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     

Toward the complete range separation of non‐hybrid exchange–correlation functional

In this study, we use a very simple scheme to achieve range separation of a total exchange–correlation functional. We have utilized this methodology to combine a short‐range pure density functional theory (DFT) functional with a corresponding long‐range pure DFT, leading to a “Range‐separated eXchange–Correlation” (RXC) scheme. By examining the performance of a range of standard exchange–correlation functionals for prototypical short‐ and long‐range properties, we have chosen B‐LYP as the short‐range functional and PBE‐B95 as the long‐range counterpart. The results of our testing using a more diverse range of data sets show that, for properties that we deem to be short‐range in nature, the performance of this prescribed RXC‐DFT protocol does resemble that of B‐LYP in most cases, and vice versa. Thus, this RXC‐DFT protocol already provides meaningful numerical results. Furthermore, we envisage that the general RXC scheme can be easily implemented in computational chemistry software packages. This study paves a way for further refinement of such a range‐separation technique for the development of better performing DFT procedures. © 2015 Wiley Periodicals, Inc.     

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     

The use of the hybrid density functional theory (DFT) approach for calculation of vibrational spectra: nitromethane

The molecular geometry of nitromethane was optimized and its force field and vibrational spectrum were calculated by the BECKE3LYP method. The accuracy of optimization of the geometry of MeNO₂ obtained by this method using the 6–311G(d,p) and 6–311++G(d,p) basis sets is not poorer than that obtained at the second-order Møller-Plesset level of perturbation theory (MP2). The vibrational frequencies of nitromethane and its d₁, d₂, and d₃ isotopomers obtained by the BECKE3LYP method are in much better agreement with the experimental data than those calculated at the MP2 level using the same basis set. The average absolute error of calculations performed without the use of any scaling factors is ～2% for frequencies; the maximum deviation is ～4%.     

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.     

Applicability of density functional theory in reproducing accurate vibrational spectra of surface bound species

The structural equilibrium parameters, the adsorption energies, and the vibrational frequencies of the nitrogen molecule and the hydrogen atom adsorbed on the (111) surface of rhodium have been investigated using different generalized‐gradient approximation (GGA), nonlocal correlation, meta‐GGA, and hybrid functionals, namely, Perdew, Burke, and Ernzerhof (PBE), Revised‐RPBE, vdW‐DF, Tao, Perdew, Staroverov, and Scuseria functional (TPSS), and Heyd, Scuseria, and Ernzerhof (HSE06) functional in the plane wave formalism. Among the five tested functionals, nonlocal vdW‐DF and meta‐GGA TPSS functionals are most successful in describing energetics of dinitrogen physisorption to the Rh(111) surface, while the PBE functional provides the correct chemisorption energy for the hydrogen atom. It was also found that TPSS functional produces the best vibrational spectra of the nitrogen molecule and the hydrogen atom on rhodium within the harmonic formalism with the error of ?2.62 and ?1.1% for the N? N stretching and Rh? H stretching frequency. Thus, TPSS functional was proposed as a method of choice for obtaining vibrational spectra of low weight adsorbates on metallic surfaces within the harmonic approximation. At the anharmonic level, by decoupling the Rh? H and N? N stretching modes from the bulk phonons and by solving one‐ and two‐dimensional Schrödinger equation associated with the Rh? H, Rh? N, and N? N potential energy we calculated the anharmonic correction for N? N and Rh? H stretching modes as ?31 cm^?1 and ?77 cm^?1 at PBE level. Anharmonic vibrational frequencies calculated with the use of the hybrid HSE06 function are in best agreement with available experiments. © 2014 Wiley Periodicals, Inc.     

Rung 3.5 density functionals: Another step on Jacob's ladder

Applications of density functional theory (DFT) to computational chemistry and solid‐state physics rely on a “Jacob's Ladder” of progressively more complicated approximations to the many‐body exchange‐correlation (XC) density functional. Accurate, computationally tractable DFT calculations on large and periodic systems remain challenging for existing XC functionals. Simple XC functionals on the three lowest rungs of Jacob's Ladder are insufficiently accurate for many properties, while fourth‐rung hybrid functionals incorporating nonlocal information can be prohibitively expensive. This perspective presents our work toward a compromise, a new class of “Rung 3.5” functionals that incorporate a linear dependence on the nonlocal one‐particle density matrix. This work reviews these functionals' formal underpinning, numerical performance, and prospects for modeling solids and surfaces. © 2012 Wiley Periodicals, Inc.     

On the performance of long‐range‐corrected density functional theory and reduced‐size polarized LPol‐n basis sets in computations of electric dipole (hyper)polarizabilities of π‐conjugated molecules

Static longitudinal electric dipole (hyper)polarizabilities are calculated for six medium‐sized π‐conjugated organic molecules using recently developed LPol‐n basis set family to assess their performance. Dunning's correlation‐consistent basis sets of triple‐ζ quality combined with MP2 method and supported by CCSD(T)/aug‐cc‐pVDZ results are used to obtain the reference values of analyzed properties. The same reference is used to analyze (hyper)polarizabilities predicted by selected exchange‐correlation functionals, particularly those asymptotically corrected. © 2012 Wiley Periodicals, Inc.     

Assessing the performance of density functional theory for the dynamic polarizabilities of amino acids: Treatment of correlation and role of exact exchange

The theoretical determination of electric response properties of the biological systems is a field where the application of density functional theory (DFT) appears to be quite promising. In this work, the performance of 41 density functional methods is evaluated in predicting dynamic polarizabilities of an experimental benchmark set of 20 proteinogenic amino acids. The behavior of a large number of density functionals, including various types of the local spin density approximation (LSDA), generalized gradient approximation (GGA), meta‐GGA (m‐GGA), hybrid‐GGA (h‐GGA), hybrid meta‐GGA (hm‐GGA), and range‐separated hybrid‐GGA (rsh‐GGA), has been assessed for the purpose. Analyzing the results of our DFT benchmarking, we found that these computationally economical methods show very diverse predictive capability and a careful selection of DFT functionals is very important in the polarizability calculations. Considering the role of exchange, correlation, dispersion and long‐range corrections, it turned out that in the LSDA class, SVWN3 gives better results than SPL and SVWN5 toward the reference values. Of the GGA methods, OPBE outperforms all other functionals. The M06‐L is the best method of m‐GGA class. The B3LYP and TPSSh are the best functionals of h‐GGA and hm‐GGA lineages, respectively. Finally, CAM‐B3LYP is the best method of rsh‐GGA functionals that predicts the most accurate polarizability for amino acids by a large margin with respect to others. Overall, the best performing functionals turn out to be hm‐GGAs TPSSh, TPSS1KCIS, M05, tau‐HCTHhyb, and h‐GGA B3LYP. Hopefully, the results of this investigation might provide the useful guidance to propose a new exchange‐correlation functional for calculating the optical properties of biomolecular materials. © 2013 Wiley Periodicals, Inc.     

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     

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.     

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     

A semiempirical long-range corrected exchange correlation functional including a short-range Gaussian attenuation (LCgau-B97)

We applied an improved long‐range correction scheme including a short‐range Gaussian attenuation (LCgau) to the Becke97 (B97) exchange correlation functional. In the optimization of LCgau‐B97 functional, the linear parameters are determined by least squares fitting. Optimizing μ parameter (0.2) that controls long‐range portion of Hartree‐Fock (HF) exchange to excitation energies of large molecules (Chai and Head‐Gordon, J Chem Phys 2008, 128, 084106) and additional short‐range Gaussian parameters (a = 0.15 and k = 0.9) that controls HF exchange inclusion ranging from short‐range to mid‐range (0.5–3 Å) to ground state properties achieved high performances of LCgau‐B97 simultaneously on both ground state and excited state properties, which is better than other tested semiempirical density functional theory (DFT) functionals, such as ωB97, ωB97X, BMK, and M0x‐family. We also found that while a small μ value (～0.2) in LC‐DFT is appropriate to the local excitation and intramolecular charge‐transfer excitation energies, a larger μ value (0.42) is desirable in the Rydberg excitation‐energy calculations. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Molecular structure and vibrational spectra of phenobaraitone by density functional theory and ab initio hartree-Fock calculations

Quantum chemistry calculations have been performed using Gaussian03 program to compute optimized geometry, harmonic vibrational frequency along with intensities in IR and Raman spectra at RHF/6-31++G** and B3LYP/6-31++G** levels for phenobarbitone (C₁₂H₁₂N₂O₃) in the ground state. The scaled harmonic vibrational frequencies have been compared with experimental FT-IR and FT-Raman spectra. Theoretical vibrational spectra of the title compound were interpreted by means of potential energy distributions (PEDs) using MOLVIB program. A detailed interpretation of the infrared spectra of the title compound is reported. On the basis of the agreement between the calculated and observed results, the assignments of fundamental vibrational modes of phenobarbitone were examined and some assignments were proposed. The theoretical spectrograms for FT-IR and FT-Raman spectra of the title compound have been constructed.     

Systematic study of vibrational frequencies calculated with the self-consistent charge density functional tight-binding method

We present a detailed study of harmonic vibrational frequencies obtained with the self-consistent charge density functional tight-binding (SCC-DFTB) method. Our testing set comprises 66 molecules and 1304 distinct vibrational modes. Harmonic vibrational frequencies are computed using an efficient analytical algorithm developed and coded by the authors. The obtained results are compared to experiment and to other theoretical findings. Scaling factor for the SCC-DFTB method, determined by minimization of mean absolute deviation of scaled frequencies, is found to be 0.9933. The accuracy of the scaled SCC-DFTB frequencies is noticeably better than for other semiempirical methods (including standard DFTB method) and approximately twice worse than for other well established scaled ab initio quantum chemistry methods (e.g., HF, BLYP, B3LYP). Mean absolute deviation for the scaled SCC-DFTB frequencies is 56 cm(-1), while standard deviation is 82 cm(-1), and maximal absolute deviation is as large as 529 cm(-1). Using SCC-DFTB allows for substantial time savings; computational time is reduced from hours to seconds when compared to standard ab initio techniques.     

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     

About the compatibility between ansatzes and constraints for a local formulation of orbital‐free density functional theory

Functional properties that are exact for the Hohenberg–Kohn functional may turn into mutually exclusive constraints at a given level of ansatz. This is exemplarily shown for the local density approximation. Nevertheless, it is possible to reach exactly the Kohn–Sham data from an orbital‐free density functional framework based on simple one‐point functionals by starting from the Levy–Perdew–Sahni formulation. The energy value is obtained from the density‐potential pair, and therefore does not refer to the functional dependence of the potential expression. Consequently, the potential expression can be obtained from any suitable model and is not required to follow proper scaling behavior.     

Time-dependent density functional theory scheme for efficient calculations of dynamic (hyper)polarizabilities

The authors present an efficient perturbative method to obtain both static and dynamic polarizabilities and hyperpolarizabilities of complex electronic systems. This approach is based on the solution of a frequency-dependent Sternheimer equation, within the formalism of time-dependent density functional theory, and allows the calculation of the response both in resonance and out of resonance. Furthermore, the excellent scaling with the number of atoms opens the way to the investigation of response properties of very large molecular systems. To demonstrate the capabilities of this method, they implemented it in a real-space (basis-set-free) code and applied it to benchmark molecules, namely, CO, H2O, and para-nitroaniline. Their results are in agreement with experimental and previous theoretical studies and fully validate their approach.