Filipp Furche

Adiabatic time-dependent density functional methods for excited state properties

This work presents theory, implementation, and validation of excited state properties obtained from time-dependent density functional theory (TDDFT). Based on a fully variational expression for the excited state energy, a compact derivation of first order properties is given. We report an implementation of analytic excited state gradients and charge moments for local, gradient corrected, and hybrid functionals, as well as for the configuration interaction singles (CIS) and time-dependent Hartree–Fock (TDHF) methods. By exploiting analogies to ground state energy and gradient calculations, efficient techniques can be transferred to excited state methods. Benchmark results demonstrate that, for low-lying excited states, geometry optimizations are not substantially more expensive than for the ground state, independent of the molecular size. We assess the quality of calculated adiabatic excitation energies, structures, dipole moments, and vibrational frequencies by comparison with accurate experimental data f...

Property-optimized Gaussian basis sets for molecular response calculations

With recent advances in electronic structure methods, first-principles calculations of electronic response properties, such as linear and nonlinear polarizabilities, have become possible for molecules with more than 100 atoms. Basis set incompleteness is typically the main source of error in such calculations since traditional diffuse augmented basis sets are too costly to use or suffer from near linear dependence. To address this problem, we construct the first comprehensive set of property-optimized augmented basis sets for elements H-Rn except lanthanides. The new basis sets build on the Karlsruhe segmented contracted basis sets of split-valence to quadruple-zeta valence quality and add a small number of moderately diffuse basis functions. The exponents are determined variationally by maximization of atomic Hartree-Fock polarizabilities using analytical derivative methods. The performance of the resulting basis sets is assessed using a set of 313 molecular static Hartree-Fock polarizabilities. The mean absolute basis set errors are 3.6%, 1.1%, and 0.3% for property-optimized basis sets of split-valence, triple-zeta, and quadruple-zeta valence quality, respectively. Density functional and second-order Møller-Plesset polarizabilities show similar basis set convergence. We demonstrate the efficiency of our basis sets by computing static polarizabilities of icosahedral fullerenes up to C(720) using hybrid density functional theory.

Gaussian basis sets of quadruple zeta valence quality for atoms H-Kr

We present Gaussian basis sets of quadruple zeta valence quality with a segmented contraction scheme for atoms H to Kr. This extends earlier work on segmented contracted split valence (SV) and triple zeta valence (TZV) basis sets. Contraction coefficients and orbital exponents are fully optimized in atomic Hartree–Fock (HF) calculations. As opposed to other quadruple zeta basis sets, the basis set errors in atomic ground-state HF energies are less than 1 mEh and increase smoothly across the Periodic Table, while the number of primitives is comparably small. Polarization functions are taken partly from previous work, partly optimized in atomic MP2 calculations, and for a few cases determined at the HF level for excited atomic states nearly degenerate with the ground state. This leads to basis sets denoted QZVP for HF and density functional theory (DFT) calculations, and for some atoms to a larger basis recommended for correlated treatments, QZVPP. We assess the performance of the basis sets in molecular HF...

An efficient implementation of second analytical derivatives for density functional methods

Abstract We present an implementation of analytical second-order geometric derivatives for density functional methods using Gaussian basis sets. Key features include a stable and efficient numerical quadrature, the direct iterative solution of the coupled perturbed Kohn–Sham equations, integral prescreening based on rigorous estimates, and exploitation of point group symmetry for all finite groups. Benchmark results indicate a moderate cubic growth of CPU and storage requirements with system size; low symmetry molecules with up to 100 heavy atoms can be treated on personal computers. The performance of gradient corrected functionals in predicting IR spectra of larger molecules is exemplified for transition metal carbonyl complexes.

The structures of small gold cluster anions as determined by a combination of ion mobility measurements and density functional calculations

A combined experimental and theoretical study of small gold cluster anions is performed. The experimental effort consists of ion mobilitymeasurements that lead to the assignment of the collision cross sections for the different cluster sizes at room temperature. The theoretical study is based on ab initiomolecular dynamics calculations with the goal to find energetically favorable candidate structures. By comparison of the theoretical results with the measured collision cross sections as well as vertical detachment energies (VDEs) from the literature, we assign structures for the small Au n − ions (n<13) and locate the transition from planar to three-dimensional structures. While a unique assignment based on the observed VDEs alone is generally not possible, the collision cross sections provide a direct and rather sensitive measure of the cluster structure. In contrast to what was expected from other metal clusters and previous theoretical studies, the structural transition occurs at an unusually large cluster size of twelve atoms.

On the density matrix based approach to time-dependent density functional response theory

The formulation of time-dependent Kohn–Sham (TDKS) response theory based on the noninteracting one-particle density matrix is reanalyzed in detail. A transparent derivation starting from a von-Neumann-type equation of motion for the TDKS one-particle density matrix is presented. The resulting scheme has a simple structure and leads to compact expressions for frequency-dependent response properties. A systematic treatment of excited states is inferred from a pole analysis of the frequency-dependent density matrix response. A variational principle for excitation energies is established. Excited state properties are straightforward by analytical derivative techniques. The theory provides a particularly suitable starting point for linear scaling implementations. Magneto-optic properties such as rotatory strengths and the rotatory dispersion are accessible from the TDKS current-density response. The formalism is gauge-invariant. Various new sum rules within the adiabatic approximation (AA) are derived. It is s...

Electron correlation methods based on the random phase approximation

In the past decade, the random phase approximation (RPA) has emerged as a promising post-Kohn–Sham method to treat electron correlation in molecules, surfaces, and solids. In this review, we explain how RPA arises naturally as a zero-order approximation from the adiabatic connection and the fluctuation-dissipation theorem in a density functional context. This is contrasted to RPA with exchange (RPAX) in a post-Hartree–Fock context. In both methods, RPA and RPAX, the correlation energy may be expressed as a sum over zero-point energies of harmonic oscillators representing collective electronic excitations, consistent with the physical picture originally proposed by Bohm and Pines. The extra factor 1/2 in the RPAX case is rigorously derived. Approaches beyond RPA are briefly summarized. We also review computational strategies implementing RPA. The combination of auxiliary expansions and imaginary frequency integration methods has lead to recent progress in this field, making RPA calculations affordable for systems with over 100 atoms. Finally, we summarize benchmark applications of RPA to various molecular and solid-state properties, including relative energies of conformers, reaction energies involving weak and covalent interactions, diatomic potential energy curves, ionization potentials and electron affinities, surface adsorption energies, bulk cohesive energies and lattice constants. RPA barrier heights for an extended benchmark set are presented. RPA is an order of magnitude more accurate than semi-local functionals such as B3LYP for non-covalent interactions rivaling the best empirically parametrized methods. Larger but systematic errors are observed for processes that do not conserve the number of electron pairs, such as atomization and ionization.

Efficient characterization of stationary points on potential energy surfaces

Traditional methods for characterizing an optimized molecular structure as a minimum or as a saddle point on the nuclear potential energy surface require the full Hessian. However, if f denotes the number of nuclear degrees of freedom, a full Hessian calculation is more expensive than a single point geometry optimization step by the order of magnitude of f. Here we present a method which allows to determine the lowest vibrational frequencies of a molecule at significantly lower cost. Our approach takes advantage of the fact that only a few perturbed first-order wave functions need to be computed in an iterative diagonalization scheme instead of f ones in a full Hessian calculation. We outline an implementation for Hartree–Fock and density functional methods. Applications indicate a scaling similar to that of a single point energy or gradient calculation, but with a larger prefactor. Depending on the number of soft vibrational modes, the iterative method becomes effective for systems with more than 30–50 a...

Developing the random phase approximation into a practical post-Kohn-Sham correlation model.

The random phase approximation (RPA) to the density functional correlation energy systematically improves upon many limitations of present semilocal functionals, but was considered too computationally expensive for widespread use in the past. Here a physically appealing reformulation of the RPA correlation model is developed that substantially reduces its computational complexity. The density functional RPA correlation energy is shown to equal one-half times the difference of all RPA electronic excitation energies computed at full and first order coupling. Thus, the RPA correlation energy may be considered as a difference of electronic zero point vibrational energies, where each eigenmode corresponds to an electronic excitation. This surprisingly simple result is intimately related to plasma theories of electron correlation. Differences to electron pair correlation models underlying popular correlated wave function methods are discussed. The RPA correlation energy is further transformed into an explicit functional of the Kohn-Sham orbitals. The only nontrivial ingredient to this functional is the sign function of the response operator. A stable iterative algorithm to evaluate this sign function based on the Newton-Schulz iteration is presented. Integral direct implementations scale as the fifth power of the system size, similar to second order Moller-Plesset calculations. With these improvements, RPA may become the long-sought robust and efficient zero order post-Kohn-Sham correlation model.

III – Density Functional Methods for Excited States: Equilibrium Structure and Electronic Spectra

This chapter discusses density functional methods for excited states. Density functional theory (DFT) is nowadays one of the most popular methods for ground state electronic structure calculations in quantum chemistry and solid state physics. A number of commercial programs are available, and DFT calculations of ground state energies, structures, and many other properties are routinely performed by nonexperts in (bio-)chemistry, physics, and materials sciences. The chapter focuses on the use of time-dependent density functional theory (TDDFT). Algorithms to compute spectra and excited state properties are also reviewed. The chapter describes the steps necessary in a TDDFT excited state calculation and some timing for typical applications are presented. Further the chapter also summarizes the performance of TDDFT excitation energies, transition moments, and excited state properties. Specific applications are surveyed that included compounds such as aromatic systems and fullerenes, porphyrins and related compounds, transition metal compounds, metal and semiconductor clusters, organic polymers, and biologically relevant systems.