Appendix

Inverse interpolation

We consider an integration as follows:

\[\begin{align} \langle X \rangle = \sum_{k} X_k w(\varepsilon_k) \end{align}\]

If this integration has conditions that

  • \(w(\varepsilon_k)\) is sensitive to \(\varepsilon_k\) (e. g. the stepfunction, the delta function, etc.) and requires \(\varepsilon_k\) on a dense \(k\) grid, and
  • the numerical cost to obtain \(X_k\) is much larger than the cost for \(\varepsilon_k\) (e. g. the polarization function),

it is efficient to interpolate \(X_k\) into a denser \(k\) grid and evaluate that integration in a dense \(k\) grid. This method is performed as follows:

  1. Calculate \(\varepsilon_k\) on a dense \(k\) grid.
  2. Calculate \(X_k\) on a coarse \(k\) grid and obtain that on a dense \(k\) grid by using the linear interpolation, the polynomial interpolation, the spline interpolation, etc.
\[\begin{align} X_k^{\rm dense} = \sum_{k'}^{\rm coarse} F_{k k'} X_{k'}^{\rm coarse} \end{align}\]
  1. Evaluate that integration in the dense \(k\) grid.
\[\begin{align} \langle X \rangle = \sum_{k}^{\rm dense} X_k^{\rm dense} w_k^{\rm dense} \end{align}\]

The inverse interpolation method (Appendix of [2]) arrows as to obtain the same result to above without interpolating \(X_k\) into a dense \(k\) grid. In this method, we map the integration weight on a dense \(k\) grid into that on a coarse \(k\) grid (inverse interpolation). Therefore, if we require

\[\begin{align} \sum_k^{\rm dense} X_k^{\rm dense} w_k^{\rm dense} = \sum_k^{\rm coarse} X_k^{\rm coarse} w_k^{\rm coarse} \end{align}\]

we obtain

\[\begin{align} w_k^{\rm coarse} = \sum_k^{\rm dense} F_{k' k} w_{k'}^{\rm dense} \end{align}\]

The numerical procedure for this method is as follows:

  1. Calculate the integration weight on a dense \(k\) grid \(w_k^{\rm dense}\) from \(\varepsilon_k\) on a dense \(k\) grid.
  2. Obtain the integration weight on a coarse \(k\) grid \(w_k^{\rm coarse}\) by using the inverse interpolation method.
  3. Evaluate that integration in a coarse \(k\) grid where \(X_k\) was calculated.

All routines in libtetrabz can perform the inverse interpolation method; if we make \(k\) grids for the orbital energy (nge) and the integration weight (ngw) different, we obtain \(w_k^{\rm coarse}\) calculated by using the inverse interpolation method.

Double delta integration

For the integration

\[\begin{align} \sum_{n n' k} \delta(\varepsilon_{\rm F} - \varepsilon_{n k}) \delta(\varepsilon_{\rm F} - \varepsilon'_{n' k}) X_{n n' k} \end{align}\]

first, we cut out one or two triangles where \(\varepsilon_{n k} = \varepsilon_{\rm F}\) from a tetrahedron and evaluate \(\varepsilon_{n' k+q}\) at the corners of each triangles as

\[\begin{align} \varepsilon'^{k+q}_{i} = \sum_{j=1}^4 F_{i j}( \varepsilon_1^{k}, \cdots, \varepsilon_{4}^{k}, \varepsilon_{\rm F}) \epsilon_{j}^{k+q}. \end{align}\]

Then we calculate \(\delta(\varepsilon_{n' k+q} - \varepsilon{\rm F})\) in each triangles and obtain weights of corners. This weights of corners are mapped into those of corners of the original tetrahedron as

\[\begin{align} W_{i} = \sum_{j=1}^3 \frac{S}{\nabla_k \varepsilon_k}F_{j i}( \varepsilon_{1}^k, \cdots, \varepsilon_{4}^k, \varepsilon_{\rm F}) W'_{j}. \end{align}\]

\(F_{i j}\) and \(\frac{S}{\nabla_k \varepsilon_k}\) are calculated as follows (\(a_{i j} \equiv (\varepsilon_i - \varepsilon_j)/(\varepsilon_{\rm F} - \varepsilon_j)\)):

_images/dbldelta.png

How to divide a tetrahedron in the case of \(\epsilon_1 \leq \varepsilon_{\rm F} \leq \varepsilon_2\) (a), \(\varepsilon_2 \leq \varepsilon_{\rm F} \leq \varepsilon_3\) (b), and \(\varepsilon_3 \leq \varepsilon_{\rm F} \leq \varepsilon_4\) (c).

  • When \(\varepsilon_1 \leq \varepsilon_{\rm F} \leq \varepsilon_2 \leq \varepsilon_3 \leq\varepsilon_4\) [Fig. 1 (a)],

    \[\begin{split}\begin{align} F &= \begin{pmatrix} a_{1 2} & a_{2 1} & 0 & 0 \\ a_{1 3} & 0 & a_{3 1} & 0 \\ a_{1 4} & 0 & 0 & a_{4 1} \end{pmatrix}, \qquad \frac{S}{\nabla_k \varepsilon_k} = \frac{3 a_{2 1} a_{3 1} a_{4 1}}{\varepsilon_{\rm F} - \varepsilon_1} \end{align}\end{split}\]
  • When \(\varepsilon_1 \leq \varepsilon_2 \leq \varepsilon_{\rm F} \leq \varepsilon_3 \leq\varepsilon_4\) [Fig. 1 (b)],

    \[\begin{split}\begin{align} F &= \begin{pmatrix} a_{1 3} & 0 & a_{3 1} & 0 \\ a_{1 4} & 0 & 0 & a_{4 1} \\ 0 & a_{2 4} & 0 & a_{4 2} \end{pmatrix}, \qquad \frac{S}{\nabla_k \varepsilon_k} = \frac{3 a_{3 1} a_{4 1} a_{2 4}}{\varepsilon_{\rm F} - \varepsilon_1} \end{align}\end{split}\]
    \[\begin{split}\begin{align} F &= \begin{pmatrix} a_{1 3} & 0 & a_{3 1} & 0 \\ 0 & a_{2 3} & a_{3 2} & 0 \\ 0 & a_{2 4} & 0 & a_{4 2} \end{pmatrix}, \qquad \frac{S}{\nabla_k \varepsilon_k} = \frac{3 a_{2 3} a_{3 1} a_{4 2}}{\varepsilon_{\rm F} - \varepsilon_1} \end{align}\end{split}\]
  • When \(\varepsilon_1 \leq \varepsilon_2 \leq \varepsilon_3 \leq \varepsilon_{\rm F} \leq \varepsilon_4\) [Fig. 1 (c)],

    \[\begin{split}\begin{align} F &= \begin{pmatrix} a_{1 4} & 0 & 0 & a_{4 1} \\ a_{1 3} & a_{2 4} & 0 & a_{4 2} \\ a_{1 2} & 0 & a_{3 4} & a_{4 3} \end{pmatrix}, \qquad \frac{S}{\nabla_k \varepsilon_k} = \frac{3 a_{1 4} a_{2 4} a_{3 4}}{\varepsilon_1 - \varepsilon_{\rm F}} \end{align}\end{split}\]

Weights on each corners of the triangle are computed as follows [(\(a'_{i j} \equiv (\varepsilon'_i - \varepsilon'_j)/(\varepsilon_{\rm F} - \varepsilon'_j)\))]:

  • When \(\varepsilon'_1 \leq \varepsilon_{\rm F} \leq \varepsilon'_2 \leq \varepsilon'_3\) [Fig. 1 (d)],

    \[\begin{align} W'_1 = L (a'_{1 2} + a'_{1 3}), \qquad W'_2 = L a'_{2 1}, \qquad W'_3 = L a'_{3 1}, \qquad L \equiv \frac{a'_{2 1} a'_{3 1}}{\varepsilon_{\rm F} - \varepsilon'_{1}} \end{align}\]
  • When \(\varepsilon'_1 \leq \varepsilon'_2 \leq \varepsilon_{\rm F} \leq \varepsilon'_3\) [Fig. 1 (e)],

    \[\begin{align} W'_1 = L a'_{1 3}, \qquad W'_2 = L a'_{2 3}, \qquad W'_3 = L (a'_{3 1} + a'_{3 2}), \qquad L \equiv \frac{a'_{1 3} a'_{2 3}}{\varepsilon'_{3} - \varepsilon_{\rm F}} \end{align}\]