Hartree-Fock method operator

In this expression, the Fa operator is the usual Fock operator of the Unrestricted Hartree-Fock method [14] ... 

In most work reported so far, the solute is treated by the Hartree-Fock method (i.e., Ho is expressed as a Fock operator), in which each electron moves in the self-consistent field (SCF) of the others. The term SCRF, which should refer to the treatment of the reaction field, is used by some workers to refer to a combination of the SCRF nonlinear Schrodinger equation (34) and SCF method to solve it, but in the future, as correlated treatments of the solute becomes more common, it will be necessary to more clearly distinguish the SCRF and SCF approximations. The SCRF method, with or without the additional SCF approximation, was first proposed by Rinaldi and Rivail [87, 88], Yomosa [89, 90], and Tapia and Goscinski [91], A highly recommended review of the foundations of the field was given by Tapia [71],... 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

In this respect, the single-configurational Hartree-Fock method looks more promising and universal when combined with accounting for the relativistic effects in the framework of the Breit operator and for correlation effects by the superposition-of-configurations or by some other method (e.g. by solving the multi-configurational Hartree-Fock-Jucys equations (29.8), (29.9)). 

Y -Hab(Hbb -E)- Hbd Haa -Hab(Hbb -E)"Hh If the Hamiltonian is a one-electron Hamiltonian, for example the Fock operator, the partitioning is done by basis functions, since the latter are usually centered on the atomic nuclei, which belong to donor (d), bridge (b) or acceptor (a). In the Hartree-Fock case, the total wave function is a Slater determinant. There may be problems with symmetry breaking in the symmetric case. Cl that includes the two localized solutions can solve this problem [29-31]. The problem is that the Hartree-Fock method gives energy advantage to a localized state, which holds true also in the unsymmetric case. 

As in the conventional Hartree-Fock method, the approximate eigenvalue I is hence essentially different from the sum of the N eigenvalues of the one-particle operator. 

Special Case when T = T = T. Let us now consider the special case when a complex symmetric operator is real, so that T = T. In this case, the operator T is also self-adjoint, T = T, and one can use the results of the conventional Hartree-Fock method 7. The eigenvalues are real, X = X. and - if an eigenvalue X is non-degenerate, the associated eigenfunction C is necessarily real or a real function multiplied by a constant phase factor exp(i a). In both cases, one has D = C 1 = C. In the conventional Hartree-Fock theory, the one-particle projector p takes the form... 

The UHF ansatz is necessary because in case of neutral solitons one has to deal with a doublet state. Thus a DODS (different orbitals for different spins) ansatz, as the UHF one, is necessary to describe the system. However, in the UHF method described so far, one Slater determinant with different spatial orbitals for electrons of different spins is applied, which is not an eigenfunction of S2, i.e. S(S+l)h2. The best way to overcome this difficulty would be to use the PHF (Projected Hartree Fock) method, also called EHF method (Extended Hartree Fock) where before the variation the correct spin eigenfunction is projected out of the DODS ansatz Slater determinant [66,67a]. Unfortunately numerical solution of the rather complicated EHF equations in each time step seems to be too tedious at present. Moreover for large systems the EHF wavefunction approaches the UHF one [68], however, this might be due to the approximations used in [67a]. Another possibility is to apply the projection after the variation using again Lowdin s projection operator [66]. Projection and annihilation techniques were... 

In fact, this is the principle role of the exchange term to cancel the unphysical self-repulsion in the Coulomb sum. It is the difference between the Hartree and the Hartree-Fock methods, and the reason why all the MOs are the eigenfunctions of the same Hartree-Fock operator, while a separate Hartree operator is needed for each MO which excludes the self-repulsion for that MO. 

It is obvious from the last chapter that much of the work of any implementation of the Hartree-Fock method consists of operations with matrices which are stored in a computer as (usually contiguous) strings of numbers. However, it is organisationally convenient to distinguish between a matrix (or a vector) and a string or array of numbers, since this distinction affects the way in which matrices are stored and manipulated. 

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