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i am trying to calculate the interaction between two excited hydrogen atoms, using degenerate perturbation theory. but comes up a few problems, any helps will be greatly appreciated.

the perturbation has the form:

H`=e^{3}/R^{3}(X_{a}X_{b}+Y_{a}Y_{b}-2Z_{a}Z_{b})

where the R is the distancee between two atoms and a , b subscripts donates atom a and b respectively.

i need to calculate the matrix elements have the form:

(<ψ_{a}ψ_{b}H` ψ_{a}ψ_{b}>

first thing is how to construct the total wave functions of the two atoms system? they add together or times together? and why? what is the physics meaning of:

ψ_{total}=ψ_{a}ψ_{b}

ψ_{total}=ψ_{a}+ψ_{b}

the second is, if using the exzact form, and ignor the radial funtion, is it possible to evaluate the matrix element? i.e

for one atom in its 2 0 0 state and the other in its 2 1 0 state. then wavefunctions equal to theirs spherical harrmonics which are:

2 0 0 state = 1/(2* square root pi )

2 1 0 state = square root 3 /(2* square root pi) * cosθ

couples to one atom in 2 1 0 state and the other in 2 1 1 state

2 1 1 state = squre root (3)/2 * 1/(squre root (2* pi)) * sinθ*e^{i∅}

the first term yields:

∫∫∫2 0 0 state * X_{a}* 2 1 0 state dr sinθ dθ d∅ *∫∫∫2 1 0 state * X_{b}* 2 1 1 state dr sinθ dθ d∅

since x= rsinθcos∅ and we have ignored the radial part, how can I evaluate dr ?

above all, am I right for constructing the matrix element like the above way?

Apologise for such a messy question, it is my first time to put equations up here, and I havent figured out how to construct a complex equation.

Thank you in advance.

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# Interaction between two excited hydrogen atoms

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