spin_squared

sisl.physics.electron.spin_squared(state_alpha, state_beta, S=None)

Calculate the spin squared expectation value between two spin states

This calculation only makes sense for spin-polarized calculations.

The expectation value is calculated using the following formula:

\[\begin{split}S^2_{\alpha,i} &= \sum_j |\langle \psi_j^\beta | \mathbf S | \psi_i^\alpha \rangle|^2 \\ S^2_{\beta,j} &= \sum_i |\langle \psi_i^\alpha | \mathbf S | \psi_j^\beta \rangle|^2\end{split}\]

where \(\alpha\) and \(\beta\) are different spin-components.

The arrays \(S^2_\alpha\) and \(S^2_\beta\) are returned.

Parameters

• state_alpha (array_like) – vectors describing the electronic states of spin-channel \(\alpha\), 2nd dimension contains the states

• state_beta (array_like) – vectors describing the electronic states of spin-channel \(\beta\), 2nd dimension contains the states

• S (array_like, optional) – overlap matrix used in the \(\langle\psi|\mathbf S|\psi\rangle\) calculation. If None the identity matrix is assumed. The overlap matrix should correspond to the system and \(k\) point the eigenvectors have been evaluated at.

Notes

state_alpha and state_beta need not have the same number of states.

Returns

list of spin squared expectation value per state for spin state \(\alpha\) and \(\beta\)

Return type

oplist