This is the first blog for this experiment github blog web. As a test.
But what should i write for the very first blog?
Let me think.
Ok
Spin contamination
What is spin contamination, it occurs when we use UHF(abbr. for Unrestrictied Hartree Fock), in which we relax the spin orbitals so that the sptial part of such orbitals do not need to be the same between the $\alpha$ and $\beta$ ones. Therefore, in some cases such as bond dissociation we can evitate the error caused by the RHF(abbr. for Restricted Hartree Fock).
So basically:
In RHF:
$$ \psi^{\alpha}_{i} = \psi^{\beta}_{i} $$
While in UHF:
$$ \psi^{\alpha}_{i} \neq \psi^{\beta}_{i} $$
However, this makes the wavefunction formed not the eigenfunction of $\hat{S}^2$. Thus people express a new form of wavefunction $\ket{\phi}$ as the unretricted state expanded by exact singlet, doublet, triplet ...
in such that:
$$ \begin{align} \langle^{1} \phi \rangle = c^{1}_{1} \ket{1} + c^{1}_{3} \ket{3} + c^{1}_{5} \ket{5} + ... \\ \langle^{2} \phi \rangle = c^{2}_{2} \ket{2} + c^{2}_{4} \ket{4} + c^{2}_{6} \ket{6} + ... \\ \end{align} $$
the exact spin states mentioned here may be refered to as configuration state functions(CSFs), obtained by making certain operations on the unstricted wavefunctions.
Thus, such expansion introduces contaminations from higher spin states. This is exactly where spin contaminations come from.
Reference:
Szabo A, Ostlund N S. Modern quantum chemistry: introduction to advanced electronic structure theory[M]. Courier Corporation, 2012.