<U|V> overlap integral of two many-electron determinant wave functions

[limorsj]

Hello,

If we let U and V be two single determinant wave functions built up of spin orbitlas ui and vj respectively, will the overlap between them be as follows:

<U|V> = Det{<ui|vi>}

Thank you

 

[cgk]

This is not the third thread you make with exactly the same question. Your question has been answered in the thread in the Quantum Physics forum.

 

[limorsj]

This is not the third thread you make with exactly the same question. Your question has been answered in the thread in the Quantum Physics forum.

Dear cgk,

According to your answer in another forum I can calculate the SAB by computing a singular value decomposition (SVD) of the occupied orbital overlap Socc. However, I would be very glad to know if the SAB can also be calculated by
SAB=<U|V> = Det{<ui|vj>},

Thank you for your support

 

[cgk]

I think so, yes. This would even get the sign right. Note that for a square matrix (as you have here) the determinant is equal to the product of singular values multiplied by a phase factor (since if SAB = U diag(sig) V (i.e., a SVD, with U and V unitary), then det(SAB) = det(U diag(sig) V) = det(U) det(diag(sig)) det(V) = e^{i phi} det(diag(sig)) = e^{i phi} prod{sig_i}, since unitary matrices have determinants with absolute value 1).

 

[limorsj]

I think so, yes. This would even get the sign right. Note that for a square matrix (as you have here) the determinant is equal to the product of singular values multiplied by a phase factor (since if SAB = U diag(sig) V (i.e., a SVD, with U and V unitary), then det(SAB) = det(U diag(sig) V) = det(U) det(diag(sig)) det(V) = e^{i phi} det(diag(sig)) = e^{i phi} prod{sig_i}, since unitary matrices have determinants with absolute value 1).[/QUOT

Thank you very much

 

[limorsj]

I think so, yes. This would even get the sign right. Note that for a square matrix (as you have here) the determinant is equal to the product of singular values multiplied by a phase factor (since if SAB = U diag(sig) V (i.e., a SVD, with U and V unitary), then det(SAB) = det(U diag(sig) V) = det(U) det(diag(sig)) det(V) = e^{i phi} det(diag(sig)) = e^{i phi} prod{sig_i}, since unitary matrices have determinants with absolute value 1).

Dear cgk,

Thank you very much for your kind answers. I was wondering whether you could help me again... I calculate the SAB of the two slater determinants. I have two sets of molecular orbitals, in both sets I have equivalent orbitals, just in different energy order and different electrons occupation. Since using SAB=<U|V> = Det{<ui|vj>} does not take into account the occupation or the order of the orbitals (changing the order will just change the sign of the determinant value), I get actually SAB=1... It seems not to calculate what I need...since SAB should reflect the probability of electron transfer between this two sets...
p.s. for my calculation I take only the occupied orbitals (I have one singly and the other doubly occupied).

Thank you