X-Ray Absorption Spectroscopy (XAS)

Introduction

The pseudo wave-functions are solutions to this generalized eigenvalue problem:

\[H \tilde{\psi}_n = \epsilon_n S \tilde{\psi}_n.\]

This can be transformed into a standard eigenvalue problem:

\[S^{-1/2} H S^{-1/2} \psi_n = \epsilon_n \psi_n,\]

where \(\psi_n = S^{1/2} \tilde{\psi}_n\) is an all-electron wave function.

XAS cross section

For the cross section, we need this quantity:

\[\langle \psi_n | x | \phi^a \rangle = \sum_i \langle \tilde{\psi}_n | \tilde{p}_i^a \rangle \langle \phi_i^a | x | \phi^a \rangle = \langle \tilde{\psi}_n | \tilde{\phi}^a \rangle,\]

where \(\phi^a\) is the core state localized on atom \(a\) and \(\tilde{\phi}^a = \sum_i \langle \phi_i^a | x | \phi^a \rangle \tilde{p}_i^a\). Now, the cross section is:

\[\sum_n |\langle \tilde{\psi}_n | \tilde{\phi}^a \rangle|^2 \delta(\epsilon_n - E) = \sum_n \langle \tilde{\phi}^a | S^{-1/2} | \psi_n \rangle \delta(\epsilon_n - E) \langle \psi_n | S^{-1/2} | \tilde{\phi}^a \rangle.\]

By introducing \(G(E) = (E - S^{-1/2} H S^{-1/2} + i \gamma)^{-1}\), we get:

\[\text{Im}[\langle S^{-1/2} \tilde{\phi}^a | G(E) | S^{-1/2} \tilde{\phi}^a \rangle].\]

Recursion method

Instead of working with the \(u_i\) functions from the Taillefumier paper, we introduce \(w_i=S^{1/2}u_i\) which are the actual functions that we need to find. We now define \(y_i\) and \(z_i\) as:

\[w_i = S z_i,\]
\[y_i = H z_i.\]

With these definitions, the recursion formula reads:

\[y_i = a_i w_i + b_{i+1} w_{i+1} + b_i w_{i-1},\]

where:

\[a_i = \langle z_i | y_i \rangle,\]

and

\[b_i = \langle z_i | y_{i-1} \rangle = \langle z_{i-1} | y_i \rangle.\]

The \(w_i\) functions should be normalized as:

\[\langle w_i | S^{-1} | w_i \rangle = \langle w_i | z_i \rangle = 1,\]

and the recursion is started with \(w_0 \propto \tilde{\phi}^a\).

Inverting the S matrix

The S (or O) operator is defined as:

\[\hat O = 1 + \sum_a \sum_{i_1 i_2} |\tilde p^a_{i_1}> O^a_{i_1 i_2}< \tilde p^q_{i_2}|\]

Where \(O^a_{i_1 i_2} = <\phi ^a_{i_1}| \phi ^a_{i_2}> - <\tilde \phi ^a_{i_1}| \tilde \phi ^a_{i_2}>\)

Assume that \(\hat O^{-1}\) can be written as

\[\hat O^{-1} = 1 + \sum_a \sum_{i_1 i_2} |\tilde p^a_{i_1}> P^a_{i_1 i_2}< \tilde p^a_{i_2}|\]

Then according to [P.J. Hasnip et al, Comp. Phys. Comm. 174 (2006) 24-29 ] the coefficients \(P^a_{i_1 i_2}\) are given by

\[P^a_{i_1 i_2} = -O^a_{i_1 j} ( 1 + B^a_{kl} O^a_{lm} )^{-1}_{j i_2}\]
\[B^a_{kl} = < \tilde p^a_{k}| \tilde p^a_{l}>\]

With summation over equal indices (except a). These formulas ignore overlap between projectors on different atoms. The accuracy of the \(\hat O^{-1}\) operator can be checked for example by doing:

\[<\tilde \phi_{i_1}| \hat O \hat O^{-1} \hat O |\tilde \phi_{i_2}> - \delta_{i_1 i_2}\]

which should be zero for all normalized, orthogonalized \(\tilde \phi\)