Orbital-Optimized Single-Configuration Methods

The GVB method is generally used in its restricted perfectly paired form, referred to as GVB-PP, which pairs the atoms as in the most important Lewis structure. The GVB-PP method introduces two simplifications. The first one is the Perfect-Pairing (PP) approximation, by which only one VB structure is generated in the calculation. The wave function may then be expressed in the simple form of Eq. [89], where each term in parentheses is a so-called geminal two-electron function, which takes the form of a singlet-coupled GVB pair 

9/fo) and is associated with one particular bond or lone pair. 

The second simplification, which is introduced for computational convenience, is the strong orthogonality constraint, whereby all the orbitals in Eq. [89] are required to be orthogonal to each other unless they are singlet 

This strong orthogonality constraint is, of course, a restriction, however, usually not a serious one, since it applies to orbitals that are not expected to overlap significantly. By contrast, the orbitals that are coupled together in a given GVB pair display, of course, a strong overlap. 

For further mathematical convenience, each geminal in Eq. [89] can be rewritten, by simple orbital rotation, as an expansion in terms of natural orbitals. 

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MOs and the configuration expansion. To be successful, we must choose the parametrization of the MCSCF wave function with care and apply an algorithm for the optimization that is robust as well as efficient. The first attempts at developing MCSCF optimization schemes, which borrowed heavily from the standard first-order methods of single-configuration Hartree-Fock theory, were not successful. With the introduction of second-order methods and the exponential parametrization of the orbital space, the calculation of MCSCF wave functions became routine. Still, even with the application of second-order methods, the optimization of MCSCF wave functions can be difficult - more difficult than for the other wave functions treated in this book. A large part of the present chapter is therefore devoted to the discussion of MCSCF optimization techniques. 

A CASSCF calculation is a combination of an SCF computation with a full Configuration Interaction calculation involving a subset of the orbitals. The orbitals involved in the Cl are known as the active space. In this way, the CASSCF method optimizes the orbitals appropriately for the excited state. In contrast, the Cl-Singles method uses SCF orbitals for the excited state. Since Hartree-Fock orbitals are biased toward the ground state, a CASSCF description of the excited state electronic configuration is often an improvement. 

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