How to do Lowdin symmetric orthonormalisation? Löwdin symmetric orthonormalisation seems to be a common practice in quantum chemistry. I come from a different background though and have to understand it and possibly implement in a computer code. Here is the definition of the transformation;

$$
|{\phi'}\rangle = S^{-1/2}|\phi\rangle,
$$

where $S$ is the overlap matrix between the non-orthogonal basis,

$$
S = \left&lt; \phi|\phi \right&gt; .
$$


My first question is how to obtain $S^{-1/2}$ analytically?
Are there any parallel algorithms available to compute it numerically?
You obtain the diagonalization of $S$ as defined
$$
S = U \cdot s  \cdot U^{\dagger}
$$
(which holds for the case of symmetric $S$, which it is) and apply the power needed
$$
S^{-\frac{1}{2}} = U \cdot s^{-\frac{1}{2}}  \cdot U^{\dagger}
$$

For more background, try the application section of Wikipedia's article on diagonalizable matrices.
As far as understanding it goes, I'm afraid not much beyond doing some model calculations on 2D or 3D vectors can be done.

I do not have a strong background in parallel computing, but all operations involved are well-known and your linear algebra framework (BLAS, anyone?) will document how to run any of this in parallel and what storage may be needed for it. Note that, typically, diagonalization is not the most expensive step of a self-consistent field calculation (Hartree-Fock or DFT) anyway, so parallelization may not be critical for this step.