Restricted open-shell Hartree–Fock (ROHF) is a variant of

Hartree–Fock method In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Hartree–Fock method often a ...

for

open shell In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom ...

molecules. It uses doubly occupied

molecular orbitals In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding ...

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 Clemens C. J. Roothaan (August 29, 1918 – June 17, 2019) was a Dutch physicist and chemist known for his development of the self-consistent field theory of molecular structure. Biography Roothaan was born in Nijmegen. He enrolled TU Delft in 19 ...

in a celebrated paper and then extended by various authors, see e.g. for in-depth discussions. As with restricted Hartree–Fock theory for closed shell molecules, it leads to

Roothaan equations The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type. It applies to closed-shell molecules or atoms where all molecular orbitals or atomic orbitals ...

written in the form of a

generalized eigenvalue problem In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...

\mathbf \mathbf = \mathbf \mathbf \mathbf

Where F is the so-called

Fock matrix In the Hartree–Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors. It is most often formed in computational chemistry when atte ...

(which is a function of C), C is a matrix of coefficients, S is the

overlap matrix In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. Linus Pauling explained the importance of orbital overlap in the molecular bond an ...

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 Unrestricted Hartree–Fock (UHF) theory is the most common molecular orbital method for open shell molecules where the number of electrons of each spin are not equal. While restricted Hartree–Fock theory uses a single molecular orbital t ...

(UHF), the ROHF wave function is a satisfactory eigenfunction of the total spin operator -

\mathbf^2

(i.e. no

Spin contamination In computational chemistry, spin contamination is the artificial mixing of different electronic spin-states. This can occur when an approximate orbital-based wave function is represented in an unrestricted form – that is, when the spatial p ...

). Developing

post-Hartree–Fock In computational chemistry, post-Hartree–Fock methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. They add electron correlation which is a more accurate way of including the repu ...

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 molecular orbitals. However, different choices of reference orbitals have shown to provide similar results, and thus many different

methods have been implemented in a variety of electronic structure packages. Many (but not all) of these post-Hartree–Fock methods are completely

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...

with respect to orbital choice (assuming that no orbitals are "frozen" and thus not correlated). The ZAPT2 version of

Møller–Plesset perturbation theory Møller–Plesset perturbation theory (MP) is one of several quantum chemistry post–Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by m ...

specifies the choice of orbitals.

References

Electronic structure methods {{Quantum-chemistry-stub