Fock-Dirac density operators

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. 

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. ... 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

The Fock operator f and the one-particle density matrix 7 commute, i.e. have common eigenfunctions. This allows an iterative construction of 7 from the eigenstates of f. The leading relativistic corrections for the Dirac-Coulomb operator are ... 

Figure 16.4 Total electronic densities of M(C2H2) with M=Ni,Pt from Hartree-Fock calculations with two-component ZORA, scalar-relativistic DKH10, and nonrelativistic Schrodinger one-electron operators subtracted from the four-component Dirac-Hartree-Fock reference densities (data taken from Ref. [880]). The molecular structure of the complexes is indicated by element symbols and lines positioned just below the atomic nuclei (top panel). Asymmetries in the plot are due to the discretization of the density on a cubic grid of points. The DKH densities have not been corrected for the picture-change effect and, hence, deviate from the four-component reference density in the closest proximity to the nuclei. But these effects can hardly be resolved on the numerical grid employed to represent the densities.

Bold quantities are operators, vectors, matrices or tensors. Plain symbols are scalars. a Polarizability a, P Spin functions a, p Dirac 4x4 spin matrices ap-jS Summation indices for basis functions F Fock operator or Fock matrix Fy, Eajd Fock matrix element in MO and AO basis Y Second hyperpolarizability yk Density matrix of order k gc Electronic g-factor... 

If effects from electron correlation on parity violating potentials shall be accounted for in a four-component framework, the situation becomes more complicated than in the Dirac Hartree-Fock case. This is related to the fact, that in four-component many body perturbation theory (MBPT) or in a four-component coupled cluster (CC) scheme the reduced density matrices on the respective computational level are required in order to determine the parity violating potentials. Since these densities were not available in analytic form, Thyssen, Laerdahl and Schwerdtfeger [153] used a finite field approach to compute parity violating potentials in a four-component framework on a correlated level. This amounts to adding the parity violating operator with different scaling factors A to the... 

The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

Although the Thomas-Fermi method is an interesting theory representing the Hamiltonian operator as the functional only of the electron density, even qualitative discussions cannot be contemplated based on this method in actual electronic state calculations. Dirac considered that this problem may be attributed to the lack of exchange energy (see Sect. 2.4), which was proposed in the same year (Fock 1930), and proposed the first exchange functional of electron density p (Dirac 1930),... 

The purpose of this contribution is to give an overview of the results which center around the atomic density function and the recovery of the periodicity. Since all the calculations are based on atomic density functions, it is appropriate to revisit the construction of these densities in some depth. First a workable definition of the density function is established in the framework of the multi-configuration Hartree-Fock method (MCHF) [2] and the spherical harmonic content of the density function is discussed. A spherical density function is established in a natural way, by using spherical tensor operators. The proposed expression can be evaluated for any multi-configuration state function corresponding to an atom in a particular well-defined state and a recently developed extension of the MCHF code [3] is used for that purpose. Three illustrative examples are given. In the next section relativistic density functions for the relativistic Dirac-Hartree-Fock method [4] are defined. The latter will be used for a thorough analysis of the influence of relativistic effects on electron density functions later on in this paper. 

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