RI approximation

The CC2 method [74] is an approximation to coupled cluster with singles and doubles (CCSD), and the excited state energies calculated have MP2 quality. An implementation that employs the resolution of identity (RI) approximation for two-electron integrals to reduce the CPU time is also available, RI-CC2 [75], which is suitable for large scale integral-direct calculations. This method has been implemented in TURBOMOLE [76],... 

This inconvenience has been solved by introduction of the resolution-of-identity (RI) approximation. 

However, the introduction of the RI approximation led to the need for large basis sets. In old R12 method, only one single basis was used for both the electronic wave function and the RI approximation. The new formulation of R12 theory presented here uses an independent basis set denoted auxiliary basis set for the RI approximation while we employ a (much) smaller basis set for the MP2 wave function (7). This auxiliary basis set makes it possible to employ standard basis sets in explicitly correlated MP2-R12 calculations. 

The similarity-transformed Hamiltonian method has so far been applied only to two-electron systems. Using closure (i.e., RI) approximations, this technique will be generalized to many-electron systems (IS). 

All of the systems were initially optimized using a much higher level of theory, in order to ensure that the OM2 method provides a realistic description of the structure. The method employed was the second-order Mpller-Plesset perturbation theory (MP2) [50] using the cc-pVDZ basis set [51]. The resolution-of-identity (RI) approximation for the evaluation of the electron-repulsion integrals implemented in Turbomole was utilized [52]. 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

Exact methods are developed for calculating second and fourth moments of chain molecules. The RIS approximation is adopted to represent the effects of hinderance potentials affecting bond rotations. IMo other approximations are invoked in deriving the statistical-mechanical averages. 

The anisotropy of the polarizability is described by the tensor invariant y2, which determines the depolarization ratio. This invariant y2 is averaged over all configurations of the chain molecule treated in the RIS approximation, interdependenca of rotations about neighboring bonds is taken into account. 

Expressions for the optical anisotropy AT of Kuhn s random link (an equivalent to the stress-optical coefficient) of stereo-irregular and multirepeat polymers are derived on the basis of the additivity principle of bond polarizabilities and the RIS approximation for rotations about skeletal bonds. Expressions for the unperturbed mean-square end-to-end distance , which are required in the calculation of Ar, are also obtained. 

A method is developed for calculating even moments of the end-to-end distance r of polymeric chains, on the basis of the RIS approximation for rotations about skeletal bonds. Expressions are obtained in a form which is applicable in principle to arbitrary k, but practical applications are limited by a tremendous increase in the order of the matrices to be treated, with increasing k. An application is made to the PE chain by using the familiar three-state model. Approximate values of the distribution function Wn (r) of the end-to-end vector r, Wn (0), and , are calculated from these even moments. 

Distribution functions for very short n-alkane chains are calculated. The results obtained implicate the discrete nature of the RIS approximation as the origin of the multimodal nature of the short-chain distribution, and suggest that the best representation of such results would be a simple unimodal curve averaging out all of the minima and retaining only the most prominent maximum. 

The conformational energies of the lower members of POM, 2,4-dioxapentane and 2,4,6-trioxaheptane are estimated by the empirical force field method. The gauche states of the Internal rotation around the skeletal C—0 bonds are successfully predicted to be of lower energies in both molecules. In order to calculate the unperturbed dimension and dipole moment of POM, RIS approximations are made by using the results obtained from the force field calculations on 2,4,6-trioxaheptane. Although these parameters are significantly different from those estimated earlier, they reproduce the observed values fairly well. 

In this study, the dipole moments at 298 K and the corresponding temperature coefficient of PDTC are calculated in the RIS approximation. The results are compared to the values determined by experiment. The results indicate that an extra stabilization energy of about 3.8 kj mol-1 must be added to these gauche states relative to trans state. This provides evidence of a large attractive sulfur gauche effect in the polysulfides, about equal in magnitude to that found for the analogous POM. 

The inter- and Intramolecular contributions to the entropy and energy of fusion are calculated for several linear aliphatic polyesters and polyamides assuming the fusion process consists of two Independent contributions the volume expansion (intermolecular contribution) and the increase in the conformational freedom of each polymer chain on melting (intramolecular contribution). The intramolecular entropy and energy contributions are obtained from the configurational partition function and Its temperature coefficient calculated for an isolated, unperturbed polymer chain using the RIS approximation. 

The magnetic susceptibilities of dimer liquid crystals such as NC-Ph—Ph—O—(CH2>n—Ph—Ph—CN(n 9, 10) are measured by a SQUID magnetometer. The results obtained are interpreted within the framework of the RIS approximation, the effect arising from the conformational anisotropy of the flexible spacer being strictly taken into account. The order parameters of the mesogenic core axis thus estimated are found to be consistent with those directly observed at just below 7N) by the ZH NMR technique using mesogen-deuterated samples. 

The unperturbed dimensions and characteristic ratios of PTMPS chains are calculated using the RIS approximation. The low experimental value of the characteristic ratio reported for this polymer is successfully Interpreted by these computations. The stiffening effect expected from introduction of thep-phenylene group into the chain backbone is apparently offset by its reducing the repulsive Interactions that would otherwise occur among the atoms or groups before and after it eiong the chain. 

The present theoretical approach to rubberlike elasticity is novel in that it utilizes the wealth of information which RiS theory provides on the spatial configurations of chain molecules. Specifically, Monte Carlo calculations based on the RIS approximation are used to simulate spatial configurations, and thus distribution functions for end-to-end separation r of the chains. Results are presented for polyethylene and polydimethylsiloxane chains most of which are quite short, in order to elucidate non-Gaussian effects due to limited chain extensibility. 

The RI approximation is based on the evaluation of individual electron repulsion integrals according to... 

In the rotational isomeric state (RIS) approximation, each bond in the chain backbone is assumed to occur in any one of a small, discrete number of... 

The SOMF operator was successfully employed in various calculations as implemented in the ORCA [103], MOLPRO [104], and AMFI [105] integral programs. Several additional approximations can be optionally employed in the ORCA program such as a one-center approximation and utilization of atomic densities (in line with the AMFI code) as well as utilization of the RI approximation for the Coulomb-like (second) term of (45) [103]... 

Thus, the computation of the SSC Hamiltonian matrix element requires calculations of six (five if one employs the fact that the SSC operator is traceless) Cartesian components of two-electron integrals. Those two-electron SSC integrals can be evaluated using an RI approximation [113]. 

---