Ground state self-consistent field method

There is far less reported experience for the HF studies of electronic excited states (ESs). Especially, highly, doubly and core hole excited (ionized) states are not often studied. It is clear that existing ground state self-consistent field (SCF) methods cannot be directly applied to excited states of the same symmetry or of the same spin multiplicity as a lower state because of the so-called variational collapse i.e., the optimization procedure will find only the lowest solution of a given symmetry or a given spin multiplicity. Therefore, such calculations for ES cannot be considered as routine. The most powerful scheme for accurate treatment of ESs is based on multireference methods [2-8]. They typically provide an accuracy of about 0.1 eV but require the expense of much computational cost. Thus, it can be quite difficult to carry out the corresponding calculations. Such methods are, however, indispensable to study systems where... 

The MC SCF method usually takes into account a minimum number of configurations capable of assuring some fundamental requirements, this step being followed by the optimization of the basis functions using the self-consistent-field method. For example, in order to describe accurately the dissociation of ground-state H2 it is only necessary to consider the two-configuration wave function... 

Hyperfine couplings, in particular the isotropic part which measures the spin density at the nuclei, puts special demands on spin-restricted wave-functions. For example, complete active space (CAS) approaches are designed for a correlated treatment of the valence orbitals, while the core orbitals are doubly occupied. This leaves little flexibility in the wave function for calculating properties of this kind that depend on the spin polarization near the nucleus. This is equally true for self-consistent field methods, like restricted open-shell Hartree-Fock (ROHF) or Kohn-Sham (ROKS) methods. On the other hand, unrestricted methods introduce spin contamination in the reference (ground) state resulting in overestimation of the spin-polarization. 

In Chapter IX we discussed one particular limit of the self-consistent field method, where the ground state was dominant. We then introduced a ground state wavefunction r) with the following properties ... 

The electronic spectrum of aniline has been analyzed, using wave-mechanical methods, by various authors - . The self-consistent field method which has been presented in section I and used in section II to describe the ground state of this molecule can also be used to represent its electronically excited states. For practical reasons it is necessary to introduce the lcao approximation and either the Hiickel or the Pariser and Parr method is used. Furthermore, the virtual orbitals approximation is added. Let us consider the eigenfunctions 77j of the self-consistent field operator A " corresponding to the ground state of aniline in the lcao approximation. They obey the equation... 

Ferre N, Assfeld X (2002) Application of the local self-consistent-field method to core-ionized and core-excited molecules, polymers, and proteins true orthogonalify between ground and excited states. J Chem Phys 117 4119-4125... 

Most of these developments may be applied most directly within the framework of the isolated molecule method, in which the reactivity indices are the charges and self-polarizabilities of the unperturbed ground state of a given molecule calculations based on the localization model (e.g. Nesbet, 1962) have made less progress, and will not be considered. It is therefore natural to enquire whether indices similar to and tt,, in Hiickel theory can still be defined, and calculated more precisely, in self-consistent field theory. The obvious questions are... 

Analytic, exact solutions cannot be obtained except for the simplest systems, i.e. hydrogen-like atoms with just one electron and one nucleus. Good approximate solutions can be found by means of the self-consistent field (SCF) method, the details of which need not concern us. If all the electrons have been explicitly considered in the Hamiltonian, the wave functions V, will be many-electron functions V, will contain the coordinates of all the electrons, and a complete electron density map can be obtained by plotting Vf. The associated energies E, are the energy states of the molecule (see Section 2.6) the lowest will be the ground state , and the calculated energy differences En — El should match the spectroscopic transitions in the electronic spectrum. 

Calculations by the self-consistent field LCAO-MO method for the ground state wave function of the pyrazine molecule indicate that the lone pairs are quite different. The lower lone pair is little delocalized (1.88 electrons on nitrogen), but the second lone pair is as delocalized as the lone pair in pyridine with 1.37 electrons on nitrogen, 0.22 electrons on hydrogen, and 0.40 electrons on carbon.63... 

---