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Additional info for Ab Initio Valence Calculations in Chemistry
18). 4) where the overlap matrix plays a vital role in the definition of the atomic orbitals. 4) will be discussed as an application of theorems derived in Chapter 9. 5 SUMMARY The unavoidably mathematical nature of the present chapter may have masked the essential steps in the solution of the independent-electron model for atomic systems. The following is a bald outline of the previous sections. (i) The model wavefunction is a single determinant of (doubly-occupied spatial) orbitals Ψ = detU^^^ (ii) ....
The notation ίεα means that the summation is to be carried out over the variables of those electrons associated with nucleus a. In this case neglect of the second term leads to a set of different atomic Hamiltonians H and associated a 56 The Molecular Orbital Schrödinger equations. and Valence Bond Methods If we solve each of the atomic Schrödinger equations by the RUF method we obtain a set of atomic orbitals for each atom. This suggests that an approximate solution of the molecular Schrödinger equation would be given by a determinant of atomic orbitals.
1 μ ι . 1 12 n 1 1* 2* n μ x (r ) ... 5) n again provided t h a t the functions μ, are a complete s e t in three dimensions. In the case of molecular wave functions the extra "spin" v a r i a b l e can be taken i n t o account by noting t h a t the two spin "functions" are a formal complete set in d i s c r e t e spin "space". Thus any function of the n s e t s of four variables ( x . , y . , z . , s . ) = x. xn) = . ^ . 1 12 n ( r 2 ) . . p . (rn) χ ζ ( s 1 ) C i ( s 2 ) · . 6) where the prime has been added to the symbols F and B to denote the expansion of the space to include spin.