Dirk Andrae

Finite nuclear charge density distributions in electronic structure calculations for atoms and molecules

Abstract The present review provides comprehensive information on finite nuclear charge density distribution models, not only for the purpose of quantum chemical electronic structure calculations for atoms and molecules, but also for other fields of atomic and molecular physics. A general discussion of the electrostatic behaviour of nuclear charge density distributions, spherical ones and non-spherical ones, is given. A large and reasonably complete set of spherical finite nucleus models, covering all models widely used in atomic and nuclear physics, is discussed in detail. Analytic expressions are given for charge density distributions, for important radial expectation values, and for their corresponding electrostatic potentials; these include new material not found in the literature. Thus, the necessary prerequisites for the use of finite nucleus models which are more realistic than the simple, frequently considered models (e.g., the ‘homogeneous’, ‘Gaussian’, and Fermi models) are fulfilled. The use of finite nucleus models in standard quantum chemical electronic structure programs is briefly reviewed. In order to detect differences between physical properties obtained with various finite nucleus models, six standardized models were selected to study and compare energy shifts (non-relativistic and relativistic) in hydrogen-like atoms. It is shown that within this set a clear differentiation of models can be made, not only from the point of view of total energy shifts but also from the point of view of energy differences and in fact even for rather low nuclear charge numbers. This could be important for future experimental as well as theoretical work on hydrogen-like atoms.

Electronic quantum fluxes during pericyclic reactions exemplified for the Cope rearrangement of semibullvalene.

The outcome of a pericyclic reaction is typically represented by arrows in the Lewis structure of the reactant, symbolizing the net electron transfer. Quantum simulations can be used to interpret these arrows in terms of electronic fluxes between neighboring bonds. The fluxes are decomposed into contributions from electrons in so-called pericyclic orbitals, which account for the mutation of the Lewis structure for the reactant into that for the product, in other valence and in core orbitals. Series of time-integrated fluxes of pericyclic electrons can be assigned to the arrows, for example 0.09-0.23 electrons for Cope rearrangement of semibullvalene, with hysteresis-type time evolutions for 27.3 fs. This means asynchronous electronic fluxes during synchronous rearrangement of all the nuclei. These predictions should become observable by emerging techniques of femto- to attosecond time-resolved spectroscopy.

Molecular knots, links, and fabrics: prediction of existence and suggestion of a synthetic route

The possible existence and formation of molecular knots, molecular links, and molecular fabrics, built from quite arbitrary monomers, is discussed. General theoretical considerations lead to the conclusion that such molecular species are likely to represent local minima on potential energy hypersurfaces and, therefore, should be stable and able to exist, if once formed. A surface template technique is suggested as a possible experimental avenue to actually form molecular knots and links. This technique may open the door to the directed and controlled synthesis of knots, thus transcending present-day methods based on self-assembly of constituting monomers or oligomers.

Fast and accurate 3D tensor calculation of the Fock operator in a general basis

Abstract The present paper contributes to the construction of a “black-box” 3D solver for the Hartree–Fock equation by the grid-based tensor-structured methods. It focuses on the calculation of the Galerkin matrices for the Laplace and the nuclear potential operators by tensor operations using the generic set of basis functions with low separation rank, discretized on a fine N × N × N Cartesian grid. We prove the C h 2 error estimate in terms of mesh parameter, h = O ( 1 / N ) , that allows to gain a guaranteed accuracy of the core Hamiltonian part in the Fock operator as h → 0 . However, the commonly used problem adapted basis functions have low regularity yielding a considerable increase of the constant C , hence, demanding a rather large grid-size N of about several tens of thousands to ensure the high resolution. Modern tensor-formatted arithmetics of complexity O ( N ) , or even O ( log N ) , practically relaxes the limitations on the grid-size. Our tensor-based approach allows to improve significantly the standard basis sets in quantum chemistry by including simple combinations of Slater-type, local finite element and other basis functions. Numerical experiments for moderate size organic molecules show efficiency and accuracy of grid-based calculations to the core Hamiltonian in the range of grid parameter N 3 ∼ 10 15 .

Recursive evaluation of expectation values for arbitrary states of the relativistic one-electron atom

An algorithm for the recursive evaluation of expectation values for a given state of the relativistic one-electron atom with a point-like nucleus of charge number Z is proposed. In contrast to previous approaches, the present algorithm is based on only a single recurrence relation, which is a recurrence relation for a special type of generalized hypergeometric series. Two sequences of values of such series are generated. Finally, the members of these two sequences are assembled to yield the expectation values. The algorithm can be considered as a relativistic analogue of the Kramers - Pasternack recursion for expectation values for states of the non-relativistic hydrogen-like atom. As an application closed-form expressions for , were derived. In addition, numerical values for are given for some representative states of one-electron atoms with Z = 1, 80 and 137.

Molecular Dynamics Simulations of Dendrimer-Encapsulated α-Keggin Ions in Trichloromethane Solution

We report on molecular dynamics simulations of dendrimer-encapsulated alpha-Keggin ions in trichloromethane solution. The simulations were done within the NVE ensemble at temperatures around T = 300 K. The eight examined systems are model compounds for dendrizymes, a hybrid material where a polyoxometalate ion (the core) is surrounded by amphiphilic cationic dendrimers (the shell) such that the complete system may exhibit enzyme-like regioselectivity and substrate selectivity, e.g., in olefin oxidation. The influence of dendrimer type, dendrimer generation, and number of dendritic cations bound by electrostatic interaction to the polyoxometalate core on the structure and dynamics of the shell has been studied. It is shown that the resulting distribution of trichloromethane molecules within the shell may serve as an indicator for the shells permeability for small molecules. The dendritic shell causes a size exclusion effect that influences the access of small molecules to the central polyoxometalate ion, i.e., to that part where the enzyme-like reaction of a dendrizyme is supposed to take place.

A comparative study of finite nucleus models for low-lying states of few-electron high-Z atoms

We studied low-lying states of lithium-like and beryllium-like ions with Z = 80 and 100 less than or equal to Z less than or equal to 120 for differential effects due to variation of the nuclear charge density distribution. The latter was represented by exponential, Gauss-type, and Fermi-type models, standardized to a common rms radius. Numerical Hartree-Fock and Dirac-Fock-Coulomb calculations were performed. Changes in total energies due to the use of different models become increasingly important for high Z in relativistic approaches. Different models may even lead to different sequences of atomic states, and thus energy differences between those states may vary by an order of magnitude

Nuclear charge density distributions in quantum chemistry

This chapter presents a discussion of nuclear structure, with emphasis to nuclear charge density distributions and their use in electronic structure calculations. First, nuclear structure is discussed on the basis of general, non-spherical nuclear charge and current density distributions. Several spherical models, widely used to represent the nuclear charge density distributions, are discussed in full detail. The important effects on single-particle functions and energies, due to the change from the point-like nucleus to an extended nucleus, are demonstrated for the one-electron atom as a simple example. The application of finite nucleus models in standard electronic structure calculations for many-electron systems is discussed, as are special topics like, e.g., higher quantum electrodynamic effects and parity non-conservation.

Numerical self-consistent field method for polyatomic molecules

A method for non-relativistic self-consistent field (SCF) electronic structure calculations for polyatomic molecules is described, which retains the linear combination of atomic orbitals ansatz for molecular orbitals (MO-LCAO), but replaces the usual algebraic expansion of atom-centred radial parts in terms of basis functions (usually some kind of Gauss-type functions) by a numerical representation on a set of radial grid points around each centre. The radial parts are optimized, according to the variation principle, until self-consistency is achieved. Even though Fourier integral transform techniques are used the method works completely in ordinary space. Intermediate quantities defined in momentum space are evaluated in closed form.

Asymptotically exact calculation of the exchange energies of one-active-electron diatomic ions with the surface integral method

We present a general procedure, based on the Holstein–Herring method, for calculating exactly the leading term in the exponentially small exchange energy splitting between two asymptotically degenerate states of a diatomic molecule or molecular ion. The general formulae we have derived are shown to reduce correctly to the previously known exact results for the specific cases of the lowest Σ and Π states of H+2. We then apply our general formulae to calculate the exchange energy splittings between the lowest states of the diatomic alkali cations K+2, Rb+2 and Cs+2, which are isovalent to H+2. Our results are found to be in very good agreement with the best available experimental data and ab initio calculations.