calculation of $Z$

Calculation of matrix elementns

$$⟨E_{\mu }^{𝒒}{\Psi }_{𝒌n}|{\Psi }_{𝒒+𝒌n^{\prime }}⟩=\sum _I{z}_{\mu I}^{𝒒\ast }⟨M_I^{𝒒}{\Psi }_{𝒌n}|{\Psi }_{𝒒+𝒌n^{\prime }}⟩$$

INFO

Regarding $|E_{\mu }^{𝒒}⟩$, it is a linear combination of $|M_I^{𝒒}⟩$, so it is not separated into inside or outside of MT for a certain $\mu$.

For the case where MPB is a function within MT:

$$⟨M_I^{𝒒}{\Psi }_{𝒌n}|{\Psi }_{𝒒+𝒌n^{\prime }}⟩=\sum _{𝑹u{u}^{\prime }}{\alpha }_{𝑹u}^{𝒌n\ast }{\alpha }_{𝑹u^{\prime }}^{𝒒+𝒌n^{\prime }}⟨M_I^{𝒒}{\phi }_{𝑹u}^{𝒌}|{\phi }_{𝑹u^{\prime }}^{𝒒+𝒌}⟩$$

$⟨M_I^{𝒒}{\phi }_{𝑹u}^{𝒌}|{\phi }_{𝑹u^{\prime }}^{𝒒+𝒌}⟩$ : variable name PPB

In the case of IPW:

$$⟨M_{𝑮}^{𝒒}{\Psi }_{𝒌n}|{\Psi }_{𝒒+𝒌n^{\prime }}⟩=\sum _{{𝑮}_1{𝑮}_2}{\beta }_{{𝑮}_1}^{𝒌n\ast }{\beta }_{{𝑮}_2}^{𝒒+𝒌n^{\prime }}{⟨M_{𝑮}^{𝒒}{P}_{{𝑮}_1}^{𝒌}|P_{{𝑮}_2}^{𝒒+𝒌}⟩}_{\mathrm{I}}$$

inside the MT

IPW part

The calculation for the region outside MT is obtained by calculating the MT part and subtracting it:

$$\begin{array}{rl}{⟨M_{𝑮}^{𝒒}{P}_{{𝑮}_1}^{𝒌}|P_{{𝑮}_2}^{𝒒+𝒌}⟩}_{\mathrm{I}}& ={\int }_{\mathrm{I}}d𝒓e^{i(-𝑮-{𝑮}_1+{𝑮}_2)\cdot 𝒓}\\ & ={\delta }_{-𝑮-{𝑮}_1+{𝑮}_2}-\sum _{𝑹}{\int }_{\mathrm{MT}}d{𝒓}_I{e}^{i(-𝑮-{𝑮}_1+{𝑮}_2)\cdot ({𝒓}_I+𝑹)}\\ & \equiv {𝒪}_{𝑮+{𝑮}_1,{𝑮}_2}\end{array}$$

It should be noted (there might be a Fourier transform factor). $𝒓=𝑹+{𝒓}_I$. The integral of $𝒓={𝒓}_I$ is written in spherical coordinates because the integration region is a sphere:

$$\begin{array}{r}{\int }_{\mathrm{MT}}d{𝒓}_I{e}^{i(-𝑮-{𝑮}_1+{𝑮}_2)\cdot ({𝒓}_I+𝑹)}=\frac{4\pi e^{i(-𝑮-{𝑮}_1+{𝑮}_2)\cdot 𝑹}}{|-𝑮-{𝑮}_1+{𝑮}_2|}{\int }_0^{R}d{r}_I{r}_I\sin (|-𝑮-{𝑮}_1+{𝑮}_2|r_I)\end{array}$$

INFO

It is also acceptable to expand the plane wave with spherical Bessel functions and consider only the contribution of $l=0$ as the result of the integration.

Total

$$\begin{array}{rl}⟨E_{\mu }^{𝒒}{\Psi }_{𝒌n}|{\Psi }_{𝒒+𝒌n^{\prime }}⟩& =\sum _{I\in \mathrm{MT}}{z}_{\mu I}^{𝒒\ast }\sum _{𝑹u{u}^{\prime }}{\alpha }_{𝑹u}^{𝒌n\ast }{\alpha }_{𝑹u^{\prime }}^{𝒒+𝒌n^{\prime }}⟨M_I^{𝒒}{\phi }_{𝑹u}^{𝒌}|{\phi }_{𝑹u^{\prime }}^{𝒒+𝒌}⟩\\ & +\sum _{𝑮}{z}_{\mu 𝑮}^{𝒒\ast }\sum _{{𝑮}_1{𝑮}_2}{𝒪}_{𝑮+{𝑮}_1,{𝑮}_2}{\beta }_{{𝑮}_1}^{𝒌n\ast }{\beta }_{{𝑮}_2}^{𝒒+𝒌n^{\prime }}\end{array}$$