Making of pseudo-atomic orbitals

The pseudo-atomic orbitals are used in the program package, OpenMX, as the primitive basis orbitals. The pseudo-atomic orbitals are generated as follows: first, the SCF calculation is performed in consideration of all electrons under a confinement potential, second, the pseudopotentials are generated, and finally, the pseudo-atomic orbitals for the confinement pseudopotentials are evaluated numerically up to a required excited state. In this section, the generation of the pseudo-atomic orbitals is illustrated. In the file, C.inp, please set the keyword, calc.type, to PAO, and run the executable file, adpack, as follows:

     % adpack C.inp
   

When the run is completed normally, then you find a file, C0.pao, in the directory, work. In this file, C0.pao, the valence electron density and the radial parts of the pseudo-atomic orbitals are output. For your adversaria, the contents of the input file and the results of all electron SCF calculation are also included. They are stored in order of log(r), r, and the valence electron density, and in order of log(r), r, and the radial part 1, the radial part 2,..., in the flexible date format, respectively. In Fig. 4, the confinement potential and the pseudo-atomic orbitals for the s-orbital are shown. From Fig. 4, we see that the pseudo-atomic orbitals are localized due to the confinement potential, and the number of nodes increases as the eigenvalue increases. The confinement potential is made by modifying the core potential as follows:

$$V_{\rm core}(r) = \left\{ \begin{array}{cl} -\frac{... ... \quad \mbox{for $r_{\rm c}<r$},\\ \end{array} \right.$$ (1)

where $b_0$, $b_1$, $b_2$, and $b_3$ are determined so that the values and the derivatives are continuous at both $r_1$ and $r_{\rm c}$. Considering that there are relations, $r_{\rm c}$=radial.cutoff.pao, $r_{\rm 1}$=$r_{\rm c} -$rising.edge, and $h$=height.of.wall, we find that the tunneling of wave function for the confinement wall becomes small as height.of.wall increases. Also, it is possible to control the shape of the rising edge aroung the wall by changing rising.edge. If you use a huge value for height.of.wall, then you might meet a case that the calculation is not completed normally, since the computational instability appears often. In such a case, the numerical instability may be avoided by enlarging the keywords, rising.edge and num.grid. As for the keyword, rising.edge, please refer the section, Input file. The file, *.pao, created here can be an input file of the program package, OpenMX.

Figure: Confinement potential and radial parts of pseudo-atomic orbitals of a carbon atoms