Multichannel quantum defect theory (MQDT) combined with the frame transformation concept is perhaps the most successful theoretical approach capable of treating electronically excited molecular systems beyond the Bo -Oppenheimer approximation. 1-4 It surpasses the traditional coupled equations approach in that it is not restricted to just a few excited states which have to be added to the treatment one by one but instead handles whole families of excited states (Rydberg series) up to the ionization thresholds and beyond. Furthermore, using the powerful concept of frame transformations, MQDT bypasses the evaluation of the electronic coupling matrix elements, which require the knowledge of numerical ab initio electronic wave functions. A quantitative comparison of the two approaches may be made by referring to two papers which appeared more than 10 years ago,5,6 where each theory was applied to the same vibronic problem. Numerous applications of MQDT to molecular spectroscopic problems have been published during the last decades, progressing from rotationally resolved absorption,4,7 emission,8 or multiphoton ionization spectroscopy and dynamics9,10 to hyperfine resolved millimeter wave and photoionization spectroscopy. 11-13 However, to our knowledge, no quantitative MQDT study of molecular line intensities of any molecule has been made so far in a situation where absolute measured line intensities were available for comparison. In this paper, we present a fully ab initio calculation of the line positions and intensities of the Q(N) (N ) 1-4) absorption transitions in molecular hydrogen H2, based on MQDT. We extract the quantum defects and channel dipole transition moments from the latest ab initio calculations of Wolniewicz and collaborators14-16 and use them in the framework of the nonadiabatic frame transformation-MQDT. We obtain overall good agreement with the recent synchrotron-based absolute intensity measurements of Glass-Maujean et al.17 It is wellknown that the energy positions of the excited levels of H2 of 1Πu - symmetry (which are the upper states reached by the Q transitions) are only little affected by vibronic interactions. By contrast, the intensities of the Q transitions are surprisingly strongly perturbed by the 1Πu - ∼ 1Πu - vibronic interactions in many instances, even when the line positions are apparently unperturbed. This fact had been established experimentally in ref 17 and is confirmed by the present calculations. 2. Determination of Quantum Defects and Channel Transition Moments from Quantum Chemical Data We use quantum defect theory in its simplest form by disregarding channel interactions between singly excited and doubly (core) excited Rydberg channels. In other words, we assume that the manifold of the 1Πu excited states of H2 represent a single unperturbed npπ Rydberg series converging to the X+ 2Σg + ground state of H2 +. Correlation between the excited electron and the ion core electron are included in an effective manner in the quantum defects because we extract the latter from highly accurate theoretical clamped nuclei (Bo - Oppenheimer) potential energy curves14,16 in which electron correlation has been fully accounted for. We use the familiar one-channel Rydberg equation written here in atomic energy units in order to extract the quantum defects for 1Πu symmetry. Unpπ(R) is the clamped nuclei curve for the npπ Rydberg state and U +(R) is the corresponding curve for the ion ground state, X+ 2Σg +. We used the data for n ) 2 (C state), 3 (D state), and 4 (D′ state) from ref 14 (see Figure 1), which yield the quantum defect curves for n ) 2, 3, and 4. The set of clamped nuclei quantum defect † Part of the “Robert W. Field Festschrift”. * To whom correspondence should be addressed. ‡ Universite´ Pierre et Marie Curie. § Universite´ de Paris-Sud. ⊥ University College London. Unpπ(R) ) U+(R) - 1 2[n - μnpπ(R)]2 (1) J. Phys. Chem. A XXXX, xxx, 000 A 10.1021/jp902846c CCC: $40.75 XXXX American Chemical Society Downloaded by AUSTRIA CONSORTIA on July 6, 2009 Published on June 23, 2009 on http://pubs.acs.org | doi: 10.1021/jp902846c curves thus obtained is subsequently represented by an energydependent polynomial of the form where ( R) ) Unpπ(R) - U+(R) is the binding energy of the Rydberg electron in the field of the core with the nuclei kept fixed at a distance R. (The fourth term on the rhs of eq 2 is discussed below.) The process of reducing the three available ab initio curves Unpπ (R) to the three functions μ(0)(R), μ(1)(R), and μ(2)(R) is illustrated in Figure 1a-c. Note that the functions μ(0)(R), μ(1)(R), and μ(2)(R) allow us to recover the potential energy curves for n ) 2-4 exactly (because for each R, three values have been represented in terms of three parameters) by means of eqs 1 and 2, but we also can predict all of the higher curves with n > 4. The reliability of this extrapolation (within reasonable limits) is suggested by the fact that the three quantum defect curves in Figure 1b nearly coincide. The last term in eq 2 corresponds to the “specific” mass effect (mass polarization term) arising from the cross term H3′ ) -(m/4M)∇1∇2 in the molecular Hamiltonian (where m is the electron mass and M is the nuclear reduced mass). This term couples the Rydberg and the core electrons and, as shown in ref 4, contributes a small mass-dependent correction to the quantum defect. In quantum chemical computations, the same term arises as part of the adiabatic corrections but unfortunately is rarely given separately when these are evaluated. Following ref 4, we take it here independent of R, and energy. We use the value μspecific ) -0.16 for the npπ1Πu channel. A similar procedure is applied to the ab initio dipole transition moments from refs 14 and 16. We first reduce the values DXfnpπ(R) (Figure 1d) to energy-normalized transition moments dXfnpπ(R) by multiplying by the square root of the density of states dν/d ) (n - μnpπ)3/2 (in au, where ν ) n - μ ) (-2)-1/2) (Figure 1e). It may be seen that the energy-normalized moments d, while exhibiting less energy dependence than the moments D shown at the top, are by no means as constant as their quantum defect counterparts represented in Figure 1b. The reasons for this will be discussed in section 6 below. We note on the other hand that the photon absorption from the ground state effectively takes place in a small region centered around the equilibrium position near 1.4 a0, where the energy dependence of the d’s is relatively mild. We finally represent these reduced moments for each R by an expression analogous to eq 2 The resulting channel dipole transition functions d(0)(R), d(1)(R), and d(2)(R) for excitation from the ground state to the pπ Rydberg channel are shown in Figure 1f. In agreement with what has just been said, the quantities d(1) and d(2) are seen to be rather small in the region around Re ) 1.4 a0.
