Dirac density

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

Fock-Dirac density matrix, 225-Framework, 379 Franck-Condon principle, 199 Free volume, 26, 27, 33... 

A Dirac density operator is defined at specified time by its matrix kernel... 

The Dirac density operator for the reference state is idempotent ... 

The determinant is best characterized by the associated Fock-Dirac density matrix.10... 

P is the first-order Fock-Dirac density matrix... 

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. ... 

The Fermi hole in turn is defined in terms of the idempotent Dirac density matrix ys(r, r ) of Eq. (36) where the orbitals < j(x) are solutions of the Har-... 

The field z(r [y]) thus defined is for the interacting system since the tensor involves the density matrix y(r,r ) of Eq. (6). With the field z(r [yj) derived similarly from the tensor tajj(r [ys])) written in terms of the idempotent Dirac density matrix > s( F) of Kohn-Sham theory, the field Z, (r) is then defined as... 

This form is obtained from the small r behavior of the Thomas-Fermi-Dirac density and an exponential decreasing behavior. This is a very simple form that approximates reasonably well the values of p2 r) obtained from the numerical resolution mentioned above. 

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. 

A particular noteworthy feature of ZORA is that even in this zeroth-order approximation there is an efficient relativistic correction for the region close to the nucleus, where the main relativistic effects come from. Excellent agreement of orbital energies and other valence shell properties with results from the Dirac equation has been obtained, and can even be improved within the so-called scaled ZORA variant [607], which takes the renormalization to the transformed large component approximately into account. The ZORA density has been compared in detail with the Dirac density by Autschbach and Schwarz [706]. 

R. McWeeny, Phys. Rev., 126,1028 (1961). Perturbation Theory for the Fock-Dirac Density Matrix. 

McWeeny, R. (1962). Perturbation theory for the Fock-Dirac density matrix. Physical Review, 126,... 

The simplest measure of MO bond order is the Coul-son charge and bond-order matrix, which is essentially the Fock-Dirac density matrix for the minimal basis of occupied valence AOs. When the first-order density matrix is expressed in terms of its NAOs (keyword DMNAO), one obtains a generalization of Coulson s charge and bond-order matrix that exhibits many parallels to the expected Hiickel-like patterns, but can be evaluated for an arbitrarily high ab initio treatment (HF, DFT, or Cl)... 

This factorization of the 2-electron density in terms of the 1-electron p is peculiar to the 1-determinant approximation it means that in this approximation everything is determined by the function p(xi x[), which is often called the Fock-Dirac density matrix (Fock, 1930 Dirac, 1930 Lennard-Jones, 1931). It is in fact clear that the reduced density matrix... 

Show that the Fock-Dirac density matrix (5.3.7), for one determinant of orthonormal spin-orbitals, has the fundamental property... 

The first order Fock-Dirac density matrix is p. From this one constructs a zero order Hamiltonian. For a system of M-electrons, is defined as... 

EFGs and other electric-field-related properties are dealt with in a somewhat different manner. The EFG and multipole moments are calculated as expectation values with the relevant operators and the electron charge density. To avoid PC errors, if the operators are the four-component versions this charge density has to be the four-component (Dirac) density. The latter differs from the two-component density [24,25] already in order which is the same leading order as the relativistic effects on the properties. In the so-called ZORA-4 (Z4) framework, the relevant operators are kept in their four-component form, and an approximate four-component electron charge density is reconstructed from the two-component ZORA density. As was shown by van Lenthe and Baerends [26], the Z4 method eliminates most of the PC errors in order c , with relatively small residual errors. In a Kohn-Sham (KS) DFT framework with two-component molecular orbitals y>, with occupations the ZORA two-component density is... 

---