We present a pedagogical derivation of the Hartree--Fock equations using the second-quantization atomic-orbital density-matrix formalism developed by Kjærgaard, Jørgensen, Olsen, Coriani, and Helgaker for AO-based response theory. The purpose is to introduce an alternative derivation of the Hartree--Fock equation, showing that the standard AO Hartree--Fock stationarity condition follows naturally from the exponential parametrization of the one-particle density matrix in a nonorthogonal AO basis. This route provides a compact bridge between elementary Hartree--Fock theory and the density-matrix machinery used in modern response theory and linear-scaling formulations.
We present a pedagogical derivation of the Hartree--Fock equations using the second-quantization atomic-orbital density-matrix formalism developed by Kjærgaard, Jørgensen, Olsen, Coriani, and Helgaker for AO-based response theory. The purpose is to introduce an alternative derivation of the Hartree--Fock equation, showing that the standard AO Hartree--Fock stationarity condition follows naturally from the exponential parametrization of the one-particle density matrix in a nonorthogonal AO basis. This route provides a compact bridge between elementary Hartree--Fock theory and the density-matrix machinery used in modern response theory and linear-scaling formulations.
Following a recent work [Gál, Phys. Rev. A64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree–Fock equations, the Hartree–Fock–Kohn–Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn–Sham equations, the Hartree–Fock–Kohn–Sham equations and the ground-state Schrödinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn–Sham formulation of the Hartree–Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn–Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N = 1, 2, …), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.
A density-matrix formalism is developed based on the one-particle density-matrix of a single-determinantal reference state. Unlike traditional density-functional-theory approaches, the v-representable problem does not appear in the proposed method, nor the need to introduce functionals defined by a constrained search. The correlation-energy functionals are nonuniversal, in the sense that they depend on the external potential. Nevertheless, model systems can still be used to derive universal energy functionals. Variational and nonvariational energy functionals are introduced that yield the target-state energy when the reference state---or its corresponding one-particle density matrix---is constructed from Brueckner orbitals. Nonvariational energy functionals yield generalized Hartree-Fock equations involving a nonlocal correlation potential and the Hartree-Fock exchange operator; these equations are obtained by imposing the Brillouin-Brueckner condition. The same equations---for the most part---are obtained from variational energy functionals using functional minimization, yielding the (kernel of the) correlation potential as the functional derivative of correlation-energy functionals. Approximations for the correlation-energy functions are introduced, including a one-particle-density-matrix variant of the local-density approximation, a variant of the Colle-Salvetti functional, and a linear combination of the two that is a variant of the correlation-energy functional within the hybrid, three-parameter, Becke-Lee-Yang-Parr density functional (B3LYP).
A system of N antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
Two possible routes are considered to arrive at a one-particle reduced density matrix formulation of electronic structure theory. In the first scheme, an extended Fock matrix H is defined that has twice the dimension of the one-particle basis set. The corresponding Green’s function, defined as the upper left block of (ω1−H)−1, yields the exact one-particle density matrix and energy. The poles of the Green’s function are precisely the ionization potentials and electron affinities of the extended Koopmans theorem. In the second scheme, a generalized Fock equation [F(ρ),ρ]=X is derived that is satisfied by the exact non-idempotent one-particle density matrix. The antisymmetric matrix X on the right-hand side is obtained from the irreducible part of the two-particle reduced density matrix, while F is the usual Fock matrix defined using the correlated one-matrix. The generalized Fock equation is a necessary condition but does not determine ρ uniquely. Alternatively, the one-matrix can be obtained from the irreducible part of the two-matrix directly, using a sum rule. The analysis leads to some additional desiderata and separability properties that may be imposed on traditional wave function based approaches. Possibilities for practical computational schemes are addressed briefly.
We present a direct basis formalism for using nonorthogonal basis sets in the second quantization framework. As an alternative to the dual basis formalism, a direct basis retains the Hermiticity relation between the creation and annihilation operators, with which the form of quantum operators -- e.g. the number operator and the Hamiltonian -- can be readily interpreted and manipulated. To tackle the difficulty of formulating quantum operators in the direct basis, we introduce the coefficient matrix and the generalized creation and annihilation operators, with which an arbitrary N-particle operator can be generated by simple matrix manipulations with the metric tensor of a general basis set. We illustrate the application of the direct basis formalism with the Hubbard Hamiltonian and a dynamical study with the Heisenberg equations of motion
We state in the following a structure theory of the density matrices from the standpoint of secind quantization. The difficulties involved in the quantum-mechanical many body problem have so far proved to be so tremendous that as yet some have been obliged to stop before exact treatments and others have retired to phenomenological model theories. There are two main tools to approach the problem, the theory of second quantization and the theory of density matrix. The former, in spite of its generality and compactness, has never revealed us its adaptability to the practical problems of quantum statistical mechanics. In fact, about the exchange effect, namely the statistical correlation of particles, the latter informs us the detailed features of it by application of the group theory, while the former lacks, at present, such an exposition. For a long time a more general and simpler formulation than the current group theoretic method has been desired in the theoretical researches of spectroscopy. A distinguished contribution has been made by P. Jordan (1934) and H. Ostertag (1937) from the standpoint of second quantization. H. Ostertag made use of the theory of hypercomplex ring and introdeced the original concept of the elementary algebra. (Part II) Since the latter part of his formulation, however, seems to be complicated, it will be better, for clearness' sake, to avoid this part. Accordingly the concepts of the elementary algebra and the symmetric algebra will be introduced as a remedy for it. Then we shall complete our theory to include the whole theory of symmetric group. These concepts will serve to clarify and generalize the previous formulations and to unify the several group theoretic standpoints of the spectroscopy. In §1 and §2 the representation of the quantized density of Bose or Fermi particles will reveal itself as the density matrix of Bose or Fermi particles or as the density matrix of many electrons with a certain term of spin multiplets. Indeed, the theory of the statistical operator, or the density matrix of J. von Neumann and P.A.M. Dirac, the equilibrium properties of which have been fully examined by one of the authors (1940), is the natural formalism of dealing with quantum gases. A contribution has been made by W. Kofink (1938) to introduce the intrinsic degrees of freedom of particles. This attempt has resulted in a fair seccess. His formulation consists in making use of the initial stages of the theory of spectroscopy, which will be reexamined in the second section and Part II. In §3, as an example at zero temperature, the Hartree-Fock equation for a prescribed orbital and spin configuration will be derived, and in §4 some properties of the reduced density matrix will be illustrated.