Page 98 - 71 the abc of dft_opt

196 APPENDIX B. RESULTS FOR SIMPLE ONE-ELECTRON SYSTEMS
>
and in terms of g , the susceptibility is
%
>
#
>∗
#
χ(x, x ; &) = n(x)n(x ) [g (x, x ; & + & 0 ) + g (x, x , −& + & 0 )] (B.9)
#
#
From this we can calculate both the static and dynamic polarizabilites:
Appendix B 5
α(0) = − 4 (B.10)
4Z
Results for simple one-electron systems and %
−16 (˜ω − 1)
Im(α(ω)) = (B.11)
4 4
Z ˜ω
where ˜ω = ω/|& 0 |
B.1 1d H atom
The Hamiltonian for this problem is
1 d 2
H = − − Zδ(x) (B.1)
2 dx 2
Integrating the Schr¨odinger equation through x = 0 yields the cusp condition:
# 0 +
= −Zφ(0) (B.2)
φ | 0 −
2
The ground-state energy is just & 0 = −Z /2, and T = −& 0 , V = 2& 0 . The normalized orbital
is:
√
φ(x) = Z exp(−Z|x|) (B.3)
and ground-state density
n(x) = Z exp(−2Z|x|) (B.4)
We can also extract continuum excited states, which are useful for TDDFT. With the usual
scattering boundary conditions:
φ s (x) = exp(ikx) + r exp(−ikx), x < 0
= t exp(ikx), x > 0 (B.5)
where r and t are given by:
iZ ik
r = , t = (B.6)
k − iZ Z + ik
√
ˆ
and k = 2&, E > 0. Furthermore, when H is given by (B.1), the solution to:
+
ˆ
>
#
(& − H)g (x, x ; &) = δ(x − x ) (B.7)
#
is the outgoing Green’s function:
 √ ! 
1 √ Ze i 2*(|x|+|x |)
> #  i 2*|x−x | ! 
g (x, x ; &) = √ e − √  . (B.8)
i 2& i 2& + Z
195