We present a new geminal product wave function ansatz where the geminals are
    not constrained to be strongly orthogonal nor to be of seniority zero. Instead,
    we introduce weaker orthogonality constraints between geminals which
    significantly lower the computational effort, without sacrificing the
    indistinguishability of the electrons. That is to say, the electron pairs
    corresponding to the geminals are not fully distinguishable, and their product
    has still to be antisymmetrized according to the Pauli principle to form a
    \textit{bona fide} electronic wave function.Our geometrical constraints
    translate into simple equations involving the traces of products of our geminal
    matrices. In the simplest non-trivial model, a set of solutions is given by
    block-diagonal matrices where each block is of size 2×2 and consists of either
    a Pauli matrix or a normalized diagonal matrix, multiplied by a complex
    parameter to be optimized. With this simplified ansatz for geminals, the number
    of terms in the calculation of the matrix elements of quantum observables is
    considerably reduced. A proof of principle is reported and confirms that the
    ansatz is more accurate than strongly orthogonal geminal products while
    remaining computationally affordable.

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