio wave functions labeled by the eigenvalues Ej. The DIM diatomic basis, {I&)),can be chosen to be a subset of (IE,))in the separated atom limit:
-::
The Journal of Physlcai Chemistry, Vol. 86, No. 7, 1982
1213
-;-0.7 -0.8 (L
c
z
OI
-0.9 -
I
W
-1.0 -1.1
-
2
0
i, 6 8 10 R ( H - H I [BOHR]
12
Figure 1. A comparison of the present H, calculations (dashed lines) wlth the exact results of Kolos and Wolniewicz (solid lines).
0.4
0.3
1
1
1
(I4i)l C {Pim)] where IEi") = limR-.mlEi), R being the internuclear distance. We refer to such a basis as an asymptotic basis. The projection of one of the states Ei onto the DIM asymptotic basis is then (EjlPIEi),where the projection operator P is defined as
P = ld)s-1(4 14) being a row vector of basis functions and s being the overlap matrix: s = (@IC$). We now apply the above analysis to the singlet and triplet Q manifolds of Hz to ascertain what sort of DIM basis may be necessary to properly describe the H4 molecule. To carry out the projection analysis, we computed Hzeigenvectorsusing the valence bond (VB)method" with 31 basis functions for the X lX,+ state and 26 basis functions for the b 32u+ state. On each H atom we used a set of Is, 29, and 2p contracted Gaussian orbitalslZplus a 2s' single Gaussian function whose exponent was optimized to yield the energy for the 2 2Sstate of H well. The computed atomic energies were -0.499815,-0.124628, and -0.124996 au for the 1 %, 2 %, and 2P states of H, respectively. Gaussian s and p orbitals midway between the two H atoms were optimized at 1.4 bohr radii to yield an energy of -1.15641 au (cf. the Kolos and Wolniewi~z'~ energy of -1.17447 au, 11 kcal/mol lower). The quality of these calculations is similar to that of Hirschfelder and Linnettl' and a comparison of the potential curves in each manifold with the corresponding Kolos and Wolniewicz c u r ~ e s is ~ shown ~ J ~ in Figure 1. (10)J. L. Schreiber and P. J. Kuntz, J. Chem. Phys., accepted for publication. (11)0. G.Balint-Kurti and R. N. Yardley, Quantum Chemistry Program Exchange, Program 335 (MULTIBOND A). (12) 5.Huzinaga, J. Chem. Phys., 42, 1293 (1966). (13) W. Kolos and L. Wolniewicz, J. Chem. Phys., 43, 2429 (1965). (14)J. 0.Hirschfelder and J. W. Linnett, J. Chem. Phys., 18, 130 (1950).
1. 2. 3. 4.
:0.2 t , \ "
0
1
2
3
Is Is Is
+
Is
+
2p 2s H-
+
H+ +
4
5
6
7
R ( H - H ) Bohr
Flgure 2. The complement of the projections of the H, 'ZP+ and 3Zu+ states onto several DIM asymptotic bases. For the first four basis functions, the orbitals on the separated atoms are listed in the bottom frame.
The projection analysis for various bases is shown in Figure 2. From the top frame it is clear that the DIM minimum basis (function 1)is quite adequate to describe the X lXg+ state. At large R, this basis is also adequate for the 3Xu+ state; however, at smaller R , the DIM basis must be augmented by the hydrogen 2Pstate. Moreover, neither the 2 2Sstate nor the ionic states (H+H-) are effective in the DIM basis. We therefore conclude that previous DIM treatments of H4, which fail to yield a high enough energy for the coplanar configuration, have a basis set deficiency. A further investigation of the H4 molecule with a basis including (at least) I2P) functions is necessary before the DIM method can be said to fail for this system. Acknowledgment. It is a pleasure to dedicate this paper to J. 0. Hirschfelder on the occasion of his 70th birthday. We also thank Mrs. G. Snoei, Mrs. K. Gfrorer, and Mrs. H. Gadewoltz for their assistance in preparing the manuscript. (15)W.Kolos and L. Wolniewicz, J. Chem. Phys., 50,3228 (1969).
