The non-linear part of the Hamiltonian

[loveislonely]

In Quantum chemistry, some times the Hamiltonian is written as the combination of 1-electron and 2-electron contributions:

 

H=[math]\sum<i|j>A[/math]+1/2[math]\sum<ij|kl>B[/math].

 

<i|j> and <ij|kl> are one and two electron repulsion integrals. In second quantization the coefficients A and B can be written as:

 

A=<I|E[math]_{ij}[/math]|J>

 

which is a linear contribution. This is easy to calculate.

 

B=[math]\sum[/math][math]_{M}[/math]<I|E[math]_{ij}[/math]|M><M|E[math]_{kl}[/math]|J>-[math]\delta[/math][math]_{jk}[/math]<I|E[math]_{il}[/math]|J>

 

where the first term is a bilinear contribution, the second term is a linear contribution.

 

My question is: is there any way to tern the bilinear contribution into linear expression to simplify the calculation? Thank you very much.

[Severian]

No, but you can remove the linear term (basically absorb it into the bilinear) by redefining your states.

[loveislonely]

No, but you can remove the linear term (basically absorb it into the bilinear) by redefining your states.

 

Thank you for your help. I was asking that because I realized for the linear contribution, I can calculate it only once for all the orbital indices (ijkl above), but for the bilinear case, I have to calculate it for every orbital index pair, which spend me ages to finish the calculation, especially when the chemical system is big. Thank you any way.