MO Functions

An accurate solution for the problem can be found by a complete multiconfiguration SCF treatment, in which the expansion coefficients of Eqs. (1) and (2) are determined simultaneously using the SCF techniques, with the usual trial and error procedure. This formulation can be developed along the lines of that given by Veil-lard and Clementi (1967) for closed-shell systems with inclusion of only two-electron excitations. 

The practical difficulties in the application of such a method to large systems cannot be overcome at present. A simpler procedure, which can lead ultimately to equivalent results, consists of the SCF determination of the molecular orbitals, with introduction of Cl at a later stage. The functions to be used in the Cl treatment, corresponding to the excited configurations, can be determined within the virtual orbital approximation. That is, the excited MO are approximated by those orbitals, not occupied in the ground state, which are obtained simultaneously with the occupied orbitals when solving the SCF problem for that state. 

For closed-shell configurations, which is the case for the ground states of most molecules such as those considered here, the SCF formulation to be applied is that developed by Roothaan (1951a). Within this procedure, the MO are found as solutions of the pseudo-eigenvalue equation 

6( denotes the orbital energy, negative for the occupied orbitals and usually positive for the virtual orbitals. The determination of the orbitals, within the expansion approximation, is carried out in matrix form, by a trial and error procedure. 

In the case of open-shell configurations, such as those corresponding, for example, to the lowest singlet and triplet excited states, one can use the SCF formulation of Birss and Fraga (1963) and Fraga and Birss (1964). The use of virtual orbitals provides, however, a simpler way of determining approximate functions for excited states. This approximation has been used in the calculations reported and reviewed here. 

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As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

Goldman, D. S., Kiessling, A. A., and Cooper, G. M. (1988). Post-transcriptional processing suggests that c-mos functions as a maternal message in mouse eggs. Oncogene 3 159-162. 

Furuno N, Nishizawa M, Okazaki K et al 1994 Suppression of DNA replication via Mos function during meiotic divisions in Xenopus oocytes. EMBO J 13 2399—2410... 

Calculations of nuclear spin-spin couplings by Karplus and co-workers (60) using valence bond functions have been gratifyingly successful. It was to be expected that valence bond functions would be more accurate than LCAO MO functions for this purpose since the former include explicitly effects of spin correlation. For interactions between nonbonded nuclei such as occur, for example, between the hydrogen atoms of CH4 or NH4+, structures of the forms (61)... 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO hr terms of basis functions as describing the MO function in the cooidinale system of... 

The atomic orbital (AO) of H needs no eomment. The spatial molecular orbital (MO) functions of O2 are linear combinations of the one-eleetron functions with leading one-eenter Coulomb asymptotics that correspond to the lowest angular momenta, /i=2,... 

Figure 5. Matching between HFR MO functions of OH and their one-centre one-electron asymptotics from Eqs. 17 and 18 outer and inner contours of the wave functions in the figure s plane enclosing the molecular axis are differing by the factor 2 contours with a S5unmetry plane enclosing the axis are for Ik, contours with a symmetry plane orthogonal to the axis are for 3a solid and dashed contoius are differing in sigp.

An alternative to the MO method for the quantum mechanical treatment of molecular systems is the so-called Valence-Bond (VB) theory where molecular wavef unctions Eire obtained as linear combinations of covalent and ionic structures. It was shown long ago 181> that for distances larger than equilibrium distances, VB approximate wave functions should be better than MO functions of the same level, and hence VB theory should find its most profitable application in the evaluation of potential surfaces and reaction paths. Although true in principle, this statement has little influence in practice this is mostly because VB theory has only recently been formulated in a nonempirical form 182-184) so that applications are only just beginning to appear. 

The simplest LCAO-MO function, in the form of equation 6.3, using the atomic orbitals of the hydrogen atom, is the linear combination, compare equation 6.1, using the Is function of Table 1.1... 

To carry out the full Hartree-Fock calculation for dihydrogen, we have to assemble the two-electron term in the molecular Hamiltonian over the Gaussian sets. For the jsto-3g) representation of the hydrogenic Is orbitals in the LCAO-MO function, this means the determination of the potential terms Vaaaa. Vabab, T abb, Vabbb and Vbhbb. with... 

Exercise 6.7. The SCF calculation of the energy of dihydrogen with bond length, orbital energies and LCAO-MO functions. 

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