SVT–OZ and notation
We derive the linear-algebra form of the SVT–OZ relation
[Seuferling et al., 1989, Shaffer et al., 2017] for a general \(M\)-component,
\(M\)-temperature plasma. First, we use the explicit \(M=2\) case as a
warm-up, then we extend this to the full formula for the \(M^2 \times M^2\)
matrix entries.
Consider \(M\) species (note that with electrons included in the
calculation, this means: M-1 ions plus electron) with number densities
\(\{n_a\}\), masses \(\{m_a\}\), and temperatures \(\{T_a\}\)
(\(a=1,\dots,M\)). In \(k\)-space, the multi-component SVT–OZ relation
can be obtained by extending the work of [Shaffer et al., 2017] through summation
over all species, yielding
\[\hat h_{ab}
=\hat c_{ab}
+\sum_{s=1}^M n_s\frac{m_{ab}\,T_{as}}{m_a\,T_{ab}}\;\hat c_{as}\hat h_{sb}
+\sum_{s=1}^M n_s\frac{m_{ab}\,T_{sb}}{m_b\,T_{ab}}\;\hat h_{as}\hat c_{sb},
\qquad a,b\in\{1,\dots,M\}.
\label{eq:svt-oz}\]
The mass-weighted cross temperatures and reduced masses are
\[\begin{split}\begin{aligned}
m_{ab}&=\frac{m_a m_b}{m_a+m_b}, &
T_{ab}&=\frac{m_a T_b + m_b T_a}{m_a+m_b}, \label{eq:def-Tab-mab}\\
T_{as}&=\frac{m_a T_s + m_s T_a}{m_a+m_s}, &
T_{sb}&=\frac{m_s T_b + m_b T_s}{m_s+m_b}.
\end{aligned}\end{split}\]
For brevity define
\[\alpha_{abs}=n_s\,\frac{m_{ab}\,T_{as}}{m_a\,T_{ab}},
\qquad
\beta_{abs}=n_s\,\frac{m_{ab}\,T_{sb}}{m_b\,T_{ab}}.
\label{eq:def-alpha-beta}\]
Moving all terms to the left, except \(\hat c_{ab}\).
(1)\[\hat h_{ab}
-\sum_{s=1}^M \alpha_{abs}\,\hat c_{as}\,\hat h_{sb}
-\sum_{s=1}^M \beta_{abs}\,\hat h_{as}\,\hat c_{sb}
=\hat c_{ab}.\]
In real space we use the HNC closure with the cross temperature
\(T_{ab}\),
\[\ln g_{ab}(r)=-\beta_{ab}V_{ab}(r)+N_{ab}(r),\qquad
\beta_{ab}=\frac{1}{k_B T_{ab}},\quad
h_{ab}=g_{ab}-1,\quad N_{ab}=h_{ab}-c_{ab}.
\label{eq:hnc-closure}\]
For isotropic fields we use the 3D radial transforms
\[\begin{aligned}
\hat f(k)&=\frac{4\pi}{k}\int_0^\infty r\,f(r)\sin(kr)\,dr,
&
f(r)&=\frac{1}{2\pi^2 r}\int_0^\infty k\,\hat f(k)\sin(kr)\,dk.
\label{eq:radialFT}
\end{aligned}\]
For two-component plasma \(M{=}2\) system
For \(M{=}2\), the four unknowns are
\[\mathrm{vec}(\hat H)=
\begin{bmatrix}\hat h_{11}&\hat h_{12}&\hat h_{21}&\hat h_{22}\end{bmatrix}^{\!\top}\!,
\qquad
\mathrm{vec}(\hat C)=
\begin{bmatrix}\hat c_{11}&\hat c_{12}&\hat c_{21}&\hat c_{22}\end{bmatrix}^{\!\top}.\]
Writing (1) for
\((a,b)=(1,1),(1,2),(2,1),(2,2)\) gives
\[\begin{split}\begin{aligned}
\hat{h}_{11} -
\alpha_{111}\, \hat{c}_{11}\, \hat{h}_{11} -
\alpha_{112}\, \hat{c}_{12}\, \hat{h}_{21} -
\beta_{111}\, \hat{h}_{11}\, \hat{c}_{11} -
\beta_{112}\, \hat{h}_{12}\, \hat{c}_{21} &=
\hat{c}_{11},\\
\hat{h}_{12} -
\alpha_{121}\, \hat{c}_{11}\, \hat{h}_{12} -
\alpha_{122}\, \hat{c}_{12}\, \hat{h}_{22} -
\beta_{121}\, \hat{h}_{11}\, \hat{c}_{12} -
\beta_{122}\, \hat{h}_{12}\, \hat{c}_{22} &=
\hat{c}_{12},\\
\hat{h}_{21} -
\alpha_{211}\, \hat{c}_{21}\, \hat{h}_{11} -
\alpha_{212}\, \hat{c}_{22}\, \hat{h}_{21} -
\beta_{211}\, \hat{h}_{21}\, \hat{c}_{11} -
\beta_{212}\, \hat{h}_{22}\, \hat{c}_{21} &=
\hat{c}_{21},\\
\hat{h}_{22} -
\alpha_{221}\, \hat{c}_{21}\, \hat{h}_{12} -
\alpha_{222}\, \hat{c}_{22}\, \hat{h}_{22} -
\beta_{221}\, \hat{h}_{21}\, \hat{c}_{12} -
\beta_{222}\, \hat{h}_{22}\, \hat{c}_{22} &=
\hat{c}_{22}.
\end{aligned}\end{split}\]
Collecting like terms yields the \(4\times4\) linear system
\[\begin{split}\underbrace{
\begin{pmatrix}
1 - (\alpha_{111} + \beta_{111})\hat{c}_{11} & -\beta_{112}\hat{c}_{21} & -\alpha_{112}\hat{c}_{12} & 0 \\
-\beta_{121}\hat{c}_{12} & 1 - \alpha_{121}\hat{c}_{11} - \beta_{122}\hat{c}_{22} & 0 & -\alpha_{122}\hat{c}_{12} \\
-\alpha_{211}\hat{c}_{21} & 0 & 1 - \alpha_{212}\hat{c}_{22} - \beta_{211}\hat{c}_{11} & -\beta_{212}\hat{c}_{21} \\
0 & -\alpha_{221}\hat{c}_{21} & -\beta_{221}\hat{c}_{12} & 1 - (\alpha_{222} + \beta_{222})\hat{c}_{22}
\end{pmatrix}
}_{\displaystyle A}
\;
\underbrace{
\begin{pmatrix}
\hat{h}_{11} \\
\hat{h}_{12} \\
\hat{h}_{21} \\
\hat{h}_{22}
\end{pmatrix}
}_{\displaystyle \mathrm{vec}(\hat H)}
=
\underbrace{
\begin{pmatrix}
\hat{c}_{11} \\
\hat{c}_{12} \\
\hat{c}_{21} \\
\hat{c}_{22}
\end{pmatrix}
}_{\displaystyle \mathrm{vec}(\hat C)}.
\label{eq:M2-matrix}\end{split}\]
Now we just need to solve this linear equation to get \(h_{ab}\)
From the \(M{=}2\) pattern to the general \(M\): a selector-\(\delta\) derivation
We now derive the general matrix entries \(A[p,q]\)
Flattening
Fix a wave number \(k\) and vectorize by lexicographic order. Using
0-based indexing,
\[p=\mathrm{idx}(a,b)=(a-1)M+(b-1),\qquad
q=\mathrm{idx}(u,v)=(u-1)M+(v-1).
\label{eq:idx}\]
Turn the single \(s\)-sum into a matrix–vector product
For a fixed row \((a,b)\) (i.e. fixed \(p\)), rewrite each term
of [eq:svt-oz-LHS] as a sum over all column
indices \((u,v)\) using Kronecker deltas that select which columns
are hit:
\[\begin{split}\begin{aligned}
\hat{h}_{ab} &= \sum_{u,v} \delta_{u,a} \delta_{v,b} \, \hat{h}_{uv}, \\
\sum_{s} \alpha_{abs} \, \hat{c}_{as} \, \hat{h}_{sb} &= \sum_{u,v} \left( \sum_{s} \alpha_{abs} \, \hat{c}_{as} \, \delta_{u,s} \, \delta_{v,b} \right) \hat{h}_{uv} = \sum_{u,v} \left( \alpha_{ab,u} \, \hat{c}_{au} \, \delta_{v,b} \right) \hat{h}_{uv}, \\
\sum_{s} \beta_{abs} \, \hat{h}_{as} \, \hat{c}_{sb} &= \sum_{u,v} \left( \sum_{s} \beta_{abs} \, \delta_{u,a} \, \delta_{v,s} \, \hat{c}_{sb} \right) \hat{h}_{uv} = \sum_{u,v} \left( \beta_{ab,v} \, \hat{c}_{vb} \, \delta_{u,a} \right) \hat{h}_{uv}.
\end{aligned}\end{split}\]
Therefore,
\[\sum_{u,v}\Big[\,
\delta_{u,a}\delta_{v,b}
-\alpha_{ab\,u}\,\hat c_{a u}\,\delta_{v,b}
-\beta_{ab\,v}\,\hat c_{v b}\,\delta_{u,a}\,\Big]\hat h_{uv}
=\hat c_{ab}.\]
Comparing with \(\sum_q A[p,q]\,H[q]=C[p]\) and
\(H[q]=\hat h_{uv}\), \(C[p]=\hat c_{ab}\), we define the
entry formula
\[\boxed{%
A\big[p,q]
=A\big[(a{-}1)M+(b{-}1),\ (u{-}1)M+(v{-}1)\big]
=\delta_{u,a}\delta_{v,b}
-\alpha_{ab\,u}\,\hat c_{a u}\,\delta_{v,b}
-\beta_{ab\,v}\,\hat c_{v b}\,\delta_{u,a}.}
\label{eq:A-elem}\]