Open-shell systems clusters

Noga, J., Valiron, P. Explicitly correlated R12 coupled cluster calculations for open shell systems. Chem. Phys. Lett. 2000, 324, 166-74. 

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). 

For the treatment of electron correlation, Cizek uses classical techniques as well as techniques based on mathematical methods of quantum field theory, namely, a coupled-cluster approach. A rapid development and deployment of these methods during the past decade was stimulated by the realization of the importance of size consistency or size extensivity in the studies of reactive chemical processes. Although truly remarkable accuracy and development have been achieved for ground states of closed-shell systems, an extension to quasidegenerate and general open-shell systems is most challenging. Cizek also works on the exploitation of these approaches to study the electronic structure of extended systems (molecular crystals, polymers107). His many interests in-... 

The Ni atoms are treated as one-electron systems in which the effects of the Ar-like core and the nine 3d electrons are replaced by a modified effective potential (MEP) as suggested by Melius et al./165/ A contracted gaussian basis set is used for Ni, which includes two functions to describe the 4s and one to describe the 4p atomic orbitals. Since a previous study/159/ found the O-Ni spacing and the vibrational frequency we insensitive to correlations in this open-shell system, the authors have adopted the SCF calculation scheme. To check the approximation of treating the Ni atoms as a one-electron system, they performed both the MEP and an all electron SCF calculation for the NisO cluster. They found that the MEP spacing is 0.37 A or 35% smaller than the all-electron value the MEP we value is 90 cm-1 or 24% smaller. Since the 3d... 

G2(MP2)-RAD, which implements restricted-open-shell versions of both coupled cluster and perturbation theories. The latter method has been shown to generally yield reliable results when applied to open-shell systems [66]. 

We emphasize that the present discussion focuses only on high-spin open-shell systems to which a single-determinant reference wavefunction is applicable. Coupled cluster techniques for low-spin cases, such as open-shell singlets, have been pursued in the literature for many years, however, and provide a fertile area of research (Refs. 158, 167-170). 

T. D. Crawford and H. F. Schaefer,/. Chem. Phys., 104, 6259 (1996). A Comparison of Two Approaches to Perturbational Triple Excitation Corrections to the Coupled-Cluster Singles and Doubles Method for High-Spin Open-Shell Systems. 

P. Neogrady, M. Urban, and I. Hubac, /. Chem. Phys., 100, 3706 (1994). Spin Adapted Restricted Hartree-Fock Reference Coupled-Cluster Theory for Open-Shell Systems. 

As a result, we may view the orbital invariant SS-MRCEPA, termed by us as SS-MRCEPA(I) (I, for invariant), as the optimal approximation to the parent SS-MRCC method, which includes all the EPV terms exactly and which utilizes only those counter terms of the equations which eliminate the lack of extensivity of the attendant non-EPV terms in an orbital invariant manner [59]. In this article, we will present a couple of invariant SS-MRCEPA methods, viz. SS-MRCEPA(O) and SS-MRCEPA(I), for general open-shell systems using spin-free unitary generator adapted cluster operators starting from explicitly spin-free full-blown parent SS-MRCC formalism. Eor a detailed discussion of the allied issues pertaining to all the SS-MRCEPA-like methods, we refer to our recent SS-MRCEPA papers [58,59] and an earlier expose by Szalay [66]. 

Nonadditive effects in open-shell clusters have been investigated only recently and relatively little information is available on their importance and physical origin. From the theoretical point of view, open-shell systems are more difficult to study since the conventional, size-consistent computational tools of the theory of intermolecular forces, like the Mpller-Plesset perturbation theory, coupled cluster theory, or SAPT, are less suitable or less developed for applications to open-shell systems than to closed-shell ones. Moreover, there are many types of qualitatively different open-shell states, exhibiting different behavior and requiring different theoretical treatments. 

Paldus, elsewhere in this book, discusses that there is as yet no generally applicable, open-shell, size-extensive, coupled cluster method, and the same holds for open-shell S APT methods. Therefore, for the computation of potentials of open-shell van der Waals molecules one has the choice between CASSCF followed by a Davidson-corrected MRCl calculation of the interaction energy, or the single reference, high spin, method RCCSD(T). When the ground state of the open-shell monomer is indeed a high spin state, then RCCSD(T) is the method of choice. With regard to the latter method we recall that a major difficulty in open-shell systems is the adaptation of the wave function to the total spin operator S for the CCSD method a partial spin adaptation was published by Knowles et al. [219,220] who refer to their method as partially spin restricted . When non-iterative triple corrections [221] are included, the spin restricted CCSD(T) method, RCCSD(T), is obtained. 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

By means of the exponential ansatz = exp(T), the Bloch equation leads directly to the coupled-cluster approach. For general open-shell systems, it is often convenient to use the normal-ordered exponential [15]... 

From the expansion (27) it can be shown that the CCA includes also those three-,. .., n-body excitations which factorize into one- and two-particle cluster amplitudes. However, despite of the benefit of CCA to incorporate selected n-body contributions to all order into the computations, setting up the equations for open-shell systems is a highly nontrivial task, in particular when compared with the usual MBPT computations. For simple shell structures, there are a number of coupled-cluster codes available today, both in the nonrelativistic as well as the relativistic framework [31-34]. 

As an example for an open-shell system, we carried out calculations for Th2. Th clusters have been attracting increasing attention due to potential nuclear energy applications. The calculated structural, electronic structure and energetic properties are summarized in Table 2. We have identified three low-lying electronic states without spin-orbit interactions, the sa dn da db sa g, sa dn da sa X, and sa dn sa 11+ states. 

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