By D. B. Cook
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Additional resources for Ab Initio Valence Calculations in Chemistry
It is quite easy to show that the set of all possible products ^ ( Γ , ) Ρ « ( Γ 2 ) form a complete set for the expansion of any continuous, normalisable function of six dimensions. M. Quinton "Contemporary British Philosophy11 in "A Critical History of Western Philosophy11, ed. J. R) k where the coefficients C, contain the information in the B, for the particular value r = R chosen. 1) and so is valid. Considering the variable r. as fixed gives a similar valid expansion for the second variable r_.
We therefore write each MO ψ. s d>. Φί = I Τηφ. 1) or, collecting the MO1s as a row vector ψ , Ψ =ψT The elements T.. of the matrix X are then optimised using the variation principle, and the optimum matrix defines the structure of the molecular orbitals in terms of the atomic orbitals. The atomic orbitals ψ are expanded in terms of a set of basis functions X , and the MO1s in turn expand as linear combinations of the AO f s. 3) where the matrix M contains the coefficients relating each MO to the atomic basis functions.
We can make Φ a l i t t l e less forbidding by noting that i t i s a determinant whose rows are defined by the λ. and whose columns are enumerated by the x. 4) . . 4) i s just the h i s t o r i c a l l y established notation for ( 3 . 3 . 3 ) . Note that i f any λ. appears more than once the function Φ i s zero. Within the orbital model then, the Pauli principle reduces to "spin-orbitals can be, at most, singly occupied" or "spatial orbitals can contain either one electron, or two electrons with opposite spin".