Hartree-Fock theory optimization techniques

The procedure for determining the electron density and the energy of the system within the DFT method is similar to the approach used in the Hartree-Fock technique. The wavefunction is expressed as an antisymmetric determinant of occupied spin orbitals which are themselves expanded as a set of basis functions. The orbital expansion coefficients are the set of variable parameters with respect to which the DFT energy expression of equation 15 is optimized. The optimization procedure gives rise to the single particle Kohn-Sham equations which are similar, in many respects, to the Roothaan-Hall equations of Hartree-Fock theory. 

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 expressions for the electronic gradient and Hessian derived in this subsection play an important role in Hartree-Fock theory and will be used repeatedly in the remainder of this chapter. Note carefully, however, that these expressions are valid only at the expansion point ic = 0. More general expressions, valid for any k, are considered in Exercise 10.1 but are not needed here. Expressions for higher-order derivatives of the energy may also be derived by the techniques of this subsection but are not considered here since they are not required for the optimization and the characterization of the wave function. 

Well-known procedures for the calculation of electron correlation energy involve using virtual Hartree-Fock orbitals to construct corresponding wavefunctions, since such methods computationally have a good convergence in many-body perturbation theory (MBPT). Although we know the virtual orbitals are not optimized in the SCF procedure. Alternatively, it is possible to transform the virtual orbitals to a number of functions. There are some techniques to do such transformation to natural orbitals, Brueckner orbitals and also the Davidson method. 

Configuration Interaction Coupled-cluster Theory Density Functional Applications Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field M0ller-Plesset Perturbation Theory Numerical Hartree-Fock Methods for Molecules Pericyclic Reactions The Diels-Alder Reaction Solvation Modeling Transition Structure Optimization Techniques. 

With only a few electrons outside a closed shell, the above equations may be solved iteratively in essentially the same manner as the Hartree-Fock equations. Usually an algebraic approximation is used in which every is expressed as a linear combination of basis functions as in Section 6.2 and this leads to matrix equations that are more suitable for computational purposes. The actual techniques of optimization are similar to those used in multiconfiguration versions of MO SCF theory (Chapter 8) the more rapidly convergent procedures require also second derivatives of the energy expression, but these may be obtained in the same way as the first derivatives. 

The next five chapters are each devoted to the study of one particular computational model of ab initio electronic-structure theory Chapter 10 is devoted to the Hartree-Fock model. Important topics discussed are the parametrization of the wave function, stationary conditions, the calculation of the electronic gradient, first- and second-order methods of optimization, the self-consistent field method, direct (integral-driven) techniques, canonical orbitals, Koopmans theorem, and size-extensivity. Also discussed is the direct optimization of the one-electron density, in which the construction of molecular orbitals is avoided, as required for calculations whose cost scales linearly with the size of the system. 

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