Recent Advances in the Theory of Chemical and Physical Systems

In this paper, we advocate the use of literate programming techniques in molecular physics and quantum chemistry. With a suitable choice of publication medium, literate programming allows both a theory and corresponding computer code to be placed in the public domain and subject to the usual “open criticism and constructive use” which form an essential ingredient of the scientific method.

The capability of the variational and perturbative-type approaches to the many-electron correlation problem – as represented by the configuration interaction (CI) and coupled cluster (CC) theories – to describe, respectively, the nondynamic and dynamic correlation effects, is emphasized, and its exploitation in the design of the so-called externally corrected CC methods, as well as in the formulation of Davidson-type corrections that are based on the CC theory, at both single reference and multireference levels, is reviewed. The performance of various methods of this type is illustrated on the DZP H4 model that consists of two interacting and slightly stretched hydrogen molecules in a trapezoidal geometry. This often studied model enables a continuous transition from the degenerate to the nondegenerate regime by varying the degree of quasidegeneracy via a single geometric parameter. In this way the role of higher-than-pair clusters, particularly in the presence of intruder states, can be explored and the performance of various approaches that exploit the complementarity of the CI and CC approaches can be evaluated.

New classes of noniterative coupled-cluster (CC) methods, which improve the results of the standard equation-of-motion (EOM) and response CC calculations for excited states dominated by two-electron transitions and excited-state potential energy surfaces along bond breaking coordinates, are reviewed. All of the methods discussed in this article are derived from the method of moments of CC equations (MMCC) and all of them are characterized by the relatively low computer costs which are similar to those characterizing the popular ground-state CCSD(T) theory. Three types of approaches are discussed: (i) the externally corrected MMCC approaches employing the con.guration interaction and multi-reference perturbation theory wave functions, (ii) the completely renormalized EOMCC methods, including their most recent extension to excited states of radicals and other open-shell systems, and (iii) the new classes of MMCC and completely renormalized EOMCC theories employing the left eigenstates of the similarity-transformed Hamiltonian used in CC/EOMCC theory.

A practical Hartree-Fock theory of atomic and molecular electronic structure is developed for individual electronically excited states that does not involve off-diagonal Lagrange multipliers. An easily implemented method for taking the orthogonality constraints into account, which has been proposed earlier by one of us, is used to impose the orthogonality of the Hartree-Fock excited state wave function of interest to states of lower energy. The applicability of systematic sequence of even-tempered basis sets with the orbital exponents, ζp, defined by the geometric series ζp = aβ ^p is examined in Hartree-Fock energy calculations for excited states which have the same spatial and spin symmetry as the ground state. It is shown that a simple reoptimization of the a and β parameters leads to a sequence of even-tempered basis sets capable of supporting high accuracy for excited state energies of some simple atoms.

We propose a practicable method for describing linear dynamics of different finite Fermi systems. The method is based on a general selfconsistent procedure for factorization of the two-body residual interaction. It is relevant for diverse density- and current-dependent functionals and, in fact, represents the self-consistent separable random-phase approximation (RPA), hence the name SRPA. SRPA allows to avoid diagonalization of high-rank RPA matrices and thus dwarfs the calculation expense. Besides, SRPA expressions have a transparent analytical form and so the method is very convenient for the analysis and treatment of the obtained results. SRPA demonstrates high numerical accuracy. It is very general and can be applied to diverse systems. Two very different cases, the Kohn-Sham functional for atomic clusters and Skyrme functional for atomic nuclei, are considered in detail as particular examples. SRPA treats both time-even and time-odd dynamical variables and, in this connection, we discuss the origin and properties of time-odd currents and densities in initial functionals. Finally, SRPA is compared with other self-consistent approaches for the excited states, including the coupled-cluster method.

The elements of the second-order reduced density matrix are pointed out to be written exactly as scalar products of specially defined vectors. Our considerations work in an arbitrarily large, but FInite orthonormal basis, and the underlying wave function is a full-CI type wave function. Using basic rules of vector operations, inequalities are formulated without the use of wave function, including only elements of density matrix.

Calculations of the first-order shell corrections of the ionization potential, δ ₁ I, electron affinity, δ ₁ A, electronegativity, δ ₁ χ, and chemical hardness, δ₁η are performed for elements from B to Ca, using the previously described Strutinsky averaging procedure in the frame of the extended Kohn-Sham scheme. A good agreement with the experimental results is obtained, and the discrepancies appearing are discussed in terms of the approximations made.

The generalized electronic diabatic (GED) approach is used to study a cis-trans isomerization process. At variance with standard Born-Oppenheimer approach, where a unique adiabatic potential energy function depending of a dihedral angle connects both isomers, a configuration interaction model permits describing isomerization process with four diabatic electronic states. These GED states form a minimal CI space; each state conserves local symmetry properties along a properly defined reaction coordinate. The diabatic states diagonalize the Coulomb Hamiltonian. The state mixing obtains via kinematic couplings, electron-phonon and spin-orbit operators. The process is mapped to a full quantum mechanical linear superposition of diabatic states.

BERTHA is a 4-component relativistic molecular structure program based on relativistic Gaussian (G-spinor) basis sets which is intended to make affordable studies of atomic and molecular electronic structure, particularly of systems containing high-Z elements. This paper reviews some of the novel technical features embodied in the code, and assesses its current status, its potential and its prospects.

An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence.

The Generalized Relativistic Effective Core Potential (GRECP) method is described, which allows to simulate Breit interaction and finite nuclear models by an economic way with high accuracy. The corresponding GRECPs for the uranium, plutonium, eka-mercury (E112), eka-thallium (E113) and eka-lead (E114) atoms are generated. The accuracy of these GRECPs and of the RECPs of other groups is estimated in atomic numerical SCF calculations with Coulomb two-electron interactions and point nucleus as compared to the corresponding all-electron Hartree-Fock-Dirac- Breit calculations with the Fermi nuclear charge distribution. Different nuclear models and contributions of the Breit interaction between different shells are studied employing all-electron four-component methods.

Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TlF and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TlF, and PbO.

A highly accurate, ab-initio approach to the relativistic calculation of spectra of multi-electron, super-heavy ions, accounting for correlation, nuclear, radiative, and relativistic effects is developed. The method is based on quantum electrodynamics (QED) perturbation theory (PT). Zeroth approximation is generated by an effective ab-initio model functional constructed on the basis of the comprehensive gauge-invariant procedure. The wave-function zeroth-order basis set is found from the Dirac equation with a potential including the core abinitio potential and the electric and polarization potentials of the nucleus (a Gaussian form is used for the charge distribution in the nucleus). The magnetic interelectronic interaction is included in the lowest (α)² term, and the Lamb shift polarization effect, in the Uehling-Serber approximation (self-energy part of the Lamb shift), is accounted for within the Ivanov-Ivanova non-perturbative procedure. Results of the calculations are presented for the energy levels, dielectronic satellite wavelengths, hyperfine structure constants, and QED corrections for 1s ² n l j- states of Li-like ions.

There is presented a consistent energy approach to the quantum electrodynamics (QED) theory of the discharge of a nucleus with emission of a γ radiation and further muon conversion, which initiates this discharge. A numerical calculation is carried out for nucleus 49,21Sc ₂₈

One area in atomic and molecular physics that computer modelling and its applications are extensively used is in ion-atom collisions. In this paper we consider computer simulations for various theoretical continuum-distorted-wave eikonal-initial-state (CDW-EIS) models used in the study of single ionization of neutral target atoms by fast highly charged ions. In our first study we examine ultra-low energy electrons for 3.6 MeV amu-1 Au53+ on helium, neon and argon. Doubly differential cross sections as a function of the longitudinal electron velocity for various transverse velocity cuts are obtained using the CDW-EIS model. A sharp asymmetric peak centred at a longitudinal velocity of zero is observed to emerge at ultra-low energies in all of the collisions studied.

The separation of the total energy into one- and two-body terms for interacting molecules at the HF-SCF level has already been proved to provide useful information. Several energy contributions for the linear water dimer molecule and for some hexamer water clusters were calculated using various basis sets in the present work. The results suggest that some partitioned energy terms derived separately for each contributing monomer (using the method of Separated Molecular Orbitals) show significant differences for the proton donor and the proton acceptor molecules in the linear structured water dimer. The sum of these terms per each subunit corresponds to a (partitioned) total energy quantity for the given subsystem. This energy contribution allows characterizing the proton donor / acceptor ability of the monomers. In this way the identification of the monomers by their proton donor / acceptor nature in some water hexamers was also investigated.

The additivity of the two and three-body potential energy surfaces (PES) is studied for the van der Waals (vdW) complex formed by Br₂ and two Helium atoms. First, the three-dimensional interaction potential for HeBr₂ molecule is calculated using a coupled-cluster (CCSD(T)) method. This surface shows a double-minimum topology in agreement with the available experimental data. In turn, an intermolecular potential energy surface for He₂Br₂ complex in the ground state is calculated at the levels of fourth-order (MP4) Møller-Plesset and coupled-cluster [CCSD(T)] approximations. It is found that results obtained by summing the above [CCSD(T)] three-body parameterized HeBr₂ interactions and the He–He interaction are in very good accord with the corresponding MP4/CSSD(T) configuration energies. Variational calculations using the above potential form are performed to calculate the bound states of the vdW complex and these results are compared with available experimental data.

We present a quantum-classical determination of stable isomers of Na*Ar_n clusters with an electronically excited sodium atom in 3p²P states. The excited states of Na perturbed by the argon atoms are obtained as the eigenfunctions of a single-electron operator describing the electron in the field of a Na⁺ Arn core, the Na⁺ and Ar atoms being substituted by pseudo-potentials. These pseudo-potentials include core-polarization operators to account for polarization and correlation of the inert part with the excited electron ^{(1, 2)} . The geometry optimization of the excited states is carried out via the basin-hopping method of Wales et al. ^{(1, 2)}. The present study confirms the trend for small Na*Ar_n clusters in 3p states to form planar structures, as proposed earlier by Tutein and Mayne ⁽⁴⁾ within the framework of a first order perturbation theory on a “Diatomics in Molecules“ type model.

We study structural and chemisorption properties of pure and doped noble metal clusters by means of first-principles density functional calculations, based on norm-conserving pseudo-potentials and numerical atomic basis sets. First, we show that, together with relativistic effiects, the level of theory, that is, the use of GGA or LDA exchange-correlation functionals, is of paramount importance to determine the onset of three dimensional structures in Au clusters, whereas for Ag or Cu clusters it is not so critical. Second, within the GGA framework, we find cage-like stable structures for neutral Au₁₈, Au₂₀, Au₃₂, Au₅₀, and Au₁₆₂.

The present work extends the family of nonconventional proton acceptors to the coinage metal Au. Based on high level computations, we demonstrate the ability of the triangle three-gold cluster to behave as nonconventional proton acceptor and hence to form hydrogen bonds with conventional hydrogen bond donors. Three molecules: formic acid, alanine, and adenine, involving O-H and N-H groups as typical conventional hydrogen bond donors, are chosen for this purpose.

Using a density matrix approach to Gutzwiller method, we present a formalism to treat ab-initio multiband Tight-Binding Hamiltonians including local Coulomb interaction in a solid, like, for e.g., the degenerate Hubbard model. We first derive the main results of our method: starting from the density matrix of the non-interacting state, we build a multi-configurational variational wave function. The probabilities of atomic configurations are the variational parameters of the method. The kinetic energy contributions are renormalized whereas the interaction contributions are exactly calculated. A renormalization of effective on-site levels, in contrast to the usual one-band Gutzwiller approach, is derived. After minimization with respect to the variational parameters, the approximate ground state is obtained, providing the equilibrium properties of a material. Academic models will illustrate the key points of our approach. Finally, as this method is not restricted to parametrized Tight-Binding Hamiltonians, it can be performed from first principles level by the use of the so-called “Linearized Muffin Tin Orbitals” technique. To avoid double counting of the repulsion, one subtracts the average interaction, already taken into account in this density functional theory within local density approximation (DFTLDA) based band structures method and one adds an interaction part “a la Hubbard”. Our method can be seen as an improvement of the more popular LDA+U method as the density-density correlations are treated beyond a standard mean field approach. First application to Plutonium will be presented with peculiar attention to the equilibrium volume, and investigations for other densities will be discussed.

We compute the velocity correlation function of electronic states close to the Fermi energy, in approximants of quasicrystals. As we show the long time value of this correlation function is small. This means a small Fermi velocity, in agreement with previous band structure studies. Furthermore the correlation function is negative on a large time interval which means a phenomenon of backscattering. As shown in previous studies the backscattering can explain unusual conduction properties, observed in these alloys, such as for example the increase of conductivity with disorder.

Boron nanostructures B₉₆ in form of α-rhombohedral nanocrystals and of nanotubes with different diameters were first calculated at the Hartree-Fock Self-Consistent-Field level of theory to generate the electron density. The obtained HF-SCF density-matrix was used as initial input to solve the Kohn-Sham equations of the density functional theory with the Becke-Lee-Yang-Parr Exchange-Correlation Potentials (B3LYP). The total B3LYP energy of each supercluster was determined. The stability of B₉₆ nanotubes at the B3LYP level seems to be similar to or higher than the stability of naturally existing α-boron.

We recently developed an all-atom free energy force field (PFF01) for protein structure prediction with stochastic optimization methods. We demonstrated that PFF01 correctly predicts the native conformation of several proteins as the global optimum of the free energy surface. Here we review recent folding studies, which permitted the reproducible all-atom folding of the 20 amino-acid trp-cage protein, the 40-amino acid three-helix HIV accessory protein and the sixty amino acid bacterial ribosomal protein L20 with a variety of stochastic optimization methods. These results demonstrate that all-atom protein folding can be achieved with present day computational resources for proteins of moderate size.