# Semi-canonicalisation vs canonicalisation of the Fock matrix and orbitals

I have seen the terms semi-canonicalized and canonicalized used in relation to the Fock matrix, density matrices, and orbitals; however, I am unsure what these terms actually describe.

For instance:

In the communication: ROHF theory made simple.

At convergence of the iterative procedure, the orbitals resulting from our optimization procedure are the same as the semicanonical orbitals previously proposed in the literature.

Or in the paper: Canonical density matrix perturbation theory.

The canonical density matrix perturbation theory can be used to calculate temperature-dependent response properties.

How do the semi-canonicalized and canonicalized terms relate to the Fock matrix, density matrices, and orbitals? Is it possible to switch between them?

For simplicity, I will stick to the restricted Hartree-Fock level of theory since the question of canonical and semi-canonical orbitals already exists there.

Let's remember the SCF equations: $${\bf F C} = {\bf SCE}$$, where $${\bf F}$$ and $${\bf S}$$ are the Fock and overlap matrices, with $${\bf C}$$ the orbital coefficients and $${\bf E}$$ the corresponding orbital energies.

Left-projecting the SCF equation by $${\bf C}^{\rm T}$$ gives $${\bf C}^{\rm T} {\bf F C} = {\bf E}$$, since $${\bf C}^{\rm T}{\bf SC}={\bf 1}$$ is the basis set version of the orbital orthonormality condition $$\langle i | j \rangle = \delta_{ij}$$.

We can identify $${\bf C}^{\rm T} {\bf F C}$$ as the Fock matrix in the molecular orbital basis, $${\bf F}^{\rm MO} = {\bf C}^{\rm T} {\bf F C}$$.

By definition, canonical orbitals diagonalize the Fock matrix: $$\boldsymbol{F}^{\text{MO}}=\left(\begin{array}{ccc} \epsilon_{1} & \cdots & 0\\ \vdots & \ddots & \vdots\\ 0 & \cdots & \epsilon_{n} \end{array}\right)$$

and typically, the first $$N$$ orbitals are occupied.

Semicanonical orbitals only diagonalize the occupied-occupied and the virtual-virtual blocks, while the occupied-virtual and virtual-occupied blocks may be nonzero: $$\boldsymbol{F}^{\text{MO}}=\left(\begin{array}{cc} \boldsymbol{\epsilon}_{o} & \boldsymbol{\Delta}_{ov}\\ \boldsymbol{\Delta}_{vo} & \boldsymbol{\epsilon}_{v} \end{array}\right)$$.

Once you have defined the orbitals via the Fock matrices, you can build density matrices.

It is in general not possible to switch between the canonical and the semicanonical forms, since the transformation to canonize semicanonical orbitals may change the orbitals in a way that is not allowed by the theory.

For instance, semicanonical orbitals are used in several self-consistent field convergence algorithms in order to precondition the descent direction. Semicanonization does not affect the energy of the wave function at the SCF level of theory, meaning you can diagonalize the Fock matrix in the occupied and virtual blocks; then, you have a pretty good estimate for the diagonal Hessian as $$\epsilon_{a}-\epsilon_{i}$$ where $$\epsilon_a$$ and $$\epsilon_i$$ denote virtual and occupied orbital diagonal values.

The semicanonical and canonical orbitals are only the same in SCF when the orbitals satisfy the SCF equations, i.e. the occupied-virtual gradients vanish, $$\boldsymbol{\Delta}_{ov}={\bf 0}$$.

PS. the second paper you linked talks about "canonical (NVT) free-energy ensembles" which is a thermodynamical concept which should not be confused with the present context of orbitals.