Valence optimized orbital

Let us finally mention two additional SS-type CC approaches that strive for the same goal. First, we mention the valence optimized orbital (VOO) CCSD method (55) and the variational VCCD method of Van Voorhis and Head-Gordon (5). Particularly the latter approach, based on the energy expectation value with the CCD cluster Ansatz, represents the upper bound to (full) FCI or FCC, and thus avoids the fallacious asymptotic behavior of standard CCSD PECs [see, e.g. the case of N2 (8,9)]. Yet, the variational requirement tends to raise the energy [cf. also Ref. (36)] and produces a rather significant overestimate in the 7 - < limit [e.g., about 45 mhartree for the VDZ model of N2, cf. Ref. ( )], not to mention the complexity of the resulting formalism which depends factorialy on the electron number. 

We can also verify that the usual ECAO-MO description (3.2) and (3.4) leads to predicted hybridizations that are generally consistent with the donor-acceptor estimates (3.8). Suppose that each H atom is associated with a valence spin-orbital of hybridized form (3.6). According to Eq. (3.2), the optimal electronic energy of bond formation is obtained by choosing the hybridization parameter k to maximize the magnitude of the interaction element... 

The number No of occupied valence SCF orbitals in a molecule is typically less than the total number Nmb of orbitals in the minimal valence basis sets of all atoms. The full valence MCSCF wavefunction is the optimal expansion in terms of all configurations that can be generated from N b molecular orbitals. Closely related is the full MCSCF wavefunction of all configurations that can be generated from Ne orbitals, where Nc is the number of valence electrons, i.e. each occupied valence orbital has a correlating orbital, as first postulated by Boys (48) and also presumed in perfect pairing models (49,50), We shall call these two types of frill spaces FORS 1 and FORS 2. In both, the inner shell remains closed. 

Such a spin-coupled wavefunction is optimized with respect to the core wavefunction (if applicable), as well as to the nonorthogonal valence bond orbitals,... 

To extend NDDO methods to elements having occupied valence d orbitals that participate in bonding, it is patently obvious that such orbitals need to be included in the formalism. However, to accurately model even non-metals from the third row and lower, particularly in hypervalent situations, d orbitals are tremendously helpful to the extent they increase the flexibility with which the wave function may be described. As already mentioned above, the d orbitals present in the SINDOl and INDO/S models make them extremely useful for spectroscopy. However, other approximations inherent in the INDO formalism make these models poor choices for geometry optimization, for instance. As a result, much effort over the last decade has gone into extending the NDDO fonnalism to include d orbitals. 

Multiconfiguration Valence Bond Methods with Optimized Orbitals... 

Figure 15. Radial charge density plot for the resonant p-type virtual orbital for dilation angles 9 = 0.0 and 6 = 90pt (0.42 radians) in e-Be scattering. The role of optimal theta in the accumulation of electron density near the nucleus is clearly seen. In the inset, the maximum is seen to occur at rmaz — 2.5 a.u., very close to that for the rmax of the outer valence 2s orbital, seen in fig. 14- Though a cursory look at the nodal pattern identifies this as a 4P orbital, the dominant contribution to the charge density distribution is mainly of 2p-iype.

The calculations reported so far are based on unconstrained mixing of all valence functions as a result, the optimized orbitals differ greatly from those pictured by Pauling, which - although usually hybrids - were strictly monocentric in character. The optimized forms resemble more closely the Coulson-Fischer orbitals of Sect.2, being distorted AOs which result in considerably increased overlap in the bond regions. In this general context, such AOs have been referred to as overlap-enhanced... 

A more balanced description thus requires multiconfiguration self-consistent field (MCSCF)-based methods, where the orbitals are optimized for each particular state or optimized for a suitable average of the desired states (state-averaged MCSCF). In semiempirical methods, however, an MCSCF procedure is normally not required due to the limited flexibility of the minimal valence atomic orbital basis commonly used in these methods. Instead, a multireference Cl method including a limited number of suitably chosen configurations will be appropriate. 

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