Improved tetrahedron method for the Brillouin-zone integration applicable to response functions

We improve the linear tetrahedron method to overcome systematic errors due to overestimations (underestimations) in integrals for convex (concave) functions, respectively. Our method is applicable to various types of calculations such as the total energy, the charge (spin) density, response functions, and the phonon frequency, in contrast with the Blöchl correction, which is applicable to only the first two. We demonstrate the ability of our method by calculating phonons in MgB${}_2$ and fcc lithium.

Figure 1

Subcell division into six tetrahedra and numbering of the tetrahedron corners; the red lines in the rightmost tetrahedron are the edges of the subcell.

Figure 2

Two kinds of approximations of the matrix element. True and approximated matrix elements $A_T$ are depicted as black dash-doted lines and red solid lines, respectively; the black points indicate the matrix elements $A_{\mathbf{k}}$ for a given value of $\mathbf{k}$; the dashed lines indicate the boundaries of the tetrahedra. (a) The liner interpolated function $A_T^1$ is always smaller (larger) than the true function $A_T$ in the case of a convex (concave) function. (b) The leveled linear function is a better approximation of the true function.

Figure 3

Points for constructing a third-order interpolation function (parallel stereogram). Red points denote the corners of $T$. The blue and green points are explained in Table 1.

Figure 4

Left: The Mg-centered Wigner-Seits cell of MgB${}_2$. Green and purple spheres indicate Mg and B atoms respectively. The $\sigma$ orbital (blue and red isosurfaces of opposite signs) and the displacement pattern of the intralayer vibrational mode of the B atoms with wave number $\mathbf{q}$ at the $A$ point (arrows) are also depicted. Right: Schematic illustration of the Fermi surfaces of the $\sigma$ bands; the red arrow indicates the momentum vector of a phonon at the $A$ point which connects occupied (O) and unoccupied (U) regions in the vicinity of the Fermi surface.

Figure 5

Left: $\mathbf{k}$ convergences of the frequencies of the intralayer vibrational mode of the B atoms at the $A$ point in the BZ for MgB${}_2$ (top) and the lower transverse acoustic mode at the $K$ point in the BZ for fcc Li at 20 GPa (bottom) with a different $\mathbf{k}$ integration method. The upward-pointing and downward-pointing triangles with red and green solid lines are the results of the linear and improved tetrahedron methods; the plus signs, multiplication signs, squares, and diamonds with blue, purple, cyan, and black dashed lines denote the results of the broadening method for widths of 0.01, 0.03, 0.05, and 0.06 Ry, respectively. Lines are guides for the eyes. Right: The frequency of these modes converged about the number of $\mathbf{k}$ at each broadening width (circles with orange line); the green solid lines indicate the converged value obtained by our method.

Figure 6

The $\mathbf{k}$ convergences of $\lambda$ (top), ${\omega }_{ln}$ (middle), and $T_C$ from McMillan's formula (bottom) of MgB${}_2$ (left) and fcc Li (right) calculated using ${\omega }_{\mathbf{q}\nu }$ and $\delta v_S/\delta R_{\tau \alpha }\left(\mathbf{q}\right)$ with different $\mathbf{k}$ integration schemes; the upward-pointing and downward-pointing triangles with red and green solid lines are the results of the linear and improved tetrahedron methods; the diamonds, plus signs, multiplication signs, squares, and circles with gray, blue, purple, black, and cyan dashed lines denote the results of broadening methods of widths 0.01, 0.02, 0.03, 0.04, and 0.05 Ry respectively. Lines are guides for the eyes.

Figure 7

Flow of the calculation of weights. We divide the tetrahedra two times to cut out regions where two Heaviside functions become one.

Figure 8

How to divide a tetrahedron in the case of ${\epsilon }_1\le 0<{\epsilon }_2$ (left), ${\epsilon }_2\le 0<{\epsilon }_3$ (center), and ${\epsilon }_3\le 0<{\epsilon }_4$ (right).