  Understanding the nature of Cooper pairs is essential to describe the
properties of superconductors. The original proposal of Bardeen, Cooper, and
Schrieffer (BCS) was based on electrons pairing with same energy and zero
center-of-mass momentum. With the advent of new superconductors, different
forms of pairing have been discussed. In particular, Cooper pairs with finite
center-of-mass momentum have received large interest. Along with such
finite-momentum pairs, pairing of electrons at different energies is also
central to understanding some superconductors. Here, we investigate the
interplay of finite-momentum and finite-energy Cooper pairs considering two
different systems: a conventional $s$-wave superconductor under applied
magnetic field and a $d$-wave finite-momentum pairing state in the absence of
magnetic field relevant to correlated superconductors. Investigating both these
systems, we find finite-energy pairs persisting independently of
finite-momentum pairing, and that they lead to odd-frequency superconducting
correlations. We contrast this finding by showing that the even-frequency
correlations are predominantly driven by zero-energy pairs for most
frequencies. We further calculate the Meissner effect and find that
odd-frequency correlations are essential for correctly describing the Meissner
effect.
