Linear Variation Theory

Computer-assisted variation theory is especially powerful when there are a large number of variables in the trial function. One of the common ways for this to occur is to assume that the trial function f i is a linear combination of a set of known functions called a basis set  

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There turns out to be a way to determine not only the energy but also the coefficients. This powerful use of variation theory is called linear variation theory. 

This form of variation theory is also best illustrated by example. Although the same idea can be applied to a trial wavefunction having any number of terms, a simple example involves the use of a two-term linear combination for the trial wavefunction  

In this example, the basis set j is composed of the two basis functions and 

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Standard linear variation theory applied to the KS Hamiltonian Eq. (2) yields a generalized eigenvalue problem for the eigenvalues e,- k and coefficients C ... 

The simple and extended Hiickel methods are not rigorous variational calculations. Although they both make use of the secular determinant technique from linear variation theory, no hamiltonian operators are ever written out explicitly and the integrations in Hij are not performed. These are semiempirical methods because they combine the theoretical form with parameters fitted from experimental data. 

The above determinant is called a secular determinant. Linear variation theory rests on equation 12.31 if the secular determinant formed from the energy and overlap integrals and the energy eigenvalues (which are the unknowns ) is equal to zero, then the equations 12.30 will be satisfied and the energy will be minimized. 

Consider what happens when a molecule is formed Two (or more) atoms combine to make a molecular system. The individual orbitals of the separate atoms combine to make orbitals that span the entire molecule. Why not use this description as a basis for defining molecular orbitals This is exactly what is done. By using linear variation theory, one can take linear combinations of occupied atomic orbitals and mathematically construct molecular orbitals. This defines the linear combination of atomic orbitals—molecular orbitals (LCAO-MO) theory, sometimes referred to simply as molecular orbital theory. 

Just as in linear variation theory, the coefficients can be determined using a secular determinant. But unlike the earlier examples using secular determinants, in this case some of the integrals are not identically zero or 1 due to orthonormality. In cases where there is an integral in terms of h(i) h(2) or vice versa, we cannot assume that the integral is identically zero. This is because the wavefunctions are centered on different atoms. The orthonormality conditions to this point are only strictly... 

Note the subscripts on each of the " P s.) We substitute the following definitions from linear variation theory into the above equation ... 

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