Second-order optimization Hartree-Fock 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 ... 

MOs and the configuration expansion. To be successful, we must choose the parametrization of the MCSCF wave function with care and apply an algorithm for the optimization that is robust as well as efficient. The first attempts at developing MCSCF optimization schemes, which borrowed heavily from the standard first-order methods of single-configuration Hartree-Fock theory, were not successful. With the introduction of second-order methods and the exponential parametrization of the orbital space, the calculation of MCSCF wave functions became routine. Still, even with the application of second-order methods, the optimization of MCSCF wave functions can be difficult - more difficult than for the other wave functions treated in this book. A large part of the present chapter is therefore devoted to the discussion of MCSCF optimization techniques. 

The purpose of the present chapter is to discuss the structure and construction of restricted Hartree-Fock wave functions. We cover not only the traditional methods of optimization, based on the diagonalization of the Fock matrix, but also second-order methods of optimization, based on an expansion of the Hartree-Fock eneigy in nonredundant orbital rotations, as well as density-based methods, required for the efficient application of Hartree-Fock theory to large molecular systems. In addition, some important aspects of the Hartree-Fock model are analysed, such as the size-extensivity of the energy, symmetry constraints and symmetry-broken solutions, and the interpretation of orbital energies in the canonical representation. 

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... 

The second main group of methods analyse the response to successive powers of the perturbation separately using analytical re-arrangements of the perturbed equations. The procedure is typified in the Coupled Perturbed Hartree-Fock39,40,41 method (CPHF). which produces variationally optimized solutions in each order. Since the results represent a solution of the variational Hartree-Fock equations to each order they satisfy the energy derivative equations for the polarisabilities and the (2n+ 1) rule for the derivatives can be used to simplify the calculations. Corrections to the perturbed HF solutions can be made through MP2 or MP4 perturbation theory. 

The-quantum-chemical methods follow mainly two lines, the Hartree-Fock (HF) approach possibly supplemented by the inclusion of electron correlation via second order Mollor-Plesset perturbation theory (MP2) or alternatively a density fimctional theory (DFT) approach. Within the conventional scheme geometry optimization is usually performed at the HF level and in a... 

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. 

Prior to this, it had already been established that even the simplest forms of DFT, based on the exchange-only Slater or Xa scheme, could give good descriptions of the electronic structure of metal complexes and a number of contemporary applications confirmed this. However, in combination with structure optimization, here at last was a quantum chemical method accurate enough for transition metal (TM) systems and yet still efficient enough to deliver results in a reasonable time. This was in stark contrast to the competition which was either based on the single-determinant Hartree-Fock approximation, which had been discredited as a viable theory for TM systems,or on more sophisticated electron correlation methods (e.g., second order Moller-Plesset theory) which are relatively computationally expensive and thus, for the same computer time, treat much smaller systems that DFT. 

An efficient approach to improve on the Hartree-Fock Slater determinant is to employ Moller-Plesset perturbation theory, which works satisfactorily well for all molecules in which the Dirac-Hartree-Fock model provides a good approximation (i.e., in typical closed-shell single-determinantal cases). The four-component Moller-Plesset perturbation theory has been implemented by various groups [519,584,595]. A major bottleneck for these calculations is the fact that the molecular spinor optimization in the SCF procedure is carried out in the atomic-orbital basis set, while the perturbation expressions are given in terms of molecular spinors. Hence, all two-electron integrals required for the second-order Moller-Plesset energy expression must be calculated from the integrals over atomic-orbital basis functions like... 

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