Second Quantized Minimal Basis Hamiltonian of H2

[Physics Learner]

Hi,
I am really new in understanding second quantization formalism. Recently I am reading this journal:
https://dash.harvard.edu/bitstream/...ectronic_Structure.pdf?sequence=1&isAllowed=y

In brief, the molecular Hamiltonian is written as
$$\mathcal{H}=\sum_{ij} h_{ij} a^{\dagger}_i a_j +\frac{1}{2} \sum_{ijkl} h_{ijkl} a^{\dagger}_i a^{\dagger}_ja_la_k$$
where, $$h_{ij}=\int{d\mathbf{x}}\,\chi^*_i(\mathbf{x})\left(-\frac{1}{2}\nabla^2-\sum_{a}{\frac{Z_a}{\mathbf{r}_{a,\mathbf{x}}}}\right)\chi_j(\mathbf{x})$$
and $$h_{ijkl}=\int{d\mathbf{x_1}\,d\mathbf{x_2}}\,\chi^*_i(\mathbf{x_1})\chi^*_j(\mathbf{x_2})r_{1,2}^{-1}\chi_k(\mathbf{x_1})\chi_l(\mathbf{x_2})$$

For $\text{H}_\text{2}$, in minimal basis, there are four spin-orbitals: $$\chi_1 = c_1\left(\phi_1+\phi_2\right)\alpha$$ $$\chi_2 = c_1\left(\phi_1+\phi_2\right)\beta$$ $$\chi_3 = c_2\left(\phi_1-\phi_2\right)\alpha$$ $$\chi_4 = c_2\left(\phi_1-\phi_2\right)\beta$$
where, $\phi_{1,2}$ are 1s orbitals of a single Hydrogen atom and $\alpha$ and $\beta$ are spin functions satisfying 'orthogonality' conditions, i.e., $$\int{d\omega}\,\alpha^*\alpha = \int{d\omega}\,\beta^*\beta = 1$$ and $$\int{d\omega}\,\beta^*\alpha = \int{d\omega}\,\alpha^*\beta = 0$$

What I am not getting is why some terms in the Hamiltonian (Eq. 14 and Eq. 16) are not there when we write it down explicitly? To be more specific, I understand that $h_{12} = h_{14} = 0$ because of their spin orthogonality, but I am curious to know why terms with $h_{13}$, $h_{24}$ etc. are missing.

Similarly, I do not understand why terms like $h_{1223}$, $h_{1224}$ are missing. I understand that terms like $h_{1232} = h_{1242} = 0$ because of their spin orthogonality.

Can anyone please explain? Thanks.

 

[DrClaude]

For $\text{H}_\text{2}$, in minimal basis, there are four spin-orbitals: $$\chi_1 = c_1\left(\phi_1+\phi_2\right)\alpha$$ $$\chi_2 = c_1\left(\phi_1+\phi_2\right)\beta$$ $$\chi_3 = c_2\left(\phi_1-\phi_2\right)\alpha$$ $$\chi_4 = c_2\left(\phi_1-\phi_2\right)\beta$$

It will be clearer if you write it as
$$\chi_1 = c_1 \varphi_g \alpha$$ $$\chi_2 = c_1 \varphi_g \beta$$ $$\chi_3 = c_2\varphi_u\alpha$$ $$\chi_4 = c_2\varphi_u\beta$$

What I am not getting is why some terms in the Hamiltonian (Eq. 14 and Eq. 16) are not there when we write it down explicitly? To be more specific, I understand that $h_{12} = h_{14} = 0$ because of their spin orthogonality, but I am curious to know why terms with $h_{13}$, $h_{24}$ etc. are missing.

Similarly, I do not understand why terms like $h_{1223}$, $h_{1224}$ are missing. I understand that terms like $h_{1232} = h_{1242} = 0$ because of their spin orthogonality.

$\varphi_g$ is orthogonal to $\varphi_u$, so $h_{13} = 0$ since $\chi_1$ involves $\varphi_g$ and $\chi_3$ involves $\varphi_u$, and so on.