Hartree-Fock-Slater method total energy calculations

The calculation of Mitroy started by calculating the Hartree—Fock approximation to the ground state 3s where we denote the states by the orbitals of the two active electrons in the configuration with the largest coefficient, in addition to the symmetry notation. The calculation used the analytic method with the basis set of Clementi and Roetti (1974) augmented by further Slater-type orbitals in order to give flexibility for the description of unoccupied orbitals. The total energy calculated by this method was —199.614 61, which should be compared with the result of a numerical Hartree—Fock calculation, —199.614 64. 

This difficulty is overcome with the aid of a projection operator by projecting out from the Slater determinant the component with the desired multiplicity 25+1, annihilating all other contaminating components. This can be done either after an already performed calculation (spin projection after variation, UHF with annihilation), or, as Lowdin has pointed out, one would expect a more negative total energy if the variation is performed with an already spin-projected Slater determinant [spin projection before variation, spin-projected extended Hartree-Fock (EHF) method]. The reason is that a spin-projected Slater determinant is a given linear combination of different Slater determinants. The variation in the expectation value of the Hamiltonian formed with a spin-projected Sater determinant thus provides equations (EHF equations), whose solutions represent the solution of this particular multiconfigura-tional SCF problem. 

DFT has come to the fore in molecular calculations as providing a relatively cheap and effective method for including important correlation effects in the initial and final states. ADFT methods have been used, but by far the most popular approach is that based on Slater s half electron transition state theory [73] and its developments. Unlike Hartree-Fock theory, DFT has no Koopmans theorem that relates the orbital energies to an ionisation potential, instead it has been shown that the orbital energy (e,) is related to the gradient of the total energy E(N) of an N-electron system, with respect to the occupation number of the 2th orbital ( , ) [74],... 

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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