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Expressing the vibrational Hamiltonian in terms of ladder operators, it can be divided in terms of order of magnitude of the coefficients or operator degree $${\cal H}\sum_n\sum_k\hbar_k\left(a^{i_k}_+a^{j_k}-a^{i_k}a^{j_k}a^{j_k}_+\right)\mbox{ where: }i_k + j_k = n$$ It can be applied a contact transformation to the Hamiltonian which can be expanded in terms of $commutators^{(1)}$ $${\cal H}^{\prime}=\sum^\infty_n\frac{(-1)^n}{n!}\left([S]\right)^nH$$ with the transformation operator S written similarly to the Hamiltonian $$S=\sum_n\sum_k\alpha_k\left(a^{i_k}_+a^{j_k}+a^{i_k}a^{j_k}_+\right)\mbox{ where: }i_k + j_k = n$$ the coefficients can be selected in order to obtain a transformed Hamiltonian which is diagonal up to the order of magnitude required. The contact transformation becomes a diagonalization procedure equivalent to perturbation theory. In contact transformation it is only necessary to obtained the commutator matrices between the operators used for the Hamiltonian and the transformation operator. In this work all the mathematical procedure developed in a general form have been truncated up to third order of magnitude. The diagonalized Hamiltonian have been used to fit experimental results for $H_{2}$, HD and $D_{2}^{(2,3)}$ spectra. The results are in accordance with the precision determined by the order of magnitude truncation.


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Author Institution: Area de Quimica Cu\`{a}ntica, Universidad Aut\'onoma Metropolitana-Iztapalapa. Av. Michoac\`{a}n y La purisima