An average-of-configuration method for using Kohn-Sham density functional theory in modeling ligand-field theory

The Amsterdam Density Functional (ADF) package has been used to constrain Kohn-Sham DFT in such a fashion that a transition from KS-DFT to ligand-field theory in the form of the parametrical d(q)() model is completely well-defined. A relationship is established between the strong-field approximation of the parametrical d(2) model for the tetrahedral complexes VCl(4)(-) and VBr(4)(-) and certain fixed-orbital ADF-computed energies. In this way values for all the parameters of the d(2)() model may be computed, thus allowing the ADF results to be expressed in terms of a KS-DFT energy matrix that can be diagonalized. This means that the KS-DFT deficiency with regard to computation of nondiagonal elements has been overcome and the KS-DFT eigenenergies have become available through the KS-DFT mimicking of the ligand-field plus repulsion model. By using mutually orthogonal strong-field energy matrices, the mimicking has been further elucidated. The computed values for the empirical parameters of VCl(4)(-) and VBr(4)(-) are in good agreement with experimental data. The spectrochemical and the nephelauxetic series have been computed by including the remaining halide complexes and the quantitatively special position of F(-)() among the halides corroborated for both series.     

Topological analysis of electron densities from Kohn-Sham and subsystem density functional theory

In this study, we compare the electron densities for a set of hydrogen-bonded complexes obtained with either conventional Kohn-Sham density functional theory (DFT) calculations or with the frozen-density embedding (FDE) method, which is a subsystem approach to DFT. For a detailed analysis of the differences between these two methods, we compare the topology of the electron densities obtained from Kohn-Sham DFT and FDE in terms of deformation densities, bond critical points, and the negative Laplacian of the electron density. Different kinetic-energy functionals as needed for the frozen-density embedding method are tested and compared to a purely electrostatic embedding. It is shown that FDE is able to reproduce the characteristics of the density in the bonding region even in systems such as the F-H-F(-) molecule, which contains one of the strongest hydrogen bonds. Basis functions on the frozen system are usually required to accurately reproduce the electron densities of supermolecular calculations. However, it is shown here that it is in general sufficient to provide just a few basis functions in the boundary region between the two subsystems so that the use of the full supermolecular basis set can be avoided. It also turns out that electron-density deformations upon bonding predicted by FDE lack directionality with currently available functionals for the nonadditive kinetic-energy contribution.     

Energy surface, chemical potentials, Kohn-Sham energies in spin-polarized density functional theory

On the basis of the zero-temperature grand canonical ensemble generalization of the energy E[N,N(s),v,B] for fractional particle N and spin N(s) numbers, the energy surface over the (N,N(s)) plane is displayed and analyzed in the case of homogeneous external magnetic fields B(r). The (negative of the) left-/right-side derivatives of the energy with respect to N, N(↑), and N(↓) give the fixed-N(s), spin-up, and spin-down ionization potentials/electron affinities, respectively, while the derivative of E[N,N(s),v,B] with respect to N(s) gives the (signed) half excitation energy to the lowest-lying state with N(s) increased (or decreased) by 2. The highest occupied and lowest unoccupied Kohn-Sham spin-orbital energies are identified as the corresponding spin-up and spin-down ionization potentials and electron affinities. The excitation energies to the lowest-lying states with N(s)±2 can be obtained as the differences between the lowest unoccupied and the opposite-spin highest occupied spin-orbital energies, if the (N,N(s)) representation of the Kohn-Sham spin-potentials is used. The cases where the convexity condition on the energy does not hold are also discussed. Finally, the discontinuities of the energy derivatives and the Kohn-Sham potential are analyzed and related.     

Description of core excitations by time-dependent density functional theory with local density approximation, generalized gradient approximation, meta-generalized gradient approximation, and hybrid functionals

Time-dependent density functional theory (TDDFT) is employed to investigate exchange-correlation-functional dependence of the vertical core-excitation energies of several molecules including H, C, N, O, and F atoms. For the local density approximation (LDA), generalized gradient approximation (GGA), and meta-GGA, the calculated X1s-->pi* excitation energies (X = C, N, O, and F) are severely underestimated by more than 13 eV. On the other hand, time-dependent Hartree-Fock (TDHF) overestimates the excitation energies by more than 6 eV. The hybrid functionals perform better than pure TDDFT because HF exchange remedies the underestimation of pure TDDFT. Among these hybrid functionals, the Becke-Half-and-Half-Lee-Yang-Parr (BHHLYP) functional including 50% HF exchange provides the smallest error for core excitations. We have also discovered the systematic trend that the deviations of TDHF and TDDFT with the LDA, GGA, and meta-GGA functionals show a strong atom-dependence. Namely, their deviations become larger for heavier atoms, while the hybrid functionals are significantly less atom-dependent.     

A density matrix-based method for the linear-scaling calculation of dynamic second- and third-order properties at the Hartree-Fock and Kohn-Sham density functional theory levels

A density matrix-based time-dependent self-consistent field (D-TDSCF) method for the calculation of dynamic polarizabilities and first hyperpolarizabilities using the Hartree-Fock and Kohn-Sham density functional theory approaches is presented. The D-TDSCF method allows us to reduce the asymptotic scaling behavior of the computational effort from cubic to linear for systems with a nonvanishing band gap. The linear scaling is achieved by combining a density matrix-based reformulation of the TDSCF equations with linear-scaling schemes for the formation of Fock- or Kohn-Sham-type matrices. In our reformulation only potentially linear-scaling matrices enter the formulation and efficient sparse algebra routines can be employed. Furthermore, the corresponding formulas for the first hyperpolarizabilities are given in terms of zeroth- and first-order one-particle reduced density matrices according to Wigner's (2n+1) rule. The scaling behavior of our method is illustrated for first exemplary calculations with systems of up to 1011 atoms and 8899 basis functions.     

Development of the cyclic cluster model formalism for Kohn-Sham auxiliary density functional theory methods

The development of the cyclic cluster model (CCM) formalism for Kohn-Sham auxiliary density functional theory (KS-ADFT) methods is presented. The CCM is a direct space approach for the calculation of perfect and defective systems under periodic boundary conditions. Translational symmetry is introduced in the CCM by integral weighting. A consistent weighting scheme for all two-center and three-center interactions appearing in the KS-ADFT method is presented. For the first time, an approach for the numerical integration of the exchange-correlation potential within the cyclic cluster formalism is derived. The presented KS-ADFT CCM implementation was applied to covalent periodic systems. The results of cyclic and molecular cluster model (MCM) calculations for trans-polyacetylene, graphene, and diamond are discussed as examples for systems periodic in one, two, and three dimensions, respectively. All structures were optimized. It is shown that the CCM results represent the results of MCM calculations in the limit of infinite molecular clusters. By analyzing the electronic structure, we demonstrate that the symmetry of the corresponding periodic systems is retained in CCM calculations. The obtained geometric and electronic structures are compared with available data from the literature.     

Adiabatic approximation of time-dependent density matrix functional response theory

Time-dependent density matrix functional theory can be formulated in terms of coupled-perturbed response equations, in which a coupling matrix K(omega) features, analogous to the well-known time-dependent density functional theory (TDDFT) case. An adiabatic approximation is needed to solve these equations, but the adiabatic approximation is much more critical since there is not a good "zero order" as in TDDFT, in which the virtual-occupied Kohn-Sham orbital energy differences serve this purpose. We discuss a simple approximation proposed earlier which uses only results from static calculations, called the static approximation (SA), and show that it is deficient, since it leads to zero response of the natural orbital occupation numbers. This leads to wrong behavior in the omega-->0 limit. An improved adiabatic approximation (AA) is formulated. The two-electron system affords a derivation of exact coupled-perturbed equations for the density matrix response, permitting analytical comparison of the adiabatic approximation with the exact equations. For the two-electron system also, the exact density matrix functional (2-matrix in terms of 1-matrix) is known, enabling testing of the static and adiabatic approximations unobscured by approximations in the functional. The two-electron HeH(+) molecule shows that at the equilibrium distance, SA consistently underestimates the frequency-dependent polarizability alpha(omega), the adiabatic TDDFT overestimates alpha(omega), while AA improves upon SA and, indeed, AA produces the correct alpha(0). For stretched HeH(+), adiabatic density matrix functional theory corrects the too low first excitation energy and overpolarization of adiabatic TDDFT methods and exhibits excellent agreement with high-quality CCSD ("exact") results over a large omega range.     

Efficient computation of the exchange-correlation contribution in the density functional theory through multiresolution

A new algorithm is presented to improve the efficiency of the computation of exchange-correlation contributions, a major time-consuming step in a density functional theory (DFT) calculation. The new method, called multiresolution exchange correlation (mrXC), takes advantage of the variation in resolution among the Gaussian basis functions and shifts the calculation associated with low-resolution (smooth) basis function pairs to an even-spaced cubic grid. The cubic grid is much less dense in the vicinity of the nuclei than the atom-centered grid and the computation on the former is shown to be much more efficient than on the latter. MrXC does not alter the formalism of the current standard algorithm based on the atom-centered grid (ACG), but instead employs two fast and accurate transformations between the ACG and the cubic grid. Preliminary results with local density approximation have shown that mrXC yields three to five times improvement in efficiency with negligible error. The extension to DFT functionals with generalized gradient approximation is also briefly discussed.     

Temperature-dependent approach to chemical reactivity concepts in density functional theory

The chemical reactivity concepts of density functional theory are studied through a unified view in the temperature-dependent approach provided by the grand canonical ensemble. This procedure leads to a more general perspective that enriches our understanding of the behavior of the average energy and its derivatives with respect to the average number of electrons, provides alternative definitions for those quantities that are “ill defined” at zero temperature, and allows one to determine the relationships among reactivity concepts at any temperature. In particular, it has been found that at high temperatures the parabolic model for reactivity indicators may be justified through the electronic entropy term in the Helmholtz free energy, and that at nonzero temperatures there is an electronic heat capacity contribution to the average energy. In summary, the unified view of the temperature-dependent approach is an important complement to the zero-temperature formulation that clarifies fundamental issues therein.     

Nuclear reactivity indices in the context of spin polarized density functional theory

In this work, the nuclear reactivity indices of density functional theory have been generalized to the spin polarized case and their relationship to electron spin polarized indices has been established. In particular, the spin polarized version of the nuclear Fukui function has been proposed and a finite difference approximation has been used to evaluate it. Applications to a series of triatomic molecules demonstrate the ability of the new functions to predict the geometrical changes due to a change in the spin multiplicity. The main equations in the different ensembles have also been presented.     

Intrinsic nucleofugality scale within the framework of density functional reactivity theory

Nucleofugality is a measure of the quality of a leaving group in substitution and elimination reactions. In a conceptual DFT context, the nucleofugality is calculated for an elaborate set of common organic leaving groups, both in the gas phase and in two organic solvents (dichloromethane and methanol). An intrinsic nucleofugality scale is constructed showing fair agreement with the classical trends in leaving group capacity in organic chemistry. The correlation of the results with acidities (tabulated pK(a) values) on one hand and experimental solvolysis reaction rate constants (kinetic parameters) on the other hand is discussed. Finally, a conceptual DFT based formula is derived, describing the influence of the solvation energy on the nucleofugality; excellent correlations were found with explicit calculations for the studied leaving groups.     

The calculation of NMR shielding tensors based on density functional theory and the frozen-core approximation

The application of the frozen-core approximation to the calculation of the shielding tensor of nuclear magnetic resonance (NMR) spectroscopy is discussed and an implementation is presented. A complete formulation of the shielding calculation within the frozen-core approximation is given, both in general terms and for the special case of density functional theory (DFT) and “gauge including atomic orbitals” (GIAOs). The practical implementation is validated by a detailed discussion of the consequences of the approximation. The general conclusion is drawn that the frozen-core approximation is a useful tool for shielding calculations—if the valence space is increased to contain at least the ns, np, (n − 1)p, (n − 1)d (fourth period and higher) shells, where n is the number of the given period in the periodic table of elements. The new method is applied to ⁷⁷Se shieldings and chemical shifts for a small number of compounds. The agreement between theory and experiment is good for relative shifts, whereas calculated absolute shieldings are generally too small by about 300–400 ppm. This difference is attributed to the relativistic contraction of the core density at the selenium atom that had been explicitly incorporated into the experimental absolute shielding scale. © 1996 John Wiley & Sons, Inc.     

Self-interaction correction in a real-time Kohn-Sham scheme: Access to difficult excitations in time-dependent density functional theory

We present a real-time Kohn-Sham propagation scheme for the self-interaction correction (SIC). The multiplicative Kohn-Sham potential is constructed in real-time and real-space based on the generalized optimized effective potential equation. We demonstrate that this approach yields promising results for a wide range of test systems, including hydrogen terminated silicon clusters, conjugated molecular chains, and molecular charge-transfer systems. We analyze the nature of excitations by calculating transition densities from the time evolution and by evaluating the time-dependent exchange-correlation potential. A properly constructed Kohn-Sham SIC potential shows a time-dependent field-counteracting behavior. These favorable characteristics of the exchange-correlation potential may be lost in approximations such as the SIC-Slater potential.     

Structure, spectroscopy, and reactivity properties of porphyrin pincers: a conceptual density functional theory and time-dependent density functional theory study

Porphyrin and pincer complexes are both important categories of compounds in biological and catalytic systems. The idea to combine them is computationally investigated in this work. By employment of density functional theory (DFT), conceptual DFT, and time-dependent DFT approaches, structure, spectroscopy, and reactivity properties of porphyrin pincers are systematically studied for a selection of divalent metal ions. We found that the porphyrin pincers are structurally and spectroscopically different from their precursors and are more reactive in electrophilic and nucleophilic reactions. A few quantitative linear/exponential relationships have been discovered between bonding interactions, charge distributions, and DFT chemical reactivity indices. These results are implicative in chemical modification of hemoproteins and understanding chemical reactivity in heme-containing and other biologically important complexes and cofactors.     

The "JK-only" approximation in density matrix functional and wave function theory

Various energy functionals applying the "JK-only" approximation which leads to two-index two-electron integrals instead of four-index two-electron integrals in the electron-electron interaction term of the electronic energy are presented. Numerical results of multiconfiguration self-consistent field calculations for the best possible "JK-only" wave function are compared to those obtained from the pair excitation multiconfiguration self-consistent (PEMCSCF) method and two versions of density matrix functional theory. One of these is derived making explicit use of some necessary conditions for N representability of the second-order density matrix. It is shown that this method models the energy functional based on the best possible "JK-only" wave function with good accuracy. The calculations also indicate that only a minor fraction of the total correlation energy is incorporated by "JK-only" approaches for larger molecules.     

Structures and vibrational spectroscopic parameters of selenoxopropanedinitrile and selenoxosilanedicarbonitrile: Theoretical study based on density functional theory method

The molecular structures and vibrational spectra in harmonic and anharmonic approximations have been studied for selenoxopropanedinitrile and selenoxosilanedicarbonitrile in the gas phase. Density functional theory method with B3LYP functional and cc‐pVTZ basis set has been employed. Optimized structural parameters and spectroscopic constants, namely, anharmonic, rotational and centrifugal distortion, rotation–vibration coupling, and Coriolis coupling parameters, are reported. Infrared vibrational and Raman frequencies are provided with complete assignments to the fundamental bands, overtones, and combination tones of the molecules. This study shows that silicon for carbon substitution affects mainly those properties that are dependent on the CSe bond. The literature for these molecules is not available and therefore the data from this work would be suitable for their characterizations as and when they are synthesized. © 2009 Wiley Periodicals, Inc. Heteroatom Chem 20:208–217, 2009; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20535