Product basis set in the 𝘎𝘞 calculation

Product basis $M$

The basis set to represent product of one-particle wave functions. This is the way to reduce the dimension of product.

$$|M_I^{𝒌}⟩=\{|{\phi }_{𝑹u}^{{𝒌}_1\ast }{\phi }_{𝑹u^{\prime }}^{{𝒌}_2}⟩,|P_{{𝑮}_1}^{{𝒌}_1\ast }{P}_{{𝑮}_2}^{{𝒌}_2}⟩\}\equiv \{{|M_I^{𝒌}⟩}_{\mathrm{MT}},{|M_{𝑮}^{𝒌}⟩}_{\mathrm{IPW}}\}$$

,where $𝒌={𝒌}_2-{𝒌}_1,𝑮={𝑮}_2-{𝑮}_1$. Some of pairs of (${𝒌}_1$,${𝒌}_2$) or (${𝑮}_1$,${𝑮}_2$) which has the same $𝒌$ or $𝑮$ are got rid of in the set of $M$.

WARNING

In this notation, $|{\phi }_{𝑹u}^{{𝒌}_1\ast }{\phi }_{𝑹u^{\prime }}^{{𝒌}_2}⟩$ DOES NOT indicate the direct product between $|{\phi }_{𝑹u}^{{𝒌}_1\ast }⟩$ and $|{\phi }_{𝑹u^{\prime }}^{{𝒌}_2}⟩$. $⟨𝒓|{\phi }_{𝑹u}^{{𝒌}_1\ast }{\phi }_{𝑹u^{\prime }}^{{𝒌}_2}⟩={\phi }_{𝑹u}^{{𝒌}_1\ast }(𝒓){\phi }_{𝑹u^{\prime }}^{{𝒌}_2}(𝒓)$

Product basis $E$

The MPB $M$ introduced above does not satisfy orthogonality. Therefore, we introduce an orthogonal basis. This basis also diagonalizes the Coulomb matrix. By doing so, the calculation of the exchange self-energy becomes easier. In equations, new product basis $E$ is represent by the liner combination of $M$, i.e.:

$$|E_{\mu }^{𝒒}⟩=\sum _I{z}_{\mu I}^{𝒒}|M_I^{𝒒}⟩.$$

Then, $E$ is satisfy the following releations.

$$⟨E_{\mu }^{𝒒}|E_{\nu }^{𝒒}⟩={\delta }_{\mu \nu },v(𝒒)|E_{\mu }^{𝒒}⟩=v_{\mu }(𝒒)|E_{\mu }^{𝒒}⟩$$

where $v(𝒒)$ is Coulomb matrix and $v_{\mu }(𝒒)$ is eigen value. The coefficient $z_{\mu I}^{𝒒}$ and $v_{\mu }(𝒒)$ are obtained by soliving following generalized eigenvalue equation:

$$\sum _J(v_{IJ}^{𝒒}-v_{\mu }(𝒒)O_{IJ}^{𝒒})z_{\mu J}^{𝒒}=0,$$

where $v_{IJ}^{𝒒}$ is Coulomb matrix represented by $M$, namely, $⟨M_I^{𝒒}|v|M_J^{𝒒}⟩$. By using $E$, Coulomb interaction operator is represented as follows:

$$v(𝒒)=\sum _{\mu }|E_{\mu }^{𝒒}⟩v_{\mu }(𝒒)⟨E_{\mu }^{𝒒}|.$$

about $⟨M_I^{𝒒}|v|M_J^{𝒒}⟩$

Since $v$ is a non-local function, the calculation of this matrix element includes cross terms of $M_{\text{MT}}$ and $M_{\text{IPW}}$.