Deperturbation

The fractionation patterns within each polyad provide a different kind of deperturbed information that is useful for refining the global V(Q)- The frac-... 

Figure 1. Unzipped polyads (bottom) from the C2H2 A 0° DF spectrum as progressions in 04 (trans bend). Each row has constant vz (CC stretch) and each column has constant 04. Integrated intensity (deperturbed Franck-Condon factor) for each polyad in the 0° DF spectrum, arranged as progressions in 04 (top). Shading indicates the value of V2 for the progression.

The combination of low-resolution and spectral unzipping into noninteracting polyads enables systematic, model-free surveys of deperturbed Franck-Condon factors, deperturbed zero-order energy levels, and trends in intramolecular vibrational redistribution (IVR) rates and pathways [3]. The H[ res,/i polyad model permits extraction of the most important resonance strengths directly from fits to a few polyads [6-8]. Once these anharmonic... 

Pazyuk, E.A., Stolyarov, A.V., Tamanis, M.Ya. and Ferber, R.S. (1993). Global deperturbation analysis from energetic, magnetic and radiative measurements Application to Te2, J. Chem. Phys., 99, 7873-7887. 

A fundamental difficulty with the theory of autoionisation is that the so-called prediagonalised state does not exist as such. This is a standard situation in perturbation theory, where the unperturbed Hamiltonian is simply a mathematical convenience, because the system cannot physically be deperturbed. It does, however, have the serious consequence that one cannot, at first sight, check the theory in any quantitative sense. Methods of overcoming this limitation will be described in chapter 8... 

A fundamental issue in the description of even the simplest, isolated autoionising resonance in the parametric approach followed by Fano [391] - and further pursued in K-matrix theory - is that the atom cannot be deperturbed, that is one cannot access the so-called prediagonalised states which are imagined to exist prior to autoionisation being included as a perturbative interaction, since the effect is anyway internal to the atom and cannot truly be turned off. This has the disadvantage that the parameters, once they have been obtained, must still be calculated from an ab initio model of the atom for a full comparison with theory. It might seem that the parametric theory cannot really be checked independently of ab initio calculations whose accuracy is hard to ascertain. 

In this book we regard the following phenomena as perturbations and attempt to display the similarities and relationships between the methods and parameters used to explain (deperturb) them ... 

The deperturbation models provide a uniform explanation of level patterns, transition intensities (Chapter 7), radiative lifetimes, linewidths, and systematic variations of coupling matrix elements with quantum numbers. 

In short, this book is an attempt to demystify perturbations by providing a minimally elegant but maximally unified view of the deperturbation process and by suggesting what can be learned from perturbations. 

It is not possible to give a unique definition of a diabatic potential curve without identifying the specific term in Hel that is excluded. The impossibility of identifying such a term and the consequent nonuniqueness of the a priori definition of diabatic curves is discussed by Lewis and Hougen (1968), Smith (1969), and Mead and Truhlar (1982) (See also Section 3.3.2). Diabatic curves may be defined empirically (Section 3.3) by assuming a deperturbation model [e.g., that HffiR) is independent of R or, at most, varies linearly with R]. 

Here the electronic wavefunctions can be taken as nonsymmetrized or symmetrized. The matrix elements are identical in either case. The ideal starting point would be a basis set, i and 2, that minimizes the values of the Eq. (3.3.1) off-diagonal matrix elements. Unfortunately, it is not possible to find solutions of the electronic Hamiltonian for which both terms of Eq. (3.3.1) are zero. Two possible types of deperturbed or zeroth-order electronic functions may be defined (see also Table 3.1) ... 

In principle, whatever the initial model, after introducing the vibronic coupling terms corresponding to the chosen type of deperturbed potential curves, the experimental energy levels are obtained by diagonalizing one or the other type of interaction matrix. One example will be discussed later (Section 3.3.4). 

If the deperturbed curves intersect and are characterized by very different molecular constants, then they are diabatic curves (see Fig. 3.5). If the crossing is avoided, adiabatic curves are involved, and one of these curves can have a double minimum. One frequently finds that, in the region of an avoided crossing, the adiabatic wavefunction changes electronic character abruptly and the derivative of the electronic wavefunction with respect to R can be large. In... 

Approximate deperturbed curves can be derived from unperturbed vibrational levels far from the energy of the curve crossing region. The overlap factor between vibrational wavefunctions is calculable numerically. (Note that a Franck-Condon factor is the absolute magnitude squared of the overlap factor.) From Eq. (3.3.5) and the experimental value of an initial trial... 

Figure 3.6 Variation of AGv for the mutually interacting b and c,1 , f states of N2. The solid and dashed lines correspond, respectively, to the observed and calculated (Lefebvre-Brion, 1969) values. The deperturbed c (v = 2) and b (v = 7) levels are nearly degenerate and interact strongly (see Table 5.4). This accounts for the largest AG anomalies.

This integral has been evaluated ab initio and found equal to 300 cm-1 (Felen-bok and Lefebvre-Brion, 1966). (The one-center part of this integral is approximately the nonzero atomic integral sp 1/ r 12]dp).) This calculated value is in fair agreement with the semiexperimental value of 450 cm-1 found by a deperturbation procedure (Jungen, 1966). Note that this electrostatic interaction is responsible not only for perturbations between states of identical symmetry but also for predissociation (Section 7.8.1) and auto-ionization (Section 8.8). 

If the diabatic coupling matrix element, He, is -independent, this d/dR matrix element between two adiabatic states must have a Lorentzian H-depen-dence with a full width at half maximum (FWHM) of 46. Evidently, the adiabatic electronic matrix element We(R) is not - independent but is strongly peaked near Rc- Its maximum value occurs at R = Rc and is equal to 1/46 = a/4He. Thus, if the diabatic matrix element He is large, the maximum value of the electronic matrix element between adiabatic curves is small. This is the situation where it is convenient to work with deperturbed adiabatic curves. On the contrary, if He is small, it becomes more convenient to start from diabatic curves. Table 3.5 compares the values of diabatic and adiabatic parameters. The deviation from the relation, We(i )max x FWHM = 1, is due to a slight dependence of He on R and a nonlinear variation of the energy difference between diabatic potentials. When We(R) is a relatively broad curve without a prominent maximum, the adiabatic approach is more convenient. When We (R) is sharply peaked, the diabatic picture is preferable. The first two cases in Table 3.5 would be more convenient to treat from an adiabatic point of view. The description of the last two cases would be simplest in terms of diabatic curves. The third case is intermediate between the two extreme cases and will be examined later (see Table 3.6). 

If the approximate deperturbed potential curves cross, they are diabatic curves. One can assume an interaction matrix element given by Eq. (3.3.5) and carry out a complete deperturbation. 

The choice of an adiabatic picture leads to difficulties when one of the potentials has a double minimum (see Fig. 3.5). The vibrational level separations of such a curve do not vary smoothly with vibrational quantum number, as do the levels of a single minimum potential. In the separate potential wells (below the barrier), the levels approximately follow two different smooth curves. However, above the potential barrier the separation between consecutive energy levels oscillates. The same pattern of behavior is found for the rotational constants below and above the potential barrier. In addition, the rotational levels above the barrier do not vary as BVJ(J + 1). An adiabatic deperturbation of the (E,F+G,K) states of H2 has been possible (Dressier et al., 1979) only because the adiabatic curves were known from very precise ab initio calculations. 

If the approximate deperturbed curves do not cross or have similar spectroscopic constants, the most convenient starting point is an adiabatic approach. Two situations must be considered ... 

The observed levels in Table 3.6 may be obtained from the diabatic potentials represented by the Te,ue,ujexe, and Re constants, which generate the deperturbed levels via Tv = Te + we (v + h) — wexe ( electronic matrix element, He = 365 cm, and vibrational overlap factors calculated using the vibrational eigenfunctions of the deperturbed diabatic potentials. Similarly, the observed levels may be computed from the adiabatic potentials, a Lorentzian interaction term [Eq. (3.3.14)] We(R) with b = 0.1014... 

A, Rc = 1.915 A, and vibrational factors calculated using eigenfunctions of the deperturbed adiabatic potentials. 

Equations (3.3.16 and 3.3.17) suggest one final indicator of whether the diabatic or adiabatic approach is preferable. The better approach is the one for which the sum over deperturbed functions involves fewer terms, especially if one term is dominant (for example, a > 2-1/2). An adiabaticity parameter... 

Second-order effects arising from the product of matrix elements involving J+ L and L+ S operators have the same form as 7J+S. In the case of H2, the second-order effect seems to be smaller than the first-order effect, but in other molecules this second-order effect will be more important than the first-order contribution to the spin-rotation constant. These second-order contributions can be shown to increase in proportion with spin-orbit effects, namely roughly as Z2, but the direct spin-rotation interaction is proportional to the rotational constant. For 2n states, 7 is strongly correlated with Ap, the spin-orbit centrifugal distortion constant [see definition, Eq. (5.6.6)], and direct evaluation from experimental data is difficult. On the other hand, the main second-order contribution to 7 is often due to a neighboring 2E+ state. Table 3.7 compares calculated with deperturbed values of 7 7eff of a 2II state may be deperturbed with respect to 2E+ by... 

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