What does the χ (chi) symbol represent in a Slater determinant? [closed]








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Context: Computational Chemistry


So, I am looking at the molecular orbital approximation, which allows the assumption that the overall wavefunction can be written as individual molecular orbitals


Ψ(r1,r2..,rn) = ψ1(r1) x ψ2(r2) x .... ψn(rn) (the numbers and n are subscripted)


However, the above equation is incorrect because it does not take into account spin. Therefore, when spin is taken into account, the molecular orbital ψ(r) is replaced by molecular spin orbitals χ(r,ω). The r and ω are often combined and denoted as xi (the i is subscripted). Does this mean that χ = ψ? If that is true, I don't understand why the symbol was changed from ψ to χ? If someone could explain. Thank you.




I think maybe there is some confusion in terminology. Maybe these notes will help you:


1. When we take spin into account, we combine the **spatial orbital** of the ith electron $ \psi\_i(\mathbf{r}) $ that depends on the position with the **spin function** (of course we have two of them), $ \alpha(\omega) $ or $ \beta(\omega)$ that depend on the spin coordinate. Formally we don't combine $ \mathbf{r} $ and $ \omega $, we take the product of $ \psi\_i(\mathbf{r}) $ and $ \alpha(\omega) $ or $ \beta(\omega) $. However, we combine these two coordinates in $ \mathbf{x} = (\mathbf{r},\omega) $.
2. The product of this new function is the **spin orbital** $ \chi(\mathbf{x}) = \psi(\mathbf{r})\alpha(\omega) $ or $ \chi(\mathbf{x}) = \psi(\mathbf{r})\beta(\omega) $ .
3. What we want to calculate is the **N-electron wave function**, this is the guy you wrote there in the left-hand side $ \Phi(\mathbf{x}\_1, ...,\mathbf{x}\_N) $. However, when I read the title of the question you talk about the Slater determinant, maybe you wanted to write the following
\begin{equation}
 \Phi(\mathbf{x}\_1, ...,\mathbf{x}\_N) = \frac{1}{\sqrt{N!}} \det
 \begin{bmatrix}{}
 \chi\_i(\mathbf{x}\_1) & \chi\_j(\mathbf{x}\_1) & \cdots & \chi\_k(\mathbf{x}\_1) \\
 \chi\_i(\mathbf{x}\_2) & \chi\_j(\mathbf{x}\_2) & \cdots & \chi\_k(\mathbf{x}\_2) \\
 \vdots & \vdots & \vdots & \vdots \\
 \chi\_i(\mathbf{x}\_N) & \chi\_j(\mathbf{x}\_N) & \cdots & \chi\_k(\mathbf{x}\_N) \\
 \end{bmatrix}
\end{equation}


So, in summary:


1. When we take spin into account, we change $ \psi $ for $ \chi $.
2. What goes in the Slater determinant is always $ \chi $.
3. The N-electron wave function is the capital $ \Phi $.


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