Volume 224, Issue 3-4 -

A new implementation of the Dynamical Thresholding (DT) coupled cluster doubles (CCD) approximation is presented. Using prescreening techniques for the intermediate quantities that occur in the CCD amplitude equations, a computationally efficient algorithm is obtained, in which one single threshold allows for control of accuracy. A critical evaluation of the accuracy and efficiency is carried out for two testcases. Furthermore, the issue of potential energy surface discontinuities and the usage of enveloping localized orbitals is discussed in detail.

Time-dependent density functional theory is one of the most widely used methods for the calculation of excited states of large molecules. However, it exhibits substantial problems with charge-transfer excited states when conventional exchange-correlation functionals are employed. Here, we introduce an additive long-range potential that can, in principle, be directly employed with any existing local, GGA or hybrid-functional, without re-fitting of the original functionals. The additive potential shifts the excitation energies to higher values and corrects for the wrong asymptotic behavior of their potential energy surfaces with respect to charge-separating coordinates. First examples of its successful application are shown.

Kohn-Sham methods with orbital-dependent exchange-correlation functionals, in particular exact-exchange methods, are discussed. The numerical stability of optimized-effective-potential methods to construct the exchange-correlation potential in the case of orbital-dependent energy functionals is considered. A numerically stable exact-exchange Kohn-Sham method is briefly presented. A new ansatz for time-dependent density-functional methods in the response regime is considered and an exact-exchange time-dependent density-functional method based on it is discussed and it is demonstrated that this methods is able to treat qualitatively correctly charge-transfer excitations.

The diffusion quantum Monte Carlo method (DMC) is capable of calculating accurately the electronic energy for molecules and molecular aggregates. An overview is given on recent developments for the optimization of the guide function that determines the accuracy of the method. Furthermore, the versatility of DMC is shown with applications to Rydberg states, transition metal compounds, and weakly interacting systems.

We describe the software package SPEX, which allows first-principles calculations of quasiparticle and collective electronic excitations in solids using techniques from many-body perturbation theory. The implementation is based on the full-potential linearized augmented-plane-wave (FLAPW) method, which treats core and valence electrons on an equal footing and can be applied to a wide range of materials, including transition metals and rare earths. After a discussion of essential features that contribute to the high numerical efficiency of the code, we present illustrative results for quasiparticle band structures calculated within the GW approximation for the electronic self-energy, electron-energy-loss spectra with inter- and intraband transitions as well as local-field effects, and spin-wave spectra of itinerant ferromagnets. In all cases the inclusion of many-body correlation terms leads to very good quantitative agreement with experimental spectroscopies.

Ab initio electron correlation calculations based on quantum-chemical methods are successfully applied to metallic systems via the method of increments, which is an expansion of the correlation energy in terms of one-body, two-body, and higher-order contributions from localized orbital groups. To deal with the two distinct problems that occur in metals, the difficulty of localization of the orbitals and the generation of clusters with neutral atoms in the center, we proposed an embedding scheme which has itself no metallic character but can mimic the metal in the internal region, where the atoms are correlated. The first application was made for solid mercury, where a very good agreement with experimental ground-state properties was achieved. Further the approach has been extended to other group 2 and 12 metals (Be, Mg, Zn, and Cd) where the metallic character is more pronounced than in mercury. Different variants of the embedding and the localized orbitals used for the incremental scheme have been developed and tested. Application of the method of increments to the investigated metallic systems allows us not only to obtain values close to experimental data but also to understand the influence of individual correlation-energy increments on cohesive properties and to clarify thereby some aspects of the structural features of the group 12 metals.

The derivation of coupled-cluster response theory for explicitly correlated wavefunctions with terms linear in the interelectronic distance r 12 (CC-R12) or a function f 12 thereof (CC-F12) is reviewed and an implementation at the CCSD(R12) and CCSD(F12) for excitation energies and the linear, quadratic and cubic reponse functions is reported for ansätze 1 and 2. We discuss the results of first applications for vertical excitation energies, excited state equilibrium structures and vibrational frequencies, polarizabilities and first and second hyperpolarizabilities with focus on present limitations and future prospects of the approach. We show in particular that the standard choice of the geminal functions and wavefunction expansion can in certain cases lead to a bias towards the unperturbed ground state and how this can be avoided by extending the geminal space. If taken care of this, in particular CCSD(F12)/ansatz 2 shows for the potential energy curves of excited states and for optical properties a similar enhanced basis set convergence of the correlation contribution due to connected double excitations as for ground state energies. For diffuse excited states and hyperpolarizbilities the correlation contribution converges with CCSD(F12)/ansatz 2 often faster than orbital relaxation contributions. The convergence of the latter can be improved by allowing single excitations into a space spanned by a complementary auxiliary basis set.

A brief review of our linear-scaling method for atomic-orbital (AO) second-order Møller-Plesset perturbation theory (MP2) is given. The key feature of our method is the rigorous preselection of numerically significant four-center two-electron integrals based on multipole-based integral estimates (MBIE) that do not only account for the exponential coupling between the Gaussian-type basis functions forming charge distributions, but also for the 1/R coupling between the charge distributions. This coupling turns for the required integral products in AO-MP2 into at least a 1/R 4 or even a 1/R 6 decay behavior. Using MBIE we attain linear scaling, which is illustrated for DNA fragments with up to 1052 atoms and 10 674 basis functions as computed on a single processor. The largest molecule calculated in our present work at the scaled-opposite spin (SOS-) AO-MP2 level is an RNA system comprising 1664 atoms and 19 182 basis functions. Furthermore, we present results for the use of Cholesky-decomposed pseudo-density matrices in Laplace-based MP2, that offers the advantage of exploiting occupied/virtual blocking both with and without auxiliary basis sets.

This work describes the implementation of an efficient two-component quasirelativistic density functional and Hartree-Fock program. The fact that the basis functions are real can be exploited if a special internal representation of operators and density matrices is used. This also leads to a considerable reduction of the effort in the closed shell case. While in most applications to open shell molecules, the noncollinear approach to define a relativistic spin density is preferable, the collinear approach finds its application in the calculation of magnetic anisotropy energies. Linear algebra steps in the SCF procedure have a higher relative weight compared to the nonrelativistic case, therefore some care was necessary to make them fast when parallelizing the code. The quasirelativistic Hamiltonians that have been implemented are the ´zeroth-order regular approximation´ (ZORA) Hamiltonian, the Douglas-Kroll-Hess Hamiltonian up to the sixth order, together with an accurate approximation to treat the picture change effect of the electron interaction, and effective core potential (ECP) matrix elements. Geometry gradients are available for the ZORA and ECP methods.

We review the progress in the recent development of the relativistic optimized effective potential (ROEP) method within spin-density functional theory. The ROEP-equations for spin-polarized systems are derived and their application to open-shell atoms using the exact exchange approximation for the exchange correlation-functional is presented. Further, we expand the ROEP framework to treat extended systems within the KKR-multiple scattering formalism. We illustrate the application of the theory to open-shell free atoms and solids.

This article reviews the periodic LMP2 method and its implementation in the CRYSCOR code. The main steps of the LMP2 calculations and the techniques employed are briefly described. Illustrative single-point calculations for three TiO 2 polymorphs: rutile, anatase and brookite in their experimental geometry are performed. It is shown that the method scales linearly with respect to the number of atoms per unit cell, and can be applied to relatively complex periodic systems. The LMP2 method in contrast to DFT positions rutile slightly lower in the energy than anatase. Brookite is found to be the most stable. However, since the energy differences are relatively small, a further investigation of the influence of the domain-sizes, geometry, zero-point vibrations, etc. on the relative stability of these systems is required.

Exchange-correlation energy functionals depending on the Kohn-Sham orbitals and eigenvalues resolve some of the well-known deficiencies of the local density and generalized gradient approximations. Such functionals can be derived in first-principles fashion by use of standard many-body techniques, if the Kohn-Sham Hamiltonian is utilized as non-interacting reference Hamiltonian. In this way one can establish an exact relation for the exchange-correlation functional, which opens several routes for the derivation of approximate functionals. Straightforward expansion in powers of the electron-electron coupling constant gives, to first order, the exact exchange of density functional theory and, to second order, a correlation functional which has the same structure as the second order Møller-Plesset term. This simplest first-principles correlation functional reproduces both the shell structure in the exact correlation potential and dispersion forces. On the other hand, it overestimates all correlation effects and is variationally instable for systems with a very small HOMO-LUMO gap. Both deficiences can be resolved by partial resummation of the Kohn-Sham perturbation expansion. In this contribution two such resummations are discussed, a minimum form designed to address the variational instability at no computational cost (based on the Epstein-Nesbet diagrams) and a more systematic variant (based on the ring diagrams).

Using the discontinuity of the chemical potential as a function of excess charge, the fundamental gaps for finite systems and the band gaps of extended solids are determined within reduced density matrix functional theory. We also present the necessary and sufficient conditions for the one-body reduced density matrix of a system with fractional charge to be ensemble N -representable. The performance of most modern day reduced density matrix functionals is assessed for the gaps and the correlation energy of finite systems. Our results show that for finite systems the PNOF, BBC3, and power functionals yield very accurate correlation energies while for a correct description of the fundamental gap the removal of self-interaction terms is essential. For extended solids we find that the power functional captures the correct band gap behavior for conventional semiconductors as well as strongly correlated Mott insulators, where a gap is obtained in absence of any magnetic ordering.

A review is given on recent work with range-separated hybrid density-functional approaches, using accurate wavefunction-based ab initio methods for the treatment of the long-range inter-electronic interaction. It is shown that in many cases results can be improved with respect to the borderline methods, since the hybrid approach is less basis-set dependent than in the pure ab initio case, and deficiencies of pure DFT for weak interactions are efficiently removed. However, there is still need for improvement of currently available short-range density functionals.

The recently developed explicitly correlated MP2-F12 and CCSD(T)-F12 x ( x = a,b) methods are reviewed. The explicit correlation treatment leads to a dramatic improvement of the basis set convergence. Extensive benchmarks for reaction energies, atomization energies, electron affinities, ionization potentials, equilibrium structures, vibrational frequencies, and intermolecular interaction energies are presented which show that for many molecular properties the intrinsic accuracy of the CCSD(T) method is already reached with double-zeta (VDZ-F12) basis sets, while triple-zeta (VTZ-F12) basis sets yield results that are very close to the complete basis set limit. The steep scaling of the MP2-F12 method with molecular size can be reduced by local approximations. This has made it possible to carry out MP2-F12 calculations for molecules with up to 100 atoms. The errors caused bjavascript:filterformular(´3´)y the local domain approximation are largely removed by the explicitly correlated terms, which account for the neglected configurations in an approximate way. Extensions to LCCSD(T)-F12 are discussed and preliminary results for LCCSD-F12 are presented.

The applicability and performance of a fully automated implementation of the incremental scheme for the evaluation of correlation contributions in wavefunction-based quantum chemical calculations on large molecules is briefly reviewed and a typical example from organic chemistry is discussed in more detail. The accuracy of relative energies is analyzed for various water hexamers at the CCSD(T)/aug-cc-pVDZ level. Furthermore the accuracy of the potential energy surface of octane is analyzed at the CCSD level of theory. Finally, CCSD(T)/aug-cc-pVTZ calculations with 1656 basis functions in C 1 symmetry are reported for (H 2 O) 18 , where a full calculation is infeasible within reasonable time.

In this article we combine the favorable properties of efficient Gaussian type orbitals basis sets, which are applied with good success in conventional electronic structure methods, and tensor product multiscale bases, which provide guaranteed convergence rates and allow for adaptive resolution. To this end, we develop and study a new approach for the treatment of the electronic Schrödinger equation based on a modified adaptive sparse grid technique and a certain particle-wise decomposition with respect to one-particle functions obtained by a nonlinear rank-1 approximation. Here, we employ a multiscale Gaussian frame for the sparse grid spaces and we use Gaussian type orbitals to represent the rank-1 approximation. With this approach we are able to treat small atoms and molecules with up to six electrons.

Recent work in the field of occupied-orbital dependent (OOD) exchange-correlation functionals in density functional theory is reviewed. The main emphasis is put on the development of so-called local hybrid functionals, and on the nontrivial self-consistent implementation of complex OOD functionals. Local hybrids employ a so-called local mixing function (LMF) to govern their position-dependent exact-exchange admixture. Recently proposed LMFs have provided local hybrids of remarkable accuracy in the computation of thermochemical data and classical reaction barriers, and with good performance for some magnetic-resonance parameters. These local hybrids mix only local and exact exchange and exhibit very few semi-empirical parameters. Further refinement and the efficient implementation of local hybrids offers the prospect of a new level of accuracy in Kohn-Sham density functional calculations. Two levels of the self-consistent implementation of OOD functional are discussed: one may either stop after the derivation of the functional derivatives with respect to the orbitals (FDOs), leading to nonlocal potentials. This is discussed for local hybrids and for general OOD functionals up to and including the complicated B05 real-space model of nondynamical correlation. Alternatively, one may append an additional transformation to local and multiplicative potentials based on the optimized effective potential (OEP) approach or of approximations to the OEP. Numerical results for various properties are reviewed briefly, ranging from nonself-consistent energies via FDO-based calculations to OEP-transformed potentials.

This paper gives an overview of the theory and some selected applications of the Multi-Configuration Time-Dependent Hartree-Fock (MCTDHF) method. This method can be seen as an explicitly time-dependent version of the Complete Active Space Self-Consistent Field (CASSCF) method of standard quantum chemistry. Strengths and shortcomings are discussed, and some comparisons with other methods of correlated electron dynamics are given.

In this work, we derive the density matrix renormalization group (DMRG) algorithm in the language of configuration interaction. Furthermore, the development of DMRG in quantum chemistry is reviewed and DMRG-specific peculiarities are discussed. Finally, we discuss new results for a dinuclear μ -oxo bridged copper cluster, which is an important active-site structure in transition-metal chemistry, an area in which we pioneered the application of DMRG.

The new, Laplace transform based multi-state local CC2 response method is compared to the previous single-state approach. The new method employs adaptive state specific local approximations for the eigenvectors of the Jacobian. As a result, it is much less dependent on the initial un-truncated CCS wavefunction and able to locate the relevant eigenstates also in difficult cases where the initial CCS eigenvectors are qualitatively wrong and where the original single-state method failed. In this paper we compare excitation energies, transition strengths, and dipole moments of the individual local approaches and the canonical reference calculation, for a set of different molecules and excited states.

The basic principles behind a theoretical method for treating infinite, periodic systems exposed to an external electrostatic field are outlined. The approach, based on the vector-potential description of the external field, leads to single-particle Hartree-Fock or Kohn-Sham equations that differ from the field-free counterparts in several aspects. In particular, solving them is only possible through a careful so-called smoothing procedure. In that case it is possible to derive a numerically stable and efficient approach. Results of model studies as well as of the first ab initio calculations are reported in order to illustrate the approach. Finally, extensions and open issues are briefly discussed.

We discussed exact solutions of the Schrödinger equation for a two-dimensional parabolic confinement potential in a homogeneous external magnetic field. It turns out that the two-electron system is exactly solvable in the sense, that the problem can be reduced to numerically solving one radial Schrödinger equation. For a denumerably infinite set of values of the effective oscillator frequency ω ~ = √( ω 0 2 +( ω c /2) 2 ) (where ω 0 is the frequency of the harmonic confinement potential and ω c is the cyclotron frequency of the magnetic field) even analytical solutions can be given. Our solutions for three electrons are exact in the strong - and the weak correlation limit. For quantum dot lattices with Coulomb-correlations between the electrons exact solutions are given, provided the lattice constant is large compared with the dot diameters. We are investigating basic physical properties of these solutions like the formation and distortion of Wigner molecules, the dependence of the correlation strength from ω 0 and ω c , and we show that in general there is no exact Kohn-Sham system for the semi-relativistic current density functional Theory.

In this article, we summarize the results of our numerical analysis for various adaptive schemes in electronic structure calculations. Starting from a general form of iterative schemes used in quantum chemistry, we discuss miscellaneous features which are of importance when passing to an adaptive version of an algorithm in order to provide rigorous estimates. The key features are maintainance of convergence under small perturbations while keeping control of the computational complexity.As a prototype for correlation methods, we consider the configuration interaction (CI) method and discuss a quasi-optimal algorithm for computation of the ground state energy. Our analysis of the coupled cluster method shows that various properties of the CI method can be transferred hereunto as well. First numerical examples demonstrate the potential of the coupled cluster residual, measured in a norm on the dual space, as an error estimator. These studies are supplemented by our result for the best N -term approximation of electron-pair correlations by wavelets.For Hartree-Fock and Kohn-Sham methods, we discuss our results of the convergence analysis for a direct minimization scheme, the best N -term approximation of orbitals and s * -compressibility of the Fock operator which altogether may serve as a basis for an adaptive version of these methods.

We present an overview of our recent developments of relativistic generalized-active-space coupled cluster and its initial applications to heavy-element systems. The new genuinely string-based method may treat coupled cluster expansions of general order and simulate multi-reference expansions by including higher excitations in properly constructed active spinor spaces. The four-component Dirac-Coulomb Hamiltonian or any approximation to it may be employed, and the underlying spinors are assumed and required to be time-reversal partners. The capability of the new approach is demonstrated and discussed in sample applications to HBr and BiH.

We propose a possible generalization of Gaussian-type orbital (GTO) bases by means of canonical tensor products. The present work focus on the application of tensor products as an alternative to conventional GTO based density fitting schemes. Tensor product approximation leads to highly nonlinear optimization problems which require sophisticated algorithms. We give a brief description of the optimization problem and algorithm. The present work extends our previous paper [S. R. Chinnamsetty, M. Espig, B. N. Khoromskij, W. Hackbusch and H.-J. Flad, J. Chem. Phys. 127 (2007), 084110], where we discussed tensor product approximations of the electron density and the Hartree potential, to orbital products which are required for the exchange part of Hartree-Fock and in post Hartree-Fock methods. We provide a detailed error analysis for the Coulomb and exchange terms in Hartree-Fock calculations. Furthermore, a comparison is given between all-electron and pseudopotential calculations.

Formulae are derived and implemented for the analytic calculation of first-order molecular properties at the level of explicitly-correlated second-order Møller-Plesset perturbation theory. In this theory, which is denoted as MP2-F12 theory, Slater-type geminals are used to expand the first-order wave function. A second-order perturbation theory correction for single excitations into a complementary auxliary basis set is also included. At the MP2-F12 level, it seems sufficient to restrict the analytic calculation of energy derivatives to the level of standard approximation A of MP2-F12 theory and to assume the extended Brillouin condition to hold. Smooth and rapid convergence towards the basis-set limit is observed for the dipole moments of a selection of small closed- and open-shell molecules when calculated at the RI-MP2-F12/2A * [T+V] + CABS singles level in augmented correlation-consistent polarized valence double-, triple-, and quadruple-zeta basis sets that have been optimized especially for use in MP2-F12 theory.