  We present equations of motion (EOMs) for general time-dependent wave
functions with exponentially parametrized biorthogonal basis sets. The
equations are fully bivariational in the sense of the time-dependent
bivariational principle (TDBVP) and offer an alternative, constraint free
formulation of adaptive basis sets for bivariational wave functions. We
simplify the highly non-linear basis set equations using Lie algebraic
techniques and show that the computationally intensive parts of the theory are
in fact identical to those that arise with linearly parametrized basis sets.
Our approach thus offers easy implementation on top of existing code with
minimal computational overhead in the context of both nuclear dynamics and
time-dependent electronic structure. Computationally tractable working
equations are provided for single and double exponential parametrizations of
the basis set evolution. The EOMs are generally applicable for any value of the
basis set parameters, unlike the approach of transforming the parameters to
zero at each evaluation of the EOMs. We show that the basis set equations
contain a well-defined set of singularities, which are identified and removed
by a simple scheme. The exponential basis set equations are implemented in
conjunction with time-dependent vibrational coupled cluster with time-dependent
modals (TDMVCC) and we investigate the propagation properties in terms of the
average integrator step size. For the systems we test, the exponentially
parametrized basis sets yield slightly larger step sizes compared to linearly
parametrized basis set.
