Keywords: pseudo-gen

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Subsections

- A..1.1 Useful formulae

## 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:

- ∇2 + V(r) - ε ψ() = 0 | (1)

can be written in spherical coordinates. We write the wavefunctions as

ψ() = Ylm(), | (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:

- + (V(r) - ε)Rnl(r)Ylm() |  |  |
- sinθ + Rnl(r) = 0. |  |  | (3)

This yields an angular equation for the spherical harmonics Ylm():

- sinθ + = l (l + 1)Ylm() | (4)

and a radial equation for the radial part Rnl(r):

- + + V(r) - ε Rnl(r) = 0. | (5)

The charge density is given by

n(r) = ΘnlYlm() = Θnl | (6)

where Θnl are the occupancies ( Θnl≤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, θ, φ):

∇ψ = ,, | (7)

Laplacian in spherical coordinates:

∇2ψ = (rψ) + sinθ + | (8)

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