External coupled-cluster perturbation theory: description and application to weakly interaction dimers. Corrections to the random phase approximation

The formalism for developing perturbation theory by using an arbitrary fixed (external) set of amplitudes as an initial approximation is presented in a compact form: external coupled-cluster perturbation theory (xCCPT). Nonperturbative approaches also fit into the formalism. As an illustration, the weakly interacting dimers Ne(2) and Ar(2) have been studied in the various ring-coupled-cluster doubles (CCD) approximations; ring, direct-ring, antisymmetrized ring, and antisymmetrized direct ring, and a second-order correction in the xCCPT approach is added. The direct approaches include the summation of just Coulomb terms with the intention of selectively summing the largest terms in the perturbation first. "Coulomb attenuation" is effected by taking the random phase approximation to define such amplitudes, whose results are then improved upon using perturbation theory. Interaction energies at the ring-CCD level are poor but the xCCPT correction employed predicts binding energies which are only a few percent from the coupled-cluster single double (triple) values for the direct ring-CCD variants. Using the MP2 amplitudes which neglect exchange, the initial Coulomb-only term, leads to very accurate Ne(2) and Ar(2) potentials. However, to accurately compute the Na(2) potential required a different initial wavefunction, and hence perturbation. The potential energy surfaces of Ne(2) and Ar(2) are much too shallow using linear coupled-cluster doubles. Using xCCPT(2) with these amplitudes as the initial wavefunction led to slightly worse results. These observations suggest that an optimal external set of amplitudes exists which minimizes perturbational effects and hence improve the predictability of methods.     

Correlation potentials for molecular bond dissociation within the self-consistent random phase approximation

Self-consistent correlation potentials for H(2) and LiH for various inter-atomic separations are obtained within the random phase approximation (RPA) of density functional theory. The RPA correlation potential shows a peak at the bond midpoint, which is an exact feature of the true correlation potential, but lacks another exact feature: the step important to preserve integer charge on the atomic fragments in the dissociation limit. An analysis of the RPA energy functional in terms of fractional charge is given which confirms these observations. We find that the RPA misses the derivative discontinuity at odd integer particle numbers but explicitly eliminates the fractional spin error in the exact-exchange functional. The latter finding explains the improved total energy in the dissociation limit.     

Long-range correlation energies from frequency-dependent weighted exchange-hole dipole polarisabilities

Long-range correlation energies are calculated using an approximation of the single-particle density-density response function of the system that leads to an expression requiring only occupied orbitals and eigenvalues. Dipole-dipole polarisabilities and isotropic leading-order dispersion coefficients obtained from this approximation are shown to be in a reasonable agreement with corresponding values from the experiment or dipole oscillator strength distributions. The localised polarisabilities were used to calculate a long-range correlation correction to a hybrid-generalised gradient approximation functional using a proper damping function at short ranges. It was found that the hybrid density-functional theory+dispersion method obtained in this way has a comparable accuracy than high-level ab initio wave function methods at a much lower computational cost. This has been analysed for a number of systems from the GMTKN30 database including subsets for noncovalently bound complexes, relative energies for sugar conformers and reaction energies and barrier heights of pericyclic reactions of some medium sized organic molecules.     

Uniqueness of the iterative solution of the optimized effective potential equation

The optimized effective potential (OEP) equation can be used in a numerically efficient self-consistent form to solve for the density functional exchange and correlation potentials, as shown in a recent paper of Kummel and Perdew [Phys. Rev. Lett. 90, 43004 (2003)]. The uniqueness of an iterative solution of the OEP equation has not yet been adequately addressed. In this paper, it is shown that no nonconstant multiplicative potentials that can contaminate an iterative solution of the OEP equation exist and, hence, that formally the exact exchange-correlation potential determined form of the OEP equation is unique to within a constant.     

Solvation effects for polymers at an interface: a hybrid self-consistent field-density functional theory approach

Using polyatomic density functional theory of Kierlik and Rosinberg, we show that Wertheim's thermodynamic perturbation theory (TPT) incorporates solvation effects in a systematic, although simplified form. We derive two approximate solvation potentials, which require the knowledge of the correlation function in the reference unbonded fluid only. The theoretical predictions are tested against many-chain Monte Carlo simulations for moderate chain lengths. The predictions of the end-to-end distance in the bulk are in a reasonable agreement with simulations for the TPT(M-1) approximation, while the simpler TPT2_e approximation leads to the solvation potential that is shorter ranged and considerably less accurate. The resulting conformations are used in the subsequent self-consistent field theory calculations of hard-sphere polymers at a hard wall. While the incorporation of the solvation effects has little impact on the density profiles, the predictions of the components of the end-to-end distance vector as a function of the distance to the wall are much improved.     

Comparison between equations-of-motion and green's function methods for the particle-hole response function

We show that, apart from a few differences, the equations-of-motion method of McKoy et al. provides the leading correction to the random phase approximation (with exchange) in the fully renormalized response function (density-density correlation function). Thus, their equations-of-motion method is shown to be equivalent to a partial summation of infinite sets of terms in the perturbation expansion of the response function.     

Electron correlation methods based on the random phase approximation

In the past decade, the random phase approximation (RPA) has emerged as a promising post-Kohn–Sham method to treat electron correlation in molecules, surfaces, and solids. In this review, we explain how RPA arises naturally as a zero-order approximation from the adiabatic connection and the fluctuation-dissipation theorem in a density functional context. This is contrasted to RPA with exchange (RPAX) in a post-Hartree–Fock context. In both methods, RPA and RPAX, the correlation energy may be expressed as a sum over zero-point energies of harmonic oscillators representing collective electronic excitations, consistent with the physical picture originally proposed by Bohm and Pines. The extra factor 1/2 in the RPAX case is rigorously derived. Approaches beyond RPA are briefly summarized. We also review computational strategies implementing RPA. The combination of auxiliary expansions and imaginary frequency integration methods has lead to recent progress in this field, making RPA calculations affordable for systems with over 100 atoms. Finally, we summarize benchmark applications of RPA to various molecular and solid-state properties, including relative energies of conformers, reaction energies involving weak and covalent interactions, diatomic potential energy curves, ionization potentials and electron affinities, surface adsorption energies, bulk cohesive energies and lattice constants. RPA barrier heights for an extended benchmark set are presented. RPA is an order of magnitude more accurate than semi-local functionals such as B3LYP for non-covalent interactions rivaling the best empirically parametrized methods. Larger but systematic errors are observed for processes that do not conserve the number of electron pairs, such as atomization and ionization.     

Wavelength-decomposition-based embedded cluster density approximation for systems with nonlocal electron correlation

Local correlation methods rely on the assumption that electron correlation is nearsighted. In this work, we develop a method to alleviate this assumption. This new method is demonstrated by calculating the random phase approximation (RPA) correlation energies in several one-dimensional model systems. In this new method, the first step is to approximately decompose the RPA correlation energy to the nearsighted and farsighted components based on the wavelength decomposition of electron correlation developed by Langreth and Perdew. The short-wavelength (SW) component of the RPA correlation energy is then considered to be nearsighted, and the long-wavelength (LW) component of the RPA correlation energy is considered to be farsighted. The SW RPA correlation energy is calculated using a recently developed local correlation method: the embedded cluster density approximation (ECDA). The LW RPA correlation energy is calculated globally based on the system's Kohn-Sham orbitals. This new method is termed λ-ECDA, where λ indicates the wavelength decomposition. The performance of λ-ECDA is examined on a one-dimensional model system: a H₂₄ chain, in which the RPA correlation energy is highly nonlocal. In this model system, a softened Coulomb interaction is used to describe the electron-electron and electron-ion interactions, and slightly stronger nuclear charges (1.2e ) are assigned to the pseudo-H atoms. Bond stretching energies, RPA correlation potentials, and Kohn-Sham eigenvalues predicted by λ-ECDA are in good agreement with the benchmarks when the clusters are made reasonably large. We find that the LW RPA correlation energy is critical for obtaining accurate prediction of the RPA correlation potential, even though the LW RPA correlation energy contributes to only a few percent of the total RPA correlation energy.     

Reduced eigensystem representation of the linear density-density response function

The linear density-density response function represents a formulation of the generalized density response of a molecular (or extended) system to arbitrary perturbing potentials. We have recently established an approach for reducing the dimension of the (in principle infinite) eigenspace representation (the moment expansion) and generalized it to arbitrary self-adjoint, positive-definite, and compact linear operators. Here, we present a modified representation—the reduced eigensystem representation—which allows to define a trivial criterion for the convergence of the approximation to the density response. By means of this novel eigensystem-like structure, the remarkable reduction of the dimensionality becomes apparent for the calculation of the density-density response function.     

Energy expressions for Kohn-Sham potentials and their relation to the Slater-Janak theorem

Direct approximation of exchange-correlation potentials is a promising approach to accurate prediction of molecular response properties. However, little is known about ways of obtaining total energies from model potentials other than by using the Levy-Perdew virial relation. We introduce and explore several alternative formulas which arise as line integrals of potentials taken along density scaling and aufbau-filling paths, and which are not limited to the exchange term. The relaxed-orbital variant of the aufbau-path energy expression is shown to be closely related to the Slater-Janak theorem. Although the Levy-Perdew relation generally yields reasonable energies for all model exchange potentials, the relaxed-orbital aufbau path gives better results for those potentials that predict accurate highest-occupied orbital eigenvalues, such as the potential of Ra?sa?nen, Pittalis, and Proetto [J. Chem. Phys. 132, 044112 (2010)]. The ideas presented in this work may guide the development of new types of density-functional approximations for exchange and correlation.     

Orbital-dependent correlation energy in density-functional theory based on a second-order perturbation approach: success and failure

We have developed a second-order perturbation theory (PT) energy functional within density-functional theory (DFT). Based on PT with the Kohn-Sham (KS) determinant as a reference, this new ab initio exchange-correlation functional includes an exact exchange (EXX) energy in the first order and a correlation energy including all single and double excitations from the KS reference in the second order. The explicit dependence of the exchange and correlation energy on the KS orbitals in the functional fits well into our direct minimization approach for the optimized effective potential, which is a very efficient method to perform fully self-consistent calculations for any orbital-dependent functionals. To investigate the quality of the correlation functional, we have applied the method to selected atoms and molecules. For two-electron atoms and small molecules described with small basis sets, this new method provides excellent results, improving both second-order Moller-Plesset expression and any conventional DFT results significantly. For larger systems, however, it performs poorly, converging to very low unphysical total energies. The failure of PT based energy functionals is analyzed, and its origin is traced back to near degeneracy problems due to the orbital- and eigenvalue-dependent algebraic structure of the correlation functional. The failure emerges in the self-consistent approach but not in perturbative post-EXX calculations, emphasizing the crucial importance of self-consistency in testing new orbital-dependent energy functionals.     

First‐order correlation‐kinetic contribution to Kohn–Sham exchange charge density function in atoms,using quantal density functional theory approach

Using the static exchange‐correlation charge density concept, the total integrated exchange‐charge density function is calculated within the nonrelativistic spin‐restricted exchange‐only (i) optimized effective potential model, and (ii) nonvariational local potential derived from the exchange‐only work potential within the quantal density functional theory, for the ground‐state isoelectronic series: Ga⁺, Zn, Cu^?; In⁺, Cd, Ag^?; and Tl⁺, Hg, Au^?. The difference between the exchange charge density function derived from these potentials is employed to evaluate the first‐order correlation‐kinetic contribution to the integrated exchange charge density. This contribution is found to be important for both the intra‐ and inter‐shell regions. Screening effects on the contribution due to the nd¹⁰ (n = 3–5) subshells are discussed through comparisons with similar calculations on Ca, Sr, and Ba, wherein nd¹⁰ electrons are absent. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Exchange and correlation energy in density functional theory

Several different versions of density functional theory (DFT) that satisfy Hohenberg–Kohn theorems are characterized by different definitions of a reference or model state determined by an N‐electron ground state. A common formalism is developed in which exact Kohn–Sham equations are derived for standard Kohn–Sham theory, for reference‐state density functional theory, and for unrestricted Hartree–Fock (UHF) theory considered as an exactly soluble model Hohenberg–Kohn theory. A natural definition of exchange and correlation energy functionals is shown to be valid for all such theories. An easily computed necessary condition for the locality of exchange and correlation potentials is derived. While it is shown that in the UHF model of DFT the optimized effective potential (OEP) exchange satisfies this condition by construction, the derivation shows that this condition is not, in general, sufficient to define an exact local exchange potential. It serves as a test to eliminate proposed local potentials that are not exact for ground states. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 521–525, 2000     

Ionization potentials and electron affinities in the Perdew-Zunger self-interaction corrected density-functional theory

Using a self-consistent implementation of the Perdew-Zunger self-interaction corrected (PZ-SIC) density-functional theory, we have calculated ionization potentials (IP) and electron affinities (EA) of first- and second-row atoms and a set of small molecules. Several exchange-correlation functionals were tested. IPs and EAs were obtained by two methods: as the difference in self-consistent field (SCF) energies of neutrals and ions (deltaSCF) and as negatives of highest-occupied orbital energies. We found that, except for local spin-density approximation, PZ-SIC worsens DeltaSCF IPs and EAs. On the other hand, PZ-SIC brings orbital eigenvalues into much better agreement with electron removal energies. The Perdew-Zunger SIC seems to over-correct many-electron systems; for molecules it performs worse than for atoms. We also discuss several common approximations to PZ-SIC such as spherical averaging of orbital densities in atoms.     

Application of the Sakurai-Sugiura projection method to core-excited-state calculation by time-dependent density functional theory

The Sakurai-Sugiura projection (SS) method was implemented and numerically assessed for diagonalization of the Hamiltonian in time-dependent density functional theory (TDDFT). Since the SS method can be used to specify the range in which the eigenvalues are computed, it may be an efficient tool for use with eigenvalues in a particular range. In this article, the SS method is applied to core excited calculations for which the eigenvalues are located within a particular range, since the eigenvalues are unique to atomic species in molecules. The numerical assessment of formaldehyde molecule by TDDFT with core-valence Becke's three-parameter exchange (B3) plus Lee-Yang-Parr (LYP) correlation (CV-B3LYP) functional demonstrates that the SS method can be used to selectively obtain highly accurate eigenvalues and eigenvectors. Thus, the SS method is a new and powerful alternative for calculating core-excitation energies without high computation costs.     

An average fock operator technique of approximate open-shell LCAO-MO-SCF calculation

The exchange part of the usual Hartree-Fock potential in the unrestricted Hartree-Fock (UHF) theory is suitablyaveraged to construct an, average one-electron model Hamiltonian which generates a set of spin-restricted one-electron orbitals in a self-consistent manner. These orbitals are then used to calculate the electronic energy of the open-shell system by using the proper functional form for the energy which handles the exchange terms correctly. The eigenvalues ofF ^av can be used for calculating either the spin-polarised or spin-averaged ionisation potentials of different orbitals at theKoopmans’ theorem level of approximation. Comparison ofE ^ac with the UHF-energy shows thatE _UHF<E ^ac in each case revealing some kind of an upper bound nature ofE ^ac. An approximate variational argument is given. Relationship of our model with the hyper-Hartree-Fock theory of slater is explored and the general problem of eliminating ‘self-interaction’ terms in average Fock-operator based theories is discussed.     

Comparative performance of exchange and correlation density functionals in determining intermolecular interaction potentials of the methane dimer

We have calculated the interaction potentials of the methane dimer for the minimum-energy D(3d) conformation using the density functional theory (DFT) with 90 density functionals chosen from the combinations of nine exchange and 10 correlation functionals. Several hybrid functionals are also considered. While the performance of an exchange functional is related to the large reduced density gradient of the exchange enhancement factor, the correlation energy is determined by the low-density behavior of a correlation enhancement factor. Our calculations demonstrate that the correlation counterpart plays an equally important role as the exchange functional in determining the van der Waals interactions of the methane dimer. These observations can be utilized to better understand the seemingly unsystematic DFT interaction potentials for weakly bound systems.     

Variational cellular method for polyatomic molecules: SF6

This is the third paper on the cellular method for polyatomic systems. We show how to deal with nonspherical Coulomb potentials. We also show how to modify the variational expression for the energy eigenvalues so as to obtain a faster convergence in the angular momentum series for the wavefunctions. We apply both techniques to the self-consistent calculation of SF₆. Contrary to what we obtained in CH₄ and SiH₄, the cellular method cannot yield the correct equilibrium interatomic distance in the present case. The calculated ionization potentials are in the correct order but are all shifted by 2–3 eV. This shift is attributed to the wrong expression for exchange correlation.