Development and applications of a relaxation-inducing cluster expansion theory for treating strong relaxation and differential correlation effects

We present in this paper a multi-reference coupled cluster (MRCC) formulation for energy differences which treats orbital relaxation and correlation effects on the same footing, by invoking a novel cluster ansatz of the valence portion of the wave operator Ω_v. Unlike in the traditional normal-ordered exponential representation of Ω_v, our new relaxation-inducing ansatz, represented symbolically as E _r(S), allows contractions between the spectator lines and also certain other special contractions. By an extensive theoretical analysis, taking as an example the case of one-hole model space (the IP problem), we demonstrate that our ansatz incorporates in a manifestly spin-free form the orbital relaxation to all orders. The traditional Thouless-type of exponential transformation via one-body excitations can induce the same effect, as is done in the valence-specific or the quasi-valence-specific MRCC formalisms, but they have to be done in the spin-orbital basis – making the spin adaptation of the problem a complicated exercise. In contrast, we use a spin-free representation of the cluster operators right from start, but expand the rank of the cluster operators by involving spectator orbitals to distinguish the various spin possibilities. The combinatorial factors entering the contracted power series in E _r(S) are chosen in such a way that they correspond to what we would have obtained if we had used a Thouless-like transformation to induce the orbital relaxation. Our working equations generally have only finite powers of the cluster operators S, resulting in a very compact formulation of the relaxation problem. Pilot numerical applications for the IP computations of HF and H₂O in the core, the inner valence and the outer valence regions show very good performance of the method vis-a-vis those obtained using the traditional normal ordered ansatz for Ω_v. The improvement in the core IP value is particularly impressive, although even for the valence regions there is an overall improvement of the IP values. Received: 3 August 1998 / Accepted: 30 September 1998 / Published online: 15 February 1999     

Development of the cyclic cluster model formalism for Kohn-Sham auxiliary density functional theory methods

The development of the cyclic cluster model (CCM) formalism for Kohn-Sham auxiliary density functional theory (KS-ADFT) methods is presented. The CCM is a direct space approach for the calculation of perfect and defective systems under periodic boundary conditions. Translational symmetry is introduced in the CCM by integral weighting. A consistent weighting scheme for all two-center and three-center interactions appearing in the KS-ADFT method is presented. For the first time, an approach for the numerical integration of the exchange-correlation potential within the cyclic cluster formalism is derived. The presented KS-ADFT CCM implementation was applied to covalent periodic systems. The results of cyclic and molecular cluster model (MCM) calculations for trans-polyacetylene, graphene, and diamond are discussed as examples for systems periodic in one, two, and three dimensions, respectively. All structures were optimized. It is shown that the CCM results represent the results of MCM calculations in the limit of infinite molecular clusters. By analyzing the electronic structure, we demonstrate that the symmetry of the corresponding periodic systems is retained in CCM calculations. The obtained geometric and electronic structures are compared with available data from the literature.     

Lanczos cluster expansion for non-extensive systems

A cluster expansion of the Lanczos recursion for non-extensive systems is developed based on the plaquette expansion for extensive systems, in which an auxiliary scaling parameter, Ω, plays the role of volume and introduces extensivity into the problem. Connected Hamiltonian moments of the non-extensive system are computed and introduced into the plaquette expansion in the usual way with Ω. The extensive energy is calculated for increasing orders of the expansion in 1/Ω and the ground state and mass gap of the finite few body problem recovered in the limit Ω → ∞. This new non-perturbative method is applied to the case of N bosons interacting harmonically in one dimension and the ground state energy and mass gap in the vacuum sector are calculated exactly.     

Secondary relaxation behavior in a strong glass

In the present paper we study the enthalpy relaxation behavior of the hyperquenched GeO(2) (HQGeO(2)) glass, one of the strongest glass systems. By applying the hyperquenching-annealing-calorimetry approach, we have found that unlike fragile glasses the strong HQGeO(2) glass relaxes in a manner that all the secondary relaxation units contribute to the primary relaxation. By analyzing dynamic properties of the secondary relaxation, we have identified two typical features of the Johari-Goldstein relaxation in the HQGeO(2) glass. First, the quantitative relationship observed here between E(beta) and T(g) agrees well with the empirical relation of the JG relaxation. Second, the characteristic relaxation time of the GeO(2) glass at T(g) is found to be about 10 s, larger than that of relatively fragile glasses. These results verify that the JG peak in strong glasses is hidden by the alpha peak in the dielectric loss curves.     

Extended rotational diffusion and orientational relaxation of symmetric top molecules in a strong dc electric field: second-rank orientational correlation functions

Second-rank orientational correlation functions (pertaining to Kerr effect relaxation and Raman scattering) are obtained using the extended rotational diffusion (J-diffusion) model of symmetric top polar molecules in a strong constant external field. It is shown that the shape of the molecule noticeably affects all second-rank correlation functions and relaxation times in the rare collision limit. In the opposite limit of frequent collisions, the quantities of interest are shown to be shape independent as a consequence of vanishingly small inertial effects. An interpolation formula for the orientation relaxation times in the intermediate regime between the rare and frequent collision limits is also given.     

Self-interaction and strong correlation in DFTB

The density functional based tight-binding (DFTB) method can benefit substantially from a number of developments in density functional theory (DFT) while also providing a simple analytical proving ground for new extensions. This contribution begins by demonstrating the variational nature of charge-self-consistent DFTB (SCC-DFTB), proving the presence of a defined ground-state in this class of methods. Because the ground state of the SCC-DFTB method itself can be qualitatively incorrect for some systems, suitable forms of the recent LDA+U functionals for SCC-DFTB are also presented. This leads to both a new semilocal self-interaction correction scheme and a new physical argument for the choice of parameters in the LDA+U method. The locality of these corrections can only partly repair the HOMO-LUMO gap and chemical potential discontinuity, hence a novel method for introducing this further physics into the method is also presented, leading to exact derivative discontinuities in this theory at low computational cost. The prototypical system NiO is used as an illustration for these developments.     

Electronic relaxation dynamics of water cluster anions

The electronic relaxation dynamics of water cluster anions, (H(2)O)(n)(-), have been studied with time-resolved photoelectron imaging. In this investigation, the excess electron was excited through the p<--s transition with an ultrafast laser pulse, with subsequent electronic evolution monitored by photodetachment. All excited-state lifetimes exhibit a significant isotope effect (tau(D)2(O)/tau(H)2(O) approximately 2). Additionally, marked dynamical differences are found for two classes of water cluster anions, isomers I and II, previously assigned as clusters with internally solvated and surface-bound electrons, respectively. Isomer I clusters with n > or = 25 decay exclusively by internal conversion, with relaxation times that extrapolate linearly with 1/n toward an internal conversion lifetime of 50 fs in bulk water. Smaller isomer I clusters (13 < or = n < or = 25) decay through a combination of excited-state autodetachment and internal conversion. The relaxation of isomer II clusters shows no significant size dependence over the range of n = 60-100, with autodetachment an important decay channel following excitation of these clusters. Photoelectron angular distributions (PADs) were measured for isomer I and isomer II clusters. The large differences in dynamical trends, relaxation mechanisms, and PADs between large isomer I and isomer II clusters are consistent with their assignment to very different electron binding motifs.     

A response-function approach to the direct calculation of the transition-energy in a multiple-cluster expansion formalism

Approximate limiting collision induced dissociation (CID) probabilities are calculated as a function of the vibrational energy and temperature for HCl, HF and O₂, infinitely diluted in an inert gas, by assuming that only the molecules with an energy that is within kT below the threshold dissociate and do so with unit probability. Rotational energy is explicitly included in defining the threshold for dissociation. At the lowest temperatures of our calculations (2500 K for HCl and HF, 3500 K for O₂) the agreement between the calculated CID probabilities and those obtained by us earlier in fitting the experimental dissociation rates is quite remarkable for HCl and HF and satisfactory for O₂ at all vibrational energies. It is therefore argued that the ladder climbing model and the weak bias model for diatomic dissociation are not different in principle if rotational energy is properly accounted for.     

Incorporation of a cluster into a lattice of point charges in the SCF-CI formalism (ideal crystal)

Urals Polytechnic Institute. Translated from Zhurnal Strukturnoi Khimii, Vol. 32, No. 4, pp. 17–21, July–August 1991.     

Ion hydration effects in aqueous solutions of strong electrolytes,according to proton magnetic relaxation measurements

The concentration dependences of proton magnetic relaxation (PMR) rates measured at different temperatures in aqueous electrolyte solutions and concentrated seawater (SW) in a wide range of salt concentrations and for different seawater salinities are presented, along with the concentration dependences of PMR rates determined in salts dissolved directly in seawater. The coordination numbers of the basic ions in seawater were determined from the complete solvation limits and compared with those measured in single-component water-salt solutions. The attaining of complete solvation limits was determined using the PMR data for ions of different hydration signs.     

Energy and capacity time correlation functions for investigation of vibrational energy relaxation

Vibrational energy relaxation of a diatomic solute in a liquid solvent is investigated by means of the generalized Langevin equation. The vibrational energy, velocity and capacity time correlation functions (TCFs) are considered. It is shown that the detailed structure of the energy TCF contains an initial fast (subpicosecond) decay segment that is followed by weak oscillations on the background of an exponential relaxation curve. The direct method for evaluating the relaxation rate constant from equilibrium molecular dynamics simulations of a flexible solute is proposed and implemented. The closed form expressions for the memory function and for the relaxation rate constant in terms of quantities accessible from the simulations are derived. The simulation results for rigid and flexible solutes are compared and analyzed.     

On the theory of electron correlation in atoms and molecules: Relation between cluster expansion theory and the correlated wave functions method

A relation between the cluster expansion theory of many electron wave functions and the correlated wave functions method is established. In this way, the theoretical basis of the method is elucidated and the approximations involved in its application become apparent. General forms of the correlated wave function, differing in certain important respects from that form usually assumed, are derived.     

Many-orbital cluster expansion for the exchange-repulsion energy in the interaction of closed-shell systems

The energy of exchange repulsion between two closed-shell systems described by determinantal wave functions has been represented as a sum of contributions arising from the interaction of two, three and four orbitals at a time. These contributions have been calculated for the interaction of two neon atoms. It has been found that in the van der Waals minimum region the two-orbital components are of secondary importance and that about 90% of the total exchange energy originates from the three-orbital interactions ofL-shell electrons. The four-orbital as well as the double-exchange terms have been found negligible. The approximate algorithms for evaluation of the exchange repulsion energy have been tested and discussed.This work was partly supported by the Polish Academy of Sciences within the project MR-I.9.     

Exact ground state for a Herndon–Simpson model via resonance–theoretic cluster expansion

The Herndon–Simpson model for a particular catacondensed polyphene chain is considered as a nontrivial many-body Hamiltonian, defined on a space with a basis of orthonormal Kekulé structures. An Explicitly correlated cluster expanded resonance–theoretic wave function is described for this model, and its quality is judged by calculation of the standard deviation for the energy expectation. The quality is found to be high. Indeed, for a particular parameter ratio within the range of experimental interest, the wave function ansatz is found to be exact. This very accurate solution is then used to gauge the quality of the common ansatz with equally weighted Kekulé structures, and it is found to be reasonably good.     

On the theory of electron correlation in atoms and molecules. II. General cluster expansion theory and the general correlated wave functions method

An exact cluster expansion of many electron wave functions is derived, beginning with a finite linear combination of Slater determinants rather than the more usual single determinant. This general cluster expansion is found to apply both in the case where all possible Slater determinants from a finite set of spin orbitals are included in the linear combination, and in the case where the number of determinants is restricted. The special properties of that finite linear combination of determinants closest to the exact wave function in the least squares sense are studied. These properties lead to the derivation of a general correlated wave functions method, illustrating again the close relationship between methods of this type and cluster expansion theory. Additional approximations, necessary for practical calculations, are set out.     

Local correlation domains for coupled cluster theory: optical rotation and magnetic-field perturbations

An approach is described for selecting local-correlation orbital domains appropriate for computing response properties such as optical rotation using frequency-dependent coupled-cluster linear-response theory. This scheme is an extension of our earlier idea [N. J. Russ and T. D. Crawford, Chem. Phys. Lett., 2004, 400, 104] based on an atom-by-atom decomposition of the coupled-perturbed Hartree-Fock (CPHF) response of the component molecular orbitals to external electric and magnetic fields. We have applied this domain-selection scheme to a series of chiral molecules, including pseudo-linear structures (hydrogen molecule helices, fluoroalkanes, and [n]triangulanes), cage-like structures (beta-pinene, methylnorbornanone, and bisnoradamantan-2-one), and aromatic rings (1-phenylethanol). We find that the crossover points between the canonical- and local-correlation approaches are larger than for the conventional Boughton-Pulay domain scheme, in agreement with our earlier analysis of dipole-polarizabilities. Localization errors are reasonably small (a few percent) for pseudo-linear structures with domain sizes of approximately six to eight atoms. Cage-like molecules are significantly more problematic, requiring natural domain sizes of ten or more to obtain the most reliable localization errors.     

Interdependence of motional processes in polymers: correlation functions for nmr relaxation

The roles of both anisotropic motion and the interdependence of multiple motions in leading to nonexponential correlation functions for NMR relaxation data are explored. A motional model is developed in which rotational motions of segments of various lengths are controlled by the formation and disappearance of a suitable conformation. Such a model gives correlation functions which can be made, through adjustment of parameters, to be almost identical to correlation functions from other, quite different, models. The ability of NMR relaxation data to identify unique motional models is thus questioned.