You need to use 3N-7 (3N-6 for linear structures) mode for the saddle point, because the missing one has an imaginary frequency  (strictly speaking this mode is not a vibration rather a translational mode, it does not belong to a bound state). For reactants you have 3N-6 (3N-5 for linear structures), all of them must be included in the ZPE and also in the calculation of G (free energy).

ΔG(TS - Reactants) = G(TS with 3N-7) - G(Reactants with 3N-6)

Péter, one question out of interest - shouldn't the mode in question still be decoupled from any motion of the center of gravity of the entire system? That's how you obtain the normal modes, after all. But then it doesn't sound like a translational motion to me.

Including the normal mode leading to reaction provides a negative free energy of activation, because the reactant and the product are two conformers of n-pentane, thus they are drastically similar in their geometries, but they have huge ΔZPVE. Other source of errors (within the harmonic approximation and applying anharmonic correction) cannot occur, all calculations were performed with state-of-art computations within the focal-point approach (e.g. we used geometries close to the CCSD(T)/cc-pVTZ ones, extrapolated ΔHF, δMP2, δCCSD, δCCSD(T) terms, based on MP2/cc-pVXZ single point calculations up to X=5 and CCSD(T)/cc-pVXZ computations up to X=4). The free-energy differences of the conformers are in agreement with the values from the litterature.

Dear Matthias,

The mentioned mode is still decoupled from the center of mass. When we do the normal mode analysis of a molecule then the Hessian-matrix (the force constants matrix in harmonic approximation, the second derivative of the potential energy) is built up in the center of mass frame. The decoupling of the vibrational modes from the motion of the center of mass is always possible .  The normal mode analysis (the diagonalization of the Hessian-matrix) provides 3N-6 real vibrational modes for a molecule which contains N-atom. The remainder modes are the 3 translational and 3 rotational modes of the whole molecule. If you have a non-translating, non-rotating molecule in the energy minimum of the potential energy surface (PES) then the rotational and translational modes are theoretically decoupled and the 3-translational and 3-rotational modes have zero value frequencies. So, in an energy minimum the Hessian-matrix has 3N-6 positive eigenvalue (all of them positive).

The frequencies in cm-1 can be obtained from the eigenvalues:

                                     vi[cm-1]=sqrt(Li)/(2pi*c)           Eq (1.)

where, Li is the i'th eigenvalue of the Hessian, c is the speed of light.

If you perform a normal mode analysis not in the minimum, rather in a saddle point of the potential energy surface (that links two potential well, two stable molecular state), then you will obtain 3N-7 positive  and only one negative eigenvalue. The one negative eigenvalue will result an imaginary frequency according to the Eq. (1.). Look at the attached figure to understand why this mode with imaginary freq can be considered as a translational mode and not a vibrational one. The saddle point in the figure is called Transition Structure A. The saddle point is a metastable point of the potential energy surface. In most of the cases the reaction follows the minimum energy path on the  PES, this path is crosses the saddle point. If you put a dividing surface to this saddle point (that is perpendicular to the reaction path and it separates the reactants from the products), then the eigenvector with negative frequency will be also perpendicular to this dividing surface. In the direction of this eigenvector the PES has a maximum (this maximum is the saddle point), but in the other 3N-7 directions the PES has a minimum at the saddle-point. That's why the 3N-7 coordinates describe a vibration, a bound state motion. But the remainder one describes the crossing of the dividing surface, it is an unbound motion, like a translation. It describes the only possibility of the escape from this metastable state, it characterizes the leaving of the saddle point. The motion in the other 3N-7 directions can not result the leaving of the saddle point, because they describe a bound motion, a periodic motion around the saddle-point.

The normal mode analysis of a moving (translating+rotating) molecule is more complicated. The translational mode can be decoupled, but the decoupling of the rotation is the big problem. In this case you need to do the normal mode analysis in the Eckart-frame (the frame which is rotating and translating together with the molecule).

Thank you for that detailed reply, Péter. So it appears I was just confused about the use of the term "translational" here, because I always associate that with a center-of-mass motion of the entire molecular frame (ie, decoupled from internal motions). Good to know that there may be other uses for that word. I guess it makes sense to highlight the unbound character of that mode.