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.


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

Example $O_2$ :

Same question for $O_2$.

  • $\begingroup$ Could you explain what you think a basis set is? I have to admit to me this post just seems very confused, so explaining what you mean by some of the terms might help others understand what you are asking. $\endgroup$
    – Ian Bush
    Commented Aug 14, 2023 at 20:21
  • $\begingroup$ Sorry, by basis set I mean atomic orbital basis not the basis set which define the decomposition of the Slater-type orbitals. I want to understand what differentiate the LCAO appromixation of the $\alpha$ and $\beta$ molecular orbitals for $RKS$, $UKS$ and $ROKS$. Then I can properly compute the different matrices/operators for the SCF loop. $\endgroup$
    – mle
    Commented Aug 15, 2023 at 7:57
  • 1
    $\begingroup$ Assuming I can simply replace KS with HF for RHF and UHF all the matrices are the same except the exchange matrix, which has separate matrices for the two spin channels. I don't really know ROHF, except to say there is no unique formulation - you may well have to tell us exactly which ROHF you are interested in - hal.science/hal-03228618v2 might help, and given my background I have to advertise doi.org/10.1080/00268977400102171, requiescat in pace Vic. $\endgroup$
    – Ian Bush
    Commented Aug 15, 2023 at 8:16
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    $\begingroup$ Just to clarify what I mean by "same" in RHF and UHF the Overlap, Kinetic Energy and Nuclear attraction matrices have the same form as they don't need the density matrix. The coulomb matrix is the same in the sense that yu us the total electron density matrix, i.e. Palpha+Pbeta in the UHF case. See any elementary quantum chemistry book, Szabo and Ostlund for example. $\endgroup$
    – Ian Bush
    Commented Aug 15, 2023 at 8:23
  • 1
    $\begingroup$ Hmmm. Strongly suggest you read a good book like Szabo and Ostlund - just looked at my edition of Jensen and it has no maths for UHF (section 3.7) which really is not good enough. But in a simple view of the world, yes one density matrix in RHF (I suggest you stick to HF for the moment) and two, one for each spin in UHF. All these matrices have size n_basis*n_basis. $\endgroup$
    – Ian Bush
    Commented Aug 15, 2023 at 9:47

1 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 $\chi_\nu$.


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