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     

Direct minimization of the energy in density functional theory

The minimization of the energy functional of the first-order density matrix γ( r , r ') is achieved using unitary transformations applied to γ. Equivalently, such transformations can be carried out also on one-electron orbitals (natural orbitals) and their occupation (integer or non-integer) numbers. The conventional local density approximation based on the electron density p( r ) is then considered as a special case. The direct minimization of the energy functional of p with respect to the parameters of the unitary transformation leads to stationary conditions that are all equivalent to the Kohn–Sham equations. Preliminary numerical tests show that the proposed algorithms for the direct minimization of the energy work in a satisfactory manner. © John Wiley & Sons, Inc.     

The local orbital energy and density functional theory

From the viewpoint of density functional theory, an expression is derived which improves the average energy of a trial density. Applications to atoms and molecules are made using wave function methods and are based on properties of the variance, which is defined as $ (\overline {\varepsilon ^2 } - (\overline \varepsilon)^2)^{1/2} $, where ? is the local orbital energy. Calculated results for both Hartree-Fock and correlated wave functions are quite encouraging.     

One- and two-photon absorptions in asymmetrically substituted free-base porphyrins: a density functional theory study

Electronic spectra and structures of a new family of free-base porphyrin (H(2)P) derivatives with 4-(diphenylamino)stilbene (DPAS) or 4,4'-bis-(diphenylamino)stilbene (BDPAS) asymmetric substituents, recently synthesized and studied by Drobizhev et al. [J. Phys. Chem. B 110, 9802 (2006)] are investigated by density functional theory (DFT) using modern density functionals and the 6-31G* basis set. The time-dependent DFT technique is applied for calculations of one- and two-photon absorption spectra, electric and magnetic dipole moments, and for prediction of electronic circular dichroism for these chiral molecules. The four-band absorption spectrum of the H(2)P molecule (Q(x), Q(y), 0-0 and 1-0 bands) is enhanced in single-bond-linked DPAS. This enhancement is explained by hyperconjugation of the almost orthogonal pi systems and by small charge-transfer admixtures. The effect is much stronger for the double-bond- and triple-bond-linked DPAS and BDPAS substituents where absorption in the Q region transforms into a two-band spectrum. These molecules with ethenyl and ethynyl bonding of the porphyrin and donor substituent show very strong two-photon absorption in the near-infrared region. DFT calculations explain this by more efficient conjugation between the H(2)P and DPAS (BDPAS) chromophores, since they are almost coplanar: "Gerade" states of the H(2)P molecule occur in the Soret region and transform into charge-transfer states with nonzero transition moments. They are responsible for the strong two-photon absorption effects. Mixing of excitations in both chromophores explains the broadening of the Soret band. Though the calculated two-photon absorption cross sections are overestimated, the qualitative trends are reproduced and help understanding the whole genesis of spectra of these asymmetrically substituted H(2)P derivatives.     

Energy decompositions according to physical space partitioning schemes: treatments of the density cumulant

This article is a continuation of our previous paper on schemes of energy decompositions of molecular systems in the real space [D. R. Alcoba et al., J. Chem. Phys. 122, 074102 (2005)] now using correlated state functions. We study, according to physical arguments, the appropriate management of the density cumulant arising from the second-order reduced density matrix at correlated level, whose contributions can be assigned to one-center or to two-center terms in the energy partitioning. Our treatments are applied within two physical space partitioning schemes: the Bader partitioning into atomic basins and the fuzzy atom procedure. The results obtained in selected molecules are analyzed and discussed in detail.     

One- and three-parameter density functional theory and high level ab initio theory study of the CH+CH disproportionation reaction

Complete basis set (CBS) ab initio computational studies were performed with the target being to explore the CH+CH potential energy surface. Several closed and open shell intermediates were located on the potential energy surface. Computed enthalpies for the branching reactions, as well as heats of formation are in excellent agreement. Although CBS computed energies are of high quality, this computational study is not capable of predicting the branching product ratio due to fact that neither the MP2 nor the 6-311G(2d,2p) basis set are sufficient to locate the reactant complexes and the transition state structures for the hydrogen and carbon transfer reactions in the reaction complexes. To properly explore the CH+CH potential energy surface a much higher ab initio theory level is required.     

On the evaluation of the non-interacting kinetic energy in density functional theory

The utility of both an orbital-free and a single-orbital expression for computing the non-interacting kinetic energy in density functional theory is investigated for simple atomic systems. The accuracy of both expressions is governed by the extent to which the Kohn-Sham equation is solved for the given exchange-correlation functional and so special attention is paid to the influence of finite Gaussian basis sets. The orbital-free expression is a statement of the virial theorem and its accuracy is quantified. The accuracy of the single-orbital expression is sensitive to the choice of Kohn-Sham orbital. The use of particularly compact orbitals is problematic because the failure to solve the Kohn-Sham equation exactly in regions where the orbital has decayed to near-zero leads to unphysical behaviour in regions that contribute to the kinetic energy, rendering it inaccurate. This problem is particularly severe for core orbitals, which would otherwise appear attractive due to their formally nodeless nature. The most accurate results from the single-orbital expression are obtained using the relatively diffuse, highest occupied orbitals, although special care is required at orbital nodes.     

Adsorption energy distribution from the Aranovich-Donohue lattice density functional theory

We propose a new methodology projected for the estimation of the adsorption energy distribution from the monolayer part of a single nitrogen adsorption isotherm determined at 77 K based on the lattice density functional theory (DFT) via the Aranovich-Donohue formalism. At first sight, the presented approach is computationally more difficult than a classical one. However, it is more flexible and comprehensible. Next, we developed a numerical program and used it for the estimation of the adsorption energy distribution from the experimental data on carbon black samples. The main nitrogen molecule-carbon black surface interaction energy can be estimated as approximately 7-8 kJ/mol, but the heterogeneity of the investigated materials differs significantly. Furthermore, we compare the results obtained from the lattice DFT via the Aranovich-Donohue formalism with the solution of the integral equation with the kernel represented by the classical monolayer localized Fowler-Guggenheim isotherm equation. The similarity between these two independent approaches is observed. The proposed methodology can be used for the investigation of the energetic heterogeneity of not only the carbonaceous materials but also the other "flat-surfaced" solids.     

Ground-state energy and one-body virial in density functional theory of atomic ions

The relation between the ground-state energy and the one-body virial is discussed. Using a result of Levy and Perdew, the exchange energy, the correlation contribution to the kinetic energy and the correlation energy are also related to the one-body virial and contribution written explicitly in terms of the density. The sum of the orbital energies is likewise expressed.     

Accelerating the convergence of the total energy evaluation in density functional theory calculations

A special feature of the Strutinsky shell correction method (SCM) [D. Ullmo et al., Phys. Rev. B 63, 125339 (2001)] and the recently proposed orbital-corrected orbital-free density functional theory (OO-DFT) [B. Zhou and Y. A. Wang, J. Chem. Phys. 124, 081107 (2006)] is that the second-order corrections are incorporated in the total energy evaluation. In the SCM, the series expansion of the total electronic energy is essentially the Harris functional with its second-order correction. Unfortunately, a serious technical problem for the SCM is the lack of the exact Kohn-Sham (KS) density rho KS(r) required for the evaluation of the second-order correction. To overcome this obstacle, we design a scheme that utilizes the optimal density from a high-quality density mixing scheme to approximate rho KS(r). Recently, we proposed two total energy density functionals, i.e., the Zhou-Wang-lambda (ZW lambda) and the Wang-Zhou-alpha (WZ alpha) functionals, for use in the OO-DFT method. If the two interpolation parameters, lambda and alpha, are chosen to allow the second-order errors of the ZW lambda and the WZ alpha functionals to vanish, these two functionals reduce to the Hohenberg-Kohn-Sham functional with its second-order correction. Again, the optimal density from a high-quality density mixing scheme is used to approximate rho KS(r) in the evaluation of lambda and alpha. This approach is tested in iterative KS-DFT calculations on systems with different chemical environments and can also be generalized for use in other iterative first-principles quantum chemistry methods.     

New formulation of the analytical electronic energy and the regional density functional theory

A certain analytical model of E is proposed to satisfy the prerequisite exact density functional theory (DFT ) formulas such as μ = ∂E/∂N = −(I + A)/2 and 2η = ∂²E/∂N² = I −A. © 1996 John Wiley & Sons, Inc.     

密度泛函理论与Hartree-Fock方法混合处理中的稳态相关能校正

本文分析了通常的Hartree-Fock(HF)相关能定义和密度泛函理论(DFT)中的相关能定义的等价性条件。认为在参考电子密度与真实密度相差很大时两种定义是不等价的, 严格的DFT相关能比HF相关能(绝对值)要大。而在DFT与HF混合处理中得到的相关能比HF相关能(绝对值)要小, 两者之差相当于稳态相关能。实际计算表明, 通过合理地选择组态, 采用有限CI可以求得这一差值。本文描绘了双原子分子H2(X^1∑g^+), HF(X^1∑^+), N2(X^1∑g^+)的势能曲线, 结果比完全CISD和MP4的曲线还要好。H2的离解能是0.17a.u., 逼近实验值0.1747a.u..。     

Four-dimensional density and energy density functional

The relations based on an external one-electron operator V( r ) are examined from two view-points, i.e., from the Hohenberg–Kohn approach and the four-dimensional density concept introduced by Wilson and Frost, and extensively studied by Parr and Politzer. The object being to obtain, with the help of the Hellmann–Feynman theorem, new formulas for the energy of atoms and molecules, and to discuss the construction of the universal energy density functional on the basis of the four-dimensional density.     

Scaling dynamical correlation energy from density functional theory correlation functionals

The scaling of dynamical correlation energy in molecules obtained by the correlation functionals of density functional theory (DFT) is examined. The approach taken is very similar to the scaled external correlation method of Brown and Truhlar but is based on the observation that DFT correlation functionals, especially the LYP, appear to represent the dynamical portion of the correlation energy in molecules. We examine whether higher accuracy in atomization energies can be gained by scaling without significant deterioration of the structural and spectroscopic properties of the molecules using four DFT functionals (BLYP, OLYP, B3LYP, and O3LYP) on 19 molecules including the six molecule AE6 database, the latter being representative of a much larger, 109 molecule training set. We show that, with molecule specific scale factors, nearly perfect agreement with experiment can be achieved in atomization energies without increasing the average errors in other molecular properties relative to the DFT calculation. We further show that it is possible to find optimal scale factors which reduce the mean unsigned error per bond to levels comparable to those of some multilevel multicoefficient methods.     

Energy decompositions according to physical space partitioning schemes

This work describes simple decompositions of the energy of molecular systems according to schemes that partition the three-dimensional space. The components of those decompositions depend on one and two atomic domains thus providing a meaningful chemical information about the nature of different bondings among the atoms which compose the system. Our algorithms can be applied at any level of theory (correlated or uncorrelated wave functions). The results reported here, obtained at the Hartree-Fock level in selected molecules, show a good agreement with the chemical picture of molecules and require a low computational cost in comparison with other previously reported decompositions.