2 Theory

The spontaneous emission probability is given by the expression:

$$A(v_{j},v_{i};J_{j},J_{i}){=}\frac{4}{3\hbar^{4}c^{3}(2J_{j}+... ..._{v_{j}J_{j}}{-} E_{v_{i}J_{i}})^{3}\vert M_{S\alpha}\vert^{2}$$ (1)

where E_vJ is the energy of the level (v,J), $M_{S\alpha}$ is the electric dipole matrix element between wavefunctions of excited S(v_j,J_j) and ground electronic X(v_i,J_i) states, and $\alpha$ indicates whether the spectroscopic branch label is P, Q or R. When the emission takes place into the X continuum states, the expression is modified as:

$$A(v_{j},e_{i};J_{j},J_{i}){=}\frac{4}{3\hbar^{4}c^{3}(2J_{j}{... ..._{v_{j}J_{j}}{-} E_{e_{i}J_{i}})^{3}\vert M_{S\alpha}\vert^{2}$$ (2)

where the kinetic energy of dissociating atoms e_i replaces the vibrational indice v_i. The total part of the emission which produces dissociation is:

$$A_{{\rm c}}(v_{j};J_{j})=\sum_{J_{i}}\int_{0}^{\infty}A(v_{j},e_{i};J_{j},J_{i}){\rm d}e_{i}.$$ (3)

The mean kinetic energy of dissociating products is defined by:

$$\bar{E}_{k}(v_{j};J_{j})=\frac {\sum_{J_{i}}\int_{0}^{\infty}... ..._{i};J_{j},J_{i}){\rm d}e_{i}} {A_{{\rm c}}(v_{j};J_{j})}\cdot$$ (4)

The total emission probability is:

$$A_{{\rm t}}(v_{j};J_{j})=A_{{\rm c}}(v_{j};J_{j})+\sum_{v_{i},J_{i}}A(v_{j},v_{i};J_{j},J_{i}).$$ (5)

To go beyond the adiabatic approximation, we express the rovibronic wavefunction of excited states as an expansion over the electronic Born Oppenheimer (B.O.) wavefunctions. It is a good approximation to limit the expansion to the 4 B, C, B' and D states and we write $\Phi_{SvJ}=\sum_{T}\Psi_{TJ}f_{STvJ}$ where S, T are dummy labels for B, C, B' and D. Each $\Psi$ is the product of the electronic B.O. wavefunction and the pure rotational nuclear wavefunction. We calculate the vibrational function f_STvJ and the energy level E_vjJj by searching the eigenvalues of the coupled equations whose diagonal terms are adiabatic potentials and off-diagonal terms are rotational and radial electronic coupling matrix elements. The formalism is described in detail by Senn et al. ([1988]).

We define

$$\rho (T)= \int (f_{STvJ}(R))^{2}{\rm d}R$$ (6)

as the fraction of T in the state SvJ (the normalisation is such that ). Except for very few cases, one value of T is preponderant and by convention, in our tables, the label S stands for the state of greatest weight.

As in Abgrall et al. ([1999]), the dipole matrix elements $M_{S\alpha}$ appearing in Eq. (2) are given by the expressions (7):

$$\begin{array}{l} M_{{\rm SP}} = \\ \scriptstyle{ (J_{j}+1)... ...\vert f_{{\rm X}v_{i}J_{i}} \rangle \} \nonumber \end{array}$$

were $M_{{\rm BX}}$, $M_{{\rm CX}}$, $M_{{\rm B'X}}$ and $M_{{\rm DX}}$ are the real values of electronic transition moments calculated for each internuclear distance in the B.O. approximation.