252 A Ab initio Computational Methods 5 0 3 3 1012345 G- [001] [110] -7012 [001) 34 [110] (a) (b) Figure A.1 Muffin-tin shape potential in(110)plane of a general bcc crystal of the lattice constant a =3au with spherical radii:(a)rMT 1au,and (b)rMT =rws 1.48au A.2 Wien 95-w2k Codes The program package WIEN [433]does not use any shape approximation to the potential.The crystal environment is divided into a region of non- overlapping atomic spheres(centred at individual atomic sites)and an inter- stitial region as can be seen in Figure A.2.In order to describe ES reliably and effectively,two different basis sets are employed:a product of linear com- bination of radial functions and spherical harmonics is used inside the spheres whereas the wave functions in the interstitial region are expanded into a linear combination of plane waves.The solution in both regions must be continuous at the spherical boundaries.Each basis function is then defined as a plane- wave in the interstitial region connected smoothly to a linear combination of atomic-like functions in the spheres,thus providing an efficient represen- tation throughout the space.A similar representation is used for potentials and charge densities.The method is called the linear augmented plane wave (LAPW)method [428,434]. S Figure A.2 Illustration of a crystal model-three atomic spheres (S1-S3)with (embedded in the interstitial region I with252 A Ab initio Computational Methods (a) (b) Figure A.1 Muffin-tin shape potential in (110) plane of a general bcc crystal of the lattice constant a = 3 au with spherical radii: (a) rMT = 1 au, and (b) rMT = rW S = 1.48 au A.2 Wien 95 – w2k Codes The program package WIEN [433] does not use any shape approximation to the potential. The crystal environment is divided into a region of non￾overlapping atomic spheres (centred at individual atomic sites) and an inter￾stitial region as can be seen in Figure A.2. In order to describe ES reliably and effectively, two different basis sets are employed: a product of linear com￾bination of radial functions and spherical harmonics is used inside the spheres whereas the wave functions in the interstitial region are expanded into a linear combination of plane waves. The solution in both regions must be continuous at the spherical boundaries. Each basis function is then defined as a plane￾wave in the interstitial region connected smoothly to a linear combination of atomic-like functions in the spheres, thus providing an efficient represen￾tation throughout the space. A similar representation is used for potentials and charge densities. The method is called the linear augmented plane wave (LAPW) method [428, 434]. S1 S2 S3 I Figure A.2 Illustration of a crystal model – three atomic spheres (S1 – S3) with potential US(r) = 2 lm Ulm(r)Ylm(ˆr) embedded in the interstitial region I with UI (r) = 2 K UK(r)eiKr