, antiadiabatic conditions are likely to occur in excitonic systems where the characteristic frequencyω could be easily of order of 1eV and the boson can therefore follow the electron eesentially without retardation. In these cases the carriers effective mass is expected to be only moderately enhanced with respect to the bare electron mass.
In Section II, the dispersive Holstein model is briefly reviewed while the numerical results are displayed in Section III. Some conclusions are drawn in Section IV.

II. HOLSTEIN MODEL WITH DISPERSIVE PHONONS
My starting point is the real space -momentum space representation of the Holstein Hamiltonian which reads creates (destroys) a tight binding electron at the i − th molecular site and t is the hopping integral related to the bare electron half bandwidth D by D = zt, z being the coordination number. ǫ is a reference electronic energy and r j is the j − th lattice site vector. N is the number of molecules in the lattice. It is understood that t differs from zero only between first neighbors sites and this poses a constraint on the i = j sum in the first addendum. a † k ( a k ) creates (destroys) a k-phonon with frequency ω k. The lattice dimensionality enters the problem through the phonon dispersion relations which have been obtained analytically by assuming first neighbors pairwise intermolecular forces both along the linear chain (1D), the square lattice (2D) and the simple cubic lattice (3D) [27]: where, c x = cosk x, c y = cosk y, c z = cosk z, s x = sink x etc. β is the intra-molecular force constant and γ is the inter-molecular first neighbors force constant. Let's define ω 2 0 = 2β/M and ω 2 1 = γ/M with M being the reduced molecular mass. N is the number of diatomic molecules in the lattice and g is the local electron-phonon coupling constant. The adiabatic parameter ishω/D,ω being a characteristic phonon frequency which we take as the zone center frequency and whose expression is: ω 2 = ω 2 0 + zω 2 1. Throughout this paper we fixhω 0 = 100meV and t = 15meV so that the antiadiabatic conditionhω/D > 1 is fulfilled in any dimensionality. Moreover, our perturbative approach requires the occurence of the condition D < g [28]. By applying the Lang-Firsov unitary transformation [29] and the subsequent 1/λ 0 expansion with λ 0 ≡ g 2 /(hω 0 D) being the ratio between polaron binding energy and electron half bandwidth [30], H of eq.(1) transforms intoH =H 0 +H P with: H 0 is diagonal except for a second order term in the electron density operator which could cause an attractive electron-electron interaction [8]. The perturbationH P displays the fundamental features of the polaronic quasiparticle as the hopping integral narrowing (first factor in eq.(3)) plus the peculiar mixing of fermionic and bosonic variables. At any electron-phonon interaction vertex m (n) phonons can be emitted from (absorbed by) the cloud surrounding the electron provided the total crystal momentum is conserved. By choosing a transformed ground state with no phonons we see that the first order dispersive contribution E (1) to the ground state polaron band arises only from the n = m = 0 term in eq.(3) hence from the zero phonon scattering process. In 3D and taking a lattice spacing a = |r i − r j | = 1, one finds where the total crystal momentum p coincides with the electron momentum due to the absence of self-energy cor-rections. The second order perturbative contribution requires summation over all intermediate states having m k-phonons more than the vacuum and one electron on a i first neighbor of the j initial site. Moreover, the final electronic position f can either coincide with j (this process does not introduce dispersive effects in the polaron band) or be a first neighbor of the i site. The latter event is clearly dimension dependent: in 1D the final site is a second neighbor of j, in 2D f can be either a second or a third neighbor of j and in 3D, also the fourth neighbor site can be reached via hopping. While the detailed study of these dispersive effects (which can become relevant in adiabatic conditions) is postponed to a next paper we turn now to compute the polaron effective mass. It should be remarked that the second order corrections decrease the mass values calculated in first order perturbative theory which therefore should be meant to provide upper bounds for the polaron mass.

III. POLARON EFFECTIVE MASSES
The polaron mass m * can be obtained according to the definition where m 0 is the bare band mass and the dispersive polaron band is given by eq.(4). The polaron binding energy has obviously no p-dispersion. Then, m * /m 0 is at first order independent of t and coincident simply with the reciprocal of the band narrowing factor. This picture holds in the strong coupling regime here assumed. In Fig.1(a), the ratios m * /m 0 for the 1D, 2D and 3D are computed versus the first neighbor intermolecular force constant ω 1. While the polaron masses strongly depend on the dimensionality d and are very large at small ω 1, the ratios become essentially d independent in the upper portion of the parameter range and tend to converge to 2. The value of the polaron binding energy λ 0 > 1 signals that the energy gain associated with the lattice deformation is larger than the kinetic energy due to the tight binding hopping in the bare band. Therefore it is energetically convenient to the electron to be dressed by the phonon cloud and become a quasiparticle. Actually, in antiadiabatic regimes, the more restrictive condition concerning the lattice deformation α 0 ≡ g/(hω 0 ) > 1 needs to be fulfilled to guarantee that our quasiparticle is a small polaron. While λ 0 and α 0 refer to a system with dispersionless phonons it is clear that both polaron binding energy and lattice deformation parameter will change after switching on the intermolecular interactions. The role of the intermolecular couplings is not simply that of increasing the characteristic phonon frequency but rather that of establishing the correct Holstein model dependence on dimensionality. Ignoring the intermolecular couplings would yield the 1D polaron band ∆E 1D larger than the 2D polaron band ∆E 2D and ∆E 2D > ∆E 3D which is clearly unphysical since the polaronic wave functions overlap is larger in higher dimensionality. This wrong trend would hold for any value of the intramolecular frequency ω 0. Then, as observed by Holstein himself in his original paper [11], the phonon dispersion is a vital ingredient of the theory and this observation motivates our numerical investigation. Because of our definitions λ 0 does not depend on d whereas α 0 is ∝ √ d. In Fig.1(b) we see that the lattice deformation α = N −1 g kh ω k −1 is in all dimensions a decreasing function of the intermolecular force constant and, in 1D, the system does not fulfill the small polaron condition at the largest ω 1 values. This case has been presented to point out how the starting condition α 1D 0 = 1.3 sets the 1D system rather in an intermediate coupling regime where a broadening of the polaron size can take place [31,32]. Under these conditions the same perturbative method based on the Lang Firsov transformation becomes questionable [33]. Belowω 1 = 48meV the polaron bandwidth inequalities ∆E 3D > ∆E 2D > ∆E 1D are not satisfied [25] as expected on general grounds hence, the dispersionless and the weakly dispersive Holstein Hamiltonian yield erroneous estimates of the effective masses. The straight line in Fig.1(b) marks therefore the lower bound for the intermolecular coupling which guarantees the validity of the model.   Fig.1(a) but with a dispersionless polaron binding energy λ0 = 10.9. (b) As in Fig.1(b) but with larger lattice deformation values. The lower bound for the validity of the model is set atω1 = 59meV.
Increasing the electron-phonon coupling, see Fig.2(a), leads to a strong mass enhancement (particularly in 3D) at small ω 1 while the mass ratios converge to 5 at large intermolecular coupling strenghts. Note (Fig.2(b)) that the polaron is small in all d throughout the whole ω 1 range hence the Lang Firsov method works well in this case. The threshold value for the validity of the Holstein model is set here atω 1 = 59meV. The inequalities m * 3D > m * 2D > m * 1D keep on being satisfied for a portion of ω 1 values above the threshold before convergence is achieved but second order perturbative terms (being larger in higher dimensionality) are expected to correct partly this trend. Figs.3 show that a stronger e-ph coupling, with λ 0 = 21.3, yields a mass ratio of ≃ 25 and shifts the thresholdω 1 at 65meV pointing out the relationship between features of the phonon spectrum and strength of the g coupling. Also in this case the po-laron size remains small throughout the whole ω 1 range (see Fig.3(b)) thus confirming the reliability of the Lang Firsov method in a strong e-ph coupling regime with antiadiabatic conditions. Next, I have varied g in the range 1 -4 (in units of hω 0 and found, at any g, the minimum intermolecular couplingω 1 (g) at which the bandwidth inequalities