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Unrestricted Hartree–Fock (UHF) theory is the most common molecular orbital method 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 where the number of electrons of each spin are not equal. While restricted Hartree–Fock theory uses a single molecular orbital twice, one multiplied by the α spin function and the other multiplied by the β spin function in the
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electro ...
, unrestricted Hartree–Fock theory uses different molecular orbitals for the α and β electrons. This has been called a ''different orbitals for different spins'' (DODS) method. The result is a pair of coupled
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 ...
, known as the Pople–Nesbet–Berthier equations. :\mathbf^\alpha\ \mathbf^\alpha\ = \mathbf \mathbf^\alpha\ \mathbf^\alpha\ :\mathbf^\beta\ \mathbf^\beta\ = \mathbf \mathbf^\beta\ \mathbf^\beta\ Where \mathbf^\alpha\ and \mathbf^\beta\ are the Fock matrices for the \alpha\ and \beta\ orbitals, \mathbf^\alpha\ and \mathbf^\beta\ are the matrices of coefficients for the \alpha\ and \beta\ orbitals, \mathbf 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 \mathbf^\alpha\ and \mathbf^\beta\ are the (diagonal, by convention) matrices of orbital energies for the \alpha\ and \beta\ orbitals. The pair of equations are coupled because the Fock matrix elements of one spin contains coefficients of both spin as the orbital has to be optimized in the average field of all other electrons. The final result is a set of molecular orbitals and orbital energies for the α spin electrons and a set of molecular orbitals and orbital energies for the β electrons. This method has one drawback. A single
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electro ...
of different orbitals for different spins is not a satisfactory eigenfunction of the total spin operator - \mathbf^2. The ground state is
contaminated Contamination is the presence of a constituent, impurity, or some other undesirable element that spoils, corrupts, infects, makes unfit, or makes inferior a material, physical body, natural environment, workplace, etc. Types of contamination ...
by excited states. If there is one more electron of α spin than β spin, the ground state is a doublet. The average value of \mathbf^2, written \langle \mathbf^2 \rangle , should be \tfrac(\tfrac + 1) = 0.75 but will actually be rather more than this value as the doublet state is contaminated by a quadruplet state. A triplet state with two excess α electrons should have \langle \mathbf^2 \rangle = 1 (1 + 1) = 2, but it will be larger as the triplet is contaminated by a quintuplet state. When carrying out unrestricted Hartree–Fock calculations, it is always necessary to check this contamination. For example, with a doublet state, if \langle \mathbf^2 \rangle = 0.8 or less, it is probably satisfactory. If it is 1.0 or so, it is certainly not satisfactory and the calculation should be rejected and a different approach taken. It requires experience to make this judgment. Even singlet states can suffer from spin-contamination, for example the H2 dissociation curve is discontinuous at the point when spin-contamination states (known as the Coulson–Fischer point). Despite this drawback, the unrestricted Hartree–Fock method is used frequently, and in preference to the
restricted open-shell Hartree–Fock 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 si ...
(ROHF) method, because UHF is simpler to code, easier to develop
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 rep ...
methods with, and returns unique functions unlike ROHF where different Fock operators can give the same final wave function. Unrestricted Hartree–Fock theory was discovered by Gaston Berthier and subsequently developed by
John Pople Sir John Anthony Pople (31 October 1925 – 15 March 2004) was a British theoretical chemist who was awarded the Nobel Prize in Chemistry with Walter Kohn in 1998 for his development of computational methods in quantum chemistry. Early ...
; it is found in almost all ab initio programs.


References

{{DEFAULTSORT:Unrestricted Hartree-Fock Electronic structure methods