For the transitions upsilon J <--> upsilon'J' in the Rayleigh-Schrodinger perturbation theory, the rovibrational wave function is written in terms of the running number m as psi(vm) = Sigma(i=0) A(i)m(i) where m = [J'(J'+1)-J(J+1)]/2, and A(i) are new functions that can be obtained easily from the rotation harmonics psi(upsilon)((i)) defined in perturbation theory. By using this ''m-representation'' of the wave function we obtained the rovibrational matrix elements as M-upsilon m(upsilon'm) = M(upsilon O)(upsilon'O)G(upsilon upsilon')(m) where the rotational factor is given by G(upsilon upsilon')(m) = Sigma(i=0) gamma(i)m(i). The coefficients gamma(i) are simple combinations of integrals of the form < A(i)\f\A'(j) > and < A(i)\A'(j) > even for the high-order coefficients (up to the fourth order). These results are valid only in the case of the (1) Sigma electronic state for any potential (either analytical or numerical), any Vibrational level v (even near dissociation), and any operator f. The numerical application to the ground state of the molecule CO shows the high degree of accuracy of the present formulation with a simple algorithm and simple analysis tools.
