  A first principles quantum formalism to describe the non-adiabatic dynamics
of electrons and nuclei based on a second quantization representation (SQR) of
the electronic motion combined with the usual representation of the nuclear
coordinates is introduced. This procedure circumvents the introduction of
potential energy surfaces and non-adiabatic couplings, providing an alternative
to the Born-Oppenheimer approximation. An important feature of the molecular
Hamiltonian in the mixed first quantized representation for the nuclei and the
SQR representation for the electrons is that all degrees of freedom, nuclear
positions and electronic occupations, are distinguishable. This makes the
approach compatible with various tensor decomposition \emph{ans\"atze} for the
propagation of the nuclear-electronic wavefunction. Here, we describe the
application of this formalism within the multi-configuration time-dependent
Hartree (MCTDH) framework and its multilayer generalization, corresponding to
Tucker and hierarchical Tucker tensor decompositions of the wavefunction,
respectively. The approach is applied to the calculation of the
photodissociation cross-section of the HeH$^+$ molecule under extreme
ultraviolet irradiation, which features non-adiabatic effects and quantum
interferences between the two possible fragmentation channels, He+H$^+$ and
He$^+$+H. These calculations are compared with the usual description based on
ab \emph{ab initio} potential energy surfaces and non-adiabatic coupling matrix
elements, which fully agree. The proof-of-principle calculations serve to
illustrate the advantages and drawbacks of this formalism, which are discussed
in detail, as well as possible ways to overcome them.
