Exact relations between the electron density and external potential for systems of interacting and noninteracting electrons

The differential virial theorem (DVT) is an explicit relation between the electron density ρ( r ), the external potential, kinetic energy density tensor, and (for interacting electrons) the pair function. The time‐dependent generalization of this relation also involves the paramagnetic current density. We present a detailed unified derivation of all known variants of the DVT starting from a modified equation of motion for the current density. To emphasize the practical significance of the theorem for noninteracting electrons, we cast it in a form best suited for recovering the Kohn–Sham effective potential v_s( r ) from a given electron density. The resulting expression contains only ρ( r ), v_s( r ), kinetic energy density, and a new orbital‐dependent ingredient containing only occupied Kohn–Sham orbitals. Other possible applications of the theorem are also briefly discussed. © 2012 Wiley Periodicals, Inc.     

Exact theorems concerning noninteracting kinetic energy density functional in D dimensions and their implications for gradient expansions

The focal point of the present work is the single-particle kinetic energy density tensor in D dimensions. This quantity enters both differential and various integral forms of the virial theorems, which are again set up in D dimensions. Major new results lie in (i) demonstrating that, by one-dimensional quadrature, it is possible to construct the Pauli potential directly from the kinetic energy tensor, without the need for functional differentiation and (ii) generating the gradient expansion for the kinetic energy tensor, in D dimensions. © 1995 John Wiley & Sons, Inc.     

Exchange energy for two closed shells generated by a bare Coulomb potential energy −<Emphasis Type="Italic">Ze</Emphasis><Superscript>2</Superscript>/<Emphasis Type="Italic">r</Emphasis> in the limit of large <Emphasis Type="Italic">Z</Emphasis>, 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 −Ze ²/r and (ii) the exchange energy density e_x(r){\varepsilon_x({\bf r})} . Some progress is also possible concerning the exchange potential V _x (r), one non-local approximation being the Slater potential 2e_x(r)/n(r){2\varepsilon_x(r)/n(r)} , with n(r) the ground state electron density. However, to complete the theory of V _x (r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.     

Self-consistent Thomas–Fermi–Dirac theory,extended by Gell-Mann and Brueckner correlation,in terms of density n and its two reduced gradients ∇2n/n and ∇n/n

We prove the following results, relevant for the density functional theory: the Thomas–Fermi–Dirac theory, generalized to include the contribution due to the high electron density result of Gell-Mann and Brueckner for the correlation energy, is shown to lead to a differential equation for the self-consistent ground-state density n( r ) in atoms and molecules in the form F(n, { ∇ n/n}², ∇²n/n)=1, where the function F is given explicitly. A straightforward extension yields a similar result for the equation determining the Pauli plus exchange–correlation potential and for the divergence of the many-electron force. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 145–149, 1998     

Exact electronic properties in the classically forbidden region of a metal surface

We derive exact properties of the inhomogeneous electron gas in the asymptotic classically forbidden region at a metal–vacuum interface within the framework of local effective potential energy theory. We derive a new expression for the asymptotic structure of the Kohn–Sham density functional theory (KS‐DFT) exchange‐correlation potential energy v_xc(r) in terms of the irreducible electron self‐energy. We also derive the exact asymptotic structure of the orbitals, density, the Dirac density matrix, the kinetic energy density, and KS exchange energy density. We further obtain the exact expression for the Fermi hole and demonstrate its structure in this asymptotic limit. The exchange‐correlation potential energy is derived to be v_xc(z → ∞) = ?α_KS,xc/z, and its exchange and correlation components to be v_x(z → ∞) = ?α_KS,x/z and v_c(z → ∞) = ?α_KS,c/z, respectively. The analytical expressions for the coefficients α_KS,xc and α_KS,x show them to be dependent on the bulk‐metal Wigner–Seitz radius and the barrier height at the surface. The coefficient α_KS,c = 1/4 is determined in the plasmon‐pole approximation and is independent of these metal parameters. Thus, the asymptotic structure of v_xc(z) in the vacuum region is image‐potential‐like but not the commonly accepted one of ?1/4z. Furthermore, this structure depends on the properties of the metal. Additionally, an analysis of these results via quantal density functional theory (Q‐DFT) shows that both the Pauli W_x(z → ∞) and lowest‐order correlation‐kinetic W(z → ∞) components of the exchange potential energy v_x(z → ∞), and the Coulomb W_c(z → ∞) and higher‐order correlation‐kinetic components of the correlation potential energy v_c(z → ∞), all contribute terms of O(1/z) to the structure. Hence correlations attributable to the Pauli exclusion principle, Coulomb repulsion, and correlation‐kinetic effects all contribute to the asymptotic structure of the effective potential energy at a metal surface. The relevance of the results derived to the theory of image states and to KS‐DFT is also discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Functional derivative of the kinetic energy functional for spherically symmetric systems

Ensemble non-interacting kinetic energy functional is constructed for spherically symmetric systems. The differential virial theorem is derived for the ensemble. A first-order differential equation for the functional derivative of the ensemble non-interacting kinetic energy functional and the ensemble Pauli potential is presented. This equation can be solved and a special case of the solution provides the original non-interacting kinetic energy of the density functional theory.     

Extended virial theorem applied to predissociation phenomena. Interpretation of the nearly degenerate G1Π ∼ I 1Π energy spectra in SiO

The extended virial theorem, obtained from a Fourier–Laplace transformation of equations of motion, is applied to the phenomenon of predissociation. The underlying resonance energy model rigorously defined by means of the theory of dilation analytic operators prescribes in detail the balance between the various energy contributions in a way analogous to the situation prevailing in a stationary system. The extended virial theorem is applied to predissociation, particularly in connection with the interpretation of the nearly degenerate G ¹Π ～? I ¹Π energy spectrum in SiO. It was previously found for the rovibronic spectra assigned to G ¹Π – X ¹Σ⁺ and I¹Π – X¹ Σ⁺, that every G – X band is accompanied by a I – X band. Whereas the I ¹Π state does not seem to undergo predissociation, the G state shows increasing predissociation with increasing vibrational quantum number. We show that the virial theorem generalized to include continuum phenomena offers an interpretation of the nearly degenerate spectra of SiO as well as of the concomitant isotope shift of SiO¹⁶ and SiO¹⁸ with respect to both resonance positions and widths.     

Study of the kinetic energy densities of electrons as applied to quantum dots in a magnetic field

There are three expressions for the kinetic energy density t( r ) expressed in terms of its quantal source, the single-particle density matrix: t _A( r ) , the integrand of the kinetic energy expectation value; t _B( r ) , the trace of the kinetic energy tensor; t _C( r ) , a virial form in terms of the ‘classical’ kinetic field. These kinetic energy densities are studied by application to ‘artificial atoms‘ or quantum dots in a magnetic field in a ground and excited singlet state. A comparison with the densities for natural atoms and molecules in their ground state is made. The near nucleus structure of these densities for natural atoms is explained. We suggest that in theoretical frameworks which employ the kinetic energy density such as molecular fragmentation, density functional theory, and information-entropic theories, one use all three expressions on application to quantum dots, and the virial expression for natural atoms and molecules. New physics could thereby be gleaned.     

Obtaining self–consistent wave functions which satisfy the virial theorem

The virial theorem has played an important role in applying quantum mechanics to chemical problems. It has served as one criterion of a satisfactory wave function and its consequences on chemical bonding, molecular structure, and substituent effects have been analyzed extensively. A common method of gaining compliance with the virial theorem is to introduce a “scale” factor which adjusts all distances by a factor η. Optimizing the scale factor through the variational principle produces a wave function satisfying the virial theorem. In the present paper it is shown that when this “scaling” procedure is applied to self-consistent wave functions, the virial theorem can be satisfied, but self-consistency is lost. Scaling generally has a small effect on the total energy, but the effects on the energy components (T, V_ne, V_ee, V_nn) can be two to three orders of magnitude larger and in the range of tens to hundreds of kcal. Consequently, for applications where the energy components are useful, it is highly desirable to obtain wave functions which satisfy the virial theorem and are self-consistent. In the present paper, a simple, inexpensive extrapolation technique is reported which requires one integral evaluation and two SCF cycles to achieve convergence. Applications to atoms and small molecules are reported.     

Bosonised DFT potential estimated from QMC calculations of the ground-state density for the inhomogeneous electron liquid in Be

To avoid the solution of numerous Kohn–Sham one-body potential equations for wave functions in density functional theory, various groups independently proposed the use of Pauli potential to bosonise the customary one-body potential theory. Here, we utilise our recent quantum Monte Carlo calculations of the ground-state electron density of the Be atom to estimate the bosonised one-body potential V_B(r) and hence extract the Pauli potential for this atom.     

Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Isotope effect of hydrogen and lithium hydride molecules. Application of the dynamic extended molecular orbital method and energy component analysis

In order to explore the isotope effect including the nuclear–electronic coupling and nuclear quantum effects under the one-particle approximation, we apply the dynamic extended molecular orbital (DEMO) method and energy component analysis to the hydrogen and lithium hydride isotope molecules. Since the DEMO method determines both electronic and nuclear wave functions simultaneously by variationally optimizing all parameters embedded in the basis sets, the virial theorem is completely satisfied and guarantees the relation of the kinetic and potential energies. We confirm the isotope effect on internuclear distances, nuclear and electronic wave functions, dipole moment, the polarizability, and each energy component. In the case of isotopic species of the hydrogen molecule, the total energy decreases from the H₂ to the T₂ molecule due to the stabilization of the nuclear–electronic potential component, as well as the nuclear kinetic one. In the case of the lithium hydride molecule, the energy lowering by replacing ⁶Li with ⁷Li is calculated to be greater than that by replacing H with D. This is mainly caused by the small destabilization of electron–electron and nuclear–nuclear repulsion in ⁷LiH compared to ⁶LiH, while the change in the repulsive components from ⁶LiH to ⁶LiD increases. Received: 24 March 1999 / Accepted: 5 August 1999 / Published online: 15 December 1999     

Theoretical treatment of predissociation of the CO (3sσ) B and (3pσ) C1Σ+ Rydberg states based on a rigorous adiabatic representation

Ab initio multireference configuration interaction calculations for adiabatic potential curves, nonadiabatic couplings 〈φ _i (R,r)|d/dR|φ _j (R,r)〉 and 〈φ _i (R,r)|d²/dR ²|φ _j (R,r)〉, and nuclear kinetic energy corrections 〈dφ _i (R,r)/dR|dφ _i (R,r)/dR〉 for the (3sσ) B and (3pσ) C¹Σ⁺ Rydberg states of the CO molecule have been carried out. The energy positions and predissociation linewidths for the observed vibrational levels of these two states have been determined in a rigorous adiabatic representation by the complex scaling method employing a basis of complex scaled harmonic vibrational functions in conjunction with the Gauss-Hermite quadrature method to evaluate the complex Hamiltonian matrix elements. The present treatment correctly reproduces the observed trends in energies and line broadening for vibrational levels of the B¹Σ⁺ state and represents an improvement over the previous treatment in literature. The errors in the determined spacings of the v = 0–4 vibrational levels of the C¹Σ⁺ state are less than 2% compared with measured data. The predissociation linewidths for the v=3,4 levels of the C¹Σ⁺ state are found to be 4.9 and 8.9 cm⁻¹, respectively, in good agreement with the observed values. Received: 23 March 1998 / Accepted: 27 July 1998 / Published online: 9 October 1998     

Kohn–Sham potentials by an inverse Kohn–Sham equation and accuracy assessment by virial theorem

In the recent study, the authors have proposed an integral equation for solving the inverse Kohn–Sham problem. In the present paper, the integral equation is numerically solved for one-dimensional model of a He atom and an H₂ molecule in the electronic ground states. For this purpose, we propose an iterative solution algorithm avoiding the inversion of the kernel of the integral equation. To quantify the numerical accuracy of the calculated exchange-correlation potentials, we evaluate the exchange and correlation energies based on the virial theorem as well as the reproduction of the exact ground-state electronic energy. The results demonstrate that the numerical solutions of our integral equation for the inverse Kohn–Sham problem are accurate enough in reproducing the Kohn–Sham potential and in satisfying the virial theorem.     

DFT studies and AIM analysis of AlN-polycycles

The structure and properties of AlN-polycycles were studied by DFT (density functional theory) method. The results of calculations were obtained at B3LYP/6-311G(d, p) level on model species. Topological parameters such as electron density, its Laplacian, kinetic electron energy density, potential electron energy density, and total electron energy density at the ring critical points (RCP) from Bader’s ‘Atoms in molecules’ (AIM) theory were analyzed in detail. These results indicate a good correlation between ρ(3, +1), G(r), H(r), and V(r) averaged values and hardness of AlN-polycycles. The aromaticity of all molecules has been studied by nucleus-independent chemical shift. There is a linear correlation between ΣNICS(0.0)_molecule values and polarizability.     

Relativistic virial theorem for atoms

A relativistic virial theorem is derived for atoms in a general manner. The virial ratio consists of the usual V/T term and a correction term W/T, where T, V, and W are the kinetic energy, the potential energy, and correction terms, respectively. Explicit forms of W are presented for four specific nuclear potential models. Numerical calculations for a uniform nuclear charge model show that the magnitude of the correction term W/T increases with increasing atomic numbers and that it modifies the ratio V/T considerably for atoms with large atomic numbers in particular. Received: 21 November 2000 / Accepted: 8 January 2001 / Published online: 3 April 2001     

Exchange energy density and exchange potential via a hartree–fock plus mp2 study of the electron liquid in the ground-state conformer of glycine

Very recent criticisms of existing exchange-correlation functionals by Wanko et al. applied to systems of biological interest have led us to reopen the question of the ground-state conformer of glycine: the simplest amino acid. We immediately show that the global minimum of the Hartree–Fock (HF) ground-state leads to a planar structure of the five non-hydrogenic nuclei, in the non-ionized form NH₂–CH₂–COOH. This is shown to lie lower in energy than the zwitterion structure NHB₃ ⁺–CH₂–COO^?, as required by experiment. Refinement of the nuclear geometry using second-order Møller–Plesset perturbation theory (MP2) is also carried out, and bond lengths are found to accord satisfactorily with experimentally determined values. The ground-state electron density for the MP2 geometry is then redetermined by HF theory and equidensity contours are displayed. The HF first-order density matrix γ( r , r ′) is then used to obtain similar exchange-energy density (ε _x ( r )) contours for the lowest conformer of glycine. At first sight, their shape looks almost the same as for the density ρ( r ), which seems to vindicate the LDA proportional to ρ( r )^3/4. However, by way of an analytically soluble model for an atomic ion, it is shown that this has to be corrected to obtain an accurate HF exchange energy E_x as the volume integral of ε _x ( r ). Finally, recognizing that for larger amino acids, the use of HF plus MP2 perturbation corrections will become prohibitive, we have used the HF information for ε _x ( r ) and ρ( r ) to plot the truly non-local exchange potential proposed by Slater, from the density matrix γ( r , r ′). This latter calculation should be practicable for large amino acids, but there adopting Becke's one-parameter form of ε _x ( r ) correcting LDA exchange. Some future directions are suggested.     

On the relationship between the electron-pair distribution function and the correlation energy of an atom

The pair distribution function h(r₁₂;r₁, γ) and the virial theorem are used to derive a general expression for the local contributions to the total correlation energy of an atom. A direct link between correlation effects and the correlation energy is obtained by use of G(r₁, γ) and Γ(r₁, y). The former is the probability associated with a given choice of r₁ and γ, while the latter describes the local contribution to the correlation energy. Explicit calculations for the ground state of helium indicate that the angular dependence of the local contribution to the correlation energy is essentially independent of r₁, whereas the local correlation energy shows a strong r₁ dependence. The maximum contribution to the correlation energy occurs at intermediate values of γ where there is close agreement between the Hartree–Fock and exact densities.     

Kinetic energy density in terms of electron density for closed-shell atoms in a bare Coulomb field

A long-term aim in density functional theory is to obtain the kinetic energy density t(r) in terms of the ground-state electron density ρ(r). Here, t(r) is written explicitly in terms of ρ(r) for an arbitrary number 𝒩 of closed shells in a bare Coulomb field. In the limit as 𝒩→∞, closed results for t(r) follow from the earlier analysis of ρ(r) by Heilmann and Lieb. [Phys. Rev. A 52 , 3628 (1995)]. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 281–283, 1998