Hartree-Fock function closed shell single determinant

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 ... 

As Bartlett/18/ has pointed out, for a closed shell single determinant Hartree-Fock (HF) function as the starting point for MBPT or CC theory, separability into neutral fragments is not automatically guaranteed, since sometimes the UHF rather than the RHF dissociates properly. When the criteria (a) and (d) above are both satisfied, the model may be called... 

We now consider the theoretical calculation of excited-state wave functions. This is more difficult than ground-state calculations because we are dealing with open-shell configurations. The Hartree-Fock equations for a state of an open-shell configuration have a more complicated form than for closed shells, and there exist close to a dozen different approaches to excited-state Hartree-Fock calculations. As noted earlier, the Hartree-Fock wave function for a closed-shell state is a single determinant, but for open-shell states, we may have to take a linear combination of a few Slater determinants to get a Hartree-Fock function that is an eigenfunction of S and Sz and has the correct spatial symmetry. 

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... 

The simplest approximation corresponds to a single-determinant wave function. The best possible approximation of this type is the Hartree-Fock (HF) molecular-orbital determinant. The HF wavefunction is constructed from the minimal number of occupied MOs (i.e., NI2 for an V-eleclron closed-shell system), each approximated as a variational linear combination of the chosen set of basis functions (vide infra). 

For multi-electron systems, it is not feasible, except possibly in the case of helium, to solve the exact atom-laser problem in 3 -dimensional space, where n is the number of electrons. One might consider using time-dependent Hartree Fock (TDHF) or the time-dependent local density approximation to represent the state of the system. These approaches lead to at least njl coupled equations in 3-dimensional space which is much more attractive computationally. For example, in TDHF the wave function for a closed shell system can be approximated by a single Slater determinant of time dependent orbitals,... 

Usually, theoretical studies on ionization processes of atoms and molecules are performed using the so-called approximation of Koopmans theorem. This theorem says that the ionization potential of an electron located on the level of a closed-shell state is equal to the opposite sign to the Hartree-Fock orbital energy e%. One obtains this result by assuming that the single determinant wave function of the ion is constructed from the same molecular orbitals as the ground-state function, except for the spin orbital of the missing electron. 

In 1978, Ludena [102] carried out a Hartree-Fock calculation by using a wave function consisting of a single Slater determinant for the closed-shell atoms, whereas he used a linear combination of the Slater determinants for the open-shell atoms. Each Slater-type orbital times a cut-off function of the form (1 — r/R) to satisfy the boundary conditions. Ludena studied pressure effects on the electronic structure of the He, Li, Be, B, C and Ne neutral atoms. The energies he obtained for the confined helium atom are slightly lower than those Gimarc obtained, especially for box radii in the range R > 1.6 au. 

The Coulson-Fischer wave function for H2 can be considered as the start of the Unrestricted Hartree-Fock (UHF) approach in quantum chemistry, which is the most general single determinant method. We shall not proceed further along this line, but instead ask ourselves if there is a way to correct the simation such that we obtain a wave function that dissociates correctly while preserving the spin and space symmetry of the wave function. The CF wave function gives acmally a hint. What happens if we simply skip the trouble-some triplet term in Eq. (22). This gives rise to a wave function that is a linear combination of two closed shell determinants ... 

A very important point is that, contrary to methods based on a Hartree-Fock zero-order wave function, those rooted in the Kohn-Sham approach appear equally reliable for closed- and open-shell systems across the periodic table. Coupling the reliability of the results with the speed of computations and the availability of analytical first and second derivatives paves the route for the characterization of the most significant parts of complex potential energy surfaces retaining the cleaness and ease of interpretation of a single determinant formalism. This is at the heart of more dynamically based models of physico-chemical properties and reactivity. 

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