# Normalization constant and Roothaan Equations

### Roothaan Hall Equations:

The Hartree-Fock equations are a set of modified Schrodinger equations:

$$f_{i}\psi_{m}=\epsilon_{m}\psi_{m}$$

where:

• The Fock operator ($$f_{i}$$) is given by (restricted case):

$$f_{i}= \hat{h}_{i}+\sum_{j=1}^{n/2}[2\hat{J}_{j}(i)-\hat{K}_{j}(i)]$$

• and the molecular orbitals are expressed as a linear combination of ($$N_{b}$$) atomic orbitals ($$\chi_{o}$$):

$$\psi_{m} =\sum_{o=1}^{N_{b}}c_{om}\chi_{o}$$

By substituting $$\psi_{m}$$, one obtains:

$$f_{i}\sum_{o=1}^{N_{b}}c_{om}\chi_{o}= \epsilon_{m}\sum_{o=1}^{N_{b}}c_{om}\chi_{o}$$

If one now multiplies from the left by $$\chi_{o'}$$, and integrates over the coordinates of particle i:

$$\sum_{o=1}^{N_{b}}c_{om}\int\chi_{o'}f_{i}\chi_{o}dr_{1}= \epsilon_{m}\sum_{o=1}^{N_{b}}c_{om}\int\chi_{o'}\chi_{o}dr_{1}$$

$$\sum_{o=1}^{N_{b}}F_{o'o}c_{om}= \epsilon_{m}\sum_{o=1}^{N_{b}}S_{o'o}c_{om}$$

where:

• $$F_{o'o}=\int\chi_{o'}f_{i}\chi_{o}dr_{1}$$

• $$S_{o'o}=\int\chi_{o'}\chi_{o}dr_{1}$$

The expression has the form of a relation between matrix elements of the product matrices FC and SC. If one introduces the diagonal matrix $$\epsilon$$ along the diagonal, the expression can be written as the matrix equality:

$$FC = SC\epsilon$$

### An Example:

To set up the Roothann equations for the HF molecule using the $$N_{b} = 2$$ basis set H1s ($$\chi_{a}$$) and F2p$$_{z}$$ ($$\chi_{a}$$) one can write the two molecular orbitals (m = a, b) as:

$$\psi_{a}=c_{Aa}\chi_{A} + c_{Ba}\chi_{B}$$

$$\psi_{b}=c_{Ab}\chi_{A} + c_{Bb}\chi_{B}$$

The following matrices are obtained:

$$F = \begin{bmatrix}F_{A}(A)&F_{A}(B)\\ F_{B}(A)&F_{B}(B)\end{bmatrix}$$

$$S = \begin{bmatrix} 1 & S \\ S & 1 \end{bmatrix}$$

$$C = \begin{bmatrix} c_{Aa} & c_{Ab} \\ c_{Ba} & c_{Bb} \end{bmatrix}$$

Then the Roothan equations ($$FC=SC\epsilon$$) are:

$$\begin{bmatrix}F_{AA}&F_{AB}\\ F_{BA}&F_{BB}\end{bmatrix} \begin{bmatrix} c_{Aa} & c_{Ab} \\ c_{Ba} & c_{Bb} \end{bmatrix} = \begin{bmatrix} 1 & S \\ S & 1 \end{bmatrix} \begin{bmatrix} c_{Aa} & c_{Ab} \\ c_{Ba} & c_{Bb} \end{bmatrix} \begin{bmatrix} \epsilon_{a} & 0 \\ 0 & \epsilon_{b} \end{bmatrix}$$

### Question:

In many textbooks and lectures, the Roothan equations are often described after an introduction to the Slater determinant.

If the total wavefunction from the HF example can be written in the form of a Slater determinant:

$$\Psi = \frac{1}{\sqrt{N!}}\begin{bmatrix} \psi_{a}(i)&\psi_{b}(i)\\ \psi_{a}(j)&\psi_{b}(j)\end{bmatrix}$$

How is the normalization constant used in the Slater determinant built into these equations?

Note: The derivation has come from Atkins' Physical Chemistry 9th Edition