BOVB Levels

Several theoretical levels are conceivable within the BOVB framework. At first, the inactive orbitals may or may not be allowed to delocalize over the whole molecule (vide supra). To distinguish the two options, a calculation with localized inactive orbitals will be labeled L , as opposed to the label D that will characterize delocalized inactive orbitals. The usefulness and physical meaning of this option will be discussed below using particular cases. 

Another optional improvement concerns the description of the ionic VB structures. At the simplest level, the active ionic orbital is just a unique doubly occupied orbital as in 10 or 11. However this description can be improved by taking care of the radial correlation (also called in-out correlation) of the two active electrons, and this can be achieved most simply by splitting the active orbital into a pair of singly occupied orbitals accommodating a spin-pair, much as in GVB theory. This is pictorially represented in 12 and 13 which represent improved descriptions of 10 and 11. 

This improved level will be referred to as S (for split ) while the simpler level will carry no special label. Combining the two optional improvements, the BOVB calculations can be performed at the L, SL, D or SD levels.t 

These levels are tested below on bond energies and/or dissociation curves of classical test cases, representative of two-electron and odd-electron bonds. 

The dissociation of difluorine is a demanding test case used traditionally to benchmark new computational methods. In this regard, the complete failure of the Hartree-Fock method to account for the F2 bond has already been mentioned. Table 1 displays the calculated energies of F2 at a fixed distance of 1.43 A, relative to the separated atoms. Note that at infinite distance, the ionic structures disappear, so that one is left with a pair of singlet-coupled neutral atoms which just corresponds to the Hartree-Fock description of the separated atoms. 

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D-BOVB and SD-BOVB Calculations ofF2 Like VBSCF, the BOVB method is improved by allowing the spectator orbitals to be delocalized in the so-called D-BOVB or SD-BOVB levels. As long as the spectator orbitals can be distinguished from the active ones by symmetry, which is the case of the lone pairs of tt symmetry in the F2 molecule, delocalizing these orbitals is easily done The user has only to specify which basis functions the spectator orbitals are allowed to be made of, much the same as in VBSCF. This is shown in Input 10.6 for a calculation of D-BOVB type, in which the tt spectator orbitals are allowed to delocalize. 

This chapter considered molecules with high enough symmetry that can assist the distinction between active and inactive orbitals. Such facility is not always present in the general case, and this poses a danger that during the BOVB orbital optimization there will occur some flipping between the sets of active and inactive orbitals. This, however, depends on the BOVB level. 

BOVB Breathing orbital valence bond. A VB computational method. The BOVB wave function is a linear combination of VB structures that simultaneously optimizes the structural coefficients and the orbitals of the structures and allows different orbitals for different structures. The BOVB method must be used with strictly localized active orbitals (see HAOs). When all the orbitals are localized, the method is referred to as L-BOVB. There are other BOVB levels, which use delocalized MO-type inactive orbitals, if the latter have different symmetry than the active orbitals. (See Chapters 9 and 10.)... 

This test calculation, referred to as "Extended SD-BOVB" in Table 3, results in an improvement of only 1.1 kcal/mol of the bonding energy relative to the standard SD level, thus confirming the assumption of near-constancy of the correlation within inactive electrons. It follows therefore that going beyond the SD-BOVB level is not necessary. 

Table 3 displays also a comparison of a full Cl calculation by Bauschlicher and Taylor [32] with the best BOVB levels using a common basis set. Once again the SD-BOVB level is entirely sufficient, while its extended version leads to a meager improvement. In any case, both levels are in excellent agreement with the full Cl results. 

By nature, the BOVB method describes properly the dissociation process. As a test case, the dissociation curve of the FH molecule was calculated at the highest BOVB level (extended SD-BOVB), and compared with a reference full Cl dissociation curve calculated by Bauschlicher et al. [33] with the same basis set. The two curves, that were compared in Ref. 12, were found to be practically indistinguishable within an error margin of 0.8 kcal/mol, showing the ability of the BOVB method to describe the bonding interaction equally well at any interatomic distance from equilibrium all the way to infinite separation [12]. 

The computed equilibrium distance and bonding energy of F2- are displayed in Table 5. To appreciate better the sensitivity of active vs inactive orbitals to the breathing orbital effect, the latter has been introduced by steps In the first step no breathing orbitals are used (La = Lr, Ra = Rr, (Pi = cpf) this VBSCF calculation is nearly equivalent to the ROHF level. In the second step, only active orbitals are included in the breathing set (La 1- Lr, Ra Rr), while in the next step full breathing is permitted (La Lr, Ra Rr, cpi cpi ). The latter wave function, at the L-BOVB level, can be represented as in 26, 27 below. 

Somewhat more significant is the effect of splitting the active orbitals, leading to structures 28 and 29 at the SD-BOVB level, where the local singlet... 

Table 6 displays some bonding energies for CI2-, as calculated at the D-BOVB level and at other theoretical levels, including Hartree-Fock and Moller-Plesset perturbation theory. Unlike the F2 case, the Moller-Plesset series converges well around the values of 24-25 kcal/mol which can be taken as references for the bonding energy in this basis set. 

The (H3C CH3)+ radical cation was selected, to test the ability of the BOVB method to describe one-electron bonds, since this bond exhibits the largest correlation effect in the series. The bonding energy, calculated at the D-BOVB level, amounts to 48.7 kcal/mol, in fair agreement with the MP4 value. 

Fig. 7 A correlation of %REcs values (%REcs = OOREcs/D ) obtained at the TCSCF+PT2 level and the BOVB level, for the C-C, 0-0, N-N, and F-F bonds, from left to right. Reproduced with ACS permission from [14]...

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