Optimization of orbitals

Dalgaard E and J0rgensen P 1978 Optimization of orbitals for multieonfigurational referenee states J. Chem. Phys. 69 3833-44 Jensen H J Aa, J0rgensen P and A gren H 1987 Effieient optimization of large seale MCSCF wave funetions with a restrieted step algorithm J. Chem. Phys. 87 451 -66... 

We consider in this Section particular aspects relating to the optimization of a CASVB wavefunction. As for most procedures involving the optimization of orbitals, special attention should be given to the choice of optimization strategy. The optimization problem is in this case non-linear, so that an exact second-order scheme is preferable in order to ensure reliable convergence. A particularly useful account of various second-order optimization schemes has been presented by Helgaker [46]. 

H.-J.WernerandW.Meyer,Aquadraticallyconvergentmulticonfigurationself-consistentfieldmeth-od with simultaneous optimization of orbitals and Cl coefficients. J. Chem. Phys. 73, 2342 (1980). 

Technically, the simultaneous optimization of orbitals and coefficients for a multistructure VB wave function can be done with the VBSCF method due to Balint-Kurti and van Lenthe [21,22], The VBSCF method has the same format as the classical VB method with an important difference. While the classical VB method uses orbitals that are optimized for the separate atoms, the VBSCF method uses a variational optimization of the atomic orbitals in the molecular wave function. In this manner the atomic orbitals adapt themselves to the molecular environment with a resulting significant improvement in the total energy and other computed properties. 

Simultaneous optimization of orbital exponent (a), internuclear distance (R) and the Cl-coefficient... 

The deviation of the CASSCF curve from the FCI curve in Fig. 2 is caused by nonstatic or dynamical correlation [1]. Although dynamical correlation is usually less geometry-dependent than static correlation, it must be included for high accuracy (see Sec. 4). One might think that it is possible to include the effects of dynamical correlation simply by extending the active space. For small molecules, this is, to some extent true, in particular when using the techniques of restricted active space SCF (RASSCF) theory [46]. Nevertheless, because of the enormous number of determinants needed to recover dynamical correlation, the simultaneous optimization of orbitals and configuration coefficients as done in MCSCF theory is not a practical approach to the accurate description of electronic systems. 

The basis functions are products of Slater and/or elliptical orbitals. They are chosen so as to contain both separated and united atom contributions. Extensive, but not necessarily total, optimization of orbital exponents is performed at each internuclear separation. Miller and Browne have discussed in detail our method of integral evaluation. The Molecular Physics Group has published an extensive table of integrals. ... 

Figure 2.10 Optimization of orbital overlap in bond formation provided by hybridization of atomic orbitals energy of the p-orbitals (diamonds), s-orbitals (squares) and the promotion energy (triangles) for B, C, N, O, and F. As the promotion energy rises, the importance of hybridization is expected to decrease. Values from ref 18.

Eq. (8.94) represents an exact expression for the quantum mechanical state of a many-electron system. Note that the expansion coefficients Cj of the N-particle basis states are directly related to the expansion coefficients of the one-particle states. It is sufficient to know either of them (and this fact is related to the observation made below that a full configuration interaction wave function does not require the optimization of orbitals see the next section). 

In practice, the simultaneous optimization of orbitals and Cl coefficients is a difficult nonlinear problem, which severely restricts the length of MCSCF expansions relative to those of Cl wave functions. By itself, the MCSCF model is therefore not suited to the treatment of dynamical correlation (for which large basis sets and long configuration expansions are needed) but it may be used in conjunction with a subsequent correlation treatment by a mote extensive multirefeience Cl wave function, providing the reference configurations and orbitals needed for such treatments. 

In Chapter 12, we study the related multiconfigurational self-consistent field (MCSCF) method, in which a simultaneous optimization of orbitals and Cl coefficients is attempted. Although the MCSCF method is incapable of providing accurate energies and wave functions, it is a flexible model, well suited to the study of chemical reactions and excited states. This chapter concentrates on techniques of optimization, a difficult problem in MCSCF theory because of the simultaneous optimization of orbitals and Cl coefficients. 

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