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Let's say I have a molecule and instead of using the Hartree-Fock procedure to find a Slater determinant wavefunction in a basis of atom-centered orbitals, I want to approximate the ground-state electronic wavefunction by finding the eigenvectors of the Hamiltonian in a specified basis (e.g. a bunch of basis functions centered at points on a grid). In a given basis, I can write (where psi are the basis functions):

\begin{bmatrix} < \psi_{1} | \hat{H} | \psi_{1} > & < \psi_{2} | \hat{H} | \psi_{1} > & ... \\ < \psi_{1} | \hat{H} | \psi_{2} > & < \psi_{2} | \hat{H} | \psi_{2} > & ... \\ ... & ... & ... \end{bmatrix}

with an NxN matrix for an N-sized set of basis functions, and generally with N eigenvectors, each an allowed definite-energy state of the system. The one corresponding to the lowest eigenvalue is my approximate ground-state wavefunction in this basis.

So far so fine, but I do have one concern. I thought this was a valid procedure but I don't think it generates a wavefunction which is antisymmetric with respect to exchange (unlike typical Slater determinant wavefunctions). Is that a problem and if so, since this procedure is correct in the limit (as basis size goes to infinity, this should generate a wavefunction which approaches exact), why is it a problem? If this wavefunction has to be antisymmetric with respect to exchange and is not, how would I make it antisymmetric? Do I need to?

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