Accurate Hartree - Fock vibrational branching ratios in 3 a , photoionisation of N 2

We report vibrational branching ratios for resonant photoionisation of N2 leading to the X *E; state of N:. Our theoretical values are obtained from an accurate solution of the adiabatic-nuclei frozen-core Hartree-Fock model of molecular photoionisation. In contrast to other theoretical results the present results are in very good agreement with experimental measurements. Differences between the present and previous calculations are discussed. The shape resonance occurring in the photoionisation of molecular nitrogen leading to the X ’1; state of the ion is known to produce significant non-Franck-Condon effects in the final vibrational state distributions. These vibrational effects were first predicted by Dehmer et a1 (1979) and have subsequently been observed experimentally by West etal (1980). The original prediction by Dehmer et a1 using the continuum multiple scattering method (CMSM) was qualitatively correct but their computed vibrational branching ratio for the v’ = 1 vibrational state relative to the v = 0 vibrational state was incorrect by approximately a factor of two. Recently Raseev et a1 (1980) have studied vibrational effects in this system using an accurate Hartree-Fock single-centre expansion method. The Y’ = l / v ’ = 0 branching ratio reported by Raseev et a1 is in much better agreement with the experimentally measured value of West et a1 (1980) than is the branching ratio obtained by Dehmer et a1 (1979). However, the results of Raseev et a1 are still low by a factor of about 25%. In the present study we have re-examined the calculations of Raseev et a1 (1980) to determine whether the difference between the computed and experimental branching ratio is due to a breakdown of the adiabatic-nuclei frozen-core Hartree-Fock model used in their study or whether the results given by Raseev et a1 were not fully converged solutions for this model. We have found two deficiencies in the calculation performed by Raseev et al. The most important shortcoming of their calculation is that Raseev et a1 only computed the electronic transition matrix elements for three internuclear separations and obtained the value of these matrix elements at all other points using a polynomial interpolation. We have found that it is important to compute the transition matrix elements at more internuclear separations since they are fairly rapidly varying functions of the internuclear separation. Another deficiency of the study of Raseev et a1 is that their potential expansion parameters are not well converged. We know from previous studies (Lucchese etal 1981) that with more accurate potential parameters the t Contribution No 6450. 0022-3700/81/200629 +06$01.50 @ 1981 The Institute of Physics L629 L630 Letter to the Editor peak photoionisation cross section in the fixed-nuclei approximation for the resonant 3o, + k o , channels of N2 is found to lie at a photon energy of 29 eV rather than at 3 1 eV as reported by Raseev et al. When these two problems are corrected we have found that the adiabatic-nuclei frozen-core Hartree-Fock v' = l / v ' = 0 ratios are in very good agreement with the experimental values of West et a1 (1980). We have repeated the calculations of Raseev et a1 (1980) correcting the two problems mentioned above. The frozen-core Hartree-Fock approximation was used to describe the interaction between the photoelectron and the ionic core. To obtain the appropriate continuum solutions we have used the Schwinger variational method (Lucchese and McKoy 1980). For the purpose of this study we have not employed the iterative technique which has been applied to other systems (Lucchese et a1 1980, Lucchese and McKoy 1981), since in our previous studies of the photoionisation of N2 we found that the exact iterative cross section is in general very close to the initial non-iterative result using only L2 basis functions (Lucchese et a1 1981). The scattering basis sets we have used for the 3a,+ k o , and 3a,+ krr, photoionisation channels of N2 are given in table 1. These basis sets consist of both Cartesian Gaussian functions defined by 4a,1,m,n,A (Y) = N ( x -A,)'(y A y ) m ( ~ -A,)"exp(-air -AI2) q5a913msA(r) =Nlr-Al' exp(-aIr-A12)Ylm(nr-A). (2) (1) and spherical Gaussian functions defined by The continuum solutions which are used to obtain the photoionisation cross section are given by $!iA(r) = #~i!l;'(r) + (rlG"-'Ula,)[D-111,(a,IU14ck~l;') (3) " , W , € R where [D-'I,, is the matrix inverse of D,, =(a,IUUG"'-'Ula,) (4) and where U is the static-exchange interaction potential minus the long range Coulomb potential of the ionic core, G"-' is the Coulomb Green's function and R is the appropriate scattering basis set given in table 1. All necessary integrals are computed by expanding all functions in truncated partial wave expansions with the resulting radial integrals put on a grid and computed using Simpson's rule. We have also constructed our continuum solutions subject to the constraint that they be orthogonal to the bound orbitals of the same symmetry. More details of this method can be found in previous papers (Lucchese et a1 1981, Lucchese et a1 1980, Lucchese and McKoy 1980). The Hartree-Fock (HF) target wavefunction was constructed from a (9s5p2d/4s3p2d) contracted Cartesian Gaussian basis set (Dunning 1970, Dunning 1971). The computed HF energy of N2 for the equilibrium nuclear separation of R = 2.068 au was E = -108.973 235 au, and the quadrupole moment for the neutral Nz molecule in this basis was -0.9923 au. The parameters used to expand the static-exchange potential were as follows: (i) 1, = 30, maximum 1 included in the expansion of the scattering functions and of (ii) I $ = 30, maximum 1 included in the expansion of the scsttering functions in the Coulomb Green's function,