Average-of-configuration open-shell Hartree-Fock
This is a short introduction to the theory behind average-of-configuration open-shell Hartree-Fock as implemented in DIRAC. For a more complete description the reader may consult chapter 3 of the PhD thesis of Jørn Thyssen [Thyssen2004] .
It should first be noted that there is no restricted open-shell Hartree-Fock (ROHF) code in DIRAC. The reason is that spin-orbit interaction couples spin and spatial degrees of freedom and make the formalism much more complicated since one cannot exploit spin symmetry alone for fixing the expansion coefficients in the reference configuration state function (CSF) which serves as the trial function.
Instead of optimizing the energy for a single open-shell state, we shall optimize the energy for a limited set of open-shell states.
Energy expression
Suppose that we have a set of
We will now find the set of orbitals which minimizes the average energy
Inserting the expansion of the solutions in terms of Slater determinants and using the fact that the expansion
coefficients
showing that the average can also be taken over the N-particle basis itself.
Introducing open shells and active electrons
The above average energy expression is a functional of the orbitals entering the Slater determinants. We will make a distinction between :
inactive orbitals, present in all Slater determinants, represented by indices
active orbitals, present in some, but not all Slater determinants, represented by indices
secondary orbitals, not present in any Slater determinant, represented by indices
We shall also employ indices
In order to generate our N-particle basis for averaging we will distribute the orbitals into a number of shells. Each shell
where we have introduced
the fractional occupation
of shellthe coupling coefficient
for for
two-electron contributions
Orbital rotations
We will use a exponential parametrization for the rotation of orbitals
where
Gradient elements and off-diagonal blocks of the Fock matrix
The generally non-zero elements of the gradient vector are:
inactive-secondary rotations:
active-secondary rotations:
inactive-active rotations:
inter-shell active-active rotations
where we have introduced
These gradient elements allow the definition of the off-diagonal elements of the Fock matrix:
inactive-secondary block:
active-secondary block:
inactive-active block:
inter-shell active-active rotations
:
Diagonal blocks of the Fock matrix
The diagonal blocks of the Fock matrix are a priori not related to gradient elements and there is therefore freedom of choice in their specification. The specific choice will not affect the total energy, but will affect orbitals energies as well as convergence of the AOC HF calculation.
In order to obtain a meaningful definition of the diagonal blocks of the Fock matrix we will consider an extension of
Koopmans’ theorem to average-of-configuration Hartree-Fock, that is, we consider average energy after removal of an electron
from a specific shell
The energy difference becomes
If we now define the diagonal block of the Fock matrix corresponding to shell
the ionization potential associated with the electron removal becomes
In the case of a degenerate shell we simply find
identical to the original Koopmans’ theorem, whereas one in the general case gets an average over the orbital energies of the shell.
Based on these observations we define the diagonal blocks of the AOC Fock matrix as
inactive-inactive block:
secondary-secondary block:
active-active block:
These are the definitions employed in DIRAC12 and onwards (and also the definition found in the thesis of Jørn Thyssen [Thyssen2004]).
In previous versions the term
Convergence problems typically occur when orbital energies between shells have similar values such that the selection of occupied orbitals for the construction of the Fock matrix becomes ambiguous. In a closed-shell system this will for instance happen when the HOMO-LUMO gap closes. The definition of the active-active block in pre-DIRAC12 version (which was in fact an unintended omittal) can in some instances lead to improved convergence. More specifically, this happens when the orbitals of an open shell and the closed shell (or another open shell) are almost degenerate. However, such situations are often symptomatic for a wrong choice of partitioning of orbitals into closed and open shells. Furthermore, the definition of the active-active block in pre-DIRAC12 versions tend to close the HOMO-LUMO gap which may hamper convergence.
Level shift
Whenever there is almost degeneracy of orbitals between different shells the recommended strategy is to exploit the freedom in the
definition of diagonal blocks of the Fock matrix and introduce a level shift
The level shift of secondary (virtual) orbitals is controlled by the keyword .LSHIFT, whereas open shells can be shifted using the keyword .OLEVEL.
Convergence issues
Open-shell systems tend to be more difficult to converge than closed-shell ones, because of additional orbital classes and more possibilities of near-degeneracies between orbital classes. It is important to understand that DIRAC will generally order orbitals
according to energy. Furthermore, DIRAC starts by filling closed-shell orbitals, then open-shell ones.
In the case of Uranium (