We investigate the exact analytic structure of fermionic many-body excited states through a case study of the harmonic problem. By analyzing highly degenerate eigenstates using a group-theoretic approach, we can reconstruct each eigenspace according to the exchange symmetry, isolating a small subset of bosonic and fermionic states from a much larger set of nonsymmetric states. Moreover, we show that the fermionic and bosonic states are in one-to-one correspondence. Furthermore, we demonstrate that the exact fermionic wave functions can be expressed in a factorized form consisting of a bosonic ground-state wave function and an antisymmetrized multivariate polynomial, an algebraic structure which we refer to as the antisymmetrizer. The antisymmetrizer naturally embodies an explicit dependence on interparticle distances, reflecting the two-body interactions in the Hamiltonian. We show that this representation is significantly more efficient than conventional multideterminant expansions, providing valuable insights into the development of approximate yet efficient representations of excited-state wave functions in more realistic chemical systems.

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• Electronic structure of atoms & molecules