  In coupled cluster methods, the electronic states are biorthonormal in the
sense that the left states are orthonormal to the right states. Here we present
an extension of this formalism to a left and right total molecular wave
function. Starting from left and right Born-Huang expansions, we derive
projected Schr\"odinger equations for the left and right nuclear wave
functions. Observables may be extracted from the resulting wave function pair
using standard expressions. The formalism is shown to be invariant under
electronic basis transformations, such as normalization of the electronic
states. Consequently, the nonadiabatic coupling elements can be expressed with
biorthonormal wave functions. Calculating normalization factors that scale as
full-CI is therefore not necessary, contrary to claims in the literature. For
nuclear dynamics, we therefore need expressions for the vector and scalar
couplings in the biorthonormal formalism. We derive these expressions using a
Lagrangian formalism.
