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arXiv:physics/0506186

The correlation energy as an explicit functional of the one-particle density matrix from a determinantal reference state

James P. Finley

Published Jun 24, 2005• 0 citations
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Abstract

Using an approach based on many body perturbation theory, the correlation energy $\cEco$ is expressed as an explicit functional of $ρ_1$, $v$, and $v_s$, where $ρ_1$ is the one-particle density matrix from the noninteracting, or reference, determinantal-state; $v$ is the external potential from the interacting, or target, state; $v_s$ 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 $v_{\text{co}}$ 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, $ρ_1$ denotes the spin-less one-particle density matrix from the determinantal reference state.

DOI
10.48550/arxiv.physics/0506186

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Using an approach based on many body perturbation theory, the correlation energy \cEco\cEco\cEco is expressed as an explicit functional of ρ1ρ_1ρ1​, vvv, and vsv_svs​, where ρ1ρ_1ρ1​ is the one-particle density matrix from the noninteracting, or reference, determinantal-state; vvv is the external potential from the interacting, or target, state; vsv_svs​ is the (kernel of the) external potential from the noninteracting determinantal-state. In other words we have \cEco[ρ1,v,vs]\cEco[ρ_1,v,v_s]\cEco[ρ1​,v,vs​]. Anther possibility is the following explicit functional: \cEco[ρ1,vco,vs]\cEco[ρ_1,v_{\text{co}},v_s]\cEco[ρ1​,vco​,vs​], where vcov_{\text{co}}vco​ is the (kernel of the) correlation potential from the noninteracting Hamiltonian. The proposed method can, in principle, be used to compute \cEco\cEco\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\cEco\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, ρ1ρ_1ρ1​ denotes the spin-less one-particle density matrix from the determinantal reference state.
The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of vsv_svs​ and ϕ1ϕ_1ϕ1​, where vsv_svs​ is the local, one-body potential from the Kohn--Sham equations, and ϕ1ϕ_1ϕ1​ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential vsv_svs​. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.
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 vvv-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).
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.
A perturbation theory is developed for the correlation energy Ec{\mathit{E}}_{\mathit{c}}Ec​[n], of a finite-density system, with respect to the coupling constant \ensuremath{\alpha} which multiplies the electron-electron repulsion operator in ${\mathit{H}}^{\mathrm{\ensuremath{\alpha}}}$=T^+\ensuremath{\alpha}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$+tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$(ri{\mathbf{r}}_{\mathit{i}}ri​). The external potential ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is constrained to keep the gound-state density n fixed for all \ensuremath{\alpha}\ensuremath{\ge}0. ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is given completely in terms of functional derivatives at full charge (\ensuremath{\alpha}=1), from which Ec{\mathit{E}}_{\mathit{c}}Ec​[${\mathit{n}}_{\ensuremath{\lambda}}$]=ec,2{\mathit{e}}_{\mathit{c},2}ec,2​[n]+ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ec,3{\mathit{e}}_{\mathit{c},3}ec,3​[n]+${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}2}$ec,4{\mathit{e}}_{\mathit{c},4}ec,4​[n]+..., where each ec,{\mathit{e}}_{\mathit{c},}ec,​j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, ${\mathit{n}}_{\ensuremath{\lambda}}$(x,y,x)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z) and \ensuremath{\lambda}=${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$. The identification of ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$Ec{\mathit{E}}_{\mathit{c}}Ec​[${\mathit{n}}_{\ensuremath{\lambda}}$], which is a high-density limit, as the second-order energy ec,2{\mathit{e}}_{\mathit{c},2}ec,2​[n] allows one to compute bounds upon ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$Ec{\mathit{E}}_{\mathit{c}}Ec​[${\mathit{n}}_{\ensuremath{\lambda}}$]; the bounds imply that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$Ec{\mathit{E}}_{\mathit{c}}Ec​[${\mathit{n}}_{\ensuremath{\lambda}}$]\ensuremath{\simeq}Ec{\mathit{E}}_{\mathit{c}}Ec​[n] for a large class of small atoms and molecules, and suggest that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$Ec{\mathit{E}}_{\mathit{c}}Ec​[${\mathit{n}}_{\ensuremath{\lambda}}$] should be of the same order of magnitude as Ec{\mathit{E}}_{\mathit{c}}Ec​[n] in finite insulators and semiconductors.Approximations to Ec{\mathit{E}}_{\mathit{c}}Ec​[n] should reflect all this. In contrast, perhaps the well-known overbinding of the local-density approximation (LDA) in molecules and solids is due, in part, to the fact that the LDA correlation energy is too sensitive to a coordinate scaling of n. Indeed, the LDA for Ec{\mathit{E}}_{\mathit{c}}Ec​[${\mathit{n}}_{\ensuremath{\lambda}}$] diverges when \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} because of the presence of the -ln(\ensuremath{\lambda}) term in the Gell-Mann and Brueckner high-density expression for the correlation energy, per particle, of a homogeneous density, which is infinite. In a sense, the derived perturbation expansion transforms the Gell-Mann and Brueckner expression into one that applies specifically to an inhomogeneous density which integrates to a finite number of electrons.
The Hohenberg-Kohn theorem and the Kohn-Sham equations, which are at the basis of the density-functional theory, are reformulated in terms of a particular many-body density, which is translational and Galilean invariant and therefore is relevant for self-bound systems. In a similar way that there is a unique relation between the one-body density and the external potential that gives rise to it, we demonstrate that there is a unique relation between that particular many-body density and a definite many-body potential. The energy is then a functional of this density, and its minimization leads to the ground-state energy of the system. As a proof of principle, the analogous of the Kohn-Sham equation is solved in the specific case of 4He^{4}\mathrm{He}4He atomic clusters, to put in evidence the advantages of this formulation in terms of physical insights.
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{H}_{1}, {H}_{2},\dots{},{H}_{M}H1​,H2​,…,HM​ (MMM arbitrarily large) with local potentials v1{v}_{1}v1​,v2{v}_{2}v2​,…\dots{}…,vM{v}_{M}vM​, but given that we do not know which $\ensuremath{\rho}$ belongs with which HHH, 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 vvv'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, aaa density will not generally be associated with any ground-state wave function (is not wave function vvv 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-vvv-representable as well as for vvv-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-vvv representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-vvv representability for some Kohn-Sham noninteracting systems.