Coupled Perturbed Kohn-Sham Equations Research Articles

Zero-field splitting parameters within exact two-component theory and modern density functional theory using seminumerical integration.

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin-dipole contribution and the response equations of the spin-orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn-Sham equations for the spin-orbit perturbation term. Second, the spin-orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin-orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin-orbit mean field Ansatz.

Analytic Gradient for Density Functional Theory Based on the Fragment Molecular Orbital Method.

The equations for the response terms for the fragment molecular orbital (FMO) method interfaced with the density functional theory (DFT) gradient are derived and implemented. Compared to the previous FMO-DFT gradient, which lacks response terms, the FMO-DFT analytic gradient has improved accuracy for a variety of functionals, when compared to numerical gradients. The FMO-DFT gradient agrees with the fully ab initio DFT gradient in which no fragmentation is performed, while reducing the nonlinear scaling associated with standard DFT. Solving for the response terms requires the solution of the coupled perturbed Kohn-Sham (CPKS) equations, where the CPKS equations are solved through a decoupled Z-vector procedure called the self-consistent Z-vector method. FMO-DFT is a nonvariational method and the FMO-DFT gradient is unique compared to standard DFT gradients in that the FMO-DFT gradient requires terms from both DFT and time-dependent density functional theory (TDDFT) theories.

A Density Functional Study of the Electronic Structure and Spin Hamiltonian Parameters of Mononuclear Thiomolybdenyl Complexes

The electron paramagnetic resonance spin Hamiltonian parameters of mononuclear thiomolybdenyl complexes based upon the tris(pyrazolyl)borate ligand, together with their molybdenyl analogues, are calculated using density functional theory. The electronic g matrix and 95Mo hyperfine matrix are calculated as second-order response properties from the coupled-perturbed Kohn-Sham equations. The scalar relativistic zero-order regular approximation (ZORA) is used with an all-electron basis and an accurate mean-field spin-orbit operator which includes all one- and two-electron terms. The principal values and relative orientations of the g and A interaction matrices obtained from the experimental spectra in a previous EPR study are compared with those obtained from unrestricted Kohn-Sham calculations at the BP86 and B3LYP level, and the latter are found to be in good quantitative agreement. A quasi-restricted approach is used to analyze the influence of the various molecular orbitals on g and A. In all complexes the ground state magnetic orbital is dX2-Y2-based and the orientation of the A matrix is directly related to the orientation of this orbital. The largest single contribution to the orientation of the g matrix arises from the spin-orbit coupling of the dYZ-based lowest-unoccupied molecular orbital into the ground state. A number of smaller, cumulative charge-transfer contributions augment the d-d contributions. A comparison of the theoretical EPR parameters obtained using both crystallographic and gas-phase geometry-optimized structures of Tp*MoO(bdt) (Tp* = hydrotris(3,5-dimethylpyrazol-1-yl)borate, bdt = 1,2-benzenedithiolate) suggests a correspondence between the metal-dithiolate fold angle and the angle of noncoincidence between g and A.

Nuclear second analytical derivative calculations using auxiliary basis set expansions

We present an efficient implementation of second order analytical derivatives (force constants) within density functional theory which applies to non-hybrid functionals. The dominant step in force constant treatments concerns the solution of the coupled perturbed Kohn–Sham equations (CPKS). We employ an auxiliary basis expansion, the RI- J technique (resolution of the identity for the interelectronic Couomb energy J), to accelerate the solution of the CPKS equations. This results in pronounced reductions of computational work and implies insignificant losses in accuracy as is shown by extensive tests. A force constant calculation of cyanocobalamin is reported as a demonstrative application.

Time-dependent density functional theory calculations of molecular static and dynamic polarizabilities, cauchy coefficients and their anisotropies with atomic numerical basis functions

Static and dynamic polarizabilities of a range of small first row compounds have been calculated with time-dependent density functional theory in the local spin-density approximation using numerical atomic basis sets. The results are compared to earlier computational work, in particular the work of Van Caillie and Amos [C. Van Caillie, R.D. Amos, Chem. Phys. Lett. 291 (1998) 71], as well as experimental values. The results for static isotropic and anisotropic polarizabilities of H 2O, N 2, CO, NH 3, and CH 4 are in good agreement with previous calculations. The results for the dynamic polarizabilities as expressed in the S(−4) Cauchy coefficients and their anisotropies for H 2O, N 2, CO 2, NH 3, CH 4, C 2H 2, C 2H 4 and C 2H 6 are also in good agreement with previous results. We have also explored the scaling of our implementation of the time-dependent coupled-perturbed Kohn–Sham equations by evaluating the static and dynamic polarizabilities of bifurcated water chains ranging from 1–20 molecules in size.

Improving harmonic vibrational frequencies calculations in density functional theory

Using a previously introduced weight scheme, microbatching, and grid compression [R. E. Stratmann, G. E. Scuseria and M. J. Frisch, Chem. Phys. Lett. 257, 213 (1996)], we significantly speed up the numerical integration of the exchange-correlation contribution to the Coupled-Perturbed Kohn–Sham equations. In addition, we find that the nature of the integrand is such that it is possible to employ substantially fewer grid points in the quadrature and to use the Gaussian very Fast Multipole Method (GvFMM) with very short multipole expansions for the Coulomb contribution, with negligible loss in accuracy. As a representative example, the computational demand for the exchange-correlation portion of a coronene (C24H12) frequency calculation with a 3-21G basis is reduced by more than one order of magnitude. The overall speed up achieved in this calculation is between a factor of 4 to 6, depending on the specific functional. We also present sample calculations using polarized bases, gradient-corrected functionals, and on even larger systems (C54H18 and C96H24), to illustrate the various effects and improvements that we have accomplished.

The determination of hyperpolarisabilities using density functional theory

The theory for the calculation of the static polarisability α and hyperpolarisability β using density functional theory is presented. In particular the computational implementation of the coupled-perturbed Kohn—Sham equations is discussed. Calculations on CH 2O and CH 3CN are reported using large basis sets and accurate quadrature using the local density approximation (S-VWN). The results suggest that DFT is a promising method for the determination of these properties. Finally it is argued that the theory of Vignale, Rasolt and Geldart can be used as a basis for the determination of frequency-dependent polarisabilities.