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The LCAO approximation.

To get the orbitals that give a Slater determinant with the lowest energy we have to solve the Fock equation. That is a partial differential equation, and these are generally extremely hard to solve. Fortunately, we can turn the Fock equation into a (generalized) matrix eigenvalue equation. Such equations are relatively easy to solve with standard techniques from numerical mathematics.

The way to turn the Fock equation into a matrix eigenvalue equation is to write the orbitals as a linear combination of know functions. So if the j_n's are the orbitals that we are trying to determine, and the c_m's are known functions, then we write

j_n =

å
m

c_mc_mn.

Because we know the c's, the unknown quantities on the right-hand-side are the coefficients c_mn. If we substitute the linear combination in the Fock equation

Fj_n = e_nj_n

we get

å
m

c_mc_mn = e_n

å
m

c_mc_mn.

We convert this into a matrix eigenvalue equation for the coefficients by multiplying from the left by c_k.

å
m

c_kFc_mc_mn =

å
m

c_kc_mc_mne_n

and integrating over the electron coordinates

å
m

ác_k|F|c_mñc_mn =

å
m

ác_k|c_mñc_mne_n.

This is the desired from, although it might be a bit difficult to see that it is a matrix eigenvalue equation. We define the Fock matrix F

F_km = ác_k|F|c_mñ,

the overlap matrix S

S_km = ác_k|c_mñ,

and the vector c⁽ⁿ⁾

(c⁽ⁿ⁾)_m = c_mn.

The equation above now becomes

å
m

F_km(c⁽ⁿ⁾)_m =

å
m

S_km(c⁽ⁿ⁾)_me_n,

or in matrix-vector notation

Fc⁽ⁿ⁾ = Sc⁽ⁿ⁾e_n.

(Strictly speaking this is a generalized matrix eigenvalue equation, because of the presence of the matrix S, but this does not really change the nature of the equation.) This eigenvalue equation is often called the Roothaan equation, and it's in this form that the Fock equation is generally solved.

When one is trying to determine the MO's of a molecule one usually takes as known functions c atomic orbitals of the atoms forming the molecule. For example, in HF there is a molecular orbital that is formed from the 1s orbital of H and the 2p_z orbital of F. (It is a convention to take the molecular axis of a linear molecule as the z axis.) This looks as follows.

Because the atomic orbitals are put in the linear combination the procedure is called the Linear Combination of Atomic Orbitals (LCAO) approximation. It is an approximation because it is not know in advance which and how many atomic orbitals to use. By taking more atomic orbitals in the linear combination the approximation can be made better. Nowadays it can be made very good.