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Suppose I have two fermionic number states in different bases, with the same particle number $N$ - call them $|\Psi\rangle$ and $|\Phi\rangle$. In the position basis, I can write the many-body wavefunctions as the Slater determinants: $$ \Psi(x_1,x_2,\dots)=\frac{1}{\sqrt{N!}}\det(\overline{\psi}) $$ $$ \Phi(x_1,x_2,\dots)=\frac{1}{\sqrt{N!}}\det(\overline{\phi}) $$ where: $$\overline{\psi}=\begin{bmatrix}\psi_1(x_1)&\psi_2(x_1)&\dots&\psi_N(x_1)\\\psi_1(x_2)&\psi_2(x_2)&\dots&\psi_N(x_2)\\\vdots&\vdots&\ddots&\vdots\\\psi_1(x_N)&\psi_2(x_N)&\dots&\psi_N(x_N)\end{bmatrix}$$ $$\overline{\phi}=\begin{bmatrix}\phi_1(x_1)&\phi_2(x_1)&\dots&\phi_N(x_1)\\\phi_1(x_2)&\phi_2(x_2)&\dots&\phi_N(x_2)\\\vdots&\vdots&\ddots&\vdots\\\phi_1(x_N)&\phi_2(x_N)&\dots&\phi_N(x_N)\end{bmatrix}$$

Suppose now that I know the $\psi_n,\phi_n$, and I wish to calculate $\langle\Psi|\Phi\rangle=\int d\mathbf{x} \Psi^*(\mathbf{x})\Phi(\mathbf{x}) $. Is there a tractable way to compute this? I guess I could use the fact that $\det(\overline{\Psi}^\dagger\overline{\Phi})=\det(\overline{\Psi}^\dagger)\det(\overline{\Phi})$ to construct the integrand, then integrate over all spatial dimensions. But this seems truly horrendous: an $N\times N$ matrix multiplication and determinant calculation, then $N$ spatial integrations. I don't believe it could actually be that complicated. Is there a simplification I'm missing that will let me tractably compute this? I am hoping it would reduce to something like this: $$ \sum_n\int dx \phi_n^*(x)\psi_n(x) $$ but properly antisymmetrized so that it is invariant under exchange of particle identities. I'm totally stumped on how to simplify to a form like that, though.

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The overlap of the two Slater determinants is given by the determinant of the matrix containing all mutual orbital overlaps. The paper "Efficient and Flexible Computation of Many-Electron Wave Function Overlaps" J. Chem. Theory Comput., 2016, 123, 1207-1219 has the derivation you are looking for and goes into great detail about an efficient implementation.

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