Linear Scaling for Metallic Systems by the Korringa-Kohn-Rostoker Multiple-Scattering Method

Abstract

A Green function (GF) linear-scaling technique based on the Korringa-Kohn-Rostoker (KKR) multiple scattering method is presented for Hohenberg-Kohn-Sham density functional calculations of metallic systems. Contrary to most other methods the KKR-GF method does not use a basis-set expansion to solve the Kohn-Sham equation for the wavefunctions, but directly determines the Kohn-Sham Green function by exploiting a reference system concept. An introduction to the KKR-GF method is given and it is shown how linear-scaling is obtained by the combined use of a repulsive reference system, which leads to sparse matrix equations, iterative solution of these equations and a spatial truncation of the Green function in the sense of Kohn’s principle of nearsightedness of electronic matter. The suitability of the technique for metallic systems with thousands of atoms is illustrated by model calculations for large supercells and its usefulness for computing on massively parallel supercomputers is discussed.

Appendix

By using Gegenbauer’s addition theorem [58] for Hankel functions the free space Green function can be written in an almost separable form as

$$G^0 ({\textbf r},\ {{\textbf r}'};\ E) = - \textrm{i} \sqrt E \sum_{l=0}^{\infty} \sum_{m=-l}^{m=l} h^{(1)}_l (r_> \sqrt E) j_l ( r_< \sqrt E) Y_{lm} ({\hat{\textbf r}}) Y_{lm} ({\hat{\textbf r}}') \: ,$$

((17-33))

where the individual terms are products of r dependent and r ^′ dependent functions. Here $Y_{lm},\ h^{(1)}_l$ and j _l are real spherical harmonics, spherical Hankel functions of the first kind and spherical Bessel functions. Expression (17-33) is in a separated form with respect to the angular variables ${\hat{\textbf r}} = {\textbf r} / r$ and ${\hat{\textbf r}}' = {{\textbf r}'} / r'$. For the radial variables separation is not complete because of the restriction $r_<\,{=}\,\min (r,r')$ and $r_>\,{=}\,\max (r,\ r')$ and thus (17-33) is called semi-separable. Often a shorthand notation is introduced by using a combined index $L=lm$ and by defining products of Hankel and Bessel functions with real spherical harmonics by $H_L ({\textbf r} ;\ E) = - \textrm{i} \sqrt E h^{(1)}_l (r \sqrt E) Y_{lm} ({\hat{\textbf r}})$ and $J_L ({\textbf r};\; E) = j_l (r \sqrt E) Y_{lm} ({\hat{\textbf r}})$. Then (17-33) can be written in a more compact form as

$$G^0 ({\textbf r},\ {{\textbf r}'};\ E) = \sum_L^{\infty} H_L ({{\textbf r}_>} ;\ E) J_L ( {{\textbf r}_<} ;\ E) \: ,$$

((17-34))

where r _> is the vector r or r ^′ with larger length and r _< the one with smaller length. The multiple-scattering expression (17-17) for the Green function in cell-centered coordinates is then obtained by using the addition theorem of spherical Hankel functions in the form

$$H_L({\textbf r} + {{\textbf R}^n} -{{\textbf R}^{n'}};\; E) = \sum_{L'}^{\infty} G^{0,nn'}_{LL'} (E) J_{L'}({\textbf r};\; E) \: ,$$

((17-35))

which is valid for $r < |{{\textbf R}^n} -{{\textbf R}^{n'}}|$, with the free space Green function matrix elements

$$G^{0,nn'}_{LL'} (E) = 4 \pi (1 - \delta_{nn'} ) \sum_{L''} \textrm{i}^{l-l'+l''} C_{LL'L''} H_{L''}({{\textbf R}^n} - {{\textbf R}^{n'}} ;\ E) \: .$$

((17-36))

Here δ _nn′ indicates that the free space Green function matrix elements vanish for ${{\textbf R}^n} = {{\textbf R}^{n'}}$ and $C_{LL'L''}$ are Gaunt coefficients defined as

$$C_{LL'L''} = \int_{4 \pi} Y_L({\hat{\textbf r}}) Y_{L'}({\hat{\textbf r}}) Y_{L''}({\hat{\textbf r}}) \textrm{d} {\hat{\textbf r}}$$

((17-37))

which vanish for $l'' > l + l'$ so that the sum in (17-36) contains a finite number of terms.

In the past confusion for the validity of the full-potential KKR-GF method was caused by the spatial restrictions necessary in (17-34) and (17-35). The restriction $r < |{{\textbf R}^n}-{{\textbf R}^{n'}}|$ necessary in (17-35) means that the distance from the center of a cell to the point farthest away on the boundary of this cell must be smaller than the distance between centers of adjacent cells. This restriction is not serious, it can always be satisfied if necessary by introducing additional cells not occupied by atoms (empty cells). The restriction for the arguments in Hankel and Bessel functions in (17-33) and (17-34) means that (17-35) can be applied directly only for $r > r'$, whereas for $r < r'$ it must be used for $H_L({{\textbf r}'} - {{\textbf R}^n} + {{\textbf R}^{n'}} ;\ E)$. This makes the double sum in (17-17) conditionally convergent and convergent results are only obtained if L and L ^′ are put to infinity in correct order. The spatial restriction has also imposed doubt on the validity of the Green function expression (17-18) for general potentials. It is however elementary to show [20, 25] that (17-18) with appropriately defined quantities directly follows from expression (17-17) if all sums are restricted to a finite number of terms. For practical calculations the question of convergence for high angular momentum contributions seems to be unimportant as the good agreement of calculated total energies and forces with those obtained by other density functional methods and experiment illustrates (see Section 17.3.4). Mathematically, convergence has been demonstrated for the so-called empty lattice with a constant non-zero potential [59] and more generally for the full potential by rather sophisticated techniques [20, 60].

The regular and irregular single-scattering wavefunctions $R^n_L({\textbf r};\; E)$ and $S^n_L({\textbf r};\; E)$ are defined by integral equations

$$R^n_L({\textbf r};\ E) = J^n_L({\textbf r};\ E) + \int_n G^0 ({\textbf r},{{\textbf r}'};\ E) v^n_{eff} ({{\textbf r}'}) R^n_L({{\textbf r}'};\ E) \textrm{d} {\textbf r}$$

((17-38))

and

$$S^n_L({\textbf r};\ E) = \sum_{L'} \beta_{LL'}^n (E) H_{L'} ({\textbf r};\ E) + \int_n G^0 ({\textbf r},{{\textbf r}'};\ E) v^n_{eff} ({{\textbf r}'}) S^n_L({{\textbf r}'};\ E) \textrm{d} {\textbf r} \: ,$$

((17-39))

where the matrix

$$\beta_{LL'}^n (E) = \delta_{LL'} - \int_n S^n_L ({\textbf r} ;\ E) v^n_{eff} ({\textbf r} ) J_{L'} ({\textbf r} ;\ E) d{\textbf r}$$

((17-40))

is defined implicitly by the irregular wavefunctions. The single-cell t matrix is defined by

$$t^n_{LL'} (E) = \int_n J_L({\textbf r};\ E) v^n_{eff} ({\textbf r} ) R^n_{L'} ({\textbf r};\ E) \textrm{d} {\textbf r} \: .$$

((17-41))

Details about the numerical treatment of the single-cell equations (17-38), (17-39), (17-40) and (17-41) can be found in [61, 62].

In periodic systems the atomic positions can be written as ${{\textbf R}^n} = {{\textbf R}^{\mu}} + {{\textbf s}^m}$ where R ^μ denotes lattice vectors and s ^m positions of the atoms in the unit cell. Then (17-19) has the form

$$G^{\mu m, \mu' m'}_{L L'} (E) = G^{r,\mu m, \mu' m'}_{L L'} (E) + \sum_{\mu'' m'' L''L'''}^{\infty} G^{r,\mu m, \mu'' m''}_{L L''}(E) \Delta t^{m''}_{L'' L'''} (E) \\ G^{\mu'' m'', \mu' m'}_{L''' L'}(E), $$

((17-42))

where due to the translational lattice invariance the t matrix difference does not depend on μ and the Green function matrix elements depend only on the difference vector ${{\textbf R}^{\mu}} - {{\textbf R}^{\mu'}}$. The infinite sum over μ ^″ can be treated by lattice Fourier transformation

$$G^{m m'}_{LL'} ({{\textbf k}};\ E) = \sum_{\mu}^{\infty} G^{\mu m,\mu' m'}_{L L'} (E) \textrm{e}^{-\textrm{i} {{\textbf k}} ({{\textbf R}^{\mu}} -{{\textbf R}^{\mu'}})}$$

((17-43))

and an analogous equation for the reference Green function matrix elements. Because of the translational invariance the index μ ^′ in (17-43) can be chosen arbitrarily, for instance as $\mu' =0$. After Fourier transformation the equation

$$G^{mm'}_{LL'}({{\textbf k}} ;\ E) = G^{r,mm'}_{LL'} ({{\textbf k}} ;\ E) + \sum^{\infty}_{m'' L'' L'''} G^{r,mm''}_{LL''} ({{\textbf k}} ;\ E) \Delta t^{m''}_{L'' L'''} (E) G^{m''m'}_{L''L'} ({{\textbf k}} ;\ E) \:,$$

((17-44))

which has the same form as (17-19), must be solved at a set of k points in reciprocal space. The results are used to approximate

$$G^{\mu m, \mu' m'}_{L L'} (E) = \frac{1}{\varOmega_{BZ}} \int G^{m m'}_{L L'} ({{\textbf k}} ;\ E) \textrm{e}^{\textrm{i} {{\textbf k}} ({{\textbf R}^{\mu}} - {{\textbf R}^{\mu'}})} \textrm{d} {{\textbf k}} \: ,$$

((17-45))

the Fourier transformation back to real space, by a sampling procedure. Here Ω _BZ is the volume of the Brillouin zone and only elements with $\mu = \mu'$ and $m=m'$ are needed for the density.