  The accurate description of nuclear quantum effects, such as zero-point
energy, is important for modeling a wide range of chemical and biological
processes. Within the nuclear-electronic orbital (NEO) approach, such effects
are incorporated in a computationally efficient way by treating electrons and
select nuclei, typically protons, quantum mechanically with molecular orbital
techniques. Herein, we implement and test a NEO coupled cluster method that
explicitly includes the triple electron-proton excitations, where two electrons
and one proton are excited simultaneously. Our calculations show that this
NEO-CCSD(eep) method provides highly accurate proton densities and proton
affinities, outperforming any previously studied NEO method. These examples
highlight the importance of the triple electron-electron-proton excitations for
an accurate description of nuclear quantum effects. Additionally, we also
implement and test the second-order approximate coupled cluster with singles
and doubles (NEO-CC2) method, as well as its scaled-opposite-spin (SOS)
versions. The NEO-SOS$'$-CC2 method, which scales the electron-proton
correlation energy as well as the opposite-spin and same-spin components of the
electron-electron correlation energy, achieves nearly the same accuracy as the
NEO-CCSD(eep) method for the properties studied. Because of its low
computational cost, this method will enable a wide range of chemical and
photochemical applications for large molecular systems. This work sets the
stage for a wide range of developments and applications within the NEO
framework.
