Analytical energy gradient evaluation in relativistic and nonrelativistic density functional calculations

The expressions of analytical energy gradients in density functional theory and their implementation in programs are reported. The evaluation of analytical energy gradients can be carried out in the fully 4-component relativistic, approximate relativistic, and nonrelativistic density functional calculations under local density approximation or general gradient approximation with or without frozen core approximation using different basis sets in our programs. The translational invariance condition and the fact that the one-center terms do not contribute to the energy gradients are utilized to improve the calculation accuracy and to reduce the computational effort. The calculated results of energy gradients and optimized geometry as well as atomization energies of some molecules by the analytical gradient method are in very good agreement with results obtained by the numerical derivative method.     

Density functional studies of AnF6 (An=U,Np, and Pu) and UF6−nCln (n=1–6) using hybrid functionals: geometries and vibrational frequencies

We have compared the performance of widely used hybrid functionals for calculating the bond lengths and harmonic vibrational frequencies of AnF₆ (An=U, Np, and Pu) and UF_6?nCl_n (n=1–6) molecules using “small‐core” relativistic effective core potentials and extended basis sets. The calculated spectroscopic constants compare favorably with experimental data for the bond lengths (average error ≤ 0.01 Å) and vibrational frequencies (average error ≤ 7 cm^?1) of the AnF₆ molecules. The experimental vibrational frequencies of the stretching modes were available for most of the UF_6?nCl_n (n=1–6) molecules. The calculated vibrational frequencies are in good agreement with the experimental data to within 4.6 cm^?1 and 7.6 cm^?1 for selected Becke1 and Lee, Yang, Parr (B1LYP), and Becke3 and Perdew, Wang (B3PW91) functionals, respectively. We conclude that one can predict reliable geometries and vibrational frequencies for the unknown related systems using hybrid density functional calculations with the RECPs. The geometries and vibrational frequencies of the UF_6?nCl_n (n=1–6) molecules that have not been determined experimentally are also presented and discussed. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 2010–2017, 2001     

Benchmarking approximate density functional theory for s/d excitation energies in 3d transition metal cations

Holthausen has recently provided a comprehensive study of density functional theory for calculating the s/d excitation energies of the 3d transition metal cations. This study did not include the effects of scalar relativistic effects, and we show here that the inclusion of scalar relativistic effects significantly alters the conclusions of the study. We find, contrary to the previous study, that local functionals are more accurate for the excitation energies of 3d transition method cations than hybrid functionals. The most accurate functionals, of the 38 tested, are SLYP, PBE, BP86, PBELYP, and PW91.     

Density functional computations of 99Ru chemical shifts: relativistic effects, influence of the density functional, and study of solvent effects on fac-[Ru(CO)3I3]-

Solvent effects on the 99Ru NMR chemical shift of the complex fac-[Ru(CO)3I3]- are investigated computationally using density functional theory. Further, benchmark calculations of the 99Ru shift for a set of ten Ru complexes have been performed in order to calibrate the computational model and to determine the importance of relativistic effects on the 99Ru nuclear magnetic shielding and on the chemical shift. A computational model for fac-[Ru(CO)3I3]- that includes both explicit solvent molecules and a continuum model is shown to yield the best agreement with experiment. Relativistic corrections are shown to be of minor importance for determining 99Ru chemical shifts. On the other hand, the nature of the density functional is of importance. In agreement with literature data for ligand trends of 99Ru chemical shifts, the chemical shift range for different solvents is also best reproduced by a hybrid functional.     

Interaction of acetonitrile with alkaline metal cations: A density functional,coupled‐cluster,and quadratic configuration interaction study

A systematic quantum chemical study of CH₃CN and its CH₃CN?M⁺ 1:1 model adducts (M⁺∈{Li⁺, Na⁺}) is presented, with respect to binding energetics, structural and vibrational force field changes. Several gradient‐corrected density functional levels of theory were employed (of both “pure” and “hybrid” character), together with the coupled cluster including double substitutions from the Hartree–Fock determinant (CCD) and quadratic configuration interaction including single and double substitutions (QCISD) methods [with the rather large 6‐311G(d,p) basis set], and their performances compared. The binding energy decompositions according to the Kitaura–Morokuma scheme and the reduced variational space self‐consistent field (RVS‐SCF) method have shown that the electrostatic plus polarization interaction terms are primarily responsible for overall stabilization, while the charge‐transfer term is negligibly small and virtually identical for both adducts. The computed harmonic vibrational frequencies for acetonitrile correlate excellently with the experimental ones (r²>0.9998 for almost all cases, while for the BLYP level, r²=1). It is shown for the first time that the experimentally observed blue shifts of the ν_CN mode are caused even by formation of 1:1 adducts, contrary to the previously accepted opinions. The CCD and QCISD, as well as the BPW91 and BP86 levels of theory predict almost excellently the ν_CN mode blue shift upon adduct formation, while the BLYP and B3LYP levels perform significantly poorer. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Molecular structure and binding energies of monosubstituted hexacarbonyls of chromium,molybdenum, and tungsten: Relativistic density functional study

Relativistic density functional calculations have been carried out for the group VI transition metal carbonyls M(CO)₅L (M=Cr, Mo, W; L=OH₂, NH₃, PH₃, PMe₃, N₂, CO, OC (isocarbonyl), CS, CH₂, CF₂, CCl₂, NO⁺). The optimized molecular structures and M(SINGLE BOND)L bond dissociation energies, as well as the metal–carbonyl bond energy of the trans CO group, have been calculated. Besides the marked dependence of the trans M(SINGLE BOND)CO bond length on the type of ligand L, such an effect on the that bond energy is also observed. For the chromium compounds, the trans Cr(SINGLE BOND)CO bond length varies from 184 to 199 pm and its bond energy from 242 to 150 kJ/mol. For the molybdenum compounds, the range is 197 to 216 pm and 253 to 128 kJ/mol and, for tungsten, 198 to 214 pm and 293 to 159 kJ/mol. The observed trends can be explained with the π acceptor strength of the L ligand. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1985–1992, 1997     

Yb、YbO电子激发态的相对论含时密度泛函理论研究

本文用基于精确二分量哈密顿（exact two—component Hamiltonian）的相对论含时密度泛函理论（time-dependent relativistic density functional theory）计算了Yb和YbO的电子激发态,并利用对称性、自然原子轨道对激发态性质和归属进行了详细分析,所得结果支持实验对YbO基态与激发态的指认．     

First-order relativistic corrections to MP2 energy from standard gradient codes: Comparison with results from density functional theory

The evaluation of the first-order scalar relativistic corrections to MP2 energy based on either direct perturbation theory or the mass–velocity and Darwin terms is discussed. In a basis set of Lévy-Leblond spinors the one- and two-electron matrix elements of the relativistic Hamiltonian can be decomposed into a nonrelativistic part and a relativistic perturbation. Thus, a program capable of calculating nonrelativistic energy gradients can be used to calculate the cross-term between relativity and correlation. The method has been applied to selected closed-shell atoms (He, Be, Ne, and Ar) and molecules (CuH, AgH, and AuH). The calculated equilibrium distances and harmonic frequencies were compared with results from first-order relativistic density functional calculations. It was found that the cross-term is not the origin of the nonadditivity of relativistic and correlation effects. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1596–1603, 1998     

Relativistic effects on geometry and electronic structure of small Pdn species (n=1, 2, 4)

Relativistic effects on the properties of small neutral Pd_n species (n=1, 2, 4) and Pd₂⁻ have been examined for the first time at the all‐electron level by performing scalar‐relativistic and nonrelativistic density functional calculations using a gradient‐corrected density functional. Relativistic effects are found to be important: They lead to a contraction of bond lengths, increase of vibrational frequencies, and a significant enhancement of binding energies. While relativistic effects are quite uniform for several states of Pd₄, they vary for the states examined for Pd₂, leading to a change of ground state due to relativity. The calculated relativistic properties of Pd₂ and Pd₂⁻ are in good agreement with available experimental data from mass spectrometry and photoelectron spectroscopy. For Pd₄ three‐dimensional structures are found to be preferred to planar ones and many nearly isoenergetic isomers exist. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 405–416, 1999     

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.     

Theoretical investigation of metal oxides for SO2 capture through first-principles calculations

Atmospheric pollution is an accelerating environmental issue. Thus, methods to mitigate this problem are highly important. In this study, metal oxide-based materials for capturing SO₂ were theoretically identified using first-principles calculations. In specific, the thermodynamic properties of MnO, Na₂O, K₂O, and Ag₂O for capturing SO₂ were assessed by computing their vibrational density of states. The rank of the maximum temperature for capturing SO₂ as sulfites in decreasing order was K₂O > Na₂O > MnO > Ag₂O at a wide range of SO₂ pressure. For Na₂O and K₂O that showed a higher range of SO₂ capture temperatures than MnO and Ag₂O as sulfites, the maximum temperature of these two compounds for capturing SO₂ as sulfates was further explored. The maximum temperature of the metal oxides for capturing SO₂ changed upon pressurization of metal oxides. The maximum temperature of each metal oxide increased with increasing positive pressure (i.e., compression) and decreased with increasing negative pressure (i.e., expansion).     

Excited states of OsO4: A comprehensive time‐dependent relativistic density functional theory study

A large number of scalar as well as spinor excited states of OsO₄, in the experimentally accessible energy range of 3–11 eV, have been captured by time‐dependent relativistic density functional linear response theory based on an exact two‐component Hamiltonian resulting from the symmetrized elimination of the small component. The results are grossly in good agreement with those by the singles and doubles coupled‐cluster linear response theory in conjunction with relativistic effective core potentials. The simulated‐excitation spectrum is also in line with the available experiment. Furthermore, combined with detailed analysis of the excited states, the nature of the observed optical transitions is clearly elucidated. It is found that a few scalar states of ³T₁ and ³T₂ symmetries are split significantly by the spin‐orbit coupling. The possible source for the substantial spin‐orbit splittings of ligand molecular orbitals is carefully examined, leading to a new interpretation on the primary valence photoelectron ionization spectrum of OsO₄. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Adsorption of small molecules on helical gold nanorods: A relativistic density functional study

We study the adsorption of a variety of small molecules on helical gold nanorods using relativistic density functional theory. We focus on Au₄₀ which consists of a central linear strand of five gold atoms with seven helical strands of five gold atoms on a coaxial tube. All molecules preferentially adsorb at a single low‐coordinated gold atom on the coaxial tube at an end of Au₄₀. In most cases, there is significant charge transfer (CT) between Au₄₀ and the adsorbate, for CO and NO₂, there is CT from the Au₄₀ to adsorbate while for all other molecules there is CT from the adsorbate to Au₄₀. Thus, Au₄₀‐adsorbate can be described as a donor–accepter complex and we use charge decomposition analysis to better understand the adsorption process. We determine the adsorption energy order to be C₅H₅N >NO₂ > CO > NH₃ > CH₂?CH₂ > CH₂?CH? CHO > NO > HC?CH > H₂S > SO₂ > HCN > CH₃OH > H₂C?O > O₂ > H₂O > CH₄ > N₂. We find that the Au? C, Au? N, Au? S, and Au? O bonds are surprisingly strong, with clear implications for reactivity enhancement of the adsorbate. The Au? H bond is relatively weak but, for interactions via an H atom that is bonded to a carbon atom (e.g., CH₄), we find that there is large charge polarization of the Au? H? C moiety and partial activation of the inert C? H bond. Although the Au? S and Au? O bonds are generally weaker than the Au? C and Au? N bonds, we find that adsorption of H₂S or H₂O causes greater distortion of Au₄₀ in the binding region. However, the degree of distortion is small and the helical structure is retained, demonstrating the stability of the helical Au₄₀ nanorod under perturbations. © 2014 Wiley Periodicals, Inc.     

A screened hybrid density functional study on energetic complexes: Metal carbohydrazide nitrates

The molecular geometry, electronic structure and thermochemistry of a series of metal carbohydrazide nitrates were investigated using the Heyd–Scuseria–Ernzerhof (HSE) screened hybrid density functional. The results show that Ca, Sr, and Ba complexes have additional coordinated oxygen atoms from the nitrate ion, which differ obviously from Cu, Ni, Co, and Mg complexes in terms of the geometric structure. Detailed NBO analyses clearly indicate that the metal–ligand interactions in Cu, Ni, and Co complexes are covalent, whereas those of Mg, Ca, Sr, and Ba complexes are ionic in nature. Furthermore, the donor–acceptor interactions result in a reduction of occupancies of σ_{C? O} and σ_{N? H} orbitals. Consequently, the bond lengths increase and the bond orders decrease. Finally, the calculated heats of formation predict that the ionic alkaline‐earth metal carbohydrazide nitrates are more stable than the covalent transition metal carbohydrazide nitrates. It agrees well with the available experimental thermal stabilities, indicating that the metal–ligand bonding character plays an important role in the stabilities of these energetic complexes. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Interaction of copernicium with gold: Assessment of applicability of simple density functional theories

Interactions of Cn (element 112) atom with small Au clusters are studied using accurate ab initio scalar relativistic coupled cluster method for correlation treatment and two‐component relativistic density functional theory (RDFT) to take account of spin‐dependent relativistic effects. The results demonstrate the failure of RDFT with simple generalized‐gradient and hybrid functionals in describing Cn–Au bonds in complex systems. © 2013 Wiley Periodicals, Inc.     

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.     

Associative versus dissociative binding of CO to 4d transition metal trimers: A density functional study

Density functional calculations were performed to determine equilibrium geometrical structures, transition states and relative energies for M(3) clusters (M = Nb, Mo, Tc, Ru, Rh, Pd, Ag) reacting with CO, leading to proposed reaction pathways. For the Nb(3), Mo(3), and Tc(3) clusters, the lowest energy structure correlates to dissociated CO, with the C and O atoms bound on opposite sides of the metal triangle. For all other trimers, the lowest energy structures maintain the CO moiety. In the case of Pd(3) and Ag(3) the dissociated geometries lie higher in energy than the sum of the separated reactants. In most cases, several multiplicities were found to be similar in energy and for Mo(3)CO and Pd(3)CO singlet-triplet minimum energy crossing points were identified. In the case of Rh(3)CO, minimum energy crossing points for the doublet, quartet, and sextet reaction pathways were determined and compared. The electron densities of pertinent M(3)CO species were investigated using Natural Bond Order calculations. It was found that the effect of the metal trimer on the energy of the pure p-type pi* antibonding orbital of carbon monoxide directly correlates with the occurrence of CO dissociation.     

Vibrational corrections to geometries of transition metal complexes from density functional theory

Zero-point vibrational corrections are computed at the BP86/AE1 level for the set of 50 transition-metal/ligand bonds that have recently been proposed as testing ground for DFT methods, because of the availability of precise experimental gas-phase geometries (Bühl and Kabrede, J Chem Theory Comput 2006, 2, 1282). These corrections are indicated to be transferable to a large extent between various density-functional/basis-set combinations, so that they can be used to estimate zero-point averaged r0g distances from re values optimized at other theoretical levels. Applying this approach to a number of popular DFT levels does not, in general, improve their overall accuracy in terms of mean and standard deviations from experiment. The hybrid variant of the meta-functional TPSS is confirmed as promising choice for computing structures of transition-metal complexes.     

Transition metal atoms on oxide supports density functional calculations

The adsorption properties of Cu, Ag, Ni, and Pd atoms on O^2?, F, and F⁺ sites of MgO, CaO, SrO, and BaO (001) surfaces have been studied by means of density functional calculations. The examined clusters were embedded in the simulated Coulomb fields that closely approximate the Madelung fields of the host surfaces. The adsorption properties have been analyzed with reference to the basicity and energy gap of the oxide support in addition to orbital interactions. While the free Ni d⁹s¹ triplet ground state is preserved on adsorption on the O^2? sites of MgO, CaO, and SrO surfaces, it is no longer preserved on the O^2? site of BaO. For all adsorbates considered, adsorption is found to be stronger on F⁺ sites compared with regular O^2? sites. While on the O^2? site, Pd and Ni form the most stable complexes, on the F site, Pd and Cu form the most stable counterparts. On the F⁺ site, the single valence electron of Cu and Ag atoms couples with the unpaired electron of the vacancy forming a covalent bond. As a result, the adsorption energies of these atoms on the F⁺ site are stronger than those on the F and O^2? sites. The adsorption properties correlate linearly with the basicity and energy gap of the oxide support in addition to orbital interactions. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

A density functional theory-based chemical potential equalisation approach to molecular polarizability

The electron density changes in molecular systems in the presence of external electric fields are modeled for simplicity in terms of the induced charges and dipole moments at the individual atomic sites. A chemical potential equalisation scheme is proposed for the calculation of these quantities and hence the dipole polarizability within the framework of density functional theory based linear response theory. The resulting polarizability is expressed in terms of the contributions from individual atoms in the molecule. A few illustrative numerical calculations are shown to predict the molecular polarizabilities in good agreement with available results. The usefulness of the approach to the calculation of intermolecular interaction needed for computer simulation is highlighted.