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State-of-the-art multi-reference energy density functional calculationsrequire the computation of norm overlaps between different Bogoliubovquasiparticle many-body states. It is only recently that the efficient andunambiguous calculation of such norm kernels has become available under theform of Pfaffians~[L. M. Robledo, Phys. Rev. C79, 021302 (2009)].  The goals of this work is (i) to propose and implement an alternative to thePfaffian method to compute unambiguously the norm overlap between arbitraryBogoliubov quasiparticle states and (ii) to extend the first point toexplicitly correlated norm kernels at play in recently developpedparticle-number-restored Bogoliubov coupled-cluster (PNR-BCC) andparticle-number-restored many-body perturbation (PNR-BMBPT) ab initiotheories~[T. Duguet and A. Signoracci, J. Phys. G44, 015103 (2017)]. Point (i)constitutes the purpose of the present paper while point (ii) is addressed in aforthcoming companion paper.  We generalize the method used in~[T. Duguet and A. Signoracci, J. Phys. G44,015103 (2017)] to obtain the norm overlap between arbitrary Bogoliubov productstates under a closed-form expression. The formula is physically intuitive,accurate, versatile and relies on elementary linear algebra operations. Itequally applies to norm overlaps between Bogoliubov states of even or oddnumber parity. Numerical applications illustrate these features and provide atransparent representation of the content of the norm overlaps. Furthermore,the closed-form expression extends naturally to correlated overlaps at play inPNR-BCC and PNR-BMBPT. As such, the straight overlap between Bogoliubov statesis the zeroth-order reduction of more involved norm kernels to be studied inthe forthcoming paper.

On the norm overlap between many-body states. I. Uncorrelated overlap between arbitrary Bogoliubov product states