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.
Using an approach based on many body perturbation theory, the correlation energy $\cEco$ is expressed as an explicit functional of
ρ1,
v, and
vs, where
ρ1 is the one-particle density matrix from the noninteracting, or reference, determinantal-state;
v is the external potential from the interacting, or target, state;
vs is the (kernel of the) external potential from the noninteracting determinantal-state. In other words we have $\cEco[ρ_1,v,v_s]$. Anther possibility is the following explicit functional: $\cEco[ρ_1,v_{\text{co}},v_s]$, where
vco is the (kernel of the) correlation potential from the noninteracting Hamiltonian. The proposed method can, in principle, be used to compute $\cEco$ in a very accurate and efficient manner, since, like the Kohn--Sham approach, there are no virtual orbitals to consider. However, in contrast to the Kohn--Sham approach, $\cEco$ is a known, explicit functional that can be approximated in a systematic manner. For simplicity, we only consider noninteracting closed-shell states and target states that are nondegenerate, singlet ground-states; so, in that case,
ρ1 denotes the spin-less one-particle density matrix from the determinantal reference state.
Density functional theory is currently the most widely applied method in electronic structure theory. The Kohn-Sham method, based on a fictitious system of non-interacting particles, is the work horse of the theory. The particular form of the Kohn-Sham wavefunction admits only idem-potent one electron density matrices whereas wavefunctions of correlated electrons in post-Hartree-Fock methods invariably have fractional occupation numbers. Here we show that by generalizing the orbital concept, and introducing a suitable dot-product as well as a probability density a non-interacting system can be chosen that can represent the one-electron density matrix of any system, even one with fractional occupation numbers. This fictitious system ensures that the exact electron density is accessible within density functional theory. It can also serve as the basis for reduced density matrix functional theory. Moreover, to aid the analysis of the results the orbitals may be assigned energies from a mean-field Hamiltonian. This produces energy levels that are akin to Hartree-Fock orbital energies such that conventional analyses based on Koopmans theorem are available. Finally, this system is convenient in formalisms that depend on creation and annihilation operators as they are trivially applied to single determinant wavefunctions.
By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that ${\ensuremath{\rho}}_{\ensuremath{\alpha}}, {\ensuremath{\rho}}_{\ensuremath{\beta}},\dots{},{\ensuremath{\rho}}_{\ensuremath{\omega}}$ are ground-level densities for interacting or noninteracting Hamiltonians
H1,H2,…,HM (
M arbitrarily large) with local potentials
v1,
v2,
…,
vM, but given that we do not know which $\ensuremath{\rho}$ belongs with which
H, the correct mapping is possible and is obtained by minimizing $\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}} [{v}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+{v}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+\ensuremath{\cdots}{v}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\omega}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})]$ with respect to optimum permutations of the $\ensuremath{\rho}$'s among the
v's. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies,
a density will not generally be associated with any ground-state wave function (is not wave function
v representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to density-functional theory is emphasized for non-
v-representable as well as for
v-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-
v representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-
v representability for some Kohn-Sham noninteracting systems.
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 formulation of the ab initio methods of nonrelativistic quantum mechanics is presented. The variational equations for the standard calculational techniques of molecular quantum mechanics are derived from a single expression for the total energy of an N-particle system. A generalization of the Hohenberg-Kohn theorem is proved, showing that the ground-state energy of a many-body Hamiltonian is a functional of the single-particle density matrix and hence of the first-order reduced-density function (the diagonal of which is the particle density of the standard theorem). The theorem is used to develop a new method to calculate the total energy and single-particle properties of the many-body problem.