This post contains some notes I took when learning ab-initio calculations

## Localized orbital (LOCA) and unitary transformation

Localized molecular orbitals are molecular orbitals which are concentrated in a limited spatial region of a molecule, for example a specific bond or a lone pair on a specific atom. They can be used to relate molecular orbital calculations to simple bonding theories, and also to speed up post-Hartree–Fock electronic structure calculations by taking advantage of the local nature of electron correlation. Localized orbitals in systems with periodic boundary conditions are known as Wannier functions. Standard ab initio quantum chemistry methods lead to delocalized orbitals that, in general, extend over an entire molecule and have the symmetry of the molecule. Localized orbitals may then be found as linear combinations of the delocalized orbitals, given by an appropriate unitary transformation.

Simply speaking, localized orbital corresponds to the ones derived from MO theory. Delocalized ones are the orbitals calculated from ab-initio program.

For molecules with a closed electron shell, in which each molecular orbital is doubly occupied, the localized and delocalized orbital descriptions are in fact equivalent and represent the same physical state. It might seem, again using the example of water, that placing two electrons in the first bond and two other electrons in the second bond is not the same as having four electrons free to move over both bonds.

However, in quantum mechanics all electrons are identical and cannot be distinguished as same or other. The total wavefunction must have a form which satisfies the Pauli exclusion principle such as a Slater determinant (or linear combination of Slater determinants), and it can be shown [1] that if two electrons are exchanged, such a function is unchanged by any unitary transformation of the doubly occupied orbitals.For molecules with an open electron shell, in which some molecular orbitals are singly occupied, the electrons of alpha and beta spin must be localized separately.[2][3] This applies to radical species such as nitric oxide and dioxygen. Again, in this case the localized and delocalized orbital descriptions are equivalent and represent the same physical state.

In summary, although the orbitals look different, the actual wavefunctions are slater determinants and the final wavefunctions are the same. To calculate LOCA in Q-Chem, I just need to add input for localized orbital analysis.

```
1
``` |
LOBA 12 |

## Open-shell molecule with HF

PSI4 implements the most popular spin specializations of Hartree–Fock theory, including:

Restricted Hartree–Fock (RHF) [Default]

Appropriate only for closed-shell singlet systems, but twice as efficient as the other flavors, as the alpha and beta densities are constrained to be identical.

Unrestricted Hartree–Fock (UHF)

Appropriate for most open-shell systems and fairly easy to converge. The spatial parts of the alpha and beta orbitals are fully independent of each other, which allows a considerable amount of flexibility in the wavefunction. However, this flexibility comes at the cost of spin symmetry; UHF wavefunctions need not be eigenfunctions of the S^2 operator. The deviation of this operator from its expectation value is printed on the output file. If the deviation is greater than a few hundredths, it is advisable to switch to a ROHF to avoid this “spin-contamination” problem.

Restricted Open-Shell Hartree–Fock (ROHF)

Appropriate for open-shell systems where spin-contamination is problem. Sometimes more difficult to converge, and assumes uniformly positive spin polarization (the alpha and beta doubly-occupied orbitals are identical).