Correlation Energy Functionals Research Articles

Open shells in reduced-density-matrix-functional theory

Reduced-density-matrix-functional theory is applied to open-shell systems. We introduce a spin-restricted formulation by appropriately expressing approximate correlation-energy functionals in terms of spin-dependent occupation numbers and spin-independent natural orbitals. We demonstrate that the additional constraint of total-spin conservation is indispensable for the proper treatment of open-shell systems. The formalism is applied to the first-row open-shell atoms. The obtained ground-state energies are in very good agreement with the exact values as well as other state of the art quantum chemistry calculations

Isobaric–Isothermal Monte Carlo Simulations from First Principles: Application to Liquid Water at Ambient Conditions

A series of first-principles Monte Carlo simulations in the isobaric-isothermal ensemble were carried out for liquid water at ambient conditions (T=298 K and p=1 atm). The Becke-Lee-Yang-Parr (BLYP) exchange and correlation energy functionals and norm-conserving Goedecker-Teter-Hutter (GTH) pseudopotentials were employed with the CP2 K simulation package to examine systems consisting of 64 water molecules. The fluctuations in the system volume encountered in simulations in the isobaric-isothermal ensemble require a reconsideration of the suitability of the typical charge-density cutoff and the regular grid-generation method previously used for the computation of the electrostatic energy in first-principles simulations in the microcanonical or canonical ensembles. In particular, it is noted that a much higher cutoff is needed and that the most computationally efficient method of creating grids can result in poor simulations. Analysis of the simulation trajectories using a very large charge-density cutoff at 1200 Ry and four different grid-generation methods point to a significantly underestimated liquid density of about 0.8 g cm-3 resulting in a somewhat understructured liquid (with a value of about 2.7 for the height of the first peak in the oxygen-oxygen radial distribution function) for BLYP-GTH water at ambient conditions. In addition, a simulation using a charge-density cutoff at 280 Ry yields a higher density of 0.9 g cm-3, showing the sensitivity of the simulation outcome to this parameter.

Toward a Monte Carlo program for simulating vapor–liquid phase equilibria from first principles

Abstract Efficient Monte Carlo algorithms are combined with the Quickstep energy routines of CP2K to develop a program that allows for Monte Carlo simulations in the canonical, isobaric-isothermal, and Gibbs ensembles using a first principles description of the physical system. Configurational-bias Monte Carlo techniques and pre-biasing using an inexpensive approximate potential are employed to increase the sampling efficiency and to reduce the frequency of expensive ab initio energy evaluations. The new Monte Carlo program has been validated through extensive comparison with molecular dynamics simulations using the programs CPMD and CP2K . Preliminary results for the vapor–liquid coexistence properties ( T = 473 K ) of water using the Becke–Lee–Yang–Parr exchange and correlation energy functionals, a triple-zeta valence basis set augmented with two sets of d-type or p-type polarization functions, and Goedecker–Teter–Hutter pseudopotentials are presented. The preliminary results indicate that this description of water leads to an underestimation of the saturated liquid density and heat of vaporization and, correspondingly, an overestimation of the saturated vapor pressure.

Fluctuation-dissipation theorem density-functional theory

Using the fluctuation-dissipation theorem (FDT) in the context of density-functional theory (DFT), one can derive an exact expression for the ground-state correlation energy in terms of the frequency-dependent density response function. When combined with time-dependent density-functional theory, a new class of density functionals results that use approximations to the exchange-correlation kernel fxc as input. This FDT-DFT scheme holds promise to solve two of the most distressing problems of conventional Kohn-Sham DFT: (i) It leads to correlation energy functionals compatible with exact exchange, and (ii) it naturally includes dispersion. The price is a moderately expensive O(N6) scaling of computational cost and a slower basis set convergence. These general features of FDT-DFT have all been recognized previously. In this paper, we present the first benchmark results for a set of molecules using FDT-DFT beyond the random-phase approximation (RPA)-that is, the first such results with fxc not equal to 0. We show that kernels derived from the adiabatic local-density approximation and other semilocal functionals suffer from an "ultraviolet catastrophe," producing a pair density that diverges at small interparticle distance. Nevertheless, dispersion interactions can be treated accurately if hybrid functionals are employed, as is demonstrated for He2 and HeNe. We outline constraints that future approximations to fxc should satisfy and discuss the prospects of FDT-DFT.

Elucidating the Vibrational Spectra of Hydrogen-Bonded Aggregates in Solution: Electronic Structure Calculations with Implicit Solvent and First-Principles Molecular Dynamics Simulations with Explicit Solvent for 1-Hexanol in n-Hexane

Fourier transform infrared spectroscopy is a popular method for the experimental investigation of hydrogen-bonded aggregates, but linking spectral information to microscopic information on aggregate size distribution and aggregate architecture is an arduous task. Static electronic structure calculations with an implicit solvent model, Car-Parrinello molecular dynamics (CPMD) using the Becke-Lee-Yang-Parr (BLYP) exchange and correlation energy functionals and classical molecular dynamics simulations for the all-atom version of the optimized parameters for liquid simulations (OPLS-AA) force field were carried out for an ensemble of 1-hexanol aggregates solvated in n-hexane. The initial configurations for these calculations were size-selected from a distribution of aggregates obtained from a large-scale Monte Carlo simulation. The vibrational spectra computed from the static electronic structure calculations for monomers and dimers and from the CPMD simulations for aggregates up to pentamers demonstrate the extent of the contribution of dangling or nondonating hydroxyl groups found in linear and branched aggregates to the "monomeric" peak. Furthermore, the computed spectra show that there is no simple relationship between peak shift and aggregate size nor architecture, but the effect of hydrogen-bond cooperativity is shown to differentiate polymer-like (cooperative) and dimer-like (noncooperative) hydrogen bonds in the vibrational spectrum. In contrast to the static electronic structure calculations and the CPMD simulations, the classical molecular dynamics simulations greatly underestimate the vibrational peak shift due to hydrogen bonding.

Reference-state one-particle density-matrix theory

A density-matrix formalism is developed based on the one-particle density-matrix of a single-determinantal reference-state. The v-representable problem does not appear in the proposed method, nor the need to introduce functionals defined by a constrained search. The correlation-energy functionals are not universal; they depend on the external potential. Nevertheless, model systems can still be used to derive universal energy-functionals. In addition, the correlation-energy functionals can be partitioned into individual terms that are -- to a varying degree -- universal; yielding, for example, an electron gas approximation. Variational and non-variational energy functionals are introduced that yield the target-state energy when the reference state -- or its corresponding one-particle density matrix -- is constructed from Brueckner orbitals. Using many-body perturbation theory, diagrammatic expansions are given for the non-variational energy-functionals, where the individual diagrams explicitly depend on the one-particle density-matrix. Non-variational energy-functionals yield generalized Hartree--Fock equations involving a non-local correlation-potential and the Hartree--Fock exchange; these equations are obtained by imposing the Brillouin--Brueckner condition. The same equations -- for the most part -- are obtained from variational energy-functionals using functional minimizations, yielding the (kernel of) correlation potential as the functional derivative of correlation-energy functionals. Approximations for the correlation-energy functions are introduced, including a one-particle-density-matrix variant of the local-density approximation (LDA) and a variant of the Lee--Yang--Parr (LYP) functional.

Towards an assessment of the accuracy of density functional theory for first principles simulations of water.

A series of Car-Parrinello (CP) molecular dynamics simulations of water are presented, aimed at assessing the accuracy of density functional theory in describing the structural and dynamical properties of water at ambient conditions. We found negligible differences in structural properties obtained using the Perdew-Burke-Ernzerhof or the Becke-Lee-Yang-Parr exchange and correlation energy functionals; we also found that size effects, although not fully negligible when using 32 molecule cells, are rather small. In addition, we identified a wide range of values of the fictitious electronic mass (micro) entering the CP Lagrangian for which the electronic ground state is accurately described, yielding trajectories and average properties that are independent of the value chosen. However, care must be exercised not to carry out simulations outside this range, where structural properties may artificially depend on micro. In the case of an accurate description of the electronic ground state, and in the absence of proton quantum effects, we obtained an oxygen-oxygen correlation function that is overstructured compared to experiment, and a diffusion coefficient which is approximately ten times smaller.

Correlation factor approach to the correlation energy functional

A global survey of the correlation factor energy functionals and its application to atomic and molecular properties is made. Its performances are compared with those of the density functional theory (DFT) correlation energy functionals, and some interesting conclusions from previous publications are reinforced here; namely, after removing the one-Slater-determinant hypothesis from the Kohn–Sham method, all DFT correlation functionals are able to provide reasonable results in any circumstance, with an additional restriction, for systems having a quasi-degenerate wave function, the DFT correlation functionals must depend explicitly on the on-top density.

A study with a complete-active-space self-consistent-field plus density functional theory combination: The low-lying bound states of N 2

The use of a large set of correlation energy functionals, in a post-self-consistent-field procedure, is analyzed for the calculation of the potential-energy curves of the ground state and some excited states in the case of the N2 molecule. The results show the improvement of the molecular properties, such as dissociation energies, equilibrium bond distances and harmonic vibrational frequencies, (when some of the tested functionals are used), together with the problems that the use of density functional theory functionals applied to multideterminantal wave functions implies.

Exact expressions for energy functional in the time‐dependent density functional theory

AbstractExact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

DFT calculations of correlation energies for excited electronic states using MCSCF wave functions

AbstractA large set of correlation energy functionals has been applied, in a post‐SCF procedure, to incorporate the correlation energy of several excited atomic states using MCSCF wave functions. The results show that the application of standard density functionals does not consider the double count of the correlation energy obtained by the MCSCF wave function. However, the use of some dependence on the two‐body density matrix improves the results, with a partial account of this correlation energy balance. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

New correlation energy functionals with explicit dependence on the number of electrons

Based upon the idea of effective number of electrons, we develop simple but accurate correlation energy functionals to be used for density functional theory calculations. We derive both a spin-independent and a spin-dependent functional. The spin-dependent one, used in conjunction with Becke’s exchange functional [A. D. Becke, Phys. Rev. A 38, 3098 (1988)], yields excellent results for thermochemistry calculations, giving an average absolute error of 2.9 kcal/mol for a test set comprised of the enthalpies of formation of the 148 molecules in the extended G2 set [L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 106, 1063 (1997); L. A. Curtiss, P. C. Redfern, K. Raghavachari, and J. A. Pople, ibid. 109, 42 (1998)] plus the total energies of the atoms H through Ar. We also discuss the problem of fractional occupation number, and we show that the corresponding principle of integer preference can be fulfilled by the procedure that we propose to build correlation energy functionals.

Mono and multiconfigurational wave functions with DFT correlation energy: the case of fluorine

It is known that the Hartree–Fock–Kohn–Sham method does not constitute an improvement over the Hartree–Fock method when the latter gives a poor description of the system, as is the case for the fluorine molecule. On the contrary, it is known that a multiconfigurational (MC) wave function gives a rather good description of this molecule. We show that adding the correlation energy computed from a variety of correlation energy functionals to the MC energy improves the potential energy curve and the spectroscopic constants significantly. We consider two different ways of including the correlation energy, either in a self-consistent (SCF) fashion or through a post-SCF procedure.

The correlation energy in terms of density moments along the adiabatic connection in the density functional theory

Abstract Hierarchies of equations for the correlation energy and kinetic energy correction functionals are reviewed. Truncation of the hierarchies at different orders leads to explicit relations for the correlation energy and its kinetic component which are expanded in terms of powers of the moments of the electron density. The dependence of the correlation energy on the adiabatic constant is presented for several atoms and ions. For some two- and four-electron systems the correlation energy and its derivative are compared with the exact results, and with the local density and the generalized gradient values.

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

Adiabatic integration formula for the correlation energy functional of the Hartree-Fock density

An adiabatic integration formula for the quantum chemistry correlation energy functional of the Hartree–Fock density, EcQC[n], is presented. The functional EcQC[n] is meant to be added to the completed Hartree–Fock energy to produce the exact ground-state energy of the system under consideration. The initial slope of the integrand in this connection formula is identified as a second-order energy and an explicit expression for the initial slope of the integrand is presented. Our expression should be useful for arriving at new improved approximations to EcQC[n]. Previous numerical results by Huang and Umrigar (1997) Phys Rev A 56:290, for two-electron densities are proved, and a generalization to more than two electrons is presented. Results obtained by means of the present density functional theory correlation energy functionals, when used to approximate the initial slope in our adiabatic integration formula for EcQC[n], are compared against exact numbers.

Density-functional study of small molecules within the Krieger-Li-Iafrate approximation

We report density-functional studies of several small molecules ($H_{2}, N_{2}, CO, H_{2}O$, and $CH_{4}$) within the Krieger-Li-Iafrate (KLI) approximation to the exact Kohn-Sham local exchange potential, using a three-dimensional real-space finite-difference pseudopotential method. It is found that exchange-only KLI leads to markedly improved eigenvalue spectra compared to those obtained within the standard local-density approximation (LDA), the generalized gradient approximation (GGA), and the Hartree-Fock (HF) method. For structural properties, exchange-only KLI results are close to the corresponding HF values. We find that the addition of LDA or GGA correlation energy functionals to the KLI exact exchange energy functional does not lead to systematic improvements.

Simulation Techniques in Parametric Hamiltonians

Using simulation techniques, a general definition of parametric Hamiltonians (Hpa), is presented. The minimal distance between a parametric and the exact Hamiltonian (Hexa) is defined in terms of the sum of the least distance between the corresponding energy functional components. A reference electronic energy functional for simulation of Hexa is proposed based on a CI energy functional. General properties of parametric Hamiltonians are analyzed considering reduction of the number of particles, integrals, and basis set, elimination of orthogonal transformations, and correlation and total energy functionals. Parametric Hamiltonians based on simulation of binding energy functionals and a simple parametrization scheme are proposed. Some preliminary tests have been performed with the parametrization of a characterized Hpa and calculations of organic (CH4, C2H2, C2H4, and C2H6) and transition metal (Ni2 and Ni5) systems.

Molecular and solid-state tests of density functional approximations: LSD, GGAs, and meta-GGAs

Results from numerical tests of nine approximate exchange–correlation energy functionals are reported for various systems—atoms, molecules, surfaces, and bulk solids. The functional forms can be divided into three categories: (1) the local spin density (LSD) approximation, (2) generalized gradient approximations (GGAs), and (3) meta-GGAs. In addition to the spin densities and their first gradients, the input to a meta-GGA includes other semilocal information such as Laplacians of the spin densities or orbital kinetic energy densities. We present a way to visualize meta-GGA nonlocality which generalizes that for GGA nonlocality, and which stresses the different meta-GGA descriptions of iso-orbital and orbital overlap regions of space. While some of the tested approximations were constructed semiempirically with many parameters fitted to chemical data, others were constructed to incorporate key properties of the exact exchange–correlation energy. The latter functionals perform well for both small and extended systems, with the best performance achieved by a meta-GGA which recovers the correct gradient expansion. While the semiempirical functionals can achieve high accuracy for atoms and molecules, they are typically less accurate for surfaces and solids. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 889–909, 1999

Improvement of Multiconfigurational Wave Functions and Energies by Correlation Energy Functionals

Several aspects of hybrid methods resulting from combining multiconfigurational (MC) wave functions with correlation energy functionals are discussed, in particular for a family of functionals recently proposed.1,2 It is found that adding the correlation at the end of a MC calculation instead of including it in the self-consistent procedure constitutes an excellent approximation. MC potential energy curves and spectroscopic constants show a significant improvement after being corrected with correlation energies from this family of functionals.