  We derive a family of equations of motion (EOMs) for evolving multi-layer
multiconfiguration time-dependent Hartree (ML-MCTDH) wavefunctions that, unlike
the standard ML-MCTDH EOMs, never require the evaluation of the inverse of
singular matrices. All members of this family of EOMs make use of alternative
static gauge conditions than that used for standard ML-MCTDH. These alternative
conditions result in an expansion of the wavefunction in terms of a set of
potentially arbitrary orthonormal functions, rather than in terms of a set of
non-orthonormal and potentially linearly dependent functions, as is the case
for standard ML-MCTDH. We show that the EOMs used in the projector splitting
integrator (PSI) and the invariant EOMs approaches are two special cases of
this family obtained from different choices for the dynamic gauge condition,
with the invariant EOMs making use of a choice that introduces potentially
unbounded operators into the EOMs. As a consequence, all arguments for the
existence of parallelizable integration schemes for the invariant EOMs can also
be applied to the PSI EOMs.
