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 m ...

, spin contamination is the artificial mixing of different

electron The electron ( or ) 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 particles because they have no kn ...

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

-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 mad ...

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, th ...

of the total spin-squared operator, ''Ŝ''², but can formally be expanded in terms of pure spin states of higher

multiplicities In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multip ...

(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 In quantum mechanics, an antisymmetrizer \mathcal (also known as antisymmetrizing operatorP.A.M. Dirac, ''The Principles of Quantum Mechanics'', 4th edition, Clarendon, Oxford UK, (1958) p. 248) is a linear operator that makes a wave function of ...

. 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 pl ...

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 Doublet is a word derived from the Latin ''duplus'', "twofold, twice as much",