  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 2x2 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.
