  In this paper we present a simple strategy for the elimination of the
translational kinetic energy contamination of the total energy in
pre-Born--Oppenheimer calculations carried out in laboratory-fixed Cartesian
coordinates (LFCCs). The simple expressions for the coordinates and the
operators are thus preserved throughout the calculations, while the
mathematical form and the parametrisation of the basis functions are chosen so
that the translational and rotational invariances are respected. The basis
functions are constructed using explicitly correlated Gaussian functions (ECGs)
and the global vector representation. First, we observe that it is not possible
to parametrise the ECGs so that the system is at rest in LFCCs and at the same
time the basis functions are square-integrable with a non-vanishing norm. Then,
we work out a practical strategy to circumvent this problem by making use of
the properties of the linear transformation between the LFCCs and
translationally invariant and center-of-mass Cartesian coordinates as well as
the transformation properties of the corresponding basis function parameter
matrices. By exploiting these formal mathematical relationships we can identify
and separate the translational contamination terms in the matrix representation
of the kinetic energy operator in the LFCC formalism. We present numerical
examples for the translational contamination and its elimination for the two
lowest rotational energy levels of the singlet hydrogen molecule, corresponding
to para- and ortho-H2, respectively, treated as four-particle quantum systems.
