Second-order optimization theory

Derive the detailed expression for the orbital Hessian for the special case of a closed shell single determinant wave function. Compare with equation (4 53) to check the result. The equation can be used to construct a second order optimization scheme in Hartree-Fock theory. What are the advantages and disadvantages of such a scheme compared to the conventional first order methods ... 

Part 1, comprised of three chapters, focuses on the fundamentals of convex analysis and nonlinear optimization. Chapter 2 discusses the key elements of convex analysis (i.e., convex sets, convex and concave functions, and generalizations of convex and concave functions), which are very important in the study of nonlinear optimization problems. Chapter 3 presents the first and second order optimality conditions for unconstrained and constrained nonlinear optimization. Chapter 4 introduces the basics of duality theory (i.e., the primal problem, the perturbation function, and the dual problem) and presents the weak and strong duality theorem along with the duality gap. Part 1 outlines the basic notions of nonlinear optimization and prepares the reader for Part 2. 

Gao et al. [27] published an extensive study, at the Hartree-Fock (HF) level, which included geometry optimization with 6-31G(d) basis and correlation treatment using the Moller-Plesset second-order perturbation theory (MP2) in the valence space. Siggel et al. [28a] calculated gas-phase acidities for methane and formic acid at the MP4/6-31 -I- G(d) level and for several other compounds at lower levels of theory (HF with 3-21 -I- G and 6-311 -I- G basis sets). All these calculations provide gas-phase acidity values that systematically differ from the experimental values. Nevertheless, the results show good linear correlation with the experimental data. 

The above simple analysis now elucidates how small contributions from 4,- and la are essentially suppressed in the [2] and Afjj p[2] indices. As a rule, these small contributions appear mainly from dynamical correlations. Eor instance, MP2 (the Moller-Plesset second-order perturbation theory) normally produce the contributions of this kind. Evidently, they have no direct relation to diradicahty and polyradicality, and the [2] and Ajj p[2] indices should be rather small without a significant contribution from non-dynamical correlation. This is a good property of the generalized indices such as (6.94) and (C8), and apparently, this is the basic reason why [2] is systematically employed in papers [9, 11, 122, 124] for analyzing the unpaired electrons in large PAHs. At the same time, the dynamical correlation cannot fully ignored, and the problem of an optimal quantification... 

Electron correlation does not appear to be of crucial importance to evaluation of the optimal angular characteristic of these complexes. The a angle changes by about 10° or less when second-order MP theory is employed. On the other hand, if one wishes to calculate the height of the energy barrier for inversion separating the two equivalent geometries, correlation does become important. For example, the SCF value of this barrier is only 45 cm for H2O HF, as compared to an MP2 value of 144 cm" [35]. 

Levine BG, Coe JD, Martinez TJ. Optimizing conical intersections without derivative coupling vectors application to multistate multireference second-order permrbation theory (MS-CASPT2). J P/tys Chem B. 2008 112 405-413. 

In actual practice, the perturbation expressions given above are better suited for displaying the structural features of the theory than for use in large-scale calculations. For nonrelativistic calculations, the most efficient approach to the ab initio calculation of NMR parameters has been variational perturbation theory (Helgaker et al. 1999). This, or equivalent approaches such as second-order propagator theory, will probably also turn out to be the optimal choice for the relativistic case. 

MC approaches [30] involve the optimization of molecular orbitals within a restricted subspace of electronic occupations provided such active space is appropriately chosen, they allow for an accurate description of static electron correlation effects. Dynamical correlation effects can also be introduced either at the perturbation theory level [complete active space with second-order perturbation theory (CASPT2), and multireference Mpller-Plesset (MR-MP2) methods] [31] or via configuration interaction (MR-CI). 

MP2, the M0ller-Plesset second-order perturbation theory, which in the case of frequency-dependent perturbations has to be replaced by an iterative optimization of the perturbation amplitudes, as described by the CC2 model (Christiansen et al. 1995)... 

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