Hamiltonian in atomic units


by Shadowz
Tags: atomic, hamiltonian, units
Shadowz
Shadowz is offline
#1
Feb18-11, 01:03 AM
P: 43
1. The problem statement, all variables and given/known data
So the question is I have to use some trial function of the form [tex]\sum c_if_i[/tex] to approximate the energy of hydrogen atom where [tex]f_i=e^{-ar}[/tex] for some number a (positive real number). Note that r is in atomic unit.

2. Relevant equations
Because r is in atomic unit, I think I should use the Hamiltonian in atomic unit, that is
[tex]H = -\frac{1}{2}\nabla^2 + \frac{1}{r}[/tex]
or should I use the spherical Hamiltonian?

I try to compute [tex]H_{ij} = \int_0^\infty f_iHf_j[/tex] but there will be the term [tex]\int_0^\infty f_1\frac{1}{r}f_2dr[/tex] which cannot be integrated (not converged). So what's wrong with the way I approach the problem?

Thank you,
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sgd37
sgd37 is offline
#2
Feb18-11, 03:06 AM
P: 205
might your coefficients be r dependent since they are in the full solution of the hydrogen like atom
asheg
asheg is offline
#3
Feb23-11, 02:59 PM
P: 51
[tex]
\int_0^\infty f_1\frac{1}{r}f_2dr
[/tex]
will converge if you change 1/r to 1/(r+eps)


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