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International Journal of Quantum Chemistry

Many-electron theory: Density functional approach generalized to treat spin eigenfunctions and relation to spinless low-order density matrices

N. H. March

Published Jan 1, 1999• 2 citations
Publisher

Abstract

Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999

DOI
10.1002/(sici)1097-461x(1999)74:2<163::aid-qua13>3.0.co;2-2

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Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.
Abstract Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, { V α eff = ∂ E e e [ { n β } ] / ∂ n α } . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.
In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of NNN antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order kkk has two sets of indices of each kkk variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order NNN over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete kkk-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly NNN spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order MMM, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.
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.
The formulation of the density matrix functional theory (DMFT) in which the correlation component U of the electron–electron repulsion energy is expressed in terms of a model two-electron density cumulant matrix (a.k.a. the 2-cumulant) and the tensor of two-electron integrals g is critically analyzed. The dependence of on both the vector of occupation numbers n and g derived from the corresponding vector (1) of natural spinorbitals, and its concomitant invariance to the nature of the spin-independent interparticle interaction potential that enters the definition of g are emphasized. In the case of spin-polarized systems, (and thus U ) is found to be a function of not only n and g, but also of the vector and the matrix, where is the spin-flip operator. The presence of spin polarization imposes additional constraints upon, including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the and operators with the eigenvalues and, respectively. This feature allows targeting electronic states with definite multiplicities, which is virtually impossible in the case of the Kohn–Sham implementation of density functional theory. Among the four possible pairing schemes for the natural spinorbitals that give rise to approximations employing with only two independent indices, three are found to result in unphysical constraints even for spin-unpolarized systems, whereas the failure of the fourth one turns out to be precipitated by the presence of spin polarization. Consequently, any implementation of DMFT based upon “two-index” is shown to be generally unsuitable for spin-polarized systems (and incapable of yielding the spin-parallel components of U for the spin-unpolarized ones). A clear distinction is made between the genuine 1-matrix functionals that are defined for arbitrary N -representable 1-matrices and general energy expressions that depend on auxiliary quantities playing the role of fictitious 1-matrices subject to additional (often unphysical) constraints.
A system of NNN 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.