Hartree-Fock theory electronic Hessian

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. 

THE ELECTRONIC HESSIAN IN ORBITAL-BASED HARTREE-FOCK THEORY... 

To discuss the form and cost of analytic gradient and Hessian evaluations, we consider the simple case of Hartree-Fock (HF) calculations. In nearly all chemical applications of HF theory, the molecular orbitals (MOs) are represented by a linear combination of atomic orbitals (LCAO). In the context of most electronic structure methods, the LCAO approximation employs a more convenient set of basis functions such as contracted Gaussians, rather than using actual atomic orbitals. Taken together, the collection of basis functions used to represent the atomic orbitals comprises a basis set. 

Geometry optimizations are systematically carried out from the less expensive ab initio methods such as Hartree-Fock (HF), to density functional theory (DFT) [26] methods, which include electron correlation in a very efficient way. Second derivatives of the energy yielding the Hessian are also needed to obtain molecular vibrations. 

---