Projection

Chemical projection

There may be several possible schemes for the chemical projection.

E1

\begin{align} w_{\mathbf{k}, \textrm{XX}'} &= \left( \hat{P}^{\mathbf{k}} \hat{P}^{\textrm{X}} \tilde{\mathbf{v}} \right)^{\dagger} \left( \hat{P}^{\mathbf{k}} \hat{P}^{\textrm{X}'} \tilde{\mathbf{v}} \right) \end{align}

Generally this scheme causes cross terms, which can be negative.

E2

\begin{gather} w_{\mathbf{k}} = \left| \hat{P}^{\mathbf{k}} \tilde{\mathbf{v}} \right|^{2} \\ w_{\textrm{X}} = \left| \hat{P}^{\textrm{X}} \tilde{\mathbf{v}} \right|^{2} \\ w_{\mathbf{k}, \textrm{X}} \equiv w_{\mathbf{k}} w_{\textrm{X}} \end{gather}

This scheme does not cause cross terms. Instead, \(w_{\mathbf{k}, \textrm{X}'}/w_{\mathbf{k}, \textrm{X}}\) does no longer depend on \(\mathbf{k}\).