I am working on a code to preform a HF procedure and I am a bit confused about the unitary transformation of the basis. My instructions are as follows.
- specify the basis and geometry (done)
- Evaluate and store overlap, Kinetic energy, nuclear attraction, and electron repulsion integrals (done)
- diagonalize $ \mathbf{S}$ and find $\mathbf{S}^{-1/2}$ which is the unitary transformation matrix.
- Use the core hamiltonian to find the initial Fock matrix: $$ \mathbf{S^{-1/2}}^{T} \mathbf{H}~\mathbf{S^{-1/2}}\equiv\mathbf{F}_0$$
- Diagonalize the Fock matrix to find the mo coefficients: $$\mathbf{F}_0 \mathbf{C}^{'}_0=\mathbf{C}^{'}_0\epsilon_0$$
- Back transform the coefficients. $$\mathbf{C^{'}_0}~\mathbf{S^{-1/2}}=\mathbf{C}_0$$
- compute the density, use the density to evaluate new fock matrix, find new coefficients repeat until convergeance.
Now step 7 kinds glazes over a lot of the work but I understand that part, so It isn't important to the question.
I am confused by the $\mathbf{S}^{-1/2}$ matrix. Is this literally one over the square root of each element of $\mathbf{S}$? Or is there some decomposition I should preform that I just do not understand. I can carry out the whole procedure but without orthonormalization my results are nonsense.