Equivalence restriction

For our SCF calculation the way to obtain such a solution is to optimize the orbitals not for the energy of determinant 7.6, but for the average energy of both determinants. This is termed the imposition of symmetry and equivalence restrictions. It involves imposing a constraint on a variational calculation, and consequently the symmetry and equivalence restricted solution will have an energy no lower than the broken symmetry solution it will usually have a higher energy. We may note that in a UHF calculation we impose neither spin nor spatial symmetry and equivalence restrictions — the -terms restricted and unrestricted were first used in exactly this context of whether to impose symmetry constraints on the wave function. 

The averaging of SCF energy expressions to impose symmetry and equivalence restrictions is a straightforward, if sometimes tedious, application of the Slater-Condon rules for matrix elements between determinants of orthonormal orbitals. This matter is discussed in detail elsewhere. The most general SCF programs can handle energy expressions of the form... 

It can be necessary and/or desirable to impose symmetry and equivalence restrictions on quantum chemical calculations or results beyond the single-configuration SCF level. For instance, most Cl programs generate natural orbitals (NOs) after computing the Cl wave function, by forming and diagonalizing the first-order reduced density matrix or 1-matrix p in... 

In practical implementations the original MO basis symmetry adapted (say, by performing a symmetry and equivalence restricted SCF calculation). The symmetrization Eq. 7.11 is then very simple to implement. Consider the block of p° from orbitals that transform according to the symmetry species (a, i) (recall that p° must be block diagonal on symmetry species)... 

The multireference results of Table 5.12 were all based on full valence CASSCF calculations with eleven electrons in twelve active orbitals. This produces a large configuration expansion (about 85 000 CSFs) so it is not possible to perform MRCI(CAS) calculations. Reference configuration lists were selected at the cyclic and linear geometries (taken from MP2 optimized structures) and then merged. The core electrons were not correlated in any of the calculations. One complication in the CASSCF calculations should be pointed out. Since the cyclic state arises as the 2B component, in C3v symmetry, of a 2E state in the D3h symmetry (equilateral triangular) structure, it would be desirable to obtain MOs with D3h symmetry and equivalence restrictions... 

In the restricted Hartree-Fock (RHF) method, two restrictions are placed on the molecular orbitals u< in equation (11). The first is chat each ui transform according to one of the irreducible representations of the point group of the molecule. The second restriction is that the space functions u come in identical pairs one with spin function a and the other with spin function /S. These are called, respectively, the symmetry and equivalence restrictions.190... 

When the second of the equivalence restrictions is removed, a single determinant wavefunction of lower energy is usually obtained. In fact, it is possible for a wave-function obtained in this way, a so-called unrestricted Hartree-Fock (UHF) wavefunction191 (perhaps more properly called a spin-unrestricted Hartree-Fock wavefunction) to go beyond the Hartree-Fock approximation and thus include some of the correlation energy. Lowdin192 describes this as a method for introducing a Coulomb hole to supplement the Fermi hole already accounted for in the RHF wavefunction. 

In general, the many-electron wave function is expressed in terms of antisymmetrized products of one-electron functions and the clamped-nucleus approximation as well as the central-field and equivalence restriction for the orbitals is used. Thus the one-electron spinor takes the form... 

For a non-closed shell system, instead of Eq. (6a) there may be a linear combination of several degenerate determinants ( code-tors ). The H.F. equations and F< will then depend on which of the H.F. methods (Section VI) has been used. The one based on the average energy of a configuration with symmetry and equivalence restrictions gives the same and F, as in Eq. (7) and for all the multiplets arising from that configuration. This also simplifies the treatment of the correlation part. 

For 77-electron spectra, one may start at least formally with H.F. based on the average energy of a configuration , with symmetry and equivalence restrictions (see Section VI). As results on atoms also indicate, the e,/s of S cores as well as core-polarization potentials will not be much affected by the valence electrons, so that one deals with just the E part of Eq. (128). 

The last group in our classification comprises two approximate SCF procedures which give wave functions that are not correct to first order. The first of them, Nesbet s method of symmetry and equivalence restrictions uses the Hamiltonian of the unrestricted method for the a-spin electrons, the number of a-spin electrons being greater than that of p-spin electrons. The p-spin electrons are forced to occupy MOs given for a-spin electrons by... 

The ab-initio study of process I, the dissociation of formaldehyde into radical products, was simultaneously done by Fink 9 ) (using Nesbet s method of symmetry and equivalence restrictions and a limited Cl) and by Hayes and Morokuma 2) (using a GSMO.CI method and the "point system in the selection of the configurations). Both sets of potential energy curves exhibit the same behavior. One is shoAvn below (Fig. 10). 

The method for calculating tb is exactly that described in the introduction to this chapter for binary and multi-valued logic. The process is one of calculating fuzzy truth restrictions for the first and second lines of the deduction on the space Ux X Uy, intersecting them to produce an equivalent restriction and then projecting the result on to Uy Thus... 

In accordance with the equivalence restriction a shell is classified by one pair of quantum numbers n and k. A shell comprises 2 k atomic spinors and is therefore 2 x -fold degenerate. A closed shell is characterized by the fact that all of these degenerate spinors enter the single Slater determinant in Dirac-Hartree-Fock theory. 

However, it has been proposed by various authors to relax some or all of the above-mentioned constraints, to gain variational freedom. For example, the different orbitals for different spins method (DODS) of Refs. [14, 15] (which is usually just called unrestricted Hartree-Fock (UHF), but in this paper we use DODS to avoid confusions) relax the spin-equivalence restriction, and hence, the HF solution is no longer an eigenfunction of S. Other authors [16-18] have advocated the use of general spin-orbitals, mixing a-spin and /9-spin parts, in conjunction with the use of projectors [19]. 

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