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     

Comparison of some representative density functional theory and wave function theory methods for the studies of amino acids

Energies of different conformers of 22 amino acid molecules and their protonated and deprotonated species were calculated by some density functional theory (DFT; SVWN, B3LYP, B3PW91, MPWB1K, BHandHLYP) and wave function theory (WFT; HF, MP2) methods with the 6-311++G(d,p) basis set to obtain the relative conformer energies, vertical electron detachment energies, deprotonation energies, and proton affinities. Taking the CCSD/6-311++G(d,p) results as the references, the performances of the tested DFT and WFT methods for amino acids with various intramolecular hydrogen bonds were determined. The BHandHLYP method was the best overall performer among the tested DFT methods, and its accuracy was even better than that of the more expensive MP2 method. The computational dependencies of the five DFT methods and the HF and MP2 methods on the basis sets were further examined with the 6-31G(d,p), 6-311++G(d,p), aug-cc-pVDZ, 6-311++G(2df,p), and aug-cc-pVTZ basis sets. The differences between the small and large basis set results have decreased quickly for the hybrid generalized gradient approximation (GGA) methods. The basis set convergence of the MP2 results has been, however, very slow. Considering both the cost and the accuracy, the BHandHLYP functional with the 6-311++G(d,p) basis set is the best choice for the amino acid systems that are rich in hydrogen bonds.     

Exchange and correlation in density functional theory and quantum chemistry

The nature of exchange, dynamic correlation (DC) and left–right correlation (LRC) is considered in density functional theory and wavefunction‐based quantum chemistry. The presence of LRC in approximate exchange density functionals is highlighted and the separation of LRC and DC is considered. For H₂, the Heitler–London approach is shown to include the essential elements of exchange and LRC. The arguments are illustrated by a comparison of Gaussian orbital s‐optimised Heitler–London and OPTX potential energy curves. They agree well near equilibrium, but differ at large distances due to the inability of the OPTX form to describe the dissociation process. LRC and DC values determined using the two approaches are compared. The influence of higher angular momentum functions in the Heitler–London approach is then investigated (commonly called self‐consistent valence bond); the agreement with OPTX degrades, leading to a larger value of LRC and a smaller value of DC at H₂ equilibrium. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Performance of density functional theory in computing nonresonant vibrational (hyper)polarizabilities

A set of exchange‐correlation functionals, including BLYP, PBE0, B3LYP, BHandHLYP, CAM‐B3LYP, LC‐BLYP, and HSE, has been used to determine static and dynamic nonresonant (nuclear relaxation) vibrational (hyper)polarizabilities for a series of all‐trans polymethineimine (PMI) oligomers containing up to eight monomer units. These functionals are assessed against reference values obtained using the Møller–Plesset second‐order perturbation theory (MP2) and CCSD methods. For the smallest oligomer, CCSD(T) calculations confirm the choice of MP2 and CCSD as appropriate for assessing the density functionals. By and large, CAM‐B3LYP is the most successful, because it is best for the nuclear relaxation contribution to the static linear polarizability, intensity‐dependent refractive index second hyperpolarizability, static second hyperpolarizability, and is close to the best for the electro‐optical Pockels effect first hyperpolarizability. However, none of the functionals perform satisfactorily for all the vibrational (hyper)polarizabilities studied. In fact, in the case of electric field‐induced second harmonic generation all of them, as well as the Hartree–Fock approximation, yield the wrong sign. We have also found that the Pople 6–31+G(d) basis set is unreliable for computing nuclear relaxation (hyper)polarizabilities of PMI oligomers due to the spurious prediction of a nonplanar equilibrium geometry. © 2013 Wiley Periodicals, Inc.     

Implementation and validation of the Lacks-Gordon exchange functional in conventional density functional and adiabatic connection methods

We present an analysis of the numerical performances of the exchange functional proposed by Lacks and Gordon, which we have implemented in the Gaussian series of programs. This functional has been built with the double aim of respecting most of the known scaling and asymptotic properties and of giving good numerical performances, especially as concerns noncovalent interactions. We have found that the coupling of the Lacks-Gordon exchange and Lee-Yang-Parr correlation functionals provides a reliable conventional density functional approach. The corresponding parameter-free adiabatic connection model, in which the ratio between Hartree-Fock and Lacks-Gordon exchange is determined a priori from purely theoretical considerations, allows us to obtain remarkable results for both covalent and noncovalent interactions in a satisfactory theoretical scheme, encompassing the free electron gas limit and most of the known scaling conditions. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 418–429, 1998     

Hierarchy of equations in the generalized density functional theory

Functional relations and equations of hierarchy in the generalized density functional theory (DFT) are derived from coordinate scaling and adiabatic connection. Local and nonlocal solutions for the noninteracting kinetic energy, exchange energy, correlation energy, and the kinetic energy correction functionals are presented. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Conclusive evidence on the insensitivity of additive rules to the combinational details of exchange and correlation functional in hybrid DFT methods

This account is a quest to evaluate the sensitivity of additive rules among total electronic energies of some triple molecular sets to the employed computational level. To achieve the goal, seven randomly generated hybrid functionals have been constructed and employed for calculation on a test set of 29 molecules containing second and third row elements. The computational results based on these nonoptimized DFT methods in conjunction with those of standard B3LYP method clearly demonstrates that in contrast to atomization or isomerization energies, additive rules among total electronic energies are insensitive to reliability or precision of computational method. This insensitivity to details of exchange‐correlation functional indicates that additive rules can be generally well considered using simple computational methods. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Electronic and optical properties of pyrrole and thiophene oligomers: A density functional theory study

The conductive mechanism of pyrrole (Py) and thiophene (Th) oligomers is investigated in the framework of density functional theory. Geometric constructions and electronic structures of neutral n‐Py/n‐Th and oxidized n‐Py^m+/n‐Th^m+ oligomers (6 ≤ n ≤ 48, 2 ≤ m ≤ 18) are reported as a function of oligomer length. The charges in the oxidized oligomers have a localized distribution along the oxidized n‐Py^m+/n‐Th^m+ oligomers, and each set of two positive charges is localized in one area. Therefore, the charge carriers in oxidized n‐Py^m+/n‐Th^m+ oligomers are bipolarons. Furthermore, the nonlinear optical properties of the n‐Py/n‐Th oligomers are investigated, for which the static polarizability α, the first polarizability β, and the second polarizability γ are calculated. When the ratio of m/n is 1/3, the static polarizability <α> and the polarizability anisotropy Δα are maximized. In addition, neutral n‐Py/n‐Th oligomers have maximum <γ> values. The values of β were determined mainly by the dipole of the molecule, while the values of γ were closely related to the aromaticity of the oligomer. The stronger the aromaticity, the bigger the γ value. All calculations indicate that the polarizability and absorption spectrum can be tuned by controlling the oxidation level, making these oligomers applicable as good nonlinear optical materials.     

The exact Fermi potential yielding the Hartree–Fock electron density from orbital‐free density functional theory

The exact expression for the Fermi potential yielding the Hartree–Fock electron density within an orbital‐free density functional formalism is derived. The Fermi potential, which is defined as that part of the potential that depends on the particles’ nature, is in this context given as the sum of the Pauli potential and the exchange potential. The exact exchange potential for an orbital‐free density functional formalism is shown to be the Slater potential.     

Incorporation of solvent effects into density functional predictions of molecular polarizabilities and hyperpolarizabilities

A study of the effect of the field, the basis set, the functional, and the cavity size on molecular polarizabilities and hyperpolarizabilities of substituted benzenes in liquid or solution is reported. The calculations have been performed using the density functional theory (DFT) within the conductor‐like screening model (COSMO). The optimized computational parameters are adopted to calculate molecular polarizabilities and hyperpolarizabilities of substituted benzenes in liquid or solution. The results show good agreement with the experimental values. From comparison of the different theoretical results, it is found that at the same theoretical level, the selection of the different solvation models may play an important role in the calculations of molecular solvation polarizability, and using the same solvation model, the effects of the different theoretical methods are relatively small. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Assessing the role of Hartree‐Fock exchange,correlation energy and long range corrections in evaluating ionization potential,and electron affinity in density functional theory

Accurate determination of ionization potentials (IPs), electron affinities (EAs), fundamental gaps (FGs), and HOMO, LUMO energy levels of organic molecules play an important role in modeling and predicting the efficiencies of organic photovoltaics, OLEDs etc. In this work, we investigate the effects of Hartree Fock (HF) Exchange, correlation energy, and long range corrections in predicting IP and EA in Hybrid Functionals. We observe increase in percentage of HF exchange results in increase of IPs and decrease in EAs. Contrary to the general expectations inclusion of both HF exchange and correlation energy (from the second order perturbation theory MP2) leads to poor prediction. Range separated Hybrid Functionals are found to be more reliable among various DFT Functionals investigated. DFT Functionals predict accurate IPs whereas post HF methods predict accurate EAs. © 2017 Wiley Periodicals, Inc.     

Regional self-interaction correction of density functional theory

We propose a new simple scheme for self-interaction correction (SIC) of exchange functionals in the density functional theory. In the new scheme, exchange energies are corrected by substituting exchange self-interactions for exchange functionals in regions of self-interaction. To classify the regions of self-interaction, we take advantage of the property of the total kinetic energy density approaching the Weizs?cker density in the case of electrons in isolated orbitals. The scheme differs from conventional SIC methods in that it produces optimized molecular structures. Applying the scheme to the calculation of reaction energy barriers showed that it provides a clear improvement in cases where the barriers are underestimated by conventional "pure" functionals. In particular, we found that this scheme even reproduces a transition state that is not given by pure functionals.     

Correlation functional in screened‐exchange density functional theory procedures

In the present study, we have explored several prospects for the further development of screened‐exchange density functional theory (SX‐DFT) procedures. Using the performance of HSE06 as our measure, we find that the use of alternative correlation functionals (as oppose to PBEc in HSE06) also yields adequate results for a diverse set of thermochemical properties. We have further examined the performance of new SX‐DFT procedures (termed HSEB‐type methods) that comprise the HSEx exchange and a (near‐optimal) reparametrized B97c (c _OS,0 = c _SS,0 = 1, c _OS,1 = −1.5, c _OS,2 = −0.644, c _SS,1 = −0.5, and c _SS,2 = 1.10) correlation functionals. The different variants of HSEB all perform comparably to or slightly better than the original HSE‐type procedures. These results, together with our fundamental analysis of correlation functionals, point toward various directions for advancing SX‐DFT methods. © 2017 Wiley Periodicals, Inc.     

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     

Structure,energetics and reactivity of ternary complexes of amino acids with Cu(II) and 2,2'-bipyridine by density functional theory. A combination of radical-induced and spin-remote fragmentations

The structures, electron distributions and dissociation energies of gas-phase ternary complexes of Cu(II) with 2,2'-bipyridine and leucine, isoleucine and lysine were addressed by density functional theory using the hybrid B3LYP functional, effective core potentials and the 6-31 + G(d) and LANL2DZ basis sets. The calculations confirm the previously suggested structures, in which amino acid carboxylates coordinate to the Cu atom by the carboxylate and alpha-amino groups in square-planar complexes. The dissociation energies for consecutive eliminations of CO(2) and alkyl radicals from isomeric singly charged complexes of leucine and isoleucine correlate with the ion relative abundances observed in collisionally activated dissociation mass spectra. Doubly charged lysine complexes show extremely low dissociation energies that are consistent with the <1 eV center-of-mass collision energies that were used in previous CAD studies. The calculated charge and spin densities point to radical-induced dissociations of singly charged complexes with an open-electron shell. In contrast, the unpaired electron is virtually inert in doubly charged, open-shell complexes that undergo charge-induced, spin-remote dissociations in the amino acid residues.     

Studies of chemical hardness and Fukui function using the exact solution of the density functional theory

Neal's procedure has been applied to determine the electron density ρ(x) for the H₂ molecule. The chemical hardness has been calculated employing the ab initio and density functional theory methods and the values are found to be reasonably good. The principle of maximum hardness (PMH) was tested. Fukui functions and the distribution of electron density around the internuclear distance were studied employing the electron density of the H₂ molecule. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 4–10, 2001     

Exploring odd-even effects of simple oligomer-like DRCNnT series: a study based on density functional theory/time-dependent density functional theory calculations

Odd-even effects of short-circuit current density and power conversion efficiency (PCE) are an interesting phenomenon in some organic solar cells. Although some explanations have been given, why they behave in such a way is still an open question. In the present work, we investigate a set of acceptor-donor-acceptor simple oligomer-like small molecules, named the DRCNnT (n = 5-9) series, to give an insight into this phenomenon because the solar cells based on them have high PCE (up to 10.08%) and show strong odd-even effects in experiments. By modeling the DRCNnT series and using density functional theory, we have studied the ground-state electronic structures of the DRCNnT (n = 5-9) series in condensed phase. The calculated results reproduce the experimental trends well. Furthermore, we find that the exciton-binding energies of the DRCNnT series may be one of the key parameters to explain this phenomenon because they also show odd-even effects. In addition, by studying the effects of alkyl branch and terminal group on odd-even effects of dipole moment, we find that eliminating one or two alkyl branches does not break the odd-even effects of dipole moments, but eliminating one or two terminal groups does. Finally, we conclude that removing one alkyl branch close to the terminal group of DRCN5T can decrease highest occupied molecular orbital (HOMO) energy (thus increasing open circuit voltage) and increase dipole moment (thus enhancing charge separation and short-circuit current). This could be a new and simple method to increase the PCE of DRCN5T-based solar cells.