Idempotent Dirac Density Matrix Research Articles

The exchange-correlation potential of DFT obtained from a semiempirically fine-tuned Hartree–Fock density for inhomogeneous electron liquids

The present authors have given an exact theory of the exchange-correlation potential V xc (r) in terms of (i) the exact ground-state electron density n(r) and (ii) the idempotent Dirac density matrix γ(r, r′) generated by the DFT one-body potential V(r), having n(r) as its diagonal element. Here, we display two approximate consequences: (a) a form of V xc (r) generated by the semiempirically fine-tuned HF density of Cordero et al. (N.A. Cordero, N.H. March, and J.A. Alonso, Phys. Rev. A 75, 052502 (2007)) and (b) the exchange-only potential V x (r) determined solely by the HF ground state density for the Be atom.

Relation between the Slater–Kohn–Sham orbitals generated by the one-body potential V(r) of DFT and the Löwdin natural orbitals for inhomogeneous electron liquids

The Löwdin natural orbitals (NO) are defined as those orbitals which bring the first-order density matrix of a correlated electron assembly into diagonal form. Another one-particle density matrix with the same diagonal elements is the single-particle idempotent Dirac density matrix generated by the one-body potential of density functional theory. Here, we compare the off diagonal form of γ expanded in terms of the Slater–Kohn–Sham (SKS) orbitals generated by V(r) with the NO expansion of Löwdin for general inhomogeneous electron liquids. In particular, the equation of motion of the correlated γ is corrected from that of γ s , both now containing the one-body potential V(r). To illustrate the theory presented here, we first construct an approximate, albeit accurate, correlated 1DM for the ground state of the He atom and display connections between the resulting NOs and the SKS orbitals. The second example we discuss, but now quite briefly, is that of the inhomogeneous electron liquid in crystalline Si, where the NO expansion is available from the literature.

Towards an exact orbital-free single-particle kinetic energy density for the inhomogeneous electron liquid in the Be atom

Holas and March [Phys. Rev. A51, 2040 (1995)] wrote the gradient of the one-body potential V(r) in terms of low-order derivatives of the idempotent Dirac density matrix built from a single Slater determinant of Kohn–Sham orbitals. Here, this is first combined with the study of Dawson and March [J. Chem. Phys. 81, 5850 (1984)] to express the single-particle kinetic energy density of the Be atom ground-state in terms of both the electron density n(r) and potential V(r). While this is the more compact formulation, we then, by removing V(r), demonstrate that the ratio t(r)/n(r) depends, though non-locally, only on the single variable n′(r)/n(r), no high-order gradients entering for the spherical Be atom.

One-body potential theory of molecules and solids modified semiempirically for electron correlation

The study of Cordero, March and Alonso (CMA) for four spherical atoms, Be, Ne, Mg and Ar, semiempirically fine-tunes the Hartree–Fock (HF) ground-state electron density by inserting the experimentally determined ionization potentials. The present Letter, first of all, relates this approach to the very recent work of Bartlett ‘towards an exact correlated orbital theory for electrons’. Both methods relax the requirement of standard DFT that a one-body potential shall generate the exact ground-state density, though both work with high quality approximations. Unlike DFT, the CMA theory uses a modified HF non-local potential. It is finally stressed that this potential generates also an idempotent Dirac density matrix. The CMA approach is thereby demonstrated to relate, albeit approximately, to the DFT exchange-correlation potential.

Relation between single-particle kinetic energy and exchange energy in DFT for the inhomogeneous electron liquid in the Be atom

In a recent study, the authors have used the semi empirical fine-tuned Hartree–Fock ground-state electron density n(r) of Cordero et al. [Phys. Rev. A 75, 052502 (2007)] for the Be atom to calculate the phase θ(r) from a non-linear pendulum-like equation. Since the density amplitude n(r)1/2 plus θ(r) determine, in turn, the idempotent Dirac density matrix γ(r, r′), we use n(r) and θ(r) first of all to calculate the exchange energy density e X (r) of the density functional theory (DFT). This enables us to obtain the Slater (Sl) approximation to the exchange-only potential. A comparison can then be made, by integrating the earlier predicted exchange-correlation force −∂V XC (r)/∂r, of V XC (r) with . Relationship to the Becke semiempirical density gradient approximation for exchange is also established. Some brief discussion of the Perdew–Burke–Ernzerhof density functional is added.

Exchange energy for two closed shells generated by a bare Coulomb potential energy −Ze 2/r in the limit of large Z, in two dimensions

Two-dimensional (2D) inhomogeneous electron assemblies are becoming increasingly important in Condensed Matter and associated technologies. Here, therefore, we contribute to the Density Functional Theory of such 2D electronic systems by calculating, analytically, (i) the idempotent Dirac density matrix γ(r, r′) generated by two closed shells for the bare Coulomb potential −Ze2/r and (ii) the exchange energy density \({\varepsilon_x({\bf r})}\) . Some progress is also possible concerning the exchange potential Vx(r), one non-local approximation being the Slater potential \({2\varepsilon_x(r)/n(r)}\) , with n(r) the ground state electron density. However, to complete the theory of Vx(r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.

Correlated and idempotent Dirac first-order density matrices with identical diagonal Fermion density: a route to extract a one-body potential energy in TDDFT

After a brief introduction to the use of the idempotent Dirac first-order density matrix (DM), its time-dependent generalization is considered. Special attention is focused on the equation of motion for the time-dependent DM, which is characterized by the one-body potential V(r, t) of time-dependent density functional theory. It is then shown how the force –∇V(r, t) can be extracted explicitly from this equation of motion. Following a linear-response treatment in which a weak potential V(r, t) is switched on to an initially uniform electron gas, the non-linear example of the two-electron spin-compensated Moshinsky atom is a further focal point. We demonstrate explicitly how the correlated DM for this model can be constructed from the idempotent Dirac DM, in this time-dependent example.

Exchange energy density in density-functional theory via the Dirac density matrix for a nonrelativistic 10-electron atomic ion compared with Becke’s proposal for a gradient-corrected local-density-approximation result

Especially in atomic systems, it is now well established that exchange energy generally dominates correlation effects. Therefore we focus here on the exchange energy density ${\ensuremath{\epsilon}}_{x}(r)$ as given in terms of the idempotent Dirac density matrix. This is then brought into contact with the one-parameter form of Becke's functional, which corrects the local-density-approximation form $\ensuremath{-}{c}_{x}{[n(r)]}^{4/3}$ with $n(r)$ as the ground-state electron density, ${c}_{x}=(3/4){e}^{2}{(3/\ensuremath{\pi})}^{1/3}$, by terms involving the dimensionless gradient ratio $|\ensuremath{\nabla}n(r)|/{n}^{4/3}(r)$. A particular nonrelativistic model of the 10-electron Ne-like atomic ions, with large atomic number $Z$, is then compared to Becke's approximation to ${\ensuremath{\epsilon}}_{x}(r)$.

Idempotent Dirac density matrix for ten-electron central field inhomogeneous electron liquids in terms of electron- and kinetic energy-densities

Abstract A differential equation for the Dirac density matrix γ ( r , r ′ ) , given ground-state electron- and kinetic energy-densities, has been derived by March and Suhai for one- and two-level occupancy. For ten-electron spin-compensated spherical systems, it is shown here that γ ≡ γ [ ρ , t g ] where ρ and t g are electron- and kinetic energy-densities. The philosophy of March and Suhai is confirmed beyond two-level filling. An important byproduct of the present approach is an explicit expression for the one-body potential of DFT in terms of the p-shell electron density.

The idempotent Dirac density matrix as a functional of the diagonal ground‐state electron density obtained from diffraction experiments

AbstractThe idempotent Dirac single‐particle (s) density matrix γs(r,r′) as a functional of a given one‐body potential V(r) was constructed to all orders in perturbation theory by March and Murray, some four and a half decades ago. While the subsequent theorems of DFT ensure that V is a functional of the ground‐state density ρ, only recently has V been written explicitly as a functional of the Dirac density matrix γs. In the present study, both functionals γs[ρ] and γs[V] are studied; first for some model potentials V and somewhat then more generally in terms of partial differential equations. This later procedure should open up a route to treat numerically molecules of biophysical interest, provided the ground‐state electron density ρ(r) is initially determined from diffraction experiments. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Single-particle kinetic energy in density-functional theory: Harmonic confinement in two and three dimensions

Using the known perturbation series for the idempotent Dirac density matrix in powers of a given one-body potential $V(\mathbf{r})$, Stoddart and March (SM) generated a corresponding series for the kinetic energy density. While the general term of the SM series is known, a summation has not been achieved to date. A contribution to solve this problem is made here by exhibiting an explicit form of the above kinetic energy density for Fermions filling an arbitrary number of closed shells, when the confinement is harmonic. This example is of considerable current interest because of ongoing experiments on ultracold atomic gases of Fermions.

Exchange-correlation potential in terms of the idempotent Dirac density matrix of DFT

A formal proof has recently been proposed [I.A. Howard, N.H. March, Chem. Phys. Lett. 402 (2005) 1.] to show that the exchange-correlation potential V xc( r) of density functional theory (DFT) is, in principle, determined by the idempotent Dirac density matrix γ( r, r′) generated by the one-body potential V( r) = V ext( r) + V Hartree( r) + V xc( r). Here we turn this ‘existence’ theorem into a concrete result by expressing the Laplacian ∇ 2 V xc( r) solely in terms of γ( r, r′), which, although idempotent and therefore not many-electron in character, has the exact ground state density ρ( r) on its diagonal γ( r, r). As a specific illustration, we calculate ρ( r) by diffusion QMC and hence γ( r, r′) for the Be atom ground-state. V xc( r) is thereby obtained with satisfactory accuracy.

Can the exchange-correlation potential of density functional theory be expressed solely in terms of HOMO and LUMO properties?

In a recent Letter with Ayers, we have demonstrated that the idempotent Dirac density matrix generated by any local potential V( r) can be derived in terms of HOMO and LUMO properties. Subsequently we have pointed out that the exchange-only potential V x( r) of density functional theory can be built from the Dirac matrix γ( r 1, r 2) and its functional derivative δ γ( r 1, r 2)/δ ρ( r). By utilizing further the equation of motion of the density matrix, we propose an answer in the affirmative to the question posed in the title of this Letter.

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.

Density matrices for itinerant and localized electrons with and without external fields

Forms of canonical (Bloch) and Dirac density matrices for free electrons with and without external applied fields are first considered. The basic tool employed is the Bloch equation with a one-electron Hamiltonian. Such an approach is used to obtain a perturbation theory to all orders for the idempotent Dirac density matrix when a common potential energy V(r) is switched on to originally free electrons. The relation to density functional theory is then considered and the exchange–correlation contribution Vxc(r) to V(r) is expressed in terms of first- and second-order density matrices following Holas and March. These latter density matrices are now for the fully interacting system and, in particular, the first-order density matrix is no longer idempotent, though it must still satisfy generalized Pauli Principle conditions. Reference is also made to a localized Wigner electron in a strong magnetic field.