2-2.The APW method

08-10

2-2.The APW method

來自專欄 A note on DFT

We will search for a basis set that uses other functions than plane waves, and that does not require the introduction of a pseudopotential. Such a basis set will have to be more efficient, but of course we do not want it to be biased. Our first example of this will be the Augmented Plane Wave (APW) basis set.

In the region far away from the nuclei, the electrons are more or less 『free』. Free electrons are described by plane waves. Close to the nuclei, the electrons behave quite as they were in a free atom, and they could be described more efficiently by atomic like functions.

$egin{eqnarray}label{eq:apw0} phi_{vec{K}}^{vec{k}}(vec{r},E)equivleft{ egin{array}{ll} frac{1}{sqrt{V}}e^{i(vec{k}+vec{K})cdotvec{r}}&vec{r}in I\ sum_{l,m}A^{alpha,vec{k}+vec{K}}_{lm}u^{alpha}_{l}(vec{r}~,E)Y^l_m(vec{r}~)&vec{r}in S_{alpha},vec{r}~equivvec{r}-vec{r}_{alpha}\ sum_{l,m}A^{eta,vec{k}+vec{K}}_{lm}u^{eta}_{l}(vec{r}~,E)Y^l_m(vec{r}~)&vec{r}in S_{eta},vec{r}~equivvec{r}-vec{r}_{eta}\ ...&... end{array} ight. end{eqnarray}$

We have to require that the plane wave outside the sphere matches the function inside the sphere over the complete surface of the sphere (in value, not in slope). So the $A^{eta,vec{k}+vec{K}}_{lm}$ should be defined as:

$egin{eqnarray} A^{eta,vec{k}+vec{K}}_{lm}=frac{4pi i^le^{i(vec{k}+vec{K})cdotvec{r}_{alpha}}}{sqrt{V}u^{eta}_{l}(vec{r}_{alpha},E)}j_l(|vec{k}+vec{K}|R_{alpha})Y^{l*}_m(hat{vec{k}+vec{K}}) end{eqnarray}$

In oreder to make the calculation reachable, we can limit l with $l_{max}equiv R_alpha K_{max}$ , where $K_{max}$ is the cut-off for the plane waves and $R_alpha$ is the muffin tin radius of atom $alpha$ .

It is not useful to make $l_{max}$ larger than the condition

$R_alpha K_{max}$ requires, and the muffin tin radii for the different atoms should not be too different: if they were, a value for $l_{max}$ that is suitable for each atom would not exist.

Flow chart of the APW method