Classical toy models for the monopole shift and the quadrupole shift

The penetration of s- and p(1/2)-electrons into the atomic nucleus leads to a variety of observable effects. The presence of s-electrons inside the nucleus gives rise to the isotope shift in atomic spectroscopy, and to the isomer shift in M?ssbauer spectroscopy. Both well-known phenomena are manifestations of the more general monopole shift. In a recent paper (Koch et al., Phys. Rev. A, 2010, 81, 032507), we discussed the existence of the formally analogous quadrupole shift: a tensor correction to the electric quadrupole interaction due to the penetration of relativistic p(1/2)-electrons into the nucleus. The quadrupole shift is predicted to be observable by high-accuracy molecular spectroscopy on a set of 4 molecules (the quadrupole anomaly). The simple physics behind all these related phenomena is easily obscured by an elaborate mathematical formalism that is required for their derivation: a multipole expansion in combination with perturbation theory, invoking quantum physics and ideally relativity. In the present paper, we take a totally different approach. We consider three classical 'toy models' that can be solved by elementary calculus, and that nevertheless contain all essential physics of the monopole and quadrupole shifts. We hope that this intuitive (yet exact) analysis will increase the understanding about multipole shift phenomena in a broader community.     

SCF perturbation theory and intermolecular interactions

A self-consistent perturbation theory is derived in the framework of Roothaan's MOLCAO procedure for closed shell systems. Contrary to previous investigations which have considered only one particle perturbations, two particle perturbation operators are considered. Expressions for the first-order density matrix and first- and second-order energy corrections are obtained. A diagram formulation of the complete perturbation expansion is presented. The results are applied to the treatment of the intermolecular interaction problem. The interaction energy is represented as a sum of several contributions: Coulomb, exchange, resonance, polarization and exchange repulsion. A semi-empirical version of the theory is suggested which explicitly involves all the physically significant energy terms and may be useful for the investigation of complex systems.     

Energy gradients with respect to atomic positions and cell parameters for the Kohn-Sham density-functional theory at the Gamma point

The application of theoretical methods based on density-functional theory is known to provide atomic and cell parameters in very good agreement with experimental values. Recently, construction of the exact Hartree-Fock exchange gradients with respect to atomic positions and cell parameters within the Gamma-point approximation has been introduced. In this article, the formalism is extended to the evaluation of analytical Gamma-point density-functional atomic and cell gradients. The infinite Coulomb summation is solved with an effective periodic summation of multipole tensors. While the evaluation of Coulomb and exchange-correlation gradients with respect to atomic positions are similar to those in the gas phase limit, the gradients with respect to cell parameters needs to be treated with some care. The derivative of the periodic multipole interaction tensor needs to be carefully handled in both direct and reciprocal space and the exchange-correlation energy derivative leads to a surface term that has its origin in derivatives of the integration limits that depend on the cell. As an illustration, the analytical gradients have been used in conjunction with the QUICCA algorithm to optimize one-dimensional and three-dimensional periodic systems at the density-functional theory and hybrid Hartree-Fock/density-functional theory levels. We also report the full relaxation of forsterite supercells at the B3LYP level of theory.     

Scale‐adaptive tensor algebra for local many‐body methods of electronic structure theory

While the formalism of multiresolution analysis, based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent‐particle level and, recently, second‐order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many‐particle) theory based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this article, we present a formalism called scale‐adaptive tensor algebra, which introduces an adaptive representation of tensors of many‐body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability of certain local correlated many‐body methods of electronic structure theory, for example, those directly based on atomic orbitals (or any other localized basis functions in general). © 2014 Wiley Periodicals, Inc.     

A second quantization formulation of multimode dynamics

A new formalism for calculating and analyzing many-mode quantum dynamics is presented. The formalism is similar in spirit to the second quantization formulation of electronic structure theory. The similarity means that similar techniques can be employed for calculating the many-mode nuclear wave function. As a consequence a new formulation of the vibrational self-consistent-field (VSCF) method can be developed. Another result is that the formalism opens up for the construction of new methods that go beyond the VSCF level. A vibrational coupled cluster (VCC) theory is constructed using the new formalism. The size-extensivity concept is introduced in the context of multimode wave functions and the size extensivity of approximate VCC methods is illustrated in comparison with the non-size-extensive vibrational configuration interaction method.     

Tensor product approximation with optimal rank in quantum chemistry

Tensor product decompositions with optimal separation rank provide an interesting alternative to traditional Gaussian-type basis functions in electronic structure calculations. We discuss various applications for a new compression algorithm, based on the Newton method, which provides for a given tensor the optimal tensor product or so-called best separable approximation for fixed Kronecker rank. In combination with a stable quadrature scheme for the Coulomb interaction, tensor product formats enable an efficient evaluation of Coulomb integrals. This is demonstrated by means of best separable approximations for the electron density and Hartree potential of small molecules, where individual components of the tensor product can be efficiently represented in a wavelet basis. We present a fairly detailed numerical analysis, which provides the basis for further improvements of this novel approach. Our results suggest a broad range of applications within density fitting schemes, which have been recently successfully applied in quantum chemistry.     

Matrix elements in configuration interaction calculations

A method is proposed for the calculation of matrix elements among various states of atoms. A set of tensor operators is the only entity in the formalism, and all formulas involve merely the vacuum expectation values of these tensor operators and the recoupling transformation coefficients. Some numerical examples are given for the Coulomb interaction matrix elements.     

Intermolecular electrostatic energies using density fitting

A method is presented to calculate the electron-electron and nuclear-electron intermolecular Coulomb interaction energy between two molecules by separately fitting the unperturbed molecular electron density of each monomer. This method is based on the variational Coulomb fitting method which relies on the expansion of the ab initio molecular electron density in site-centered auxiliary basis sets. By expanding the electron density of each monomer in this way the integral expressions for the intermolecular electrostatic calculations are simplified, lowering the operation count as well as the memory usage. Furthermore, this method allows the calculation of intermolecular Coulomb interactions with any level of theory from which a one-electron density matrix can be obtained. Our implementation is initially tested by calculating molecular properties with the density fitting method using three different auxiliary basis sets and comparing them to results obtained from ab initio calculations. These properties include dipoles for a series of molecules, as well as the molecular electrostatic potential and electric field for water. Subsequently, the intermolecular electrostatic energy is tested by calculating ten stationary points on the water dimer potential-energy surface. Results are presented for electron densities obtained at four different levels of theory using two different basis sets, fitted with three auxiliary basis sets. Additionally, a one-dimensional electrostatic energy surface scan is performed for four different systems (H2O dimer, Mg2+-H2O, Cu+-H2O, and n-methyl-formamide dimer). Our results show a very good agreement with ab initio calculations for all properties as well as interaction energies.     

Spheroidal multipolar expansions for intermolecular energy in crystal theory. I. Formalism

In view of further applications to molecular crystal theory, spheroidal expansion of intermolecular potential involving. anisotropic molecules is investigated, including multipole moments up to octupole. The spheroidal formalism is worked out for the electrostatic potential outside a non-spherica charge distribution as well as for interactions between two anisotropic molecules in the non-overlap assumption. The case where either molecule is axia symmetric is presented as a reduced form of the general formulas displayed.     

Tensor formalism in anharmonic calculations

A new method is presented to compute cartesian tensors in the expansion of curvilinear internal coordinates. Second- and higher-order coefficients are related to the metrics of the space of displacements. Components of the metric tensor are taken from existing tables of inverse kinetic energy matrix elements or, when rotations are involved, derived from general invariance conditions of scalars within a molecule. This leads to a tensor formalism particularly convenient in dealing with curvilinear coordinates in anharmonic calculations of vibrational frequencies. Formulae are given for elements of the potential energy matrix, related to quadratic and cubic force constants in terms of Christoffel symbols. The latter quantities are also used in the expansion of redundancy relations, with explicit coefficients given up to the third order.     

Energy transfer dynamics and kinetics of elementary processes (promoted) by gas‐phase CO2‐N2 collisions: Selectivity control by the anisotropy of the interaction

In this work, we exploit a new formulation of the potential energy and of the related computational procedures, which embodies the coupling between the intra and intermolecular components, to characterize possible propensities of the collision dynamics in energy transfer processes of interest for simulation and control of phenomena occurring in a variety of equilibrium and nonequilibrium environments. The investigation reported in the paper focuses on the prototype CO₂–N₂ system, whose intramolecular component of the interaction is modeled in terms of a many body expansion while the intermolecular component is modeled in terms of a recently developed bonds‐as‐interacting‐molecular‐centers' approach. The main advantage of this formulation of the potential energy surface is that of being (a) truly full dimensional (i.e., all the variations of the coordinates associated with the molecular vibrations and rotations on the geometrical and electronic structure of the monomers, are explicitly taken into account without freezing any bonds or angles), (b) more flexible than other usual formulations of the interaction and (c) well suited for fitting procedures better adhering to accurate ab initio data and sensitive to experimental arrangement dependent information. Specific attention has been given to the fact that a variation of vibrational and rotational energy has a higher (both qualitative and quantitative) impact on the energy transfer when a more accurate formulation of the intermolecular interaction (with respect to that obtained when using rigid monomers) is adopted. This makes the potential energy surface better suited for the kinetic modeling of gaseous mixtures in plasma, combustion and atmospheric chemistry computational applications. © 2016 Wiley Periodicals, Inc.     

Basis set study of classical rotor lattice dynamics

The reorientational relaxation of molecular systems is important in many phenomenon and applications. In this paper, we explore the reorientational relaxation of a model Brownian rotor lattice system with short range interactions in both the high and low temperature regimes. In this study, we use a basis set expansion to capture collective motions of the system. The single particle basis set is used in the high temperature regime, while the spin wave basis is used in the low temperature regime. The equations of motion derived in this approach are analogous to the generalized Langevin equation, but the equations render flexibility by allowing nonequilibrium initial conditions. This calculation shows that the choice of projection operators in the generalized Langevin equation (GLE) approach corresponds to defining a specific inner-product space, and this inner-product space should be chosen to reveal the important physics of the problem. The basis set approach corresponds to an inner-product and projection operator that maintain the orthogonality of the spherical harmonics and provide a convenient platform for analyzing GLE expansions. The results compare favorably with numerical simulations, and the formalism is easily extended to more complex systems.     

Super-fermion representation of quantum kinetic equations for the electron transport problem

We discuss the use of super-fermion formalism to represent and solve quantum kinetic equations for the electron transport problem. Starting with the Lindblad master equation for the molecule connected to two metal electrodes, we convert the problem of finding the nonequilibrium steady state to the many-body problem with non-hermitian liouvillian in super-Fock space. We transform the liouvillian to the normal ordered form, introduce nonequilibrium quasiparticles by a set of canonical nonunitary transformations and develop general many-body theory for the electron transport through the interacting region. The approach is applied to the electron transport through a single level. We consider a minimal basis hydrogen atom attached to two metal leads in Coulomb blockade regime (out of equilibrium Anderson model) within the nonequilibrium Hartree-Fock approximation as an example of the system with electron interaction. Our approach agrees with exact results given by the Landauer theory for the considered models.     

A numerical method for computing dispersion constants

We reformulate and discuss a previously proposed variational numerical technique for the computation of dispersion coefficients. The method extends the Full CI idea to the perturbation equation for the intermolecular interaction, by expanding the perturbative solution in a small number of tensor products of suitably chosen Full CI vectors. Some new expansion vectors are proposed and their convergence properties are tested by performing computations on HF and H₂O. Last, a natural state analysis of the solution is performed via an orthogonal transformation of the original expansion vectors and it is found that a single couple of natural states strongly dominates the expansion.     

Effective multipoles and Yukawa electrostatics in dressed molecule theory

In this paper we derive the multipolar expansion of the screened Coulomb potential in electrolyte solutions with molecular solvent. The solute and solvent molecules can have arbitrary sizes, shapes, and internal charge distributions. We use the exact statistical mechanical definition of renormalized charge distributions coming from "dressed molecule theory" to determine the effective multipoles of a molecule immersed in an electrolyte. The effects of many-body correlations are fully included in our formally exact theory. We restrict ourselves to sufficiently dilute solutions so the screened Coulomb potential decays for large distances like a Yukawa function, exp(-kappa r)/r, where r is the distance and 1/kappa is the decay length (it is normally different from the Debye length). The resulting "Yukawa electrostatics" differ in many respects from ordinary, unscreened electrostatics. The "Yukawa charge" of a molecule (the lowest order moment in the multipolar expansion) is in general not equal to its Coulombic charge and it is not the integral of the renormalized charge distribution of the molecule. Moreover, as shown in this paper, the multipolar expansion of the Yukawa potential does not correspond, contrary to the case of the Coulomb potential, to its asymptotic expansion for large r. As a consequence, the charge term in the multipolar expansion is not the leading term in the asymptotic expansion. Instead, for large r values, multipoles of all orders contribute to the leading asymptotic term. Thus, the electrostatic potential from, for example, an electroneutral solvent molecule in an electrolyte solution has generally the same range as that from an ion. The proper asymptotic expansion for electrostatic interactions in electrolytes is derived. It is briefly shown how the multipole expansion formalism can also be applied in the Poisson-Boltzmann approximation for primitive model electrolytes.     

Introduction of the Floquet-Magnus expansion in solid-state nuclear magnetic resonance spectroscopy

In this article, we present an alternative expansion scheme called Floquet-Magnus expansion (FME) used to solve a time-dependent linear differential equation which is a central problem in quantum physics in general and solid-state nuclear magnetic resonance (NMR) in particular. The commonly used methods to treat theoretical problems in solid-state NMR are the average Hamiltonian theory (AHT) and the Floquet theory (FT), which have been successful for designing sophisticated pulse sequences and understanding of different experiments. To the best of our knowledge, this is the first report of the FME scheme in the context of solid state NMR and we compare this approach with other series expansions. We present a modified FME scheme highlighting the importance of the (time-periodic) boundary conditions. This modified scheme greatly simplifies the calculation of higher order terms and shown to be equivalent to the Floquet theory (single or multimode time-dependence) but allows one to derive the effective Hamiltonian in the Hilbert space. Basic applications of the FME scheme are described and compared to previous treatments based on AHT, FT, and static perturbation theory. We discuss also the convergence aspects of the three schemes (AHT, FT, and FME) and present the relevant references.     

Dielectric Boundary Forces in Numerical Poisson-Boltzmann Methods: Theory and Numerical Strategies

Continuum modeling of electrostatic interactions based upon the numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, robust and efficient methodologies to compute solvation forces must be developed. In this study, we have first reviewed the theory for the computation of dielectric boundary forces based on the definition of the Maxwell stress tensor. This is followed by a new formulation of the dielectric boundary force suitable for the finite-difference Poisson-Boltzmann methods. We have validated the new formulation with idealized analytical systems and realistic molecular systems.     

Irreducible tensor operator methods in intermediate-field coupling

The algebra of irreducible tensor operators is developed in the intermediate-field coupling case. The Wigner-Eckart theorem is formulated for a simple irreducible tensor operator as well as for the Kronecker and scalar products of these operators. The expressions required for the calculation of Coulomb repulsion, crystal field splitting, spin-orbit interaction, and Zeeman effect are given in detail. Recent applications to various problems in spectroscopy and magnetism of transition metal compounds are referred to.