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.

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

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.