Restricted open-shell Hartree–Fock explained

Restricted open-shell Hartree–Fock (ROHF) is a variant of Hartree–Fock method for open shell molecules. It uses doubly occupied molecular orbitals as far as possible and then singly occupied orbitals for the unpaired electrons. This is the simple picture for open shell molecules but it is difficult to implement.The foundations of the ROHF method were first formulated by Clemens C. J. Roothaan in a celebrated paper ^[1] and then extended by various authors, see e.g.^{[2] [3] [4]} for in-depth discussions.

As with restricted Hartree–Fock theory for closed shell molecules, it leads to Roothaan equations written in the form of a generalized eigenvalue problem

F C = S C \epsilon

Where F is the so-called Fock matrix (which is a function of C), C is a matrix of coefficients, S is the overlap matrix of the basis functions, and

\epsilon is the (diagonal, by convention) matrix of orbital energies. Unlike restricted Hartree–Fock theory for closed shell molecules, the form of the Fock matrix is not unique. Different so-called canonicalisations can be used leading to different orbitals and different orbital energies, but the same total wavefunction, total energy, and other observables.

In contrast to unrestricted Hartree–Fock (UHF), the ROHF wave function is a satisfactory eigenfunction of the total spin operator -

S ^{2} (i.e. no Spin contamination).

Developing post-Hartree–Fock methods based on a ROHF wave function is inherently more difficult than using a UHF wave function, due to the lack of a unique set of molecularorbitals.^[5] However, different choices of reference orbitals have shown to provide similar results,^[6] and thus many different post-Hartree–Fock methods have been implemented in a variety of electronic structure packages.Many (but not all) of these post-Hartree–Fock methods are completely invariant with respect to orbital choice (assuming that no orbitals are "frozen" andthus not correlated).^[7] The ZAPT2 version of Møller–Plesset perturbation theory specifies the choice of orbitals.^[8]

Notes and References

1. Roothaan. C. C. J. . 1960 . Self-consistent field theory for open shells of electronic systems . Reviews of Modern Physics . 32 . 2 . 179–185 . 10.1103/RevModPhys.32.179. 1960RvMP...32..179R .
2. Book: 978-0-387-08535-7 . 1978 . R. . Carbó. J. M.. Riera . 5 . Springer . 10.1007/978-3-642-93075-1_1. A General SCF Theory. Lecture Notes in Chemistry . 1–4. Historical Review .
3. Book: McWeeny , R.. Methods of Molecular Quantum Mechanics . 2nd . 1992 . Academic Press . 978-0-470-01187-4.
4. Book: Plakhutin , B. N.. K. D. . Sen . Reviews of Modern Quantum Chemistry . Word Scientific . 16–42 . 1 . 2002 . 978-981-02-4889-5.
5. Glaesemann . Kurt R. . Schmidt . Michael W. . On the Ordering of Orbital Energies in High-Spin ROHF† . The Journal of Physical Chemistry A . 114 . 33 . 8772–8777 . 2010 . 20443582 . 10.1021/jp101758y. 2010JPCA..114.8772G .
6. Book: Jensen , F.. Introduction to Computational Chemistry . 2nd . Wiley . 2007 . 978-0-471-98425-2.
7. 10.1063/1.471951 . On the energy invariance of open-shell perturbation theory with respect to unitary transformations of molecular orbitals . 1996 . Crawford . T. Daniel . Schaefer . Henry F. . Lee . Timothy J. . The Journal of Chemical Physics . 105 . 3 . 1060. 1996JChPh.105.1060C .
8. 18298140. 2008. Wheeler. S. E. On the convergence of Z-averaged perturbation theory. The Journal of Chemical Physics. 128. 7. 074107. Allen. W. D. Schaefer Hf. 3rd. 10.1063/1.2828523. 2008JChPh.128g4107W.