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Subsections

A..1 Nonrelativistic case

Let us assume that the charge density n(r) and the potential V(r) are spherically symmetric. The Kohn-Sham (KS) equation:

$\displaystyle \left(\vphantom{-{\hbar^2\over 2m}\nabla^2 + V(r)-\epsilon}\right.$ - $\displaystyle {\hbar^2\over 2m}$$\displaystyle \nabla^{2}_{}$ + V(r) - $\displaystyle \epsilon$$\displaystyle \left.\vphantom{-{\hbar^2\over 2m}\nabla^2 + V(r)-\epsilon}\right)$$\displaystyle \psi$($\displaystyle \bf r$) = 0 (1)
can be written in spherical coordinates. We write the wavefunctions as

$\displaystyle \psi$($\displaystyle \bf r$) = $\displaystyle \left(\vphantom{{R_{nl}(r)\over r}}\right.$$\displaystyle {R_{nl}(r)\over r}$$\displaystyle \left.\vphantom{{R_{nl}(r)\over r}}\right)$Ylm($\displaystyle \hat{{\bf r}}$), (2)
where n is the main quantum number l = n - 1, n - 2,..., 0 is angular momentum, m = l, l - 1,..., - l + 1, - l is the projection of the angular momentum on some axis. The radial KS equation becomes:
$\displaystyle \left(\vphantom{-{\hbar^2\over 2m} {1\over r} {d^2 R_{nl}(r)\over dr^2}
+(V(r)-\epsilon) {1\over r} R_{nl}(r)
}\right.$ - $\displaystyle {\hbar^2\over 2m}$$\displaystyle {1\over r}$$\displaystyle {d^2 R_{nl}(r)\over dr^2}$ + (V(r) - $\displaystyle \epsilon$)$\displaystyle {1\over r}$Rnl(r)$\displaystyle \left.\vphantom{-{\hbar^2\over 2m} {1\over r} {d^2 R_{nl}(r)\over dr^2}
+(V(r)-\epsilon) {1\over r} R_{nl}(r)
}\right)$Ylm($\displaystyle \hat{{\bf r}}$)      
  - $\displaystyle {\hbar^2\over 2m}$$\displaystyle \left(\vphantom{{1\over\mbox{sin}\theta}{\partial\over\partial\th...
...\mbox{sin}^2\theta}
{\partial^2Y_{lm}(\hat{\bf r})\over\partial\phi^2}
}\right.$$\displaystyle {1\over\mbox{sin}\theta}$$\displaystyle {\partial\over\partial\theta}$$\displaystyle \left(\vphantom{\mbox{sin}\theta{\partial Y_{lm}(\hat{\bf r})
\over\partial\theta}}\right.$sin$\displaystyle \theta$$\displaystyle {\partial Y_{lm}(\hat{\bf r})
\over\partial\theta}$$\displaystyle \left.\vphantom{\mbox{sin}\theta{\partial Y_{lm}(\hat{\bf r})
\over\partial\theta}}\right)$ + $\displaystyle {1\over\mbox{sin}^2\theta}$$\displaystyle {\partial^2Y_{lm}(\hat{\bf r})\over\partial\phi^2}$$\displaystyle \left.\vphantom{{1\over\mbox{sin}\theta}{\partial\over\partial\th...
...\mbox{sin}^2\theta}
{\partial^2Y_{lm}(\hat{\bf r})\over\partial\phi^2}
}\right)$$\displaystyle {1\over r^3}$Rnl(r) = 0.     (3)

This yields an angular equation for the spherical harmonics Ylm($ \hat{{\bf r}}$):

- $\displaystyle \left(\vphantom{{1\over\mbox{sin}\theta}{\partial\over\partial\th...
...\mbox{sin}^2\theta}
{\partial^2Y_{lm}(\hat{\bf r})\over\partial\phi^2}
}\right.$$\displaystyle {1\over\mbox{sin}\theta}$$\displaystyle {\partial\over\partial\theta}$$\displaystyle \left(\vphantom{\mbox{sin}\theta{\partial Y_{lm}(\hat{\bf r})
\over\partial\theta}}\right.$sin$\displaystyle \theta$$\displaystyle {\partial Y_{lm}(\hat{\bf r})
\over\partial\theta}$$\displaystyle \left.\vphantom{\mbox{sin}\theta{\partial Y_{lm}(\hat{\bf r})
\over\partial\theta}}\right)$ + $\displaystyle {1\over\mbox{sin}^2\theta}$$\displaystyle {\partial^2Y_{lm}(\hat{\bf r})\over\partial\phi^2}$$\displaystyle \left.\vphantom{{1\over\mbox{sin}\theta}{\partial\over\partial\th...
...\mbox{sin}^2\theta}
{\partial^2Y_{lm}(\hat{\bf r})\over\partial\phi^2}
}\right)$ = l (l + 1)Ylm($\displaystyle \hat{{\bf r}}$) (4)
and a radial equation for the radial part Rnl(r):

- $\displaystyle {\hbar^2\over 2m}$$\displaystyle {d^2 R_{nl}(r)\over dr^2}$ + $\displaystyle \left(\vphantom{ {\hbar^2\over 2m} {l(l+1)\over r^2} + V(r)-\epsilon
}\right.$$\displaystyle {\hbar^2\over 2m}$$\displaystyle {l(l+1)\over r^2}$ + V(r) - $\displaystyle \epsilon$$\displaystyle \left.\vphantom{ {\hbar^2\over 2m} {l(l+1)\over r^2} + V(r)-\epsilon
}\right)$Rnl(r) = 0. (5)
The charge density is given by

n(r) = $\displaystyle \sum_{{nlm}}^{}$$\displaystyle \Theta_{{nl}}^{}$$\displaystyle \left\vert\vphantom{{R_{nl}(r)\over r}Y_{lm}(\hat r)}\right.$$\displaystyle {R_{nl}(r)\over r}$Ylm($\displaystyle \hat{r}$)$\displaystyle \left.\vphantom{{R_{nl}(r)\over r}Y_{lm}(\hat r)}\right\vert^{2}_{}$ = $\displaystyle \sum_{{nl}}^{}$$\displaystyle \Theta_{{nl}}^{}$$\displaystyle {R^2_{nl}(r)\over 4\pi r^2}$ (6)
where $ \Theta_{{nl}}^{}$ are the occupancies ( $ \Theta_{{nl}}^{}$$ \le$2l + 1) and it is assumed that the occupancies of m are such as to yield a spherically symmetric charge density (which is true only for closed shell atoms).

A..1.1 Useful formulae

Gradient in spherical coordinates (r,$ \theta$,$ \phi$):

$\displaystyle \nabla$$\displaystyle \psi$ = $\displaystyle \left(\vphantom{{\partial\psi\over\partial r},
{1\over r}{\parti...
...theta},
{1\over r \mbox{sin}\theta}
{\partial\psi\over\partial\phi}
}\right.$$\displaystyle {\partial\psi\over\partial r}$,$\displaystyle {1\over r}$$\displaystyle {\partial\psi\over\partial\theta}$,$\displaystyle {1\over r \mbox{sin}\theta}$$\displaystyle {\partial\psi\over\partial\phi}$$\displaystyle \left.\vphantom{{\partial\psi\over\partial r},
{1\over r}{\parti...
...theta},
{1\over r \mbox{sin}\theta}
{\partial\psi\over\partial\phi}
}\right)$ (7)
Laplacian in spherical coordinates:

$\displaystyle \nabla^{2}_{}$$\displaystyle \psi$ = $\displaystyle {1\over r}$$\displaystyle {\partial^2\over\partial r^2}$(r$\displaystyle \psi$) + $\displaystyle {1\over r^2\mbox{sin}\theta}$$\displaystyle {\partial\over\partial\theta}$$\displaystyle \left(\vphantom{\mbox{sin}\theta{\partial\psi\over\partial\theta}}\right.$sin$\displaystyle \theta$$\displaystyle {\partial\psi\over\partial\theta}$$\displaystyle \left.\vphantom{\mbox{sin}\theta{\partial\psi\over\partial\theta}}\right)$ + $\displaystyle {1\over r^2\mbox{sin}^2\theta}$$\displaystyle {\partial^2\psi\over\partial\phi^2}$ (8)


next up previous contents
Next: A..2 Fully relativistic case Up: A. Atomic Calculations Previous: A. Atomic Calculations   Contents
Paolo Giannozzi 2017-10-23