Perturbation theory optimization methods

This section outlines the principles of optimization methods that are based on material density perturbations with the purpose of (1) illustrating another area for the application of perturbation theory formulations, and (2) promoting the use of these potentially powerful perturbation-based optimization methods. The perturbation theory foundations of optimization methods, and their relation with the variational formulation of these methods, have already been described in previous reviews (/, 56). Our presentation is restricted to a specific type of control variable—the material densities— and is given in terms of sensitivity functions. Moreover, we present only the conditions for the optimum and do not consider optimization algorithms. 

Abstract. We present a quantum-classieal determination of stable isomers of Na Arii clusters with an electronically excited sodium atom in 3p P 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 . The geometry optimization of the excited states is carried out via the basin-hopping method of Wales et al. The present study confirms the trend for small Na Arn clusters in 3p states to form planar structures, as proposed earlier by Tutein and Mayne within the framework of a first order perturbation theory on a "Diatomics in Molecules" type model. 

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]. 

Most multireference methods described to date have been limited to second order in perturbation theory. As analytic gradients are not yet available, geometry optimization requires recourse to more tedious numerical approaches (see, for instance. Page and Olivucci 2003). While some third order results have begun to appear, much like the single-reference case, they do not seem to offer much improvement over second order. 

The method of CHF-PT-EB CNDO has been used for several organometallic complexes. This method utilizes a coupled Hartree-Fock (CHF) scheme10 applied through the perturbation theory (PT) of McWeeny,11"13 and an extended basis (EB), complete neglect of differential overlap (CNDO/2) wavefunction. The exponents of the basis set are optimized with respect to experimental polarizabilities and second hyperpolarizabili-ties.14,15 A detailed description of the CNDO/2 method may be found in Ref. 16. 

We employed Hartree-Fock (HF) method for the geometry optimizations and the JT potential calculation. Radom has reviewed computational studies on various molecular anions that includes only first raw elements [21], It has been concluded that reliable structural predictions may be made from single-determinant MO calculations with double-zeta basis sets. Furthermore, we applied second-order M0ller-Plesset (MP2) perturbation theory for the optimized geometries with HF... 

In general, when perturbation theory applies, one can devise an elegant method] obtaining the optimal solution to problems such as those in photodissociation, f " optimal fields derived in the perturbative domain [92, 126, 129-131] do noth by much from the fields derived via the more general brute-force opti methods. 

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