Nonlocal Wigner-like correlation energy density functional: parametrization and tests on two-electron systems

Reparametrization of Wigner's correlation energy density functional yields a very close fit to the correlation energies of the helium isoelectronic sequence. However, a quite different reparametrization is required to obtain an equally close fit to the isoelectronic sequence of Hooke's atom. In an attempt to avoid having to reparametrize the functional for different choices of the one-body potential, we propose a parametrization that depends on global characteristics of the ground-state electron density as quantified by scale-invariant combinations of expectation values of local one-body operators. This should be viewed as an alternative to the density-gradient paradigm, allowing one to introduce the nonlocal dependence of the density functional on the density in a possibly more effective way. Encouraging results are obtained for two-electron systems with one-body potentials of the form r(zeta) with zeta=-12,+12,1, which span the range between the Coulomb potential (zeta=-1) and the Hooke potential (zeta=2).     

Exact expressions for ensemble functionals from particle number dependence

Some properties of exact ensemble density functionals can be determined by examining the particle number dependence of ground state ensemble density matrices for systems where the integer ground state energies satisfy a convexity condition. The results include the observation that the integral of the product of the functional derivative and Fukui function of functionals that can be expressed as the trace of an operator is particle number independent for particle numbers between successive integers and the integral itself is equal to the difference between functionals evaluated at successive integer particle numbers. Expressions that must be satisfied by 2nd and higher order functional derivatives are formulated and equations that must be satisfied point by point in space are derived. Using the analytic Hooke's atom model, it is shown that commonly used correlation functional approximations do not bear any resemblance to a spatially dependent expression derived from the exact second order functional derivative of the correlation functional. It is also shown that two expressions for the mutual Coulomb energy are not equal when approximate exchange and correlation functionals are used.     

Effect of the one-body potential on interelectronic correlation in two-electron systems

The correlation energies of the helium isoelectronic sequence (IS) and of Hooke's IS are very similar and are both weakly increasing upon increasing the nuclear charge/force constant, respectively. However, their separation into radial and angular correlations shows interesting differences. First, for intermediate (and high) values of the force constant radial correlation in Hooke's IS is surprisingly low. Second, both systems exhibit a decrease in the relative contribution of radial versus angular correlation upon strengthening the one-body attractive potential; however, unlike the helium IS, in Hooke's IS the radial correlation energy increases in absolute value upon strengthening the attractive one-body potential. The contribution of radial correlation to the Coulomb hole is examined and the asymptotic behavior at both strong and weak attractive potentials is considered. Radial correlation in Hooke's IS is found to constitute about 9.3% of the total correlation energy when the spring constant approaches the limit k-->infinity, but 100% of the total correlation energy for k-->0. Our results highlight both the similarities and the differences between the helium and Hooke's ISs.     

A DFT variational approach to Hooke''s atom based on local-scaling transformations

The local-scaling transformation version of density functional theory, LS-DFT, is employed in order to construct energy functionals for Hooke's atom. The components of the energy are analyzed and the resulting exchange and correlation potentials are compared with the exact ones. In addition, the representation of the exact one-particle density in terms of the various components of the total energy density is discussed.     

Exact density functionals for two-electron systems in an external magnetic field

In principle, the extension of density functional theory (DFT) to Coulombic systems in a nonvanishing magnetic field is via current DFT (CDFT). Though CDFT is long established formally, relatively little is known with respect to any generally applicable, reliable approximate E(XC) and A(XC) functionals analogous with the workhorse approximate functionals (local density approximation and generalized gradient approximation) of ordinary DFT. Progress can be aided by having benchmark studies on a solvable correlated system. At zero field, the best-known finite system for such purposes is Hooke's atom. Recently we extended the exact ground state solutions for this two-electron system to certain combinations of nonzero external magnetic fields and confinement strengths. From those exact solutions, as well as high-accuracy numerical results for other field and confinement combinations, we construct the correlated electron density and paramagnetic current density, the exact Kohn-Sham orbitals, and the exact DFT and CDFT exchange-correlation energies and potentials. We compare with results from several widely used approximate functionals, all of which exhibit major qualitative failures, whether in CDFT or in naive application of ordinary DFT. We also illustrate how the CDFT vorticity variable nu is a computationally difficult quantity which may not be appropriate in practice to describe the external B field effects on E(XC) and A(XC).     

Electron correlation in Hooke's law atom in the high-density limit

Closed-form expressions for the first three terms in the perturbation expansion of the exact energy and Hartree-Fock energy of the lowest singlet and triplet states of the Hooke's law atom are found. These yield elementary formulas for the exact correlation energies (-49.7028 and -5.807 65 mE(h)) of the two states in the high-density limit and lead to a pair of necessary conditions on the exact correlation kernel G(w) in Hartree-Fock-Wigner theory.     

Open-shell reduced density matrix functional theory

Open-shell reduced density matrix functional theory is established by investigating the domain of the exact functional. For spin states that are the ground state, a particularly simple set is found to be the domain. It cannot be generalized to other spin states. A number of conditions satisfied by the exact density matrix functional is formulated and tested for approximate functionals. The exact functional does not suffer from fractional spin error, which is the source of the static correlation error in dissociated molecules. We prove that a simple approximation (called the Buijse-Baerends functional, Mu?ller or square root functional) has a non-positive fractional spin error. In the case of the H atom the error is zero. Numerical results for a few atoms are given for approximate density and density matrix functionals as well as a recently developed range-separated combination of both.     

Solving the Schrodinger equation for helium atom and its isoelectronic ions with the free iterative complement interaction (ICI) method

The Schrodinger equation was solved very accurately for helium atom and its isoelectronic ions (Z=1-10) with the free iterative complement interaction (ICI) method followed by the variational principle. We obtained highly accurate wave functions and energies of helium atom and its isoelectronic ions. For helium, the calculated energy was -2.903,724,377,034,119,598,311,159,245,194,404,446,696,905,37 a.u., correct over 40 digit accuracy, and for H(-), it was -0.527,751,016,544,377,196,590,814,566,747,511,383,045,02 a.u. These results prove numerically that with the free ICI method, we can calculate the solutions of the Schrodinger equation as accurately as one desires. We examined several types of scaling function g and initial function psi(0) of the free ICI method. The performance was good when logarithm functions were used in the initial function because the logarithm function is physically essential for three-particle collision area. The best performance was obtained when we introduce a new logarithm function containing not only r(1) and r(2) but also r(12) in the same logarithm function.     

Long‐range corrected functionals satisfy Koopmans' theorem: Calculation of correlation and relaxation energies

In this article, we show that the long‐range‐corrected (LC) density functionals LC‐BOP and LCgau‐BOP reproduce frontier orbital energies and highest‐occupied molecular orbital (HOMO)—lowest‐unoccupied molecular orbital (LUMO) gaps better than other density functionals. The negative of HOMO and LUMO energies are compared with the vertical ionization potentials (IPs) and electron affinities, respectively, using CCSD(T)/6‐311++G(3df,3pd) for 113 molecules, and we found LC functionals to satisfy Koopmans' theorem. We also report that the frontier orbital energies and the HOMO‐LUMO gaps of LC‐BOP and LCgau‐BOP are better than those of recently proposed ωM05‐D (Lin et al., J. Chem. Phys. 2012, 136 , 154109). We express the exact IP in terms of orbital relaxation, and correlation energies and hence calculate the relaxation and correlation energies for the same set of molecules. It is found that the LC functionals, in general, includes more relaxation effect than Hartree–Fock and more correlation effect than the other density functionals without LC scheme. Finally, we scan μ parameter in LC scheme from 0.1 to 0.6 bohr^?1 for the above test set molecules with LC‐BOP functional and found our parameter value, 0.47 bohr^?1, is usefully applicable to our tested systems. © 2013 Wiley Periodicals, Inc.     

Variational constraints on perturbed eigenvalues and their perturbation expansions. II. Perturbational inequalities

A large family of interlocking perturbational inequalities is derived by variational considerations for the stationary states of all systems described by a Hamiltonian linear in a real perturbing parameter λ. These inequalities constrain in many different ways the perturbation expansions of both exact and variational eigenvalues for these systems; analogous inequalities are derived for the components of the eigenvalues. A special feature of the analysis consists of obtaining inequalities applicable to the separate sums of even- and odd-order perturbation energies. For lowest states of each symmetry and for positive definite perturbing operator, the interlocking effect of the inequalities becomes extremely restrictive. The inequalities are illustrated with several numerical calculations for different systems and states of the helium isoelectronic sequence. The direction of the inequalities is found to be unaffected by low-order truncation, thus rendering them applicable to low-order perturbation expansions. The inequalities are used to study the efficacy of low-order perturbation theory for two- to ten-electron atomic isoelectronic sequences, and to determine the functional dependence upon λ of the eigenvalues and their components for arbitrary atomic isoelectronic sequences.     

Solving the electron-nuclear Schrodinger equation of helium atom and its isoelectronic ions with the free iterative-complement-interaction method

Our previous paper [J. Chem. Phys. 127, 224104 (2007)] revealed that the Schrodinger equation in the fixed-nucleus approximation could be very accurately solved for helium atom and its isoelectronic ions (Z=1-10) with the free iterative-complement-interaction (ICI) method combined with the variation principle. In this report, the quantum effect of nuclear motion has further been variationally considered by the free ICI formalism for the Hamiltonian including mass-polarization operator. We obtained -2.903 304 557 729 580 294 733 816 943 892 697 752 659 273 965 a.u. for helium atom, which is over 40 digits in accuracy, similarly to the previous result for the fixed-nucleus level. Similar accuracy was also obtained for the helium isoelectronic ions. The present results may be regarded to be the nonrelativistic limits. We have further analyzed the physics of the free ICI wave function by applying it to an imaginary atom called "eneon," [e(-)e(10+)e(-)](8+), in which both of the quantum effect of nuclear motion and the three-particle collisions are differently important from the helium and its isoelectronic ions. This revealed the accurate physics automatically generated by the free ICI formalism.     

Comparison shopping for a gradient-corrected density functional

Gradient corrections to the local spin density (LSD ) approximation for the exchange-correlation energy are making density functional theory as useful in quantum chemistry as it is in solid-state physics. But which of the many gradient-corrected density functionals should be preferred a priori? We make a graphical comparison of the gradient dependencies of some popular approximations, discussing the exact formal conditions which each obeys and identifying which conditions seem most important. For the exchange energy, there is little formal or practical reason to choose among the Perdew-Wang 86, Becke 88, or Perdew-Wang 91 functionals. But, for the correlation energy, the best formal properties are displayed by the nonempirical PW 91 correlation functional. Furthermore, the real-space foundation of PW 91 yields an insight into the character of the gradient expansion which suggests that PW 91 should work especially well for solids. Indeed, while improving dissociation energies over LSD , PW 91 remains the most “local” of the gradient-corrected exchange-correlation functionals and, thus, the least likely to overcorrect the subtle errors of LSD for solids. To show that our analysis of spin-unpolarized functionals is sufficient, we also compute spin-polarization energies for atoms, finding PW 91 values only slightly more negative than LSD values. © 1996 John Wiley & Sons, Inc.     

New functionals for correlation energy deduced in the framework of the correlation factor approach

The general characteristics of two-body density functionals (TBDF) are explored and two new correlation energy functionals are derived using the correlation factor approach. The optimization of the parameters entering the above functionals requires exact and accurate atomic correlation energies (ACE). We revised the ACE values in the literature and obtained a new set of “exact” ACE for atoms with 2 ≤ Z ≤ 10. Unfortunately, there exist some inaccuracies in the ACE values of the second-row atoms, which make unsuitable the inclusion of them in the optimization. The ACE calculated for the first period with the above functionals, using the optimized sets of parameters, are in excellent agreement with the exact ones, while the corresponding values calculated for the second-row atoms are between the precision margins estimated by us for the exact values. © 1997 John Wiley & Sons, Inc.     

Doping effects on structural and electronic properties of ladderanes and ladder polysilanes: a density functional theory investigation

Doping effects on the structural and electronic properties of ladderanes and ladder polysilanes have been studied using density functional theory. Two types of doping: substitution with isoelectronic atoms or heteroatoms (or radicals), have been used to design low band gap ladderanes. It is found that the B-doped [n]-ladderanes and 1,2 P-doped [n]-silaladderanes exhibit a very noticeable bent conformation, whereas the 1,2 and 1,3 N-doped ladderanes, P-doped ladderanes, and silaladderanes keep the relatively straight ladder shapes. The isoelectronic atom doping reduces the HOMO-LUMO (H-L) gaps of [n]-ladderanes but increases those of [n]-silaladderanes with n > 5. The present results show that isoelectronic atom doping is not an effective way to decrease the H-L gaps of ladderanes and silaladderanes. Heteroatom doping has a more pronounced effect than the isoelectronic atom doping. The HOMOs of heteroatom-doped ladderanes and silaladderanes are destabilized and LUMOs are stabilized, leading to significant reduction of H-L gaps. Most of the B-, N-, and P-doped [n]-silaladderanes we designed have H-L gaps below 1.5 eV. Therefore, it is expected that these silaladderanes are promising candidates of conductive or semiconductive materials. The heteroatom doping is a viable approach to reduce H-L gaps for the silaladderanes. In addition, it is found that nine different density functionals, including B3LYP, SVWN LDA, four pure GGAs, and three hybrid GGAs, as well as the time-dependent B3LYP method, all lead to the same predictions on the H-L gaps of ladderanes, silaladderanes, as well as their doped derivatives.     

Exact Kohn-Sham versus Hartree-Fock in momentum space: Examples of two-fermion systems

The question of how density functional theory (DFT) compares with Hartree-Fock (HF) for the computation of momentum-space properties is addressed in relation to systems for which (near) exact Kohn-Sham (KS) and HF one-electron matrices are known. This makes it possible to objectively compare HF and exact KS and hence to assess the potential of DFT for momentum-space studies. The systems considered are the Moshinsky [Am. J. Phys. 36, 52 (1968)] atom, Hooke's atom, and light two-electron ions, for which expressions for correlated density matrices or momentum densities have been derived in closed form. The results obtained show that it is necessary to make a distinction between true and approximate DFTs.     

A correlation-energy functional from a correlation-factor model

In this work, a way to approximate the correlation energy functional starting from a model correlation factor is shown. The problem is addressed by using formally exact properties of the second-order density matrix and actual values of correlation energies for atoms. An Ansatz for the correlation factor is proposed that allows one to derive some known and some new correlation energy density functionals. Results for atomic systems show the reliability of the approach. © 1994 John Wiley & Sons, Inc.