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 method is named after Douglas ...
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 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 mat ...
:
where
is the so-called
Fock matrix
The Fock matrix is defined by the Fock operator. In its general form the Fock operator writes:
:\hat F(i) = \hat h(i)+\sum_^ hat J_j(i)-\hat K_j(i)/math>
Where ''i'' runs over the total ''N'' spin orbitals. In the closed-shell case, it can be si ...
(which is a function of
),
is a matrix of coefficients,
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. The general principle for orbital overlap is that, the greater the overlap between ...
of the basis functions, and
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
wave function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
, total
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
, and other
observables
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum me ...
.
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 ...
(UHF), the ROHF wave function is a satisfactory eigenfunction of the total spin operator,
(i.e. no
spin contamination is observed).
Developing
post-Hartree–Fock (post-HF) 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 post-HFk methods have been implemented in a variety of electronic structure packages. Many (but not all) of these post-HF methods are completely
invariant 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 me ...
specifies the choice of orbitals.
References
Electronic structure methods
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