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A generalized multistructural description of the ground state of ozone and water molecules
THEO CHEM ELSEVIER
Journal of Molecular Structure (Theochem) 335 (1995) 51-57
A generalized multistructural description of the ground state of ozone and water molecules Wely Brasil Floriano, Solange Regina Blaszkowski, Marco Antonio Chaer Nascimento” Znstituto de Quimica. Departamento de Fisico-Quimica, Universidade Federal do Rio de Janeiro, Cidade Universitriria. CT, Bloc0 A, sala 412, Rio de Janeiro, RJ 21949-900, Brazil
Received 25 February 1994;accepted 13 July 1994
Abstract Generalized multistructural (GMS) wavefunctions are presented for the ground state of ozone and water molecules at the equilibrium geometry. For the ozone molecule the results confirm the diradical character of the ground state and show that the contribution of the zwitterionic structures is negligible. The reasons for the discrepancy between the MO and VB descriptions of the molecule are also presented. For the water molecule the GMS results also show that once orbital optimization is considered there is no need to include polar structures in a VB-type expansion. Also, a great deal of correlation energy can be recovered using very simple and compact GMS wavefunctions.
1. Introduction There are many molecules which cannot be represented by a single chemical structure. For those, the natural way to describe their electronic wavefunction is as a linear combination of bonding structures. This is the idea behind the classical valence bond (VB) theory [l-8]. However, the classical VB theory suffers from the fact that the atomic orbitals cannot readjust their shapes as the molecule is being formed. Besides, since only the occupied orbitals of each atom are generally considered, excited electronic states are usually not well represented. Orbital optimization simultaneously with the variational determination of the coefficients in the linear combination of structures has been considered by Voter and Goddard [9], Gerrat and
* Corresponding author.
co-workers [lo] and McWeeny [ll]. Voter and Goddard examined the case of a two-structure wavefunction for the particular case where the structures are represented at either the HartreeFock (HF) or generalized valence-bond (perfectpairing) [GVB(PP)] level, but are related through a reflection operation. Gerrat and co-workers [lo] and McWeeny [ 1l] considered the more general case of multiconfigurational wavefunctions but the structures are restricted to be represented by VB-type wavefunctions. While in the Voter and Goddard approach all the orbitals are optimized, in the spin-coupled [lo] and ab initio VB [I I] approaches, the optimization procedure is carried out in general for the valence electrons (about 6-8 electrons), the remaining orbitals being kept frozen. Another approach to introduce orbital shape readjustment, in the standard VB method, was proposed by Gallup et al. [12]. In this approach no orbital optimization is directly
0166-1280/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0166-1280(94)03982-8
52
W.B. Floriano et aLlJournal of Molecular Struciure (Theochem)
attempted but, instead, many configurations are admitted from start, as in a configuration interaction (CI) calculation We have recently proposed a generalized multistructural (GMS) wavefunction [ 13- 171
335 (1995) 51-57
to test how efficient the GMS wavefunctions can be in recovering correlation energy. Another interesting point, now related to the classical VB theory, is the investigation of the importance of the polar structures is the description of neutral molecules, when orbital optimization is taken into account.
(1) where $i represents the wavefunction of the ith “structure” and ci its weight in the total wavefunction. These coefficients are obtained variationally by solving the equations (~*GMSIH
-
E~*GMs)
=
0
(4
Here the word “structure” is employed in a very broad sense, as previously discussed [13-171. In the GMS wavefunction defined in Eq. (1) there are no restrictions whatsoever to the form of the wavefunctions qi. Each one of the structures presented in Eq. (1) can be obtained at the HF, MCSCF (GVB, etc.), CI or any plausible combination of these methods. All the orbitals are optimized at either the HF or MCSCF level and although orbitals belonging to the same structure are forced to remain orthogonal no such restriction exists for orbitals of different “structures”. As also previously discussed [ 13- 171, the GMS wavefunctions are compact, easy to interpret but very efficient at recovering electronic correlation energy. In this paper we present a GMS description of the ground state of ozone and water molecules. Besides their tremendous importance in everyday life, these molecules were chosen because a large amount of data, obtained with other methods, are available for comparison. The classical VB and the molecular orbital (MO) descriptions of ozone differ considerably and the relative importance of the zwitterionic structures has not been well defined yet. This discrepancy can be clearly resolved by constructing a GMS wavefunction consisting of the two polar (VB-type) structures plus a diradical one as suggested by the GVB calculations. The water molecule provides another very interesting application for the GMS wavefunction. Results of very large MO-C1 calculations are available for this molecule, which can be used
2. Computational details For the ozone molecule the calculations were carried out at the experimental equilibrium (r, = 1.278 A, 0, = 116.8 A) [18], using the Dunning [ 191 double-zeta (DZ) contraction (9sSp/3s2p) of Huzinaga’s primitive gaussian basis sets [20] augmented with a d polarization function (Q = 0.85). The GMS wavefunction for the ozone molecule was obtained at the GVB (3/PP) level of calculation [21], correlating the three bond-pairs, for structures (I) and (II), and the two bond-pairs plus the unpaired electrons, for structure (III): +
+
-o/o\o.,/“\o.
/“\o-
0
*GMS~)
(III)
(III
(I) =
Q~‘III
+ C2($1
+&I)
(3)
The classical VB description of the ground state of the ozone molecule corresponds to a resonance hybrid of the zwitterionic structures (I) and (II) [22], while (III) corresponds to a diradical structure first proposed by Goddard et al. [23]. To solve for the charge-localized structures we used a procedure analogous to the one employed by Bagus and Schaefer [24]. A Mulliken population analysis performed on the converged chargelocalized structure indicates a transference of 0.71e from the central to the negatively charged oxygen atom (see Table 1). Since we are dealing with a homonuclear molecule, there are no ambiguities in the process of distributing the overlap population between the nuclei, and the result of the population analysis can be taken as a good indication of the degree of charge localization. Calculations for the water molecule were also
W.B. Floriano et ai./Joumal
of Molecular Structure (Theochemi
Table 1 Mulliken population analysis for GVB (3/PP) wavefunctions of ozone Atom
Diradical (III)
Zwitterion (I)
0,
8.05 7.89 8.05
8.71 7.56 7.91
02 03
carried out at the experimental equilibrium geometry (re = 0.958 A, 8, = 104.5 A) [25] using a 6-31G (d,p) basis set with & = 0.85 and $, = 0.6 [26]. The GMS wavefunction was constructed as the superposition of structures (IV)-(VII),
H/O\”
H
H(VI
(IV)
(VI)
%dAd = cI+ (IV) +
0
(VII) C2$(V)
+ C3(1DvI + lClvr1)
(4)
each one represented at the GVB (2/PP) level of calculation. Because oxygen is much more electronegative than. hydrogen we have not considered polar structures with the oxygen atom positively charged. The GVB (2/PP) wavefunction for structure (IV) correlates the two bond-pairs while for the other structures the wavefunctions correspond to correlating the bond pair plus an electron pair in the molecular plane at the oxygen atom. The same procedure used for ozone was adopted for the water molecule in solving for structures (V)-(VII). In this case, a Mulliken population analysis indicates a transference of 0.65e to the oxygen atom.
3. Results and discussion The ozone molecule has been the subject of numerous theoretical and experimental investigations, most certainly owing to its extremely
335 (19951 51-57
53
important role in atmospheric chemistry (see, for example, Ref. [27]; Refs. [28]-[32] are the most recent studies). In spite of that, from the theoretical point of view, there remain some uncertainties concerning, for instance, the energy separation between the CZv ground state open-chain and the cyclic (D3J forms of ozone [30]. Also, there is considerable disagreement between all the theoretical predictions and the experimental result for the X ‘Ai -+’ B2 transition energy. However, regarding this molecule, our main concern in this paper is to quantify the importance of the zwitterionic structures (I and II) for the description of the ground state, in an attempt to reconcile the MO and VB pictures. The HF description of ozone is known to be completely inadequate, leading incorrectly to a triplet ground state, with the first singlet state approximately 2 eV higher in energy [33]. Goddard et al. [33] using the GVB method, which combines some attributes of both the VB and MO methods, were the first to describe properly the diradical character of the ozone singlet ground state. Subsequently Hay et al. [34] presented results of CI calculations based upon the previous GVB wavefunctions [33]. The diradical character of the ground state was confirmed but, interestingly, the only excited state with some ionic character was found at an energy approximately 5 eV above the ground state. In order to take into account this diradical character in the VB description Harcourt and Rosso [35] introduced structure III in their VB expansion (long bond between non-adjacent atoms). Their all-valence electrons VB calculations showed that the long-bond structure was indeed the dominant one but the contribution of zwitterionic structures was quite significant (see Table 2). The same conclusion was achieved by projecting the HF wavefunction onto a basis of VB functions [36-381, although, as noticed by Hiberty and Ohanessian [37], the contribution of the zwitterionic structures tend to become more important for larger basis sets. However, the contribution of the zwitterionic structures obtained by the projection method [37] was always found to be smaller than that obtained by the standard VB calculation [35]. These results are apparently in agreement.
54
W.B. Floriano et al./Journal of Molecular Structure (Theochem)
335 (1995) 51-57
Table 2 Structural weights for the ground state of ozone Calculation Hiberty and Leforestier [38] (STO-3G) Hiberty and Ohanessian [39] (4-3 1G) Harcourt and Rosso [37] Present work GMS (DZP)
0.184 0.231 0.3084 1.4(-4)
However, the MO-C1 and VB calculations should be expected to converge to the same description of the molecule, as the number of terms in the respective expansions increases. Nevertheless, the GVB + CI description of ozone ground state does not show any significant contribution of configurations which could be associated with zwitterionic structures. These appear to be important only for higher electronic excited states [34]. However, from the VB point of view, except for a hypervalent structure [35], there are not too many other neutral structures (energetically equivalent to III) which could be included in the expansion in order to drastically reduce the importance of the zwitterionic structures. However, there were still some problems, of a technical and chemical nature, which prevented us of including that kind of structure in the GMS expansion [36]. In an attempt to understand those differences, we performed a series of calculations at the HF and GVB levels and for structures (I), (II) and (III). The GVB functions were used to construct the GMS wavefunction in Eq. (3). Table 3 shows the results of those calculations. The first point to be noticed from the GMS results is that once the orbitals are optimized the contribution of the zwitterionic structures becomes negligible. Also from Table 3 it can be seen that the energy difference between the HF X ‘Ai and the zwitterionic structures is small (about 0.7 eV), indicating that, at the HF level of calculation, structures (I) + (II) should mix appreciably with (I), as noticed by Hiberty and Ohanessian [39]. However, since the HF wavefunction gives a bad description of the ground state, its projection onto a VB basis will certainly overestimate the contribution of the zwitterionic structures. A comparison between our GMS and Harcourt and Rosso standard VB calculations [37] clearly indicates that in the latter
0.184 0.231 0.3084 1.4(-4)
0.593 0.476 0.7934
1.000
calculations the zwitterionic structures are just providing for orbital shape readjustment. That also explains why the contribution of the zwitterionic structures to the MO projected VB function [39] was always found to be smaller than to the standard VB wavefunction [37]. Even though the HF wavefunction gives the wrong description of the molecule, the shapes of the resulting valence orbitals are more appropriate to describe bond formation than the atomic (VB) ones and therefore less contribution of the ionic structures is needed. The water molecule has also been the subject of intensive theoretical and experimental investigation. A recent survey of the latest ab initio and post-Hartree-Fock calculations for this molecule has been compiled by Levine [40]. As mentioned before, the water molecule was chosen as a test for the efficiency of the GMS wavefunction to recover correlation energy and also to examine the importance of the ionic structures in a VB-type expansion when orbital optimization is taken into account. In the usual treatment of electronic correlation effects, a HF or MCSCF wavefunction is used as the zero-order function for a perturbation calculation (MBPT), or as a reference wavefunction either for a CI or for a coupled-cluster (CC) calculation. The VB approach, however, starts from Table 3 Total energies for the ozone molecule State
Structure
Level of calculation
Energy (a.u.)
I or II I or II III I + II + III
HF HF GVB(3/2) GVB(3/2) GMS
-224.30578 -224.28005 -224.34138 -224.41476 -224.41567
X ‘A,
X ‘A, X ‘A,
W.B. Floriano et aLlJournal of Molecular Structure (Theochem)
classical chemical structures, where the atom’s individuality is to a greater extent preserved. Orbital shape readjustments and electronic correlation effects are introduced by the superposition of chemical structures representing different coupling and distribution of electrons among the various atoms. This approach is conceptually simpler and the results are easier to interpret, contrary to the ones obtained from large scale CI and CC calculations or higher-order perturbative treatments. However, the VB approach turns out to be computationally more complex than the MO-based methods because a large number of structures involving non-orthogonal orbitals is generally needed. There are many high-quality MO-type calculations [40] for the water molecule but, to the best of our knowledge, no analysis has ever been made in order to establish the contribution, to the ground state, of configurations which could be associated with ionic VB structures. Also no attempts of projecting any of the MO-type wavefunctions available into a VB basis have been reported. Apparently the only all-electron VB calculation for the water molecule is the one performed by Peterson and Pfeiffer [41]. The importance of the ionic structures, in their VB wavefunction containing 10 covalent and 39 ionic structures, was found to be 62%, quite a large contribution. For small molecules it is known that a large fraction of the correlation energy, relative to a HF wavefunction and a given basis set, can be recovered by performing a CI calculation including all single and double excitations (CISD) from the reference configuration. In fact, for the water molecule Saxe et al. [42] have found that a CISD calculation accounts for 94.7% of the total correlation energy, when a DZ quality basis set is used. However, the recovery of the residual correlation energy requires an extremely large number of configurations which makes the final wavefunction quite difficult to interpret. However, from the VB results of Peterson and Pfeiffer [41] it is clear that many more “structures” (or coupling schemes) are needed in order to obtain results comparable to the MO-C1 ones. However, increasing the basis of VB structures requires a great deal of extra computational effort. Alternatively
55
335 (1995) 51-57
Table 4 Total energies for the Hz0 molecule Calculation
GMS” GVB(2/PP) GVB-RCI GVB-Fb Full-C1 DZ basisC DZP basisdl
Number of spatial configurations
Energy (a.u.)
8 18 36
-16.13614 -76.16483 -16.22611
256 473 6 740 280
-76.15786 -16.25662
a Only structures (IV) and (V) are being considered. b Full-C1 in the space of the GVB orbitals. ’ Ref. [42]. d Ref. [43].
the simple picture provided by the chemical structures could be preserved, but optimizing their orbitals in a self-consistent way. In addition the dominant correlation effects can be easily incorporated into the wavefunctions describing each one of the structures. The final wavefunction is obtained variationally and consists of the superposition of the wavefunctions representing the different structures. This is the idea behind the GMS wavefunction. It is also possible to perform a simultaneous optimization of the {$i} and {Ci}. However, if the overlap between the wavefunction is small there is no need to optimize them in the presence of each other. Because each structure can be represented by a correlated wavefunction containing the dominant (and clearly identifiable) effects, it is expected that a very small number of structures will be needed, in the final wavefunction, in order to describe the less important (residual effects). As mentioned before, the recovery of the residual, but not always negligible effects, by the more usual approaches, requires a large amount of computational effort. A series of GMS calculations were performed, representing the structures (IV)-(VII) at the HF, GVB (2/PP) and GVB + CI levels of calculation. The results obtained at the HF level indicated a negligible contribution (c < 10e3) of the ionic structures. Therefore only the covalent structures IV and V were considered at the higher levels of calculation. Table 4 shows the GMS results compared with some other selected calculations. It is important to emphasize that many other (and
56
W.B. Floriano et al./Journal of Molecular Structure (Theochem)
even more accurate) results are available in the literature [40]. However, the ones selected are the most appropriate to our discussion. From the results of Table 4 it is clear that the GMS wavefunctions can be extremely efficient at recovering correlation energy. At the GVB-RCI (the GVBRCI calculations correspond to relaxing the perfect-pairing (PP) condition) level of calculation, the GMS wavefunction furnishes a better description than the DZ full-C1 calculation [42]. In fact, this is not a fair comparison inasmuch as for the GMS wavefunctions a DZP basis set was used. However, if the results from the GMS-GVBF calculations are compared with the ones obtained from a full-C1 using a DZP quality basis set [43], it can be seen that the GMS result is only 29 mh (0.78 eV) above the CI results in spite of the tremendous difference in the number of configurations used in the two calculations.
4. Conclusion The GMS results obtained for the ozone and water molecules clearly indicate that once orbital optimization is considered, there is little or practically no need to consider ionic structures in a VB-type expansion wavefunction. Besides the practical aspect of having to deal with a smaller number of configurations, this also eliminates the need of including some highly unphysical structures. This same conclusion emerges from ab initio VB and spin-coupled VB calculations. For the particular case of the ozone molecule the GMS wavefunction provides a clear explanation for the discrepancy observed in the MO and standard VB descriptions of the molecule. Even more interesting, the GMS wavefunction has proven to be extremely efficient at recovering correlation energy without losing the simple picture provided by chemical structures.
Acknowledgments The authors thank CNPq and FINEP for linancial support. Two of us (WBF and SRB) also acknowledge CAPES for graduate scholarships.
335 (1995) 51-57
References [l] W. Heitler and F. London, Z. Phys., 44 (1927) 455. [2] G.Rumer, Gijttingen Nach., (1932) 337. [3] L. Pauling, Proc. Natl. Acad. Sci. U.S.A., 18 (1932) 293. [4] L. Pauling and G.W. Wheland, J. Chem. Phys., 1 (1933) 362. [5] L. Pauling and J. Sherman, J. Chem. Phys., 1 (1933) 606. [6] H. Eyring, J. Am. Chem. Sot., 54 (1932) 3191. [7] J.C. Slater, Phys. Rev., 37 (1931) 481; 38 (1931) 1109. [8] L. Pauling, J. Am. Chem. Sot., 53 (1931) 1367; 53 (1931) 3225. [9] A.F. Voter and W.A. Goddard III, J. Chem. Phys., 75 (1981) 3638; Chem. Phys., 57 (1981) 253. [lo] J. Gerrat, Adv. At. Mol. Phys., 7 (1971) 141. N.C. Pyper and J. Gerrat, Proc. R. Sot. London, Ser. A, 355 (1977) 407. D.L. Cooper, J. Gerrat and M. Raimondi, Adv. Chem. Phys., 69 (1987) 319. J. Gerrat and M. Raimondi, Proc. R. Sot. London, Ser. A, 371 (1980) 525. [l l] R. McWeeny, Int. J. Quantum Chem., 34 (1988) 25. [12] G.A. Gallup, R.L. Vance, J.R. Collins and J.M. Norbeck, Adv. Quantum Chem., 16 (1982) 229. [13] E. Hollauer and M.A.C. Nascimento, Chem. Phys. Lett., 184 (1991) 470. [14] C.E. Bielschowsky, M.A.C. Nascimento and E. Hollauer, Phys. Rev. A, 45 (1992) 4622. [15] M.A.C. Nascimento, in Collision Processes of Ion, Positron, Electron and Photon Beams with Matter, World Scientific, Singapore, 1992, p. 247. [16] E. Hollauer and M.A.C. Nascimento, Chem. Phys., 174 (1993) 79. [17] E. Hollauer and M.A.C. Nascimento, J. Chem. Phys., 99 (1993) 1207. [18] T. Tanaka and Y. Morino, J. Mol. Spectrosc., 33 (1970) 538. [19] T.H. Dunning and P.J. Hay, in H.F. Schaefer III (Ed.), Modern Theoretical Chemistry, Vol. 3, Plenum, New York, 1977, p. 1. [20] S. Huzinaga, J. Chem. Phys., 42 (1965) 1293. [21] F.W. Brobowickz and W.A. Goddard III, in H.F. Schaefer III (Ed.), Modern Theoretical Chemistry, Vol. 3, Plenum, New York, 1977, Chapter 6. [22] G.W. Wheland, Resonance in Organic Chemistry, Wiley, New York, 1955. [23] W.A. Goddard III, T.H. Dunning, Jr., W.J. Hunt and P.J. Hay, Act. Chem. Res., 6 (1973) 368. [24] P.S. Bagus and H.F. Schaefer, J. Chem. Phys., 56 (1972) 224. [25] R.L. Cook, F.C. DeLucia and P. Helminger, J. Mol. Spectrosc., 53 (1974) 62. [26] W.J. Hehre, R. Ditchfield and J.A. Pople, J. Chem. Phys., 56 (1972) 2257. [27] K. Ohono, K. Morokuma and H. Hosoya (eds.), The Quantum Chemistry Literature Data Bank, JAICI, Tokyo, 1990. [28] S. Xantheas, ST. Elbert and K. Ruedenberg, J. Chem. Phys., 93 (1990) 7519.
W.B. Floriano et al./Journal of Molecular Structure (Theochem) 335 (1995) 51-57 S. Xantheas, G.J. Atchity, S.T. Elbert and K. Ruedenberg, J. Chem. Phys., 94 (1991) 8054. [29] S.M. Anderson, J. Morton and K. Mauesberger, J. Chem. Phys., 93 (1990) 3826. [30] T.J. Lee and G.E. Scuseria, J. Chem. Phys., 93 (1990) 489. T.J. Lee, Chem. Phys. Lett., 169 (1990) 529. [31] K.A. Peterson, R.C. Mayrhofer, E.L. Sibert III and R.C. Woods, J. Chem. Phys., 94 (1991) 414. [32] W. Kock, G. Frenkling, G. Steffen, D. Reinen, M. Jansen and W. Assenmacher, J. Chem. Phys., 99 (1993) 1271. [33] W.A. Goddard III, T.H. Dunning, Jr., W.J. Hunt and P.J. Hay, Act. Chem. Res., 6 (1973) 368. [34] P.J. Hay, T.H. Dunning, Jr., and W.A. Goddard III, J. Chem. Phys., 62 (1975) 3912. [35] D.L. Cooper, J. Gerratt and M. Raimondi, J. Chem. Sot., Perkin Trans. 2, (1989) 1187.
51
[36] S.R. Blaszkowski, M.A.C. Nascimento, C. Amovili and R. McWeeny, in preparation. [37] R.D. Harcourt and W. Rosso, Can. J. Chem., 56 (1978) 1093. [38] P.C. Hiberty and C. Leforestier, J. Am. Chem. Sot., 100 (1978) 2012. [39] P.C. Hiberty and G. Ohanessian, J. Am. Chem. Sot., 104 (1982) 66. [40] I.N. Levine Quantum Chemistry, Prentice-Hall, Englewood Cliffs, NJ, 1991, pp. 474 and 510. [41] C. Peterson and G.V. Pfeiffer, Theor. Chim. Acta, 26 (1972) 321. [42] P. Saxe, H.F. Schaefer III and NC. Handy, Chem. Phys. Lett., 79 (1981) 202. [43] C.W. Bauschicher, Jr., and P.R. Taylor, J. Chem. Phys., 85 (1986) 2779.
Journal of Molecular Structure (Theochem) 335 (1995) 51-57
A generalized multistructural description of the ground state of ozone and water molecules Wely Brasil Floriano, Solange Regina Blaszkowski, Marco Antonio Chaer Nascimento” Znstituto de Quimica. Departamento de Fisico-Quimica, Universidade Federal do Rio de Janeiro, Cidade Universitriria. CT, Bloc0 A, sala 412, Rio de Janeiro, RJ 21949-900, Brazil
Received 25 February 1994;accepted 13 July 1994
Abstract Generalized multistructural (GMS) wavefunctions are presented for the ground state of ozone and water molecules at the equilibrium geometry. For the ozone molecule the results confirm the diradical character of the ground state and show that the contribution of the zwitterionic structures is negligible. The reasons for the discrepancy between the MO and VB descriptions of the molecule are also presented. For the water molecule the GMS results also show that once orbital optimization is considered there is no need to include polar structures in a VB-type expansion. Also, a great deal of correlation energy can be recovered using very simple and compact GMS wavefunctions.
1. Introduction There are many molecules which cannot be represented by a single chemical structure. For those, the natural way to describe their electronic wavefunction is as a linear combination of bonding structures. This is the idea behind the classical valence bond (VB) theory [l-8]. However, the classical VB theory suffers from the fact that the atomic orbitals cannot readjust their shapes as the molecule is being formed. Besides, since only the occupied orbitals of each atom are generally considered, excited electronic states are usually not well represented. Orbital optimization simultaneously with the variational determination of the coefficients in the linear combination of structures has been considered by Voter and Goddard [9], Gerrat and
* Corresponding author.
co-workers [lo] and McWeeny [ll]. Voter and Goddard examined the case of a two-structure wavefunction for the particular case where the structures are represented at either the HartreeFock (HF) or generalized valence-bond (perfectpairing) [GVB(PP)] level, but are related through a reflection operation. Gerrat and co-workers [lo] and McWeeny [ 1l] considered the more general case of multiconfigurational wavefunctions but the structures are restricted to be represented by VB-type wavefunctions. While in the Voter and Goddard approach all the orbitals are optimized, in the spin-coupled [lo] and ab initio VB [I I] approaches, the optimization procedure is carried out in general for the valence electrons (about 6-8 electrons), the remaining orbitals being kept frozen. Another approach to introduce orbital shape readjustment, in the standard VB method, was proposed by Gallup et al. [12]. In this approach no orbital optimization is directly
0166-1280/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0166-1280(94)03982-8
52
W.B. Floriano et aLlJournal of Molecular Struciure (Theochem)
attempted but, instead, many configurations are admitted from start, as in a configuration interaction (CI) calculation We have recently proposed a generalized multistructural (GMS) wavefunction [ 13- 171
335 (1995) 51-57
to test how efficient the GMS wavefunctions can be in recovering correlation energy. Another interesting point, now related to the classical VB theory, is the investigation of the importance of the polar structures is the description of neutral molecules, when orbital optimization is taken into account.
(1) where $i represents the wavefunction of the ith “structure” and ci its weight in the total wavefunction. These coefficients are obtained variationally by solving the equations (~*GMSIH
-
E~*GMs)
=
0
(4
Here the word “structure” is employed in a very broad sense, as previously discussed [13-171. In the GMS wavefunction defined in Eq. (1) there are no restrictions whatsoever to the form of the wavefunctions qi. Each one of the structures presented in Eq. (1) can be obtained at the HF, MCSCF (GVB, etc.), CI or any plausible combination of these methods. All the orbitals are optimized at either the HF or MCSCF level and although orbitals belonging to the same structure are forced to remain orthogonal no such restriction exists for orbitals of different “structures”. As also previously discussed [ 13- 171, the GMS wavefunctions are compact, easy to interpret but very efficient at recovering electronic correlation energy. In this paper we present a GMS description of the ground state of ozone and water molecules. Besides their tremendous importance in everyday life, these molecules were chosen because a large amount of data, obtained with other methods, are available for comparison. The classical VB and the molecular orbital (MO) descriptions of ozone differ considerably and the relative importance of the zwitterionic structures has not been well defined yet. This discrepancy can be clearly resolved by constructing a GMS wavefunction consisting of the two polar (VB-type) structures plus a diradical one as suggested by the GVB calculations. The water molecule provides another very interesting application for the GMS wavefunction. Results of very large MO-C1 calculations are available for this molecule, which can be used
2. Computational details For the ozone molecule the calculations were carried out at the experimental equilibrium (r, = 1.278 A, 0, = 116.8 A) [18], using the Dunning [ 191 double-zeta (DZ) contraction (9sSp/3s2p) of Huzinaga’s primitive gaussian basis sets [20] augmented with a d polarization function (Q = 0.85). The GMS wavefunction for the ozone molecule was obtained at the GVB (3/PP) level of calculation [21], correlating the three bond-pairs, for structures (I) and (II), and the two bond-pairs plus the unpaired electrons, for structure (III): +
+
-o/o\o.,/“\o.
/“\o-
0
*GMS~)
(III)
(III
(I) =
Q~‘III
+ C2($1
+&I)
(3)
The classical VB description of the ground state of the ozone molecule corresponds to a resonance hybrid of the zwitterionic structures (I) and (II) [22], while (III) corresponds to a diradical structure first proposed by Goddard et al. [23]. To solve for the charge-localized structures we used a procedure analogous to the one employed by Bagus and Schaefer [24]. A Mulliken population analysis performed on the converged chargelocalized structure indicates a transference of 0.71e from the central to the negatively charged oxygen atom (see Table 1). Since we are dealing with a homonuclear molecule, there are no ambiguities in the process of distributing the overlap population between the nuclei, and the result of the population analysis can be taken as a good indication of the degree of charge localization. Calculations for the water molecule were also
W.B. Floriano et ai./Joumal
of Molecular Structure (Theochemi
Table 1 Mulliken population analysis for GVB (3/PP) wavefunctions of ozone Atom
Diradical (III)
Zwitterion (I)
0,
8.05 7.89 8.05
8.71 7.56 7.91
02 03
carried out at the experimental equilibrium geometry (re = 0.958 A, 8, = 104.5 A) [25] using a 6-31G (d,p) basis set with & = 0.85 and $, = 0.6 [26]. The GMS wavefunction was constructed as the superposition of structures (IV)-(VII),
H/O\”
H
H(VI
(IV)
(VI)
%dAd = cI+ (IV) +
0
(VII) C2$(V)
+ C3(1DvI + lClvr1)
(4)
each one represented at the GVB (2/PP) level of calculation. Because oxygen is much more electronegative than. hydrogen we have not considered polar structures with the oxygen atom positively charged. The GVB (2/PP) wavefunction for structure (IV) correlates the two bond-pairs while for the other structures the wavefunctions correspond to correlating the bond pair plus an electron pair in the molecular plane at the oxygen atom. The same procedure used for ozone was adopted for the water molecule in solving for structures (V)-(VII). In this case, a Mulliken population analysis indicates a transference of 0.65e to the oxygen atom.
3. Results and discussion The ozone molecule has been the subject of numerous theoretical and experimental investigations, most certainly owing to its extremely
335 (19951 51-57
53
important role in atmospheric chemistry (see, for example, Ref. [27]; Refs. [28]-[32] are the most recent studies). In spite of that, from the theoretical point of view, there remain some uncertainties concerning, for instance, the energy separation between the CZv ground state open-chain and the cyclic (D3J forms of ozone [30]. Also, there is considerable disagreement between all the theoretical predictions and the experimental result for the X ‘Ai -+’ B2 transition energy. However, regarding this molecule, our main concern in this paper is to quantify the importance of the zwitterionic structures (I and II) for the description of the ground state, in an attempt to reconcile the MO and VB pictures. The HF description of ozone is known to be completely inadequate, leading incorrectly to a triplet ground state, with the first singlet state approximately 2 eV higher in energy [33]. Goddard et al. [33] using the GVB method, which combines some attributes of both the VB and MO methods, were the first to describe properly the diradical character of the ozone singlet ground state. Subsequently Hay et al. [34] presented results of CI calculations based upon the previous GVB wavefunctions [33]. The diradical character of the ground state was confirmed but, interestingly, the only excited state with some ionic character was found at an energy approximately 5 eV above the ground state. In order to take into account this diradical character in the VB description Harcourt and Rosso [35] introduced structure III in their VB expansion (long bond between non-adjacent atoms). Their all-valence electrons VB calculations showed that the long-bond structure was indeed the dominant one but the contribution of zwitterionic structures was quite significant (see Table 2). The same conclusion was achieved by projecting the HF wavefunction onto a basis of VB functions [36-381, although, as noticed by Hiberty and Ohanessian [37], the contribution of the zwitterionic structures tend to become more important for larger basis sets. However, the contribution of the zwitterionic structures obtained by the projection method [37] was always found to be smaller than that obtained by the standard VB calculation [35]. These results are apparently in agreement.
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W.B. Floriano et al./Journal of Molecular Structure (Theochem)
335 (1995) 51-57
Table 2 Structural weights for the ground state of ozone Calculation Hiberty and Leforestier [38] (STO-3G) Hiberty and Ohanessian [39] (4-3 1G) Harcourt and Rosso [37] Present work GMS (DZP)
0.184 0.231 0.3084 1.4(-4)
However, the MO-C1 and VB calculations should be expected to converge to the same description of the molecule, as the number of terms in the respective expansions increases. Nevertheless, the GVB + CI description of ozone ground state does not show any significant contribution of configurations which could be associated with zwitterionic structures. These appear to be important only for higher electronic excited states [34]. However, from the VB point of view, except for a hypervalent structure [35], there are not too many other neutral structures (energetically equivalent to III) which could be included in the expansion in order to drastically reduce the importance of the zwitterionic structures. However, there were still some problems, of a technical and chemical nature, which prevented us of including that kind of structure in the GMS expansion [36]. In an attempt to understand those differences, we performed a series of calculations at the HF and GVB levels and for structures (I), (II) and (III). The GVB functions were used to construct the GMS wavefunction in Eq. (3). Table 3 shows the results of those calculations. The first point to be noticed from the GMS results is that once the orbitals are optimized the contribution of the zwitterionic structures becomes negligible. Also from Table 3 it can be seen that the energy difference between the HF X ‘Ai and the zwitterionic structures is small (about 0.7 eV), indicating that, at the HF level of calculation, structures (I) + (II) should mix appreciably with (I), as noticed by Hiberty and Ohanessian [39]. However, since the HF wavefunction gives a bad description of the ground state, its projection onto a VB basis will certainly overestimate the contribution of the zwitterionic structures. A comparison between our GMS and Harcourt and Rosso standard VB calculations [37] clearly indicates that in the latter
0.184 0.231 0.3084 1.4(-4)
0.593 0.476 0.7934
1.000
calculations the zwitterionic structures are just providing for orbital shape readjustment. That also explains why the contribution of the zwitterionic structures to the MO projected VB function [39] was always found to be smaller than to the standard VB wavefunction [37]. Even though the HF wavefunction gives the wrong description of the molecule, the shapes of the resulting valence orbitals are more appropriate to describe bond formation than the atomic (VB) ones and therefore less contribution of the ionic structures is needed. The water molecule has also been the subject of intensive theoretical and experimental investigation. A recent survey of the latest ab initio and post-Hartree-Fock calculations for this molecule has been compiled by Levine [40]. As mentioned before, the water molecule was chosen as a test for the efficiency of the GMS wavefunction to recover correlation energy and also to examine the importance of the ionic structures in a VB-type expansion when orbital optimization is taken into account. In the usual treatment of electronic correlation effects, a HF or MCSCF wavefunction is used as the zero-order function for a perturbation calculation (MBPT), or as a reference wavefunction either for a CI or for a coupled-cluster (CC) calculation. The VB approach, however, starts from Table 3 Total energies for the ozone molecule State
Structure
Level of calculation
Energy (a.u.)
I or II I or II III I + II + III
HF HF GVB(3/2) GVB(3/2) GMS
-224.30578 -224.28005 -224.34138 -224.41476 -224.41567
X ‘A,
X ‘A, X ‘A,
W.B. Floriano et aLlJournal of Molecular Structure (Theochem)
classical chemical structures, where the atom’s individuality is to a greater extent preserved. Orbital shape readjustments and electronic correlation effects are introduced by the superposition of chemical structures representing different coupling and distribution of electrons among the various atoms. This approach is conceptually simpler and the results are easier to interpret, contrary to the ones obtained from large scale CI and CC calculations or higher-order perturbative treatments. However, the VB approach turns out to be computationally more complex than the MO-based methods because a large number of structures involving non-orthogonal orbitals is generally needed. There are many high-quality MO-type calculations [40] for the water molecule but, to the best of our knowledge, no analysis has ever been made in order to establish the contribution, to the ground state, of configurations which could be associated with ionic VB structures. Also no attempts of projecting any of the MO-type wavefunctions available into a VB basis have been reported. Apparently the only all-electron VB calculation for the water molecule is the one performed by Peterson and Pfeiffer [41]. The importance of the ionic structures, in their VB wavefunction containing 10 covalent and 39 ionic structures, was found to be 62%, quite a large contribution. For small molecules it is known that a large fraction of the correlation energy, relative to a HF wavefunction and a given basis set, can be recovered by performing a CI calculation including all single and double excitations (CISD) from the reference configuration. In fact, for the water molecule Saxe et al. [42] have found that a CISD calculation accounts for 94.7% of the total correlation energy, when a DZ quality basis set is used. However, the recovery of the residual correlation energy requires an extremely large number of configurations which makes the final wavefunction quite difficult to interpret. However, from the VB results of Peterson and Pfeiffer [41] it is clear that many more “structures” (or coupling schemes) are needed in order to obtain results comparable to the MO-C1 ones. However, increasing the basis of VB structures requires a great deal of extra computational effort. Alternatively
55
335 (1995) 51-57
Table 4 Total energies for the Hz0 molecule Calculation
GMS” GVB(2/PP) GVB-RCI GVB-Fb Full-C1 DZ basisC DZP basisdl
Number of spatial configurations
Energy (a.u.)
8 18 36
-16.13614 -76.16483 -16.22611
256 473 6 740 280
-76.15786 -16.25662
a Only structures (IV) and (V) are being considered. b Full-C1 in the space of the GVB orbitals. ’ Ref. [42]. d Ref. [43].
the simple picture provided by the chemical structures could be preserved, but optimizing their orbitals in a self-consistent way. In addition the dominant correlation effects can be easily incorporated into the wavefunctions describing each one of the structures. The final wavefunction is obtained variationally and consists of the superposition of the wavefunctions representing the different structures. This is the idea behind the GMS wavefunction. It is also possible to perform a simultaneous optimization of the {$i} and {Ci}. However, if the overlap between the wavefunction is small there is no need to optimize them in the presence of each other. Because each structure can be represented by a correlated wavefunction containing the dominant (and clearly identifiable) effects, it is expected that a very small number of structures will be needed, in the final wavefunction, in order to describe the less important (residual effects). As mentioned before, the recovery of the residual, but not always negligible effects, by the more usual approaches, requires a large amount of computational effort. A series of GMS calculations were performed, representing the structures (IV)-(VII) at the HF, GVB (2/PP) and GVB + CI levels of calculation. The results obtained at the HF level indicated a negligible contribution (c < 10e3) of the ionic structures. Therefore only the covalent structures IV and V were considered at the higher levels of calculation. Table 4 shows the GMS results compared with some other selected calculations. It is important to emphasize that many other (and
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even more accurate) results are available in the literature [40]. However, the ones selected are the most appropriate to our discussion. From the results of Table 4 it is clear that the GMS wavefunctions can be extremely efficient at recovering correlation energy. At the GVB-RCI (the GVBRCI calculations correspond to relaxing the perfect-pairing (PP) condition) level of calculation, the GMS wavefunction furnishes a better description than the DZ full-C1 calculation [42]. In fact, this is not a fair comparison inasmuch as for the GMS wavefunctions a DZP basis set was used. However, if the results from the GMS-GVBF calculations are compared with the ones obtained from a full-C1 using a DZP quality basis set [43], it can be seen that the GMS result is only 29 mh (0.78 eV) above the CI results in spite of the tremendous difference in the number of configurations used in the two calculations.
4. Conclusion The GMS results obtained for the ozone and water molecules clearly indicate that once orbital optimization is considered, there is little or practically no need to consider ionic structures in a VB-type expansion wavefunction. Besides the practical aspect of having to deal with a smaller number of configurations, this also eliminates the need of including some highly unphysical structures. This same conclusion emerges from ab initio VB and spin-coupled VB calculations. For the particular case of the ozone molecule the GMS wavefunction provides a clear explanation for the discrepancy observed in the MO and standard VB descriptions of the molecule. Even more interesting, the GMS wavefunction has proven to be extremely efficient at recovering correlation energy without losing the simple picture provided by chemical structures.
Acknowledgments The authors thank CNPq and FINEP for linancial support. Two of us (WBF and SRB) also acknowledge CAPES for graduate scholarships.
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References [l] W. Heitler and F. London, Z. Phys., 44 (1927) 455. [2] G.Rumer, Gijttingen Nach., (1932) 337. [3] L. Pauling, Proc. Natl. Acad. Sci. U.S.A., 18 (1932) 293. [4] L. Pauling and G.W. Wheland, J. Chem. Phys., 1 (1933) 362. [5] L. Pauling and J. Sherman, J. Chem. Phys., 1 (1933) 606. [6] H. Eyring, J. Am. Chem. Sot., 54 (1932) 3191. [7] J.C. Slater, Phys. Rev., 37 (1931) 481; 38 (1931) 1109. [8] L. Pauling, J. Am. Chem. Sot., 53 (1931) 1367; 53 (1931) 3225. [9] A.F. Voter and W.A. Goddard III, J. Chem. Phys., 75 (1981) 3638; Chem. Phys., 57 (1981) 253. [lo] J. Gerrat, Adv. At. Mol. Phys., 7 (1971) 141. N.C. Pyper and J. Gerrat, Proc. R. Sot. London, Ser. A, 355 (1977) 407. D.L. Cooper, J. Gerrat and M. Raimondi, Adv. Chem. Phys., 69 (1987) 319. J. Gerrat and M. Raimondi, Proc. R. Sot. London, Ser. A, 371 (1980) 525. [l l] R. McWeeny, Int. J. Quantum Chem., 34 (1988) 25. [12] G.A. Gallup, R.L. Vance, J.R. Collins and J.M. Norbeck, Adv. Quantum Chem., 16 (1982) 229. [13] E. Hollauer and M.A.C. Nascimento, Chem. Phys. Lett., 184 (1991) 470. [14] C.E. Bielschowsky, M.A.C. Nascimento and E. Hollauer, Phys. Rev. A, 45 (1992) 4622. [15] M.A.C. Nascimento, in Collision Processes of Ion, Positron, Electron and Photon Beams with Matter, World Scientific, Singapore, 1992, p. 247. [16] E. Hollauer and M.A.C. Nascimento, Chem. Phys., 174 (1993) 79. [17] E. Hollauer and M.A.C. Nascimento, J. Chem. Phys., 99 (1993) 1207. [18] T. Tanaka and Y. Morino, J. Mol. Spectrosc., 33 (1970) 538. [19] T.H. Dunning and P.J. Hay, in H.F. Schaefer III (Ed.), Modern Theoretical Chemistry, Vol. 3, Plenum, New York, 1977, p. 1. [20] S. Huzinaga, J. Chem. Phys., 42 (1965) 1293. [21] F.W. Brobowickz and W.A. Goddard III, in H.F. Schaefer III (Ed.), Modern Theoretical Chemistry, Vol. 3, Plenum, New York, 1977, Chapter 6. [22] G.W. Wheland, Resonance in Organic Chemistry, Wiley, New York, 1955. [23] W.A. Goddard III, T.H. Dunning, Jr., W.J. Hunt and P.J. Hay, Act. Chem. Res., 6 (1973) 368. [24] P.S. Bagus and H.F. Schaefer, J. Chem. Phys., 56 (1972) 224. [25] R.L. Cook, F.C. DeLucia and P. Helminger, J. Mol. Spectrosc., 53 (1974) 62. [26] W.J. Hehre, R. Ditchfield and J.A. Pople, J. Chem. Phys., 56 (1972) 2257. [27] K. Ohono, K. Morokuma and H. Hosoya (eds.), The Quantum Chemistry Literature Data Bank, JAICI, Tokyo, 1990. [28] S. Xantheas, ST. Elbert and K. Ruedenberg, J. Chem. Phys., 93 (1990) 7519.
W.B. Floriano et al./Journal of Molecular Structure (Theochem) 335 (1995) 51-57 S. Xantheas, G.J. Atchity, S.T. Elbert and K. Ruedenberg, J. Chem. Phys., 94 (1991) 8054. [29] S.M. Anderson, J. Morton and K. Mauesberger, J. Chem. Phys., 93 (1990) 3826. [30] T.J. Lee and G.E. Scuseria, J. Chem. Phys., 93 (1990) 489. T.J. Lee, Chem. Phys. Lett., 169 (1990) 529. [31] K.A. Peterson, R.C. Mayrhofer, E.L. Sibert III and R.C. Woods, J. Chem. Phys., 94 (1991) 414. [32] W. Kock, G. Frenkling, G. Steffen, D. Reinen, M. Jansen and W. Assenmacher, J. Chem. Phys., 99 (1993) 1271. [33] W.A. Goddard III, T.H. Dunning, Jr., W.J. Hunt and P.J. Hay, Act. Chem. Res., 6 (1973) 368. [34] P.J. Hay, T.H. Dunning, Jr., and W.A. Goddard III, J. Chem. Phys., 62 (1975) 3912. [35] D.L. Cooper, J. Gerratt and M. Raimondi, J. Chem. Sot., Perkin Trans. 2, (1989) 1187.
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[36] S.R. Blaszkowski, M.A.C. Nascimento, C. Amovili and R. McWeeny, in preparation. [37] R.D. Harcourt and W. Rosso, Can. J. Chem., 56 (1978) 1093. [38] P.C. Hiberty and C. Leforestier, J. Am. Chem. Sot., 100 (1978) 2012. [39] P.C. Hiberty and G. Ohanessian, J. Am. Chem. Sot., 104 (1982) 66. [40] I.N. Levine Quantum Chemistry, Prentice-Hall, Englewood Cliffs, NJ, 1991, pp. 474 and 510. [41] C. Peterson and G.V. Pfeiffer, Theor. Chim. Acta, 26 (1972) 321. [42] P. Saxe, H.F. Schaefer III and NC. Handy, Chem. Phys. Lett., 79 (1981) 202. [43] C.W. Bauschicher, Jr., and P.R. Taylor, J. Chem. Phys., 85 (1986) 2779.