Fock-Dirac operator

In this paper, the general theory developed in Part I is applied to the Hartree-Fock Scheme for a transformed many-electron Hamiltonian. It is shown that, if the transformation is a product of one-electron transformations, then the Fock-Dirac operator as well as the effective Hamiltonian undergo similarity transformations of the one-electron type. The special properties of the Hartree-Fock scheme for a real self-adjoint Hamiltonian based on the bi-variational principle are discussed in greater detail. 

This study was started in order to find out whether one could find meaningful complex eigenvalues in the Hartree-Fock scheme for a transformed Hamiltonian in the method of complex scaling. This problem was intensely discussed at the 1981 Tarfala Workshop in the Kebnekaise area of the Swedish mountains. It was found that, if the many-electron Hamiltonian undergoes a similarity transformation U which is a product of one-electron transformations u - as in the method of complex scaling - then the Fock-Dirac operator p as well as the effective Hamiltonian Heff undergo one-electron similarity... 

In this way, it is hence possible to separate the orbital parts and the spin parts of the Fock-Dirac operator in a simple way. The... 

Recalling the notation d = -i, one obtains according to (2.8) for the Fock-Dirac operator... 

In summary, conventional relativistic ECP s provide an efficient mean to calculate molecular properties up to and including the third row transition elements in cases where the spin-orbit coupling is weak. ECP s can also be used together with explicit relativistic no-pair operators. Such ECP s are somewhat more precise at at the atomic level, but of essentially the same quality as conventional relativistic ECP s in molecular applications. It should also be possible to combine the ECP formalism with full Fock-Dirac methods, but this has yet not been done. 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

Thus in general the elements of the group G = S x T do not commute with the effective one-electron Hamiltonian (2.25). Some of these operations may, however, commute with 9tefr and in such a case they form a subgroup of G. That subgroup characterizes the GHF solution under study in the sense that the corresponding Fock-Dirac matrix is invariant under the elements g of the invariance group of the GHF solution,... 

One avoids such unphysical states if one interprets the Dirac operator as acting in a Fock space, with a vacuum, in which no electrons and no positrons are occupied. Annihilation (creation) operators for negative-energy states are then interpreted as positron creation (annihilation) operators. 

That all the operators in are invariant under the transformation t is obvious except perhaps for which depends on the orbitals. Vf, Eq. (8), can be written in terms of the Fock-Dirac density p(x<,X/). The summation in p(X<,x ) covers a set of molecular orbitals which form a basis for certain irreducible representations. This is invariant under any unitary transformation. ... 

Equation (4.210) is most informative since, basing on the idem potency property of Eq. (4.203), through multiplying it on the right with Fock-Dirac density operator. 

Here n is the number of (in the restricted Hartree-Fock case, doubly filled) bands and J and K are again the Coulomb and exchange operators, respectively, formed now with the aid of the four-component relativistic HF COs instead of the nonrelativistic one-component HF COs. In deriving equation (1.110) the time-dependent factor of has been eliminated by diiferentiating with respect to time contained in the original Dirac operator... 

While the expansion of hpq follows readily from the development for the Dirac operator above, the electron-electron interaction integrals must be considered separately. We also want to develop a Kramers-restricted Dirac-Hartree-Fock (KR-DHF) theory, but first we develop expressions for the general, Kramers-unrestricted case. In the developments below we follow the practice of giving only the basis function index in the integrals. 

Dirac-Fock (DF) level, ab initio calculations, P,T-odd interactions, 253 Dirac matrices, P,T-odd interaction operator, 251-253... 

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

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