Long-time tails of translational and rotational Brownian motion in a suspension of hard spheres

The long-time translational and rotational Brownian motion of a sphere in a suspension of hard spheres is studied on the basis of the linearized Navier-Stokes equations and the fluctuation-dissipation theorem. It is shown that for the rotational long-time coefficient an effective medium conjecture is incorrect. There are short-range velocity correlations that decay at the same rate as the macroscopic flow pattern used in the effective medium conjecture. An estimate of the short-range correction is made on the basis of the pair term in the cluster expansion of the rotational admittance.     

Applications of a quantum virial theorem to kronig and penney's model and to a diatomic molecule in static approximation

Kronig and Penney's model is used to test a new formulation of quantum virial theorem for periodic systems. It is shown that the corrective term to the usual formulation of the virial theorem corresponding to boundary conditions is of importance and gives correct energy values for the system. Application of the virial theorem to a diatomic molecule also shows an important boundary condition correction if the distance of the nuclei is slightly different from equilibrium position, in accordance with the above correction.     

Fully retarded van der Waals interaction between dielectric nanoclusters

The van der Waals (dispersion) interaction between an atom and a cluster or between two clusters at large separation is calculated by considering each cluster as a point particle, characterized by a polarizability tensor. For the extreme limit of very large separation, the fully retarded regime, one needs to know just the static polarizability in order to determine the interaction. This polarizability is evaluated by including all many-body (MB) intracluster atomic interactions self-consistently. The results of these calculations are compared with those obtained from various alternative methods. One is to consider each cluster as a collection of many atoms and evaluate the sum of two-body interatomic interactions, a common assumption. An alternative method is to include three-body atomic interactions as a MB correction term in the total energy. A comparison of these results reveals that the contribution of the higher-than-three-body MB interactions is always attractive and non-negligible even at such a large separation, in contrast to common assumptions. The procedure employed is quite general and is applicable, in principle, to any shape or size of dielectric cluster. We present numerical results for clusters composed of atoms with polarizability consistent with silica, for which the higher-than-three-body MB correction term can be as high as 42% of the atomic pairwise sum. This result is quite sensitive to the anisotropy and orientation of the cluster, in contrast to the result found in the additive case. We also present a power law expansion of the total van der Waals interaction as a series of n-body interaction terms.     

Shell effects on symmetric fragmentations of alkali-metal clusters

Symmetric fragmentation of multiply charged alkali-metal clusters consisting of several tens of atoms is studied. The energy variation during the fragmentation process is calculated using the theory ofshell corrections, in which total energy is written as a sum of the liquid-droplet and shell correction terms. It is found that the variation of the shell correction term is much larger than that of the liquid-droplet one if the parent cluster is metastable. Fragmentation into nearly-magic cluster is most favored regardless of parent size since the barrier height for fragmentation is mainly determined by the shell configuration of fragments rather than that of the parent.     

Relativistic virial theorem for atoms

A relativistic virial theorem is derived for atoms in a general manner. The virial ratio consists of the usual V/T term and a correction term W/T, where T, V, and W are the kinetic energy, the potential energy, and correction terms, respectively. Explicit forms of W are presented for four specific nuclear potential models. Numerical calculations for a uniform nuclear charge model show that the magnitude of the correction term W/T increases with increasing atomic numbers and that it modifies the ratio V/T considerably for atoms with large atomic numbers in particular. Received: 21 November 2000 / Accepted: 8 January 2001 / Published online: 3 April 2001     

Unlinked cluster effects in limited CI calculations of molecular properties

The unlinked cluster effects in limited CI calculations of dipole moments and polarizabilities are estimated by using the approximate corrections due to Davidson and Siegbahn. The results obtained for FH, H₂O, NH₃, and CH₄ indicate that the limited CI results for molecular electric properties need to be corrected for the erratic treatment of unlinked clusters.     

Sixth-order many-body perturbation theory. I. Basic theory and derivation of the energy formula

The general expression for the sixth-order Møller-Plesset (MP6) energy, E(MP6), has been dissected in the principal part A and the renormalization part R. Since R contains unlinked diagram contributions, which are canceled by corresponding terms of the principal part A, E(MP6) has been derived solely from the linked diagram terms of the principal part A. These have been identified by a simple procedure that starts by separating A into connected and disconnected cluster operator diagrams and adding terms associated with the former fully to the correlation energy. After closing all open disconnected cluster operator diagrams, one can again distinguish between connected and disconnected energy diagrams, of which only the former lead to linked diagram representations and, therefore, contributions to E(MP6). The connected diagram parts of A have been collected in four energy terms E(MP6)₁, E(MP6)₂, E(MP6)₃, and E(MP6)₄. The sum of these terms has led to an appropriate energy formula for E(MP6) in terms of first- and second-order cluster operators. © 1996 John Wiley & Sons, Inc.     

Symmetry-adapted-cluster/symmetry-adapted-cluster configuration interaction methodology extended to giant molecular systems: ring molecular crystals

The symmetry adapted cluster (SAC)/symmetry adapted cluster configuration interaction (SAC-CI) methodology for the ground, excited, ionized, and electron-attached states of molecules was extended to giant molecular systems. The size extensivity of energy and the size intensivity of excitation energy are very important for doing quantitative chemical studies of giant molecular systems and are designed to be satisfied in the present giant SAC/SAC-CI method. The first extension was made to giant molecular crystals composed of the same molecular species. The reference wave function was defined by introducing monomer-localized canonical molecular orbitals (ml-CMO's), which were obtained from the Hartree-Fock orbitals of a tetramer or a larger oligomer within the electrostatic field of the other part of the crystal. In the SAC/SAC-CI calculations, all the necessary integrals were obtained after the integral transformation with the ml-CMO's of the neighboring dimer. Only singles and doubles excitations within each neighboring dimer were considered as linked operators, and perturbation selection was done to choose only important operators. Almost all the important unlinked terms generated from the selected linked operators were included: the unlinked terms are important for keeping size extensivity and size intensivity. Some test calculations were carried out for the ring crystals of up to 10 000-mer, confirming the size extensivity and size intensivity of the calculated results and the efficiency of the giant method in comparison with the standard method available in GAUSSIAN 03. Then, the method was applied to the ring crystals of ethylene and water 50-mers, and formaldehyde 50-, 100-, and 500-mers. The potential energy curves of the ground state and the polarization and electron-transfer-type excited states were calculated for the intermonomer distances of 2.8-100 A. Several interesting behaviors were reported, showing the potentiality of the present giant SAC/SAC-CI method for molecular engineering.     

Use of a size-consistent energy functional in many electron theory for closed shells

If the ground state wave-function ψ_gr is written as ψ_gr = Φ_o+X, withX as the correlation part satisfying (φ_o¦x) = 0, andx expressed as an expansion in terms of pair, pair-pair etc. cluster functions, then the expectation value of the energyE = (ψ_gr¦H¦ψ_gr)/(ψ_gr¦ψ_gr) has the property that the normalization term in the denominator completely cancels the unlinked part of the numerator, as noted by Sinanoglu. We use Cizek's coupled-pair ansatz ψ_gr = exp(T ₂)Φ₀ for transcribing Sinanoglu's expansion in a many-body language to study the behaviour of the size-consistent (linked) energy functional thus generated. For calculating the matrix-elements of the cluster components ofT, we use two recipes: (1) a variational determination of the cluster components using Euler's principle for the energy functional akin in spirit to the Varied Portion Approach (VPA) of Sinanoglu and (ii) a nonvariational determination of the cluster components using the conventional coupled-cluster theory. Results are presented for model test systems and are compared with variational CI and nonvariational coupled-cluster values. It has been observed that the values obtained from the size-consistent energy functional from the cluster components obtained from methods (i) and (ii) are quite close and both compare well with the nonvariational coupled-cluster results. Some useful simplifications afforded by the VPA are also indicated. A brief perspective of the method vis-a-vis other related theories is also given.     

Some unusual features of the configuration of polymer loops in a melt

A diversity of structural features can be imposed on a melt of polymer loops by controlling the linking number between each pair of loops. For the simple case of unlinked loops, considered in this paper, new statistical weighting factors have been found which depend on the global configurational properties of writhe and self inductance. The separate effect of these terms is extreme: the writhe term favours rod-like statistics, whereas the inductance favours globular configurations. Together the two terms tend to cancel each other in a way which is reminiscent of the screened potential in polymer chain melts. The result is that the exponent which describes the size of an unlinked loop in a melt of unlinked loops is perturbed from the gaussian result of 0.5 to 0.45.     

Relativistic corrections to the diamagnetic term of the nuclear magnetic shielding: analysis of contributions from localized orbitals

We have calculated the relativistic corrections to the diamagnetic term of the nuclear magnetic shielding constants for a series of molecules containing heavy atoms. An analysis of the contributions from localized orbitals is performed. We establish quantitatively the relative importance of inner core and valence shell molecular orbitals in each correcting term. Contributions from the latter are much less important than those from the former. The calculated values of the correction sigma(L-PSO), first derived within the linear response elimination of small component formalism, show a power-law dependence on the nuclear charge approximately Z(3.5), in contrast with the approximately Z(3.1) behavior of the mass-velocity external-field correction to the paramagnetic term previously reported.     

Approximate Calculations of Ionization Potential in Electronic Cluster Models

A simple formulation of generalized Koopmans' theorem is suggested for multiconfigurational wave functions. In a coupled cluster approximation (CCD variant), -electron estimates have been obtained for the first ionization potential in a series of linear polyenes and polyacenes. The calculated data satisfactorily reproduce the electron correlation effects characteristic of the ionized states of molecules.     

Ab initio calculations on halogen-bonded complexes and comparison with density functional methods

A systematic theoretical investigation on a series of dimeric complexes formed between some halocarbon molecules and electron donors has been carried out by employing both ab initio and density functional methods. Full geometry optimizations are performed at the Moller-Plesset second-order perturbation (MP2) level of theory with the Dunning's correlation-consistent basis set, aug-cc-pVDZ. Binding energies are extrapolated to the complete basis set (CBS) limit by means of two most commonly used extrapolation methods and the aug-cc-pVXZ (X = D, T, Q) basis sets series. The coupled cluster with single, double, and noniterative triple excitations [CCSD(T)] correction term, determined as a difference between CCSD(T) and MP2 binding energies, is estimated with the aug-cc-pVDZ basis set. In general, the inclusion of higher-order electron correlation effects leads to a repulsive correction with respect to those predicted at the MP2 level. The calculations described herein have shown that the CCSD(T) CBS limits yield binding energies with a range of -0.89 to -4.38 kcal/mol for the halogen-bonded complexes under study. The performance of several density functional theory (DFT) methods has been evaluated comparing the results with those obtained from MP2 and CCSD(T). It is shown that PBEKCIS, B97-1, and MPWLYP functionals provide accuracies close to the computationally very expensive ab initio methods.     

Valence bond corrected single reference coupled cluster approach

Summary An extension of the single reference coupled cluster method truncated to 1- and 2-body cluster components (CCSD) to quasidegenerate systems, where 3-and 4-body connected cluster components play an important role, is proposed. The basic idea is to extract the information concerning the 3- and 4-body clusters from some independent source, similarly as was implicitly done in the so-called ACPQ or ACC(S)D methods, and correct accordingly the absolute term in the CCSD equations. As a source of these approximate 3- and 4-body clusters, simple valence bond (VB) type wave functions are employed, since they are capable of describing electronic structure of various molecular systems for a wide range of nuclear conformations including their dissociation. The cluster analysis of these VB wave functions, that provides the desired information concerning the connected 3- and 4-body cluster components, is outlined and the explicit form of required correction terms to the CCSD equations is given.     

Ivar Waller 90 years on June 11, 1988

Linkage properties of the diagrammatic representation of the energies obtained in the multireference many-body perturbation calculations with respect to the incompleteness or completeness of the model space are discussed. The case of not completely degenerate model space is considered for which a comparison with the standard single-reference many-body perturbation expansion is possible. The Hose–Kaldor type of graphical representation of the perturbation expansion for the effective Hamiltonian is used in this comparison. It is shown that for an incomplete model space the perturbation expansion is not size-extensive. In this case, for a truncated expansion of the effective Hamiltonian, the energies obtained by diagonalization of the effective Hamiltonian matrix are represented by both linked and unlinked irreducible contributions. The unlinked ones do not appear when the complete model space is used.     

The main beam correction term in kinetic energy release from metastable peaks

The correction term for the precursor ion signal width in determination of kinetic energy release is reviewed, and the correction term is formally derived. The derived correction term differs from the traditionally applied term. An experimental finding substantiates the inaccuracy in the latter. The application of the “T ‐value” to study kinetic energy release is found preferable to kinetic energy release distributions when the metastable peaks are slim and simple Gaussians. For electronically predissociated systems, a “borderline zero” kinetic energy release can be directly interpreted in reaction dynamics with strong curvature in the reaction coordinate.     

Numerical Hartree-Fock and MCSCF calculations on diatomic copper: calibration of basis sets

NHF and NMCSCF results for Cu₂ are compared with calculations employing basis set expansions. We find that nearly all previous SCF calculations using Gaussian basis sets have underestimated the bond length by about the same amount (0.03 Å) as that attributed to the unlinked cluster and relativistic corrections. The error is shown to be due to deficiencies in the 3d primitive set which yield sizable basis set superposition errors.     

On basis set superposition error corrected stabilization energies for large n-body clusters

In this contribution, we propose an approximate basis set superposition error (BSSE) correction scheme for the site-site function counterpoise and for the Valiron-Mayer function counterpoise correction of second order to account for the basis set superposition error in clusters with a large number of subunits. The accuracy of the proposed scheme has been investigated for a water cluster series at the CCSD(T), CCSD, MP2, and self-consistent field levels of theory using Dunning's correlation consistent basis sets. The BSSE corrected stabilization energies for a series of water clusters are presented. A study regarding the possible savings with respect to computational resources has been carried out as well as a monitoring of the basis set dependence of the approximate BSSE corrections.