Basis set of the overlap, kinetic, electronic matrices for RKS, UKS and ROKS code: O2 and NO2 cases

Question

This question is a follow-up of this post.

The basis set to approximate the overlap, kinetic, electron repulsion and proton-electron attraction matrices is to be choosed by the user depending on the method. What I want is to code a STO-6G UKS code. Let's take two examples so that I can understand and then implement these ideas afterwards :

Example $NO_2$ :

The valence orbitals of $N$ : $2p_x$, $2p_y$ and $2p_z$ and the valence orbitals of $O$ : $2p_x$, $2p_y$ and $2p_z$ with one of the p orbitals filled.

I will not take the frozen approximation of the s-orbitals.

I denote : $\chi^{orbital}_O$ and $\chi^{orbital}_N$ the spatial part of the atomic orbitals for orbital $\in \{1s, 2s, 2p_x, 2p_y, 2p_z \}$.

Then what is the spatial-spin orbital basis set of the overlap, kinetic, Coulomb (Exchange) and proton attraction operators/matrices for each electronic structure methods by only using 1s, 2s and 2p orbitals?

What I think atm :

RKS case

The basis are the same for $\alpha$ and $\beta$ spin orbitals. The the energy is simply the total energy of the $\alpha$ molecular orbitals mutiplied by two since the density is supposed to be the same for $\alpha$ and $\beta$ molecular orbitals.

ROKS/UKS case

I think that the matrices of one of the spin will be at least one size larger.

Example $O_2$ :

Same question for $O_2$.

Answer 1

There seems to be a lot of confusion in the question.

Then what is the spatial-spin orbital basis set of the overlap, kinetic, Coulomb (Exchange) and proton attraction operators/matrices for each electronic structure methods by only using 1s, 2s and 2p orbitals?

In RKS, UKS and ROKS the overlap, kinetic energy, and nuclear attraction matrices are $S_{\mu \nu} = \langle \mu | \nu \rangle$, $T_{\mu \nu} = -\frac 1 2 \langle \mu | \nabla^2 | \nu \rangle$, and $V_{\mu \nu} = -\sum_A \langle \mu | \frac {Z_A} {r_{A}} | \nu \rangle$ . They only depend on the orbital basis set. Note that you would typically choose a symmetric form for ${\bf T}$, which you can do with integration by parts. Note also that I've used the common shorthand here where e.g. $ | \nu \rangle = | \chi_\nu \rangle$

It does not matter how you choose your basis functions. Instead of atomic orbitals, you can also construct these matrices for numerical basis functions.

Similarly, the Coulomb matrix only depends on the basis set; it does not contain any spin dependence, because it tells you about Coulombic repulsion of the electrons.

Only the exchange matrices carry spin dependence: you get exchange matrices for both spins ${\bf K}^\alpha$ and ${\bf K}^\beta$. The structure of these matrices is the same as that of all of the above matrices, that is, $K^\alpha_{\mu \nu}$ is the element of the spin-$\alpha$ exchange matrix that corresponds to basis functions $\chi_\mu$ and $\chi_\nu$.