# Number of atomic and molecular orbitals in different Hartree-Fock references

For a given molecule and electronic configuration,

• Are the number of contracted atomic orbitals (AOs) different in a restricted open-shell Hartree-Fock (ROHF) calculation from an RHF calculation? What about unrestricted Hartree-Fock (UHF)?
• Are the number of molecular orbitals (MOs) different in a ROHF calculation from an RHF calculation? What about UHF?

For example, if we are using the STO-3G basis set for a water molecule, we get 1 + 1 + 5 = 7 contracted functions for AOs in an RHF calculation, leading to 7 MOs. Are any of these numbers different in an ROHF calculation? What about UHF?

Essentially, what are the core differences between AOs and MOs within each approach?

For water molecule using sto-3g basis I obtained the orbitals in the following order For RHF: Occupied (A1) (A1) (B2) (A1) (B1) Virtual (A1) (B2) For ROHF: Occupied (A1) (A1) (B2) (A1) (B1) Virtual (A1) (B2) But if we take alcohols for an e.g., the number of such orbitals is not the same. I want to know how they are calculated, so that atleast I could do them by hand. For the RHF one, it is nicely explained in "Modern Quantum chemistry" by Szabo-Ostlund. I want some similar information on ROHF and UHF (if possible).

Straight answer to your question: whatever method you use, RHF, ROHF or UHF, your calculations will use the same number of basis functions (7 contracted functions for your water example). How will they be used and combined, however, is different. In your comment on the question you talk about molecular orbitals - it's important to keep these conceptually separate from the basis set you are using to construct the orbitals.

How many functions do you use for an orbital in ROHF? In general, UHF/ROHF approaches use a separate molecular orbital for each electron, so you would use basis functions twice to describe an unrestricted pair compared to a restricted pair.

In other words, basis functions aren't "split" between the electrons when treated as an unrestricted pair; rather, two orbitals are calculated using the same basis functions, one corresponding to the alpha electron and another for the beta electron.

So, if you compare UHF with RHF, you'll be using the same basis functions to construct twice the number of molecular orbitals.

Let's take an example to illustrate it: $\mathrm{O_2}$.

For singlet $\mathrm{O_2}$ ($\mathrm{S=0}$), there are 8 alpha and 8 beta electrons. Using STO-3G, we'll have 10 contracted functions, those corresponding to $\mathrm{1s}$, $\mathrm{2s}$ and 3 x $\mathrm{2p}$ for each oxygen atom, each of them composed of 3 primitive gaussians.

• In RHF, we can combine these 10 basis functions in a single Slater determinant, producing 10 molecular orbitals. The first 8 of these will be doubly occupied, the last 2 will be vacant.

• In ROHF, we have exactly the same as for RHF, because ROHF is exactly the same as RHF for molecules with $\mathrm{S=0}$.

• In UHF, we use the same 10 basis functions, but we use them to build two sets of 10 molecular orbitals - 10 for the alpha electrons and 10 for the beta electrons - in a single Slater determinant. We occupy them with 8 alpha electrons and 8 beta electrons, 1 electron per orbital, with 2 alpha and 2 beta orbitals left vacant.

For triplet $\mathrm{O_2}$ ($\mathrm{S=1}$), there are 9 alpha and 7 beta electrons. We have the same basis functions as before.

• In UHF, we do the same as before, and we also get 10 alpha and 10 beta molecular orbitals in a single Slater determinant. Occupation changes, though, so now we have 9 occupied and 1 vacant alpha orbitals and 7 occupied and 3 vacant beta orbitals.

• In ROHF, and since now we have an unequal number of alpha and beta electrons, we have a more complex construction of the Fock operator - one that is not unique. Essentially, we'll calculate alpha and beta electrons separately under the hood, and then combine them; and we'll be treating full, partially full and empty orbitals as three coupled sets. We'll produce 10 orbitals, of which we'll doubly fill the first 7 and partially fill the following 2, leaving 1 unoccupied.

In conclusion: the number of basis functions in all cases is the same: 10 contracted functions comprising 30 primitive gaussians. However, these basis functions are used a different number of times for RHF, UHF and ROHF; RHF uses them once in a simple Slater determinant, UHF uses them twice to form a Slater determinant with a two sets of orbitals, one for alpha and the other for beta electrons, and ROHF uses them twice but ends up producing a single set of full, partially or non-occupied orbitals that share their spatial part.