Hartree-Fock determinant, effective

If the Hartree-Fock determinant dominates the wavefunction, some of the occupation numbers will be close to 2. The corresponding MOs are closely related to the canonical Hartree-Fock orbitals. The remaining natural orbitals have small occupation numbers. They can be analysed in terms of different types of correlation effects in the molecule . A relation between the first-order density matrix and correlation effects is not immediately justified, however. Correlation effects are determined from the properties of the second-order reduced density matrix. The most important terms in the second-order matrix can, however, be approximately defined from the occupation numbers of the natural orbitals. Electron correlation can be qualitatively understood using an independent electron-pair model . In such a model the correlation effects are treated for one pair of electrons at a time, and the problem is reduced to a set of two-electron systems. As has been shown by Lowdin and Shull the two-electron wavefunction is determined from the occupation numbers of the natural orbitals. Also the second-order density matrix can then be specified by means of the natural orbitals and their occupation numbers. Consider as an example the following simple two-configurational wavefunction for a two-electron system ... 

In previous sections we have discussed the relativistic effect on the electron density of one-electron atoms and on atomic orbitals, about which many research papers have appeared (see Ref. [1051] for one of the first examples). We shall now consider the case of molecules where polarization effects play a role. The relativistic effect on the electron density of an extended molecule can be conveniently demonstrated by inspection of electron density differences. Figure 16.3 depicts such a density difference for the plane of the atomic nuclei of the mono-acetylene complex Pt(C2H2) [880]. The densities have been calculated from simple Hartree-Fock determinants according to Eq. (8.206) as the sum of squared molecular spinors. The relativistic contraction of the electron density in the core of the platinum atom appears as a huge tower of the dif-... 

The effect of single, double, etc. excitation operators acting on the Hartree-Fock determinant can then be expressed in both notations as... 

They are therefore the single de-excitation, double de-excitation and so forth operators and their effect is best described by letting them act on the Hartree-Fock determinant as a bra state... 

As often in quantum chemistry, the starting point in equation (1) is a single Slater determinant 0) which is most often but not necessarily chosen as the Hartree-Fock determinant. Electron correlation effects are then introduced by acting with the exponential operator expfT) on 0) with 7 as the cluster operator. The latter consists of a sum of all possible excitation operators... 

The optimization of the Haitree-Fock wave function may be carried out using any of the standard techniques of numerical analysis. However, for many purposes, it is better to use an alternative scheme, which mote directly reflects the physical contents of the Hartree-Fock state. Thus, from the discussion in Section S.l, we recall that an antisymmetric product of spin orbitals represents a state where each electron behaves as an independent particle (but subject to Fermi correlation as discussed in Section S.2.8). This observation suggests that the optimal determinant - that is, the Hartree-Fock determinant in (5.4.3) - may be found by solving a set of effective one-electron Schrbdinger equations for the spin orbitals. Such an approach is indeed possible the effective one-electron Schrddinger equations are called the Hartree-Fock equations and the associated Hamiltonian is the Fock operator... 

Usually, geometries of transition states are significantly more sensitive with respect to method than are stmctures of stable species. Since electron correlation effects are of particular importance for these stmctures, the determination of transition states at the Hartree-Fock level should be avoided. It is recommended to compare the stmctural parameters of transition states obtained from different methods (for instance DFT and MP2) in order not to be misled. 

To ensure this, the-many-body wavefunction can be written as a Slater determinant of one particle wavefunctions - this is the Hartree Fock method. The drawbacks of this method are that it is computationally demanding and does not include the many-body correlation effects. 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

It may be concluded thus that the Half-Projected Hartree-Fock model proposed more than two decades ago for introducing some correlation effects in the ground state wave-function [1,2], could be employed advantageously for the direct determination of the lowest triplet and singlet excited states, in which Ms = 0. This procedure could be especially suitable for the singlet excited states of medium size molecules for which no other efficient procedure exists. 

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