Energy expression open-shell system

The Hamiltonian, Hea, which is called the Hartree-Fock-Roothan operator is a 1-electron operator whose application yields the energy of an electron moving in the average field of the other electrons and nuclei. In principle an SCF theory approach will lead to a well-defined expression for Hett for closed and open shell systems (188, 189), and with the aid of modern computers Hm integrals can be evaluated numerically even for transition metal complexes. This type of ab initio calculation has been reported for a reasonable number of organometallic complexes of first-row transition elements by Hillier, Veillard, and their co-workers (48, 49, 102, 103, 111-115 58, 68, 70, 187, 228, 229). 

As with the closed-shell case, this matrix should be constructed from the derivative integrals in the atomic-orbital basis. Indeed, it is possible to solve the entire set of equations in the AO basis if desired. From these equations, it can be seen that properties such as dipole moment derivatives can be obtained at the SCF level as easily for open-shell systems as is the case for closed-shell systems. Analytic second derivatives are also quite straightforward for all types of SCF wavefunction, and consequently force constants, vibrational frequencies and normal coordinates can be obtained as well. It is also possible to use the full formulae for the second derivative of the energy to construct alternative expressions for the dipole derivative. 

Kobayashi, Sasagane and Yamaguchi" have developed the theory of the time-dependent spin-restricted Hartree-Fock method for application to open shell systems (TDROHF). The expression for the cubic hyperpolarizability is obtained from the quasi-energy derivative (QED) method. The theory is applied to the investigation of the frequency-dependent y susceptibility of the Li, Na, K and N atoms. 

For the open-shell d3 V(Cp)2 system, with the 4Z (a S2) ground level, the SCF orbital energies may be similarly expressed without ambiguity, yielding AeSCF =... 

By minimizing the energy of d>, in Eq. (3.12), we obtain a set of coupled integro-differential equations, the Hartree-Fock equations, which may be expressed in the following form for closed-shell systems (for open-shell cases see Szabo and Ostlund, 1989) ... 

The last contribution to correlation energy in Eq. (2.40) is fi , defined by Vosko et al. for both closed-shell and open-sheU systems [38] using different expressions [39]. The default option of this term in the Gaussian suite of programs [40] for closed-shell systems is... 

The Single Determinant Method (SD) uses the full multi-determinant nature of an open-shell state as a linear combination of Slater determinants. This yields first order expressions for the multiplet energies which are linear combinations of SD energies. In order to calculate these quantities, both spatial and spin symmetry are fully exploited. This method was first developed for the calculation of optical transitions [37], and has been recently applied to the calculation of exchange coupling constants in a number of different systems [5n]. 

The open-shell states are of ungerade symmetry and all the minimal-basis CSFs have different space and spin symmetries see Section 5.2.2. The open-shell CSFs therefore represent our final approximate states for the model system. The energies of these states are obtained from the expressions... 

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