Exchange kernel of density functional response theory from the common energy denominator approximation (CEDA) for the Kohn–Sham Green's function

A complete and explicit expression for the exchange kernel f _x of density functional response theory (DFRT) is derived in terms of the occupied Kohn-Sham (KS) orbitals _i. It is based on the common energy denominator approximation (CEDA) for the KS Green's function (O. V. Gritsenko and E. J. Baerends, Phys. Rev. A 64, 042506 (2001)). The kernel f _x ^CEDA is naturally subdivided into the Slater f _S ^CEDA and the 'response' f _resp ^CEDA parts, which are the derivatives of the Slater _S and response _resp potentials, respectively. While f _S ^CEDA is obtained with a straightforward differentiation of _S , some terms of f _resp ^CEDA are obtained from the solution of linear equations for the corresponding derivatives. All components of f _x ^CEDA are explicitly expressed in terms of the products _i ^* _j of the occupied KS orbitals taken at the positions r ₁ and r ₂, as well as the potentials of these products at r ₃. The coefficients in these expressions are obtained by inversion of the matrix, associated with the overlap matrix of the products _i ^* _j and _k ^* _l . Terms are indicated, which generate in an external electric field an ultra-nonlocal potential _x , counteracting an external field, and possible approximations to f _x ^CEDA are considered.     

Assessment of the Van Voorhis–Scuseria exchange–correlation functional for predicting excitation energies using time-dependent density functional theory

We assess the performance of the Van Voorhis–Scuseria exchange–correlation functional (VSXC), a kinetic-energy-density-dependent exchange–correlation functional recently developed in our group, for calculating vertical excitation energies using time-dependent density functional theory in a benchmark set of molecules. Overall, VSXC performs very well, with accuracy similar to that of hybrid functionals such as the hybrid Perdew–Burke–Ernzerhof functional and Becke's three parameter hybrid method with the Lee, Yang, and Parr correlation functional, which contain a portion of Hartree–Fock exchange. Received: 29 December 1999 / Accepted: 5 June 2000 / Published online: 11 September 2000     

Restricted open-shell Kohn–Sham theory: N unpaired electrons

We present an energy expression for restricted open-shell Kohn–Sham theory for N unpaired electrons. It is shown that it is possible to derive an explicit energy expression for all low-spin multiplets of systems that exhibit neither radial nor cylindrical symmetry. The approach was implemented in the CPMD code and tested for iron complexes.     

The importance of Colle–Salvetti for computational density functional theory

The purpose of this presentation is to show the importance of the Colle–Salvetti (Theor Chim Acta 37:329, 1975) paper in the development of modern computational density functional theory. To do this we cover the following topics (1) the Bright Wilson understanding (2) the Kohn–Sham equations (3) local density exchange (4) the exchange-hole (5) generalised gradient approximation for exchange (Becke and Cohen) (6) left–right correlation and dynamic correlation (7) the development of the Lee–Yang–Parr dynamic correlation functional from the Colle–Salvetti paper (8) the early success of GGA DFT. Finally we observe that the the BLYP and OLYP exchange-correlation functionals are not semi-empirical; this may explain their great success.     

Some questions on the exchange contribution to the effective potential of the Kohn–Sham theory

The current success of Density Functional Theory applications hinges upon the availability of explicitly density-dependent functionals to self-consistently solve a set of one-electron equations, the Kohn–Sham (KS) equations, which determine the occupied orbitals and its associated electronic density. In KS theory, a local exchange potential is proposed as part of an effective potential. This potential is compared to the exchange operator of the Hartree–Fock theory, which is of a non-local nature. The present paper discusses the variational framework of the KS equations, and the equivalence between both exchange potentials within a correlation-free theory. The common difficulties of current local exchange functionals to correctly simulate the non-locality of the exchange energy density in chemical systems are also analyzed and explained through an exactly solvable model. We give then numerical arguments and conclude by analyzing the performance of various commonly used approximations to exchange functionals.     

Conventional strain energies of azetidine and phosphetane: Can density functional theory yield reliable results?

The conventional strain energies for azetidine and phosphetane are determined within the isodesmic, homodesmotic, and hyperhomodesmotic models. Optimum equilibrium geometries, harmonic vibrational frequencies, and corresponding electronic energies and zero‐point vibrational energies are computed for all pertinent molecular systems using self‐consistent field theory, second‐order perturbation theory, and density functional theory and using the correlation consistent basis sets cc‐pVDZ, cc‐pVTZ, and cc‐pVQZ. Single point fourth‐order perturbation theory, CCSD, and CCSD(T) calculations using the cc‐pVTZ and the cc‐pVQZ basis sets are computed using the MP2/cc‐pVTZ and MP2/cc‐pVQZ optimized geometries, respectively, to ascertain the contribution of higher order correlation effects and to determine if the quadruple‐zeta valence basis set is needed when higher order correlation is included. In the density functional theory study, eight different functionals are used including B3LYP, wB97XD, and M06‐2X to determine if any functional can yield results similar to those obtained at the CCSD(T) level. © 2012 Wiley Periodicals, Inc.     

On the Presumptive Similarity of Kohn–Sham and Brueckner Orbitals

For states of many-electron systems disclosing various degrees of quasidegeneracy, we have carried out comparative studies of Kohn–Sham orbitals (KSO) generated for several xc-potentials, Brueckner orbitals (BO) represented by the Brueckner-coupled cluster orbitals, and Hartree–Fock (HF) orbitals by means of criteria directly related to the orbital structure which are based on relative distance indices for various pairs of equidimensional subspaces defined by the KSO and BO basis sets. We have found that both for weak and strong quasidegeneracy there are systems for which the KSO–BO distances are larger than the BO–HF ones. For strongly quasidegenerate states it is found that the distance indices are the largest for hybrid potentials, and that the subspaces spanned by KSOs are closer to those spanned by HF orbitals than by BOs. Hence, our results do not support the recently formulated expectations concerning the similarity of Brueckner orbitals and Kohn–Sham orbitals, including those corresponding to purely local exchange-correlation potentials.     

Structure–antioxidant activity relationships of dendrocandin analogues determined using density functional theory

Structural Chemistry - Quantum-chemical calculations based on the density functional theory (DFT) at the B3LYP/6–311?+?+?G(2d,2p)//B3LYP/6–31G(d,p) level were employed...     

Intermolecular potentials of the methane dimer calculated with Møller-Plesset perturbation theory and density functional theory

We have calculated the intermolecular interaction potentials of the methane dimer at the minimum-energy D(3d) conformation using the Hartree-Fock (HF) self-consistent theory, the correlation-corrected second-order M?ller-Plesset (MP2) perturbation theory, and the density functional theory (DFT) with the Perdew-Wang (PW91) functional as the exchange or the correlation part. The HF calculations yield unbound potentials largely due to the exchange-repulsion interaction. In the MP2 calculations, the basis set effects on the repulsion exponent, the equilibrium bond length, the binding energy, and the asymptotic behavior of the calculated intermolecular potentials have been thoroughly studied. We have employed basis sets from the Slater-type orbitals fitted with Gaussian functions (STO-nG) (n=3-6) [Quantum Theory of Molecular and Solids: The Self-Consistent Field for Molecular and Solids (McGraw-Hill, New York, 1974), Vol. 4], Pople's medium size basis sets of Krishnan et al. [J. Chem. Phys. 72, 650 (1980)] [up to 6-311++G(3df,3pd)] to Dunning's correlation consistent basis sets [J. Chem. Phys. 90, 1007 (1989)] (cc-pVXZ and aug-cc-pVXZ) (X=D, T, and Q). With increasing basis size, the repulsion exponent and the equilibrium bond length converge at the 6-31G** basis set and the 6-311++G(2d,2p) basis set, respectively, while a large basis set (aug-cc-pVTZ) is required to converge the binding energy at a chemical accuracy (approximately 0.01 kcal/mol). Up to the largest basis set used, the asymptotic dispersion coefficient has not converged to the destined C6 value from molecular polarizability calculations. The slow convergence could indicate the inefficacy of using the MP2 calculations with Gaussian-type functions to model the asymptotic behavior. Both the basis set superposition error (BSSE) corrected and uncorrected results are presented to emphasize the importance of including such corrections. Only the BSSE corrected results systematically converge to the destined potential curve with increasing basis size. The DFT calculations generate a wide range of interaction patterns, from purely unbound to strongly bound, underestimating or overestimating the binding energy. The binding energy calculated using the PW91PW91 functional and the equilibrium bond length calculated using the PW91VP86 functional are close to the MP2 results at the basis set limit.     

Calculation of weakly polar interaction energies in polypeptides using density functional and local Møller-Plesset perturbation theory

The interaction energies of ubiquitous weakly polar interactions in proteins are comparable with those of hydrogen bonds, consequently, they stabilize local, secondary, and tertiary structures. However, the most widely-used density functionals fail to describe the weakly polar interactions. Thus, it is important to find and test functionals which adequately describe and quantify the energetics of such interactions. For this purpose, interaction energies in the hydrophobic core of rubredoxin (PDB id: 1rb9) and in the S22 subset of the JSCH-2005 benchmark database were computed with the BHandHLYP and PWPW91 functionals and with the pseudospectral implementation of the local MP2 (PS-LMP2) method. The cc-pVDZ, cc-pVTZ(-f), cc-pVTZ, cc-pVQZ(-g), aug-cc-pVDZ, aug-cc-VTZ(-f), and aug-cc-pVTZ basis sets were used for the calculations. In the S22 subset the PS-LMP2 results were extrapolated to the complete basis set limit. Furthermore, the a posteriori counterpoise method of Boys and Bernardi was used to correct the basis set superposition errors in the calculation of interaction energies. Calculations using the BHandHLYP functional, both for the various weakly polar interactions in rubredoxin and for the dispersion interactions in the S22 subset, were in good agreement with those using the coupled cluster (CCSD(T)) and the resolution of identity MP2 (RIMP2) methods and clearly outperformed both the PWPW91 functional and the PS-LMP2 method. The results for the S22 hydrogen bonded subset, obtained with PWPW91 calculations, were closest to those of the reference high level calculations. For the "mixed" (hydrogen bonded and dispersive) interactions in the S22 subset, results obtained with the BHandHLYP and PS-LMP2 calculations agreed well with the reference calculations.     

Electronic excitation of sulfur-organic compounds – performance of time-dependent density functional theory

 To define the scope and limitations of the time-dependent density functional theory (TDDFT) method, spectral absorption data of a series of about 100 neutral or charged sulfur-organic compounds with up to 24 non-hydrogen atoms and up to four sulfur atoms were calculated in the near-UV, visible and IR regions. Although the theoretical vertical transition energies correspond only approximately to experimental absorption band maxima, the mean absolute deviation was calculated to be 0.21 eV (1600 cm⁻¹). The main absorption features of various compounds with monocoordinated or dicoordinated sulfur atoms are well reproduced. As far as possible TDDFT results were compared with those of semiempirical Zerner's intermediate neglect of differential overlap (ZINDO/S) and of Pariser–Parr–Pople (PPP) calculations. TDDFT also works well in cases where the semiempirical methods fail. Limitations of TDDFT were encountered with calculations of spectral absorptions of dye molecules. The “vinylene shift” of polymethine dyes is not reproduced by TDDFT. Whereas electronic excitation energies delocalized polar and betainic chromophores are reasonably well reproduced, excitation energies of charge-transfer-type and charge-resonance-type transitions of weakly interacting composite chromophores are significantly underestimated. Received: 30 October 2000 / Accepted: 29 November 2000 / Published online: 22 May 2001     

Intermolecular potentials of the silane dimer calculated with Hartree-Fock theory, Møller-Plesset perturbation theory, and density functional theory

We have calculated the intermolecular interaction potentials of the silane dimer at the D3d conformation using the Hartree-Fock (HF) self-consistent theory, the correlation-corrected second-order M?ller-Plesset (MP2) perturbation theory, and the density functional theory (DFT) with 108 functionals chosen from the combinations of 9 exchange and 12 correlation functionals. Single-point coupled cluster [CCSD(T)] calculations have also been carried out to calibrate the correlation effect. The HF calculations yield unbound potentials largely because of the exchange-repulsion interaction. In the MP2 calculations, the basis set effects on the repulsion exponent, the equilibrium bond length, the binding energy, and the asymptotic behavior of the calculated intermolecular potentials have been thoroughly studied. We have employed basis sets from the Slater type orbitals fitted with Gaussian functions (STO-nG, n = 3 approximately 6), Pople's medium size basis sets [up to 6-311++G(3df,3pd)], to Dunning's correlation consistent basis sets (cc-pVXZ and aug-cc-pVXZ, X = D, T, Q). With increasing basis size, the repulsion exponent and the equilibrium bond length converge at the 6-31G** basis set and the 6-311++G(3d,3p) basis set, respectively, while a large basis set (aug-cc-pVTZ) is required to converge the binding energy at a chemical accuracy ( approximately 0.05 kcal/mol). Up to the largest basis set used, the asymptotic dispersion coefficient has not converged to the expected C6 value from molecular polarizability calculations. We attribute the slow convergence partly to the inefficacy of using the MP2 calculations with Gaussian type functions to model the asymptotic behavior. Both the basis set superposition error (BSSE) corrected and uncorrected results are presented to emphasize the importance of including such corrections. Only the BSSE corrected results systematically converge to the expected potential curve with increasing basis size. The DFT calculations generate a wide range of interaction patterns, from purely unbound to strongly bound, underestimating or overestimating the binding energy. The binding energies calculated using the OPTXHCTH147, PBEVP86, PBEP86, PW91TPSS, PW91PBE, and PW91PW91 functionals and the equilibrium bond lengths calculated using the MPWHCTH93, TPSSHCTH, PBEVP86, PBEP86, PW91TPSS, PW91PBE, and PW91PW91 functionals are close to the MP2 results using the 6-311++G(3df,3pd) basis set. A correlation between the calculated DFT potentials and the exchange and correlation enhancement factors at the low-density region has been elucidated. The asymptotic behaviors of the DFT potentials are also analyzed.     

Improved supermolecular second order Møller-Plesset intermolecular interaction energies using time-dependent density functional response theory

The supermolecular second order Moller-Plesset (MP2) intermolecular interaction energy is corrected by employing time-dependent density functional (TDDFT) response theory. This is done by replacing the uncoupled second order dispersion contribution contained in the supermolecular MP2 energy with the coupled dispersion energy obtained from the TDDFT approach. Preliminary results for the rare gas dimers He2, Ne2, and Ar2 and a few structures of the (HF)2 and (H2O)2 dimers show that the conventional MP2 interaction energies are considerably improved by this procedure if compared to coupled cluster singles doubles with perturbative triples [CCSD(T)] interaction energies. However, the quality of the interaction energies obtained in this way strongly depends on the exchange-correlation potential employed in the monomer calculations: It is shown that an exact exchange-only potential surprisingly often performs better than an asymptotically corrected hybrid exchange-correlation potential. Therefore the method proposed in this work is similar to the method by Cybulski and Lytle [J. Chem. Phys., 127, 141102 (2007)] which corrects the supermolecular MP2 energies with a scaled dispersion energy from time-dependent Hartree-Fock. The results in this work are also compared to the combination of density functional theory and intermolecular perturbation theory.     

A density functional theory investigation of CrSin (n=1–6) clusters

Clusters of the form CrSi_n (n=1–6) were investigated computationally using a density functional approach. In particular, geometry optimizations were carried out under the constraint of well-defined point group symmetries at the B3LYP level employing a pseudopotential method in conjunction with double zeta basis sets. In this article, the resulting total energies, Mulliken atomic net populations, overlap populations, fragmentation energies and geometries of CrSi_n (n=1–6) are presented and discussed, together with natural populations and natural electron configurations. In addition, we comment on the charge transfer within the clusters. From this analysis, the 3d orbital of the Cr atom in CrSi_n (n=1–6) cluster absorbs electrons. From this tendency, conclusions are drawn with respect to the electronic populations and the chemical bond between Si and Cr as well as Si and Si.     

The Nature of Nonclassical Carbonyl Ligands Explained by Kohn–Sham Molecular Orbital Theory

When carbonyl ligands coordinate to transition metals, their bond distance either increases (classical) or decreases (nonclassical) with respect to the bond length in the isolated CO molecule. C−O expansion can easily be understood by π-back-donation, which results in a population of the CO's π*-antibonding orbital and hence a weakening of its bond. Nonclassical carbonyl ligands are less straightforward to explain, and their nature is still subject of an ongoing debate. In this work, we studied five isoelectronic octahedral complexes, namely Fe(CO)₆²⁺, Mn(CO)₆⁺, Cr(CO)₆, V(CO)₆⁻ and Ti(CO)₆²⁻, at the ZORA-BLYP/TZ2P level of theory to explain this nonclassical behavior in the framework of Kohn–Sham molecular orbital theory. We show that there are two competing forces that affect the C−O bond length, namely electrostatic interactions (favoring C−O contraction) and π-back-donation (favoring C−O expansion). It is a balance between those two terms that determines whether the carbonyl is classical or nonclassical. By further decomposing the electrostatic interaction ΔV_elstat into four fundamental terms, we are able to rationalize why ΔV_elstat gives rise to the nonclassical behavior, leading to new insights into the driving forces behind C−O contraction.     

Application of capillary affinity electrophoresis and density functional theory to the investigation of valinomycin–lithium complex

Capillary affinity electrophoresis (CAE) and quantum mechanical density functional theory (DFT) have been applied to the investigation of interactions of valinomycin (Val), a macrocyclic dodecadepsipeptide antibiotic ionophore, with lithium cation Li⁺. Firstly, from the dependence of effective electrophoretic mobility of Val on the Li⁺ ion concentration in the background electrolyte (BGE) (methanolic solution of 50 mM chloroacetic acid, 25 mM Tris, pH_MeOH 7.8, 0–40 mM LiCl), the apparent binding (stability) constant (K_b) of Val–Li⁺ complex in methanol was evaluated as log K_b = 1.50 ± 0.24. The employed CAE method include correction of the effective mobilities measured at ambient temperature, at different input power (Joule heating) and at variable ionic strength of the BGEs to the mobilities related to the reference temperature 25 °C and to the constant ionic strength 25 mM. Secondly, using DFT calculations, the most probable structures of the non-hydrated Val–Li⁺ and hydrated Val–Li⁺·3H₂O complex species were predicted.     

How to obtain accurate results with molecular iodine and density functional theory?

Molecular iodine is found in many organic synthetic methodologies, both as reagent and as catalyst. Naturally, many groups have carried out computational studies involving this element. However, the choice of computational method proves to be more challenging than for other non-metals. We quantify herein the errors that some common theory levels can introduce in terms of both structural and energetic deviations. We also evaluate multiple post hoc corrections, namely, vibrational entropy, dispersion, and counter-poise corrections. Our results indicate the triple- $$ basis sets are essential to obtain quality results and that post hoc corrections are overall detrimental.