Origin of the basis set

The spherical-wave basis functions[13] used in this work are eigenfunctions of the Helmholtz equation

$$(\nabla^2 + q^2) \chi({\bf r}) = 0,$$ (1)

subject to boundary conditions such that the solutions $\chi({\bf r})$ are nonvanishing only inside a sphere of radius $R$ and vanishing whenever $\vert{\bf r}\vert \ge R$. The eigenfunctions are

$$\chi(r,\theta,\phi) = \left\{ \begin{array}{ll} j_{\ell}(q_{... ...,\phi), & r < R, \nonumber\\ 0, & r \ge R, \end{array}\right.$$

where $(r,\theta,\phi)$ are spherical polar coordinates with origin at the center of the sphere, $\ell$ is a non-negative integer and $m$ is an integer satisfying $-\ell \le m \le \ell$. $j_{\ell}(x)$ is the spherical Bessel function of order $\ell$, and $Y_{\ell m}(\theta,\phi)$ is a spherical harmonic. The eigenvalue $q_{n \ell}$ is determined from the $n$th zero of $j_{\ell}(x)$ where

$$j_{\ell}(q_{n \ell} R) = 0.$$ (2)

We note that each eigenfunction in Eq. (2) has an energy of $\hbar^2 q_{n \ell}^2/(2m_{\mathrm{e}})$, hence it is possible to use the concept of cutoff energy to restrict the number of $q_{n \ell}$ in the expansion of a wavefunction.

The real spherical-wave basis functions used in this work are

$$\chi_{\alpha,n \ell m}({\bf r}) = \left\{ \begin{array}{ll} ... ...f r}-{\bf R}_{\alpha}\vert \ge r_{\alpha} , \end{array}\right.$$

where $\alpha$ signifies a basis sphere with radius $r_{\alpha}$ and centered at ${\bf R}_{\alpha}$. $\overline{Y}_{\ell m} (\theta,\phi)$ are the real linear combinations of the spherical harmonics. By construction, all basis functions within a basis sphere are orthogonal to one another. In general, more than one basis sphere is needed to expand a wavefunction

$$\psi({\bf r}) = \sum_{\alpha, n \ell m} c_{\alpha, n \ell m} \chi_{\alpha, n \ell m}({\bf r}),$$ (3)

where $c_{\alpha, n \ell m}$ are the associated coefficients. For most systems tested in this work, we have used one basis sphere per atom, where the basis spheres are centered on the atoms. For some systems we have increased the number of basis spheres by placing basis spheres between the atoms. In principle it is possible to use two or more basis spheres of different radii centered on the same atom, but this arrangement has not been studied in this work. We note that even though the basis functions belonging to different basis spheres are generally nonorthogonal, one of the main advantages of this basis set is that it is possible to analytically evaluate[13] the overlap matrix elements

$$S_{\alpha, n \ell m; \beta, n' \ell' m'} = \int d{\bf r}\ \c... ...\alpha, n \ell m}({\bf r}) \chi_{\beta, n' \ell' m'}({\bf r}),$$ (4)

and kinetic energy matrix elements

$$T_{\alpha, n \ell m; \beta, n' \ell' m'} = -\frac{\hbar^2}{2... ... \ell m}({\bf r}) \nabla^2 \chi_{\beta, n' \ell' m'}({\bf r}).$$ (5)

We also note that the matrix elements for the nonlocal pseudopotentials in the Kleinman-Bylander[19] form can also be evaluated analytically by first expanding the projectors in the spherical-wave basis set.