https://samytoel.github.ioJust to quickly jog down something in case I forgetSamLiao's random notes/山姆廖的随便乱写http://www.rssboard.org/rss-specificationpython-feedgenhttps://github.githubassets.com/favicons/favicon.svghttps://samytoel.github.ioMon, 09 Mar 2026 01:46:47 +0000SamLiao's random notes/山姆廖的随便乱写60SamLiao's random notes/山姆廖的随便乱写https://samytoel.github.io/post/Hello%20world.htmlThis 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. 。https://samytoel.github.io/post/Hello%20world.htmlMon, 09 Mar 2026 00:39:42 +0000