What's the difference between RHF and UHF?

Question

I am doing a course on electronic structure methods and am somewhat puzzled by the content (I have a physics background). From what I understand Hartree-Fock is a method where you only use the Slater determinant for the ground state of occupied spin orbitals. Then configuration interaction (CI) is an extension of this where you include unoccupied spin orbitals for your Slater determinants. But if Hartree-Fock considers the ground state energy, then how can you have unrestricted Hartree-Fock?

Answer 1

The names RHF and UHF refer to the invariance of Hartree-Fock wave functions with respect to certain symmetry operations. Fukutome has worked out the details. Also, Stuber gives the details in his PhD work. I.e., a rRHF ( = real Restricted Hartree-Fock) wave function is invariant under the $\hat{S}^2, \hat{S}_z, \hat{K}, \hat{\theta}$, where $\hat{S}$ is the spin operator, $\hat{K}$ the complex conjugation operator and $\hat{\theta}$ the time reversal operator. rUHF is invariant only w.r.t. the operators $\hat{S}_z$ and $\hat{K}$. So the operators form a group and the labels rRHF, rUHF, $\ldots$ are used to group wave functions.

Configuration Interaction (CI) is the interaction of a predefined set of Slater-Determinants (SD) or Configuration State functions (CSF) with each other. I.e., one forms a CI wave function $$ \Psi_\mathrm{CI} = \Phi_0 + \sum\limits_{r}^{a} \Phi_{r}^{a} + \sum\limits_{rs}^{ab} \Phi_{rs}^{ab} + \ldots $$ $\Phi_0$ is the reference wave function and $\Phi_r^a$ denotes the wave function constructed from the reference by exciting one electron from orbital $r$ to orbital $a$. The choice of $\Phi_0$, however, is up to you. Oftentimes a rUHF solution is chosen. But similarly, rRHF is possible.

Ground state in your nomenclature just refers to the lowest energy HF solution of a given symmetry. But in some cases, the wave function can be unstable w.r.t. the loss of some symmetries. E.g., you can calculate an rRHF wave function for the disscociated $\ce{H2}$ molecule and you will find it to be a (local) minimum. Using this wave function as a starting point for an rUHF calculation, you will quickly discover the local rRHF minimum to be a local rUHF maximum. Hence an optimization will lower the energy. This is closely connected to the terms singlet or triplet instability. For a more complete discussion please refer to books as, e.g., Molecular Electronic-Structure Theory by Helgaker, Jørgensen and Olsen.

In the mentioned case of $\ce{H2}$ dissociation it is also quite logical. As indicated already by others, RHF implies one set of spatial orbitals for all electrons of $\alpha$ and $\beta$ spin or, to put it differently, double occupation of each molecular orbital (MO). Triplet states cannot be represented with an RHF wave function! But while dissociating $\ce{H2}$ it is unreasonable to assume that the electron spins on the atoms stay paired forever. One would rather expect both atoms to be completely identical at infinite distance and thus to end up with two hydrogen atoms instead of a $\ce{H+H-}$ system. And this is what happens when you use UHF.

The complex versions of HF become interesting especially when dealing with relativistic effects. Due to the Pauli matrices the Hamiltonian becomes complex and hence the MOs become complex as well. By this the wave function is not symmetric w.r.t. to $\hat{K}$ anymore and hence belongs to a different group in the scheme of Fukutome and Stuber.

Answer 2

The restricted and unrestricted qualifiers refer to whether certain properties of the orbitals you use to make the HF Slater determinant are allowed to change. Specifically, RHF requires that every spatial orbital is used to form both an $\alpha$ and $\beta$ spin orbital, whereas UHF lifts this restriction and allows every spin orbital to be formed from a different spatial orbital.

A great resource that explains Hartee-Fock, from the very basics to the in-depth details of calculations, is Modern Quantum Chemistry by Szabo and Ostlund.

Edit: The main reason we do UHF calculations is that restricted determinants lead to poor results for systems where there are unpaired electrons. A good example of this is lithium. We could describe it by the restricted determinant $|\psi_{1s}\bar\psi_{1s}\psi_{2s}$>. There are two electrons in the same $\psi_{1s}$ spatial orbital, but each electron experiences a different effective potential (the $\alpha$ spin electron in $1s$ has an exchange interaction with the $\alpha$ spin electron in $2s$). So, we would expect that the $1s$ electrons would be better described by their own spatial wavefunctions which are consistent with the potential they feel. In this case, the new unrestricted determinant $|\psi_{1s}^{\alpha}\bar\psi_{1s}^{\beta}\psi_{2s}^{\alpha}$>, where $\psi_{1s}^{\alpha}$ and $\bar\psi_{1s}^{\beta}$ have different spatial components, gives a lower energy for the ground state of lithium.

This is detailed in chapter 2 of the book above.