We discuss the one-particle density-matrix functional theory for the N-particle ground-state problem. Using the variational principle we obtain a set of self-consistent equations which could be useful for practical calculations. The derived relation between the eigenvalues of the density matrix and the chemical potential agrees with that of Gilbert. The present theory is compared with the Kohn–Sham and the Hartree–Fock approach to the ground state problem. It is found that both approaches can be obtained from the present theory under certain assumptions.
We discuss the one-particle density-matrix functional theory for the N-particle ground-state problem. Using the variational principle we obtain a set of self-consistent equations which could be useful for practical calculations. The derived relation between the eigenvalues of the density matrix and the chemical potential agrees with that of Gilbert. The present theory is compared with the Kohn–Sham and the Hartree–Fock approach to the ground state problem. It is found that both approaches can be obtained from the present theory under certain assumptions.
In parallel with standard density-functional theory, we study the energy of the ground state of a finite many-body system as a functional of the one-particle density matrix. We show that the formulation of a variational principle that is valid beyond the Hartree-Fock limit requires that two-body correlations be included not only in the ground-state energy but also in the constraints. As an illustration, we apply a linear-response argument to derive formulas for first-order corrections to the Hartree-Fock density matrix. Further analysis suggests an approach in terms of the density matrix of an independent-particle system, which can be introduced by the application of an alternative variational principle. This approach is reminiscent of Kohn-Sham theory, but the effective external potential is not required to be local. This variational method can be implemented in a systematic fashion by means of the linked-cluster expansion. In an appendix we study a variant of the Hohenberg-Kohn theorem for nonlocal potentials.
The self consistent version of the density functional theory is presented, which allows to calculate the ground state and dynamic properties of finite multi-electron systems. An exact functional equation for the effective interaction, from which one can construct the action functional, density functional, the response functions, and excitation spectra of the considered systems, is outlined. In the context of the density functional theory we consider the single particle excitation spectra of electron systems and relate the single particle spectrum to the eigenvalues of the corresponding Kohn-Sham equations. We find that the single particle spectrum coincides neither with the eigenvalues of the Kohn-Sham equations nor with those of the Hartree-Fock equations.
The Hohenberg–Kohn theorem implies the existence of an energy functional based solely on the first-order reduced density matrix of the ground state of an atomic or molecular system. Application of the variational principle for the functional EvM[γ] generates a set of coupled Euler equations for the representation coefficients and spin orbitals of a rank-M approximation to the exact ground-state density matrix. Defining the (assumed Hermitian) kernel FM[γ;x′,x]≡δEvM/δγ (x,x′), the equations in an arbitrary representation for the approximate density matrix, γ (x,x′) =ΣijMψ1(x) γijψj* (x′) are the following: ℱdξFM[γ;x′,ξ]ψi(ξ) =μψi(x′); i=1, 2,...,M; ℱdξFM[γ;x′,ξ] ΣMiψi(ξ) γ ij=ΣMiψi(x′) λij; j=1, 2, ...,M. The quantity μ is the chemical potential of the system of interest and the λij are a set of M2 Lagrange multipliers constraining the orthonormality of the spin orbital basis {ψ}. The coefficients γij must be chosen such that FM has the degenerate eigenvalue spectrum, FiiM=μ, i=1, 2,...,M, for all partially occupied orbitals. The spin orbitals, for a fixed set of coefficients, must be simultaneously determined so that λ is Hermitian. Provided all occupation numbers lie on the open interval (0,1), the stationary density matrix itself obeys the eigenvalue equation ℱdξFM[γ;x,ξ]γ (ξ,x′) =μγ (x,x′), and for any stationary density matrix the following commutation rules are valid: [FM[γ], γ]=0; [λ, γ]=0. The matrices γ and λ consequently can be brought simultaneously to diagonal form, and the canonical representation of the energy functional provides an eigenvalue equation determining the natural spin orbitals. The value of the chemical potential is μ=εi/ni, i=1, 2, ...,M; ε=diag (λ); from which follows the distribution function governing the occupation numbers of a stationary density matrix. Two limiting forms of the variational principle are examined, the exact and Hartree–Fock functionals, and related previous work by Gilbert is discussed. The physical content of the equations is illuminated by identification of the chemical potential as the negative of the electronegativity; Sanderson’s Principle of Electronegativity Equalization follows.
A system of electrons in a local or nonlocal external potential can be described with one-matrix functional theory (1MFT), which is similar to density-functional theory (DFT) but takes the one-particle reduced density matrix (one-matrix) instead of the density as its basic variable. Within 1MFT, Gilbert derived [Phys. Rev. B 12, 2111 (1975)] effective single-particle equations analogous to the Kohn-Sham (KS) equations in DFT. The self-consistent solution of these 1MFT-KS equations reproduces not only the density of the original electron system but also its one-matrix. While in DFT it is usually possible to reproduce the density using KS orbitals with integer (0 or 1) occupancy, in 1MFT reproducing the one-matrix requires in general fractional occupancies. The variational principle implies that the KS eigenvalues of all fractionally occupied orbitals must collapse at self-consistency to a single level. We show that as a consequence of the degeneracy, the iteration of the KS equations is intrinsically divergent. Fortunately, the level-shifting method, commonly introduced in Hartree-Fock calculations, is always able to force convergence. We introduce an alternative derivation of the 1MFT-KS equations that allows control of the eigenvalue collapse by constraining the occupancies. As an explicit example, we apply the 1MFT-KS scheme to calculate the ground state one-matrix of an exactly solvable two-site Hubbard model.
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.
We propose a framework to construct the ground-state energy and density matrix of an N-electron system by solving a self-consistent set of single-particle equations. The method can be viewed as a nontrivial extension of the Kohn-Sham scheme (which is embedded as a special case). It is based on separating the Green's function into a quasiparticle part and a background part, and expressing only the background part as a functional of the density matrix. The calculated single-particle energies and wave functions have a clear physical interpretation as quasiparticle energies and orbitals.