Idempotent Density Matrix Research Articles

Linear Scaling Pseudo Fermi-Operator Expansion for Fractional Occupation.

Recursive Fermi-operator expansion methods for the calculation of the idempotent density matrix are valid only at zero electronic temperature with integer occupation numbers. We show how such methods can be modified to include fractional occupation numbers of an approximate or pseudo Fermi-Dirac distribution and how the corresponding entropy term of the free energy is calculated. The proposed methodology is demonstrated and evaluated for different electronic structure methods, including density functional tight-binding theory, Kohn-Sham density functional theory using numerical orbitals, and quantum chemistry Hartree-Fock theory using Gaussian basis functions.

Quantum crystallography: A perspective.

Extraction of the complete quantum mechanics from X-ray scattering data is the ultimate goal of quantum crystallography. This article delivers a perspective for that possibility. It is desirable to have a method for the conversion of X-ray diffraction data into an electron density that reflects the antisymmetry of an N-electron wave function. A formalism for this was developed early on for the determination of a constrained idempotent one-body density matrix. The formalism ensures pure-state N-representability in the single determinant sense. Applications to crystals show that quantum mechanical density matrices of large molecules can be extracted from X-ray scattering data by implementing a fragmentation method termed the kernel energy method (KEM). It is shown how KEM can be used within the context of quantum crystallography to derive quantum mechanical properties of biological molecules (with low data-to-parameters ratio). © 2017 Wiley Periodicals, Inc.

Communication: Constrained search formulation of the ground state energy as a functional of an idempotent one-matrix

Despite the fact that idempotent one-particle reduced density matrices are pervasive in quantum chemistry, the understanding of a general energy functional of such idempotent density matrices for the ground state energy has been lacking. By a constrained search, we show the structure of the general functional, illuminating the contributions from various terms. For the examples of the "best idempotent density matrix" and Kohn-Sham idempotent density matrices, we contrast the functional forms and suggest how the best idempotent density matrix approach may be a good starting point for further development.

A local energy method based on the reduced density matrix equations

In the Frost local energy method a “best” wavefunction is determined by minimizing the variance of the local energy. Certain simplifications in Frost's method may be obtained by working with the hierarchy equations for reduced density matrices which are obtained by successive integrations over the Schrödinger equation. We define a reduced local energy matrix which is a function of only two variables, in contrast to Frost's local energy which is a function of the number of particles in the system. Considering the Hartree-Fock case a procedure for obtaining a “best” reduced density matrix by minimizing the variance of the reduced local energy matrix is presented. This results in a set of iterative matrix equations for the determination of the normalized idempotent density matrix. Our results which are based on one equation in two variables are compared to the related efforts of Scott who worked directly with the N orbital energies of the Hartree-Fock equations.

An approximate density matrix energy functional

Several authors have shown that there exists an energy functional based on the one-matrix of an atomic or molecular ground state; this functional has been contrasted with energy functionals based on the two-matrix. Based on earlier work of Slater and Lowdin it is shown that the commutation rules satisfied by an approximate energy functional are just those satisfied by the exact functional. Thus there exists a known energy functional which is dependent on the one-matrix. For an idempotent density matrix this is the energy functional proposed originally by Slater.

Methods for obtaining an electron-density matrix from X-ray diffraction data

A method is given for obtaining from X-ray diffraction data an electron density which reflects the antisymmetry of the N-electron wave function. This is achieved by ensuring Hartree-Fock representability via idempotency of the one-body density matrix. Alternate formulisms for achieving an idempotent density matrix are presented. Numerical examples illustrative of the theory are displayed.

Exchange potential via functional differentiation of the Dirac idempotent density matrix

Following two illustrative examples, a quite general form for the functional derivative of the Dirac density matrix $\ensuremath{\gamma}({\mathbf{r}}_{1},{\mathbf{r}}_{2})$ with respect to the electron density $\ensuremath{\rho}(\mathbf{r})$ is proposed. In turn, this functional derivative is related to a further three-point object, which reduces to the linear response function ${\ensuremath{\chi}}_{0}({\mathbf{r}}_{1},\mathbf{r})$ when ${\mathbf{r}}_{2}$ tends to ${\mathbf{r}}_{1}$. The main application is then to derive the exchange-only potential of density functional theory. To avoid heavy numerical work in molecules and clusters, an approximation is suggested, with major calculational simplifications as a probable consequence.

Idempotent density matrix derived from a local potential [formula omitted] in terms of HOMO and LUMO properties

In a recent paper one of us has exposed a link between the idempotent Dirac density matrix built from occupied Kohn–Sham orbitals and the frontier orbitals. An alternative argument is here presented, from the equation of motion of the density matrix generated by any local potential V( r ) .

Corrections to Slater exchange potential in terms of Dirac idempotent density matrix: With an approximate application to Be-like positive atomic ions for large atomic number

In earlier studies, we have considered the exchange energy density εx(r) in terms of the Dirac density matrix ρ1(r,r′) for the nonrelativistic limit of large atomic number Z in (i) the Be-like series with configuration (1s)2(2s)2 and (ii) the Ne-like series with closed K+L shells. Subsequently the work of Della Sala and Görling [J. Chem. Phys. 115, 5718 (2001)] has appeared, in which an integral equation for the exchange potential vx(r) is given in terms of the idempotent Dirac density matrix, based on the admittedly drastic approximation that the Hartree–Fock and the Kohn–Sham determinants are equal. Here a formally exact generalization of the integral equation is set up and an approximate solution is presented for the Be series at large Z.

Ten-Electron Central Field Problem: An Inhomogeneous Electron Liquid

Recent work has been carried out on the exchange energy density epsive;x(r) of a ten-electron atomic ion in the (bare Coulomb) limit of large atomic number Z [Howard, I. A. et al (2000). Phys. Rev. A, 62, 062512]. This analytical study of epsive; x(r) was made possible by the existence of a closed form of the first-order (idempotent) density matrix (IDM). Here, some generalizations are effected to a central potential energy V(r) which (a) localizes the ten electrons and (b) yields closed K and L shells for these ten electrons occupying the lowest eigenstates with spin compensation. In particular, it is shown that p-shell properties alone determine the IDM in this example of a confined inhomogeneous electron liquid.

Nonrelativistic exchange-energy density and exchange potential in the lowest order of the1/Zexpansion for ten-electron atomic ions

The Dirac exchange energy ${E}_{x}$ has a density ${\ensuremath{\varepsilon}}_{x}(r)$ in a spherically symmetric ten-electron atomic ion that is determined by the idempotent first-order density matrix (1 DM). In turn, this 1 DM has as its leading term in the $1/Z$ expansion, with Z the atomic number, the bare Coulomb result. This latter quantity is known analytically from the study of March and Santamaria [Phys. Rev. A 38, 5002 (1988)]. Here ${\ensuremath{\varepsilon}}_{x}(r)$ is calculated analytically, and presented graphically for $Z=92$ in this large-Z limit. The Slater approximation to the exchange potential ${V}_{x}(\mathbf{r}),$ namely, ${V}_{\mathrm{Sla}}(\mathbf{r})=2{\ensuremath{\varepsilon}}_{x}(r)/\ensuremath{\rho}(r)$ with $\ensuremath{\rho}(r)$ the ground-state electron density, is also plotted for $Z=92.$ In the large-Z limit, ${V}_{x}(\mathbf{r})$ can be obtained by functional differentiation of the resultant exchange energy, and is expressed in terms of electron and kinetic-energy densities. Numerical calculations, based on ${V}_{\mathrm{Sla}}(\mathbf{r})$ plus approximate corrections, are also presented.

Exchange, idempotent density matrices and beyond

After a brief historical introduction, emphasis is placed on the treatment of exchange using density matrices that are explicitly idempotent. This embraces both Hartree–Fock and density functional ‘exchange-only’ methods. For an atom, the exchange energy density can be evaluated asymptotically, and unlike the local density approximation it involves not just one length, ρ(r)−1/3, but two, the second being the distance from the nucleus. It is also emphasized, following March and Santamaria [N.H. March, R. Santamaria, Phys. Chem. Liquids 19 (1984) 187] that non-local generalizations of kinetic and exchange energy densities can be related via idemopotency of the Dirac, first-order, density matrix. Examples are then given, starting with the two-level Be atom and its isoelectronic ions, where both Hartree–Fock and density functional exchange-only idempotent density matrices can be expressed in terms of the diagonal density and its gradient, gradρ. This example is followed by the example of a metal surface, but now within a jellium-like-model framework. Some explicit results are again given for such models. Attention is then shifted to a formally exact expression for the exchange-correlation potential, using the differential virial theorem following Holas and March [A. Holas, N.H. March, Phys. Rev. A 51 (1995) 2040]. After a brief physical discussion of the result, in terms now of interacting (as well as non-interacting reference) density matrices; both the now non-idempotent first-order density matrix and the interacting electron pair correlation function, the separation of the Holas–March exchange potential into a sum of exchange and correlation pieces is effected, following the approach of Levy and March [M. Levy, N.H. March, Phys. Rev. A 55 (1997) 1885], in which the strength of the electron–electron interactions is scaled appropriately. Finally, the exact equation of motion for the first-order density matrix is treated, and it is stressed here that this is satisfied exactly by the Hartree–Fock idempotent density matrix. Of course, the correct interacting matrix γ satisfies γ2<γ, where γ now includes the full electron–electron interactions. An example of γ as a function of the electron density is given for He-like atomic ions in the limit of large atomic number.

Potential-locality constraint in determining an idempotent density matrix from diffraction experiment

Potential-locality constraint in determining an idempotent density matrix from diffraction experiment

Density functional theory via density matrices

After a short survey of currently available routes for constructing an idempotent firstorder density matrix as a function(al) of its diagonal element, the electron density p(r); recent progress will be surveyed with specific reference to the exchange potential in (a) the Be atom and (b) in the jellium and Bardeen models of a metal surface. In both examples the off-diagonal density matrix is expressed in terms of its diagonal density only. Finally, a brief discussion will be given of the way the exact exchange-correlation potential can be written in terms of higher-order density matrices, the idempotency condition on the first-order density matrix γ now being replaced by the inequality γ2 < γ. © 1995 John Wiley & Sons, Inc.

Density-matrix refinement for molecular crystals.

The fitting of an idempotent density matrix to x-ray structure factors is discussed. If the density matrix is identified with a single Slater determinant of orbitals, then a model wave function can be recovered, enabling the computation of two-electron properties as well as the usual one-electron properties derived from the electron density. Hence this approach is potentially superior to the usual multipole expansion used in charge-density analysis, although the current formalisms are only useful for molecular crystals of very small molecules. A procedure is described, and applications to one theoretical example (methylamine) and one experimental example (formamide) are presented.

Density matrix equation for crystals

AbstractAn idempotent density matrix in which the orbitals are expanded in Bloch sums is used in the density matrix equation for ρ1(1,1′) to obtain an equation appropriate for a crystal. The general equation is presented as well as its simplification for the single‐cell approximation and the nearest‐neighbor approximation.

Analytic construction of Hartree-Fock density matrices

A method is presented for variational calculation of the energy and the spin densities derived from a single-determinant wavefunction. Sum and difference coordinates [unk]R = (1/2)([unk]r(1) + [unk]r(1)) and [unk]r = [unk]r(1) - [unk]r(1) are introduced, and the density matrix P([unk]r(1),[unk]r(1)) is expanded in partial waves in the new coordinate frame: [Formula: see text] The functions h(L)(epsilon,r) are bound or continuum hydrogenic functions with energy epsilon.It is shown that the spin densities depend on the s partial waves only, and a Euler equation for these partial waves is derived: [Formula: see text] in which U([unk]R) is the electrostatic potential, a(epsilon) = h(0)(epsilon,0), and mu is chosen to normalize the spin density to N electrons. Further, the electronic energy can be expressed in terms of the s partial waves and the constant mu in the above equations.The idempotent density matrix that ensues from a particular choice of functions {B(00)[unk](epsilon,[unk]R)} is generated by choosing partial density waves {B(LM)(epsilon,[unk]R), L > 0} so that tr[P(2)(1 - P)(2)] is minimized.

Notes on density matrix model for coherent x‐ray diffraction

AbstractIt is shown how an idempotent density matrix provides a proper model for interpretation of the x‐ray coherent diffraction experiment.

One-electron properties as variational parameters

The algorithm of Clinton et al. [Phys. Rev. 177, 7 (1969)] for the direct determination of an idempotent density matrix from one-electron expectation values is adapted to perform energy-variational calculations in which the expectation values serve as variational parameters. As an illustration, constrained variational calculations are performed on lithium hydride. A technique is presented to insure that from virtually any idempotent initial guess density matrix, the algorithm will converge to that idempotent density matrix having the desired expectation values which is ’’closest’’ to the initial guess.

Constrained density matrix study of simple basis sets for helium

The method of Clinton et al. for the direct determination of an idempotent density matrix subject to constraints is used to examine in detail the ability of two basis sets, each consisting of a pair of 1s functions to predict one- and two-electron properties for the helium atom. The two bases differ in that one has variationally optimized exponents. Idempotent density matrices constrained to give each of eleven different expectation values computed from the accurate Pekeris wavefunction are constructed for each basis. These constrained densities are then evaluated in terms of the accuracy of the other expectation values compared to the Pekeris values. As the basis becomes energetically optimized, one-electron properties improve, as expected for an independent-particle model, but all computed two-electron properties become less accurate. Several criteria for over-all quality of the electron densities are examined. Although the density can change significantly in specific regions while the energy changes only slightly, the energy is found to remain a prime indicator of average quality. There is evidence that the accuracy with which a free-variational density already predicts a given property may provide a prior indication of the effectiveness of that property as a constraint in a constrained-variational calculation.