Hartree-Fock approximation zeroth-order Hamiltonian

When using the Mpller-Plesset partitioning of the Hamiltonian, the zeroth order Hamiltonian Ho is defined by the Hartree-Fock approximation. For a nondegenerate ground state the matrix elements Vij of the one-particle part of the interaction in Eq. (44) are then given by... 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

A simple way to implement n-particle space truncation is to use the uncorrelated wave function (which as noted above is a very substantial fraction of the exact wave function) to classify terms in the n-particle space. If we consider the Hartree-Fock determinant, for example, we can construct all CSFs in the full n-particle space by successively exciting one, two,.., electrons from the occupied Hartree-Fock MOs to unoccupied MOs. For cases in which a multiconfigurational zeroth-order wave function is required, the same formal classification can be applied. Since only singly and doubly excited CSFs can interact with the zeroth-order wave func tion via the Hamiltonian in Eq. (1), it is natural to truncate the n-particle expansion at this level, at least as a first approximation. We thus obtain single and double excitations from Hartree-Fock (denoted SDCI) or its multiconfigurational reference analog, multireference Cl (MRCI). 

In a recent publication we have investigated this first order approximation to the particle-hole self energy for the choice y>) = o ) for the reference state tp) and starting from a Hartree-Fock zeroth order [21]. This particular approximation to the particle-hole self energy is referred to as First Order Static Excitation Potential (FOSEP). In terms of the matrix elements of the Hamiltonian the FOSEP approximation of the primary block H reads... 

A one-component quasirelativistic DFT method, also a part of the ADF package [92], was extensively used in the calculations for transition element and actinide compounds. (Earlier, the quasirelativistic Hartree-Fock-Slater (QR HFS) method was widely used for such calculations [93]). In this method, the Hamiltonian contains relativistic corrections already in the zeroth-order and is therefore called the zeroth-order regular approximation (ZORA) [94, 95]. The spin operator is also included in the ZORA Fock operator [96]. Other popular quasirelativistic 2c-DFT methods are based on the DKH approximation [97, 98] and implemented in many program packages. The following codes should also be mentioned of Rbsch [99, 100], Ziegler [101], and Case and Young [102]. They were, however, not used for the heaviest elements. A review on relativistic DFT methods for solids can be found in [103]. 

The problem of the exchange term in Hartree-Fock equations has been treated in different ways. The HF-Slater (HFS) method was used in [15]. Numerical SCF calculations of ground-state total energies in relativistic and nonrelativistic approximations are compared in [16, 17]. HFS wavefunctions served as zeroth-order eigenfunctions to compute the relativistic Hamiltonian. In [18], seven contributions to the total energy (including magnetic interaction, retardation, and vacuum polarization terms) are detailed. 

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