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computational chemistry Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of mo ...
, spin contamination is the artificial mixing of different
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary partic ...
ic spin-states. This can occur when an approximate orbital-based
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
is represented in an unrestricted form – that is, when the spatial parts of α and β
spin-orbital In atomic theory and quantum mechanics, an atomic orbital is a Function (mathematics), function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electr ...
s are permitted to differ. Approximate wave functions with a high degree of spin contamination are undesirable. In particular, they are not
eigenfunctions In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, t ...
of the total spin-squared operator, ''Ŝ''2, but can formally be expanded in terms of pure spin states of higher multiplicities (the contaminants).


Open-shell wave functions

Within Hartree–Fock theory, the wave function is approximated as a
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 spin-orbitals. For an open-shell system, the mean-field approach of Hartree–Fock theory gives rise to different equations for the α and β orbitals. Consequently, there are two approaches that can be taken – either to force double occupation of the lowest orbitals by constraining the α and β spatial distributions to be the same (
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 simp ...
, ROHF) or permit complete variational freedom (
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). In general, an ''N''-electron Hartree–Fock wave function composed of ''N''α α-spin orbitals and ''N''β β-spin orbitals can be written as :\Psi^(\mathbf_\sigma(1)\cdots\mathbf_\sigma(N)) = \mathcal\left(\psi_^(\mathbf_\alpha_)\cdots\psi_^(\mathbf_\alpha_) \psi_^(\mathbf_\beta_)\cdots\psi_^(\mathbf_\beta_)\right). where \mathcal is the antisymmetrization operator. This wave function is an eigenfunction of the total spin projection operator, ''Ŝ''z, with eigenvalue (''N''α − ''N''β)/2 (assuming ''N''α ≥ ''N''β). For a ROHF wave function, the first 2''N''β spin-orbitals are forced to have the same spatial distribution: :\psi^_(\mathbf_) = \psi^_(\mathbf_),\ \ \ 1\leq j\leq N_. There is no such constraint in an UHF approach.


Contamination

The total spin-squared operator commutates with the nonrelativistic
molecular Hamiltonian In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation p ...
so it is desirable that any approximate wave function is an eigenfunction of ''Ŝ''2. The eigenvalues of ''Ŝ''2 are ''S''(''S'' + 1) where ''S'' can take the values 0 ( singlet), 1/2 ( doublet), 1 ( triplet), 3/2 (quartet), and so forth. The ROHF wave function is an eigenfunction of ''Ŝ''2: the expectation value ''Ŝ''2 for a ROHF wave function is : \langle S^\rangle_ = \langle S^\rangle_ =\left(\frac\right)\left(\frac+1\right). However, the UHF wave function is not: the expectation value of ''Ŝ''2 for an UHF wave function is : \langle S^\rangle_ = \langle S^\rangle_ + N_ - \sum_^, \langle\psi_^, \psi_^\rangle, ^. The sum of the last two terms is a measure of the extent of spin contamination in the unrestricted Hartree–Fock approach and is always non-negative – the wave function is usually contaminated to some extent by higher order spin eigenstates unless a ROHF approach is taken. Naturally, there is no contamination if all electrons are the same spin. Also, there is often no contamination if the number of α and β electrons is the same. A small basis set could also constrain the wavefunction sufficiently to prevent spin contamination. Such contamination is a manifestation of the different treatment of α and β electrons that would otherwise occupy the same molecular orbital. It is also present in
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 ...
calculations that employ an unrestricted wave function as a reference state (and even some that employ a restricted wave function) and, to a much lesser extent, in the unrestricted Kohn–Sham approach to
density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-bo ...
using approximate exchange-correlation functionals.


Elimination

Although the ROHF approach does not suffer from spin contamination, it is less commonly available in
quantum chemistry computer programs Quantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock (HF) and some post-Hartree–Fock methods. They may also include density functional theory (DF ...
. Given this, several approaches to remove or minimize spin contamination from UHF wave functions have been proposed. The annihilated UHF (AUHF) approach involves the annihilation of first spin contaminant of the density matrix at each step in the self-consistent solution of the Hartree–Fock equations using a state-specific Löwdin annihilator. The resulting wave function, while not completely free of contamination, dramatically improves upon the UHF approach especially in the absence of high order contamination. Projected UHF (PUHF) annihilates all spin contaminants from the self-consistent UHF wave function. The projected energy is evaluated as the expectation of the projected wave function. The spin-constrained UHF (SUHF) introduces a
constraint Constraint may refer to: * Constraint (computer-aided design), a demarcation of geometrical characteristics between two or more entities or solid modeling bodies * Constraint (mathematics), a condition of an optimization problem that the solution ...
into the Hartree–Fock equations of the form λ(''Ŝ''2 − ''S''(''S'' + 1)), which as λ tends to infinity reproduces the ROHF solution. All of these approaches are readily applicable to unrestricted
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 ...
.


Density functional theory

Although many
density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-bo ...
(DFT) codes simply calculate spin-contamination using the Kohn–Sham orbitals as if they were Hartree–Fock orbitals, this is not necessarily correct.


References

{{reflist Computational chemistry Quantum chemistry