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We present a new geminal product wave function ansatz where the geminals arenot constrained to be strongly orthogonal nor to be of seniority zero. Instead,we introduce weaker orthogonality constraints between geminals whichsignificantly lower the computational effort, without sacrificing theindistinguishability of the electrons. That is to say, the electron pairscorresponding to the geminals are not fully distinguishable, and their producthas still to be antisymmetrized according to the Pauli principle to form a\textit{bona fide} electronic wave function.Our geometrical constraintstranslate into simple equations involving the traces of products of our geminalmatrices. In the simplest non-trivial model, a set of solutions is given byblock-diagonal matrices where each block is of size 2x2 and consists of eithera Pauli matrix or a normalized diagonal matrix, multiplied by a complexparameter to be optimized. With this simplified ansatz for geminals, the numberof terms in the calculation of the matrix elements of quantum observables isconsiderably reduced. A proof of principle is reported and confirms that theansatz is more accurate than strongly orthogonal geminal products whileremaining computationally affordable.

2D-Block Geminals: a non 1-orthogonal and non 0-seniority model with reduced computational complexity