MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture # 18 Supplement Contents A. A Model for the Perturbations and Fine Structure of the Π States of CO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 B. Factorization of Perturbation Parameters . . . . . . . . . . . . . . . . . 2 C. The Electronic Perturbation Parameters . . . . . . . . . . . . . . . . . 3 A. A Model for the Perturbations and Fine Structure of the Π States of CO This paper reports the results of an analysis of the numerous perturbations of the CO a3Π and A1Π states, which belong to the (σ1s)2 (σ∗1s)2 (σ2s)2 (σ∗2s)2 (π2p)4 (σ2p) (π∗2p) electronic con-figuration (abbreviated σπ∗), by the a�3Σ+, e3Σ−, d3Δi, I1Σ− and D1Δ states, which belong to the (σ1s)2 (σ∗1s)2 (σ2s)2 (σ∗2s)2 (π2p)3 (σ2p)2 (π∗2p) configuration (abbreviated π3π∗). As many data as possible from the absorption, emission, and radio frequency spectra involving these states were combined in a uniform and systematic analysis. In several cases it was possible to use earlier data from the work of Ger¨o and Szabo[3] to augment the modern studies. The vibronic levels fitted are listed in Table I. The letters adjacent to each level indicate the sources of the data. The 47 horizontal rows contain the combinations of vibronic levels treated by degenerate perturbation theory. The choice of which levels to include in each of the groups was usually determined by selecting all of the nearest and strongest perturbers of a given a3Π or A1Π level. (The vibronic levels of the A and a states conveniently never occur at the same energy.) Therefore, the fitted a3Π and A1Π constants reported here should be regarded as fully deperturbed with respect to all of the nearest interacting levels. Extensive use was made of a nonlinear least squares fitting procedure which has been described earlier. The elements of the effective Hamiltonian matrix are given by Wicke et al.[2] 15.76 Lecture # 18 Supplement Page 2 The analysis of perturbations of the CO a3Π and A1Π states has led to two important conclusions. (i) For perturbations between vibronic levels belonging to a given pair of electronic states the perturbation matrix element is the product of a vibrational factor with a constant electronic factor. (ii) Simple, single-configuration arguments successfully predict the sign and relative magnitude of each of the electronic factors for perturbations between levels of each pair of electronic states. Thus all of the perturbations considered here can be related to two constants which are joint properties of the electronic configurations. It is now possible to calculate the interaction energy between any rovibronic level of one electronic configuration with those of the other configuration whenever the vibrational wave-functions are known. B. Factorization of Perturbation Parameters All perturbation parameters are listed in Table II. The electronic part of each perturbation param-eter was obtained by dividing the spin-orbit parameter, A or α, by the vibrational overlap �v|v��, or by dividing the rotation-electronic parameter, β, by 8πh 2cµ �v|r−2|v�� ≡ �v|B|v�� = Bvv� . For extremely weak perturbations, the beta parameters, denoted with an asterisk, were calculated by multiplying an average value of the assumed constant electronic factor by the proper vibrational factor. Also included in Table II are the r–centroids, �v|r|v��/ �v|v�� which indicate the internu-clear distance of maximum vibrational overlap (stationary phase point). For perturbations between any pair of electronic states, the r–centroid is nearly constant. For a given perturbing state the r–centroids are approximately 0.3 ˚A greater for perturbations of a3Π than of A1Π. This indicates that the overlap is greater at a larger internuclear distance for perturbations of the a3Π state than for the A1Π state. This result can be rationalized by examining the potential energy curves for CO. The inner wall of the A1Π state runs close to and parallel to the inner walls of the perturbing a�, e, d, D, and I states, whereas the curves of the perturbing states cross the a3Π state on the outer wall. For the pairs of states for which multiple determinations of the electronic part of the perturbation parameters were possible, the independence of this constant electronic factor with respect to v, v� is demonstrated in Table II. Thus, it is possible to partially evaluate second order effects by summations over all known vibrational levels of a particular electronic symmetry. Second-order constants for the a3Π state are calculated and discussed by Wicke et al.[2]5.76 Lecture # 18 Supplement Page 3 C. The Electronic Perturbation Parameters The relative signs and magnitudes of the electronic perturbation parameters have been derived considering only two electronic configurations: σπ∗ (σ1s)2(σ∗1s)2(σ2s)2(σ∗2s)2 (π2p)4(σ2p)(π∗2p) a3Π, A1Π π3π∗ (σ1s)(σ∗1s)2(σ2s)2(σ∗2s)2 (π2p)3(σ2p)2(π∗2p) a�3Σ+, e3Σ−, d3Δi, D1Δ, I1Σ−, 1Σ+ . The treatment which follows considers the six 2p electrons to the right of the line and neglects those to the left as core. The general procedures used to obtain wavefunctions in a Hund’s case a basis set and to calculate matrix elements are analogous to those given by Condon and Shortley,[4] Chapter 6. The Hamiltonian operators and symmetry operator, σv, used here are those given by Hougen[5] and are consistent with the phase convention of Condon and Shortley.[4] The complete wavefunctions of definite overall parity contain electronic, rotational, and vi-brational parts. Consider first the electronic part. In order to maintain a consistent arbitrary phase throughout the calculation, a standard order for the complete set of individual spin-orbital quantum numbers must be defined. The standard order for the 2p orbitals chosen here was (1+1− − 1+ − 1−0+0−1∗−1∗− − 1∗+ − 1∗−) where the numerals denote the values of m�, ± de-notes ms = ±12