Fock-Dirac density matrix

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... 

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

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

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... 

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... 

From the point of view of interpretation it is preferable to write the orbital form of the Fock-Dirac matrix in terms of the number density matrix... 

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 ... 

In fact in this resides the power of the density matrix formalism reducing a many-body problem to the single particle density matrix, abstracted from the single Slater determinant of Eq. (4.190) called also as Fock-Dirac matrix... 

This p is exactly the 1-electron density matrix of determinant ) (it is usually named the Dirac or Dirac-Fock matrix). Following Ldwdin [37], we can term it the fundamental idempotent. The r—matrices basically differ from the Dirac-Fock matrix in its algebraic nature since... 

What is important here is that similar objects can be constructed for RPA and TDDFT. At this stage we need only one basic result of the reduced density matrix (RDM) theory for determinant states. Let p and p be two possible Dirac-Fock matrices of the respective nonorthogonal determinants, ) and l>. Then it is possible to present... 

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. 

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