MC-SCF theory

Lengsfield B H III 1980 General second-order MC-SCF theory a density matrix directed algorithm J. Chem. Phys. 73 382... 

MC-SCF treatments written in terms of coupled Fock equations [44], The simplest examples are the two-configuration SCF theory [45] used in and pi+2 atomic mixing [46], or bonding-antibonding molecular problems [47], and more generally the Clementi-Veillard electron-pair MC-SCF theory [48],... 

While the (one-particle) Brillouin condition BCi has been known for a long time, and has played a central role in Hartree-Fock theory and in MC-SCF theory, the generalizations for higher particle rank were only proposed in 1979 [38], although a time-dependent formulation by Thouless [39] from 1961 can be regarded as a precursor. 

This is nothing but the Brillouin condition of MC-SCF theory. Explicitly, in an... 

If one does not care to describe the correlation cusp correctly, but uses a Cl-like expansion of the wave function, the just-mentioned singularity would not show up, so it is not surprising that so far it has not plagued any numerical calculation. In particular, everything remains regular in the framework of Hartree-Fock or MC-SCF theory. 

An interesting example of relativistic MC-SCF theory is the isoelectronic Be-like series. While the non-relativistic correlation energy is linear in Z (due to the degeneracy), the Z-dependence of the relativistic... 

The main challenge as to an improved theory of electron correlation as a basis of accurate numerical quantum chemistry have been mentioned in this review, namely (a) the explicit treatment of the correlation cusp, (b) the formulation of methods that scale with a low power of the number of particles, (c) the consistent combination of MC-SCF-theory for the nondynamic and coupled-cluster methods for the dynamic correlation. 

Our first objective is to illustrate how thinking in VB language solves many of the objectivity problems in the practical application of MC-SCF theory to problems in chemical reactivity. Thus the essential elements of MC-SCF theory will be reviewed with the objective of conveying sufficient information that the practical and conceptual aspects of MC-SCF theory can be understood. We shall also show that there exists a completely rigorous transformation to Valence Bond Space. Thus we will illustrate that one has the option of thinking in VB space (a natural one for the chemist) but doing all iht numerical work in MC-SCF space. 

Let us now briefly summarize the essentials of MC-SCF theory. It is helpful to take an entirely different approach to the usual formulation of closed shell SCF theory. In fact, it proves to be useful to think of the MC-SCF process in a similar way to geometry optimization. Thus we shall view the orbital and Cl coefficient variables that occur in MC-SCF in the same way as the internal geometrical variables in a geometry optimization. Technically in order to do this we must assume that we have an orthogonal set of starting orbitals o and an orthogonal set of Cl vectors K>. The MC-SCF Cl expansion (for state K) is written as... 

Attempts have been made to make the choice of core/valence/virtual orbital partition into a black box. In particular the unrestricted natural orbitals can provide a good starting point for MC-SCF as as suggested by Pulay[17]. We now explore this point in some detail since it also gives us some additional insight into MC-SCF theory. As discussed by Pulay[17] in his paper one can understand the problem with a two electron example. In UHF wavefunction for a two electron localized bond, one seeks the optimum energy of a wavefunction of the form... 

The iterative procedure based on (8.3.16) is in fact the exact analogue, in MC SCF theory, of the Roothaan closed-shell SCF method the singular-value decomposition corresponds to solution of the matrix eigenvalue problem, and the updating of the X matrix corresponds to the updating of the G matrix in each case a single iteration would serve, were it not for the need to revise the electron interaction terms in every cycle. 

So far, the possibility of optimizing the orbitals in the presence of a perturbation (i.e. of making self-consistent property calculations) has been considered only at the Hartree-Fock level. In many cases, however, it is necessary to use a many-determinant wavefunction, either because the IPM ground state is degenerate or because electron-correlation effects are too important to be ignored and it is then desirable to optimize both Cl coefficients and orbitals as in MC SCF theory (Section 8.6). To formulate the perturbation equations, both coefficients and orbitals will be expanded in terms of a perturbation parameter and the orders will be separated the zeroth-order equations will be the MC SCF equations in the absence of the perturbation, while the first-order equations will determine the (optimized) response of the wavefunction, and will thus permit the calculation of second-order properties. Important progress had been made in this area (Jaszunski, 1978 Daborn and Handy, 1983), for particular types of perturbation and Cl function. In fact, however, the equations in their most general form have been known for many years (Moccia, 1974), and are implicit in the stationary-value... 

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