Dunham expansion

Representation of molecular spectra by fitting formulas Dunham expansion of energy levels... 

Quite often, rotational-vibrational spectra of molecules are analyzed by means of empirical formulas. A convenient formula for diatomic molecules is the Dunham expansion (Dunham, 1932 Ogilvie and Tipping, 1983)... 

The following notation has been introduced in Eq. (4.92) As denote coefficients of terms linear in the Casimir operators, A.s denote coefficients of terms linear in the Majorana operators, Xs denote coefficients of terms quadratic in the Casimir operators, Ks denote coefficients of terms containing the product of one Casimir and one Majorana operator, and Zs denote coefficients of terms quadratic in the Majorana operators. This notation is introduced here to establish a uniform notation that is similar to that of the Dunham expansion, where (Os denote terms linear in the vibrational quantum numbers, jcs denote terms that are quadratic in the vibrational quantum numbers and y s terms which are cubic in the quantum numbers (see Table 0.1). Results showing the improved fit using terms bilinear in the Casimir operators are given in Table 4.8. Terms quadratic in the Majorana operators, Z coefficients, have not been used so far. A computer code, prepared by Oss, Manini, and Lemus Casillas (1993), for diagonalizing the Hamiltonian is available.2... 

The algebraic vibrational analysis should be compared with the vibrational analysis carried out using the Dunham expansion. The quality of the fit of Table 4.8 is equivalent to that of a Dunham expansion with cubic terms... 

Calculations of vibrational spectra of bent triatomic molecules with second order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.9. These results should again be compared with those of a Dunham expansion with cubic terms [Eq. (0.1)]. An example of such an expansion for the bent S02 molecule is given in Table 0.1. Note that because the Hamiltonian (4.96) has fewer parameters, it establishes definite numerical relations between the many Dunham coefficients similar to the so-called x — K relations (Mills and Robiette, 1985). For example, to the lowest order in l/N one has for the symmetric XY2 case the energies E(vu v2, V3) given by... 

Even if one restricts one s attention to vibrations and rotations of molecules, there are a variety of Lie algebras one can use. In some applications, the algebras associated with the harmonic oscillator are used. We mention these briefly in Chapter 1. We prefer, however, even in zeroth order to use algebras associated with anharmonic oscillators. Since an understanding of the algebraic methods requires a comparison with more traditional methods, we present in several parts of the book a direct comparison with both the Dunham expansion and the solution of the Schrodinger equation. 

The analysis of spectroscopic data for bound states of diatomic molecules gives accurate potential curves if one follows the semi-classical Rydberg-Klein-Rees method. For a review of this see Ref. 126). It is sufficient to note that this gives the two values of r as a function of potential energy by considering the dependence of the total spectroscopic energy on the vibrational and rotational quantum numbers n and J. A somewhat simpler procedure, and the only one plicable to polyatomic molecule, is to use the Dunham expansion of the potential 127). 

The fact that matrix elements of the fundamental band are dependent on the rotational quantum numbers j, f cannot be ignored. As a consequence, many more B coefficients must now be evaluated which, in principle, poses no special problem. The volume of data needed renders the task awkward. Molecular spectroscopists have for generations coped with similar problems which were solved with the so-called Dunham expansion in terms of j(j+1). Specifically, for our purpose, the lowest-order expansion for the fundamental band looks like [63]... 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

In integrable systems, the periodic orbits are not isolated but form continuous families, which are associated with so-called resonant tori. In action-angle variables, the Hamiltonian depends only on the action variables, similar to the Dunham expansion, ... 

In the energy range 0-16,000 cm-1, the vibrational Hamiltonian of this molecule can be modeled by a Dunham expansion without anharmonic resonances of the classical form [112]... 

Figure 4. Vibrogram of C2HD calculated with = 2000 cm-1 from all the vibrational energy levels predicted by the Dunham expansion corresponding to the Hamiltonian (3.12) obtained by Herman and co-workers by fitting to high-resolution spectra [112], The periods of the bulk periodic orbits of Table I obtained numerically for the classical Hamiltonian (3.12) are superimposed as circles. On the right-hand side, the main labels (n4,ns) of the periodic orbits are given.

The effective Hamiltonian by Abbouti Temsamani and Herman [123] is composed of a diagonal part given by a Dunham expansion with all the x s and g s as well as y244, and of a nondiagonal part including the following resonances [123] ... 

Just above the saddle energy, the quantization can be performed by the usual perturbation theory applied to scattering systems as described by Miller and Seideman [24], This equilibrium point quantization uses Dunham expansions of the form (2.8) with imaginary coefficients. This method is valid for relatively low-lying resonances above the saddle, up to the point where anhar-monicities become so important that the Dunham expansion is no longer applicable (see the discussion in Section II.B). 

In a more recent work, Joens [158] has assigned the structures of the Hartley band using a Dunham expansion, that is equilibrium point quantization. The lifetime predicted by his analysis is extremely short, equal to 3.2 fs, while the symmetric stretching period is of 30 fs. Recall, however, that the interpretations in terms of equilibrium point expansions and in terms of periodic orbits are strictly complementary only for regular regimes. 

R. Schinke In the case of HNO and HO2, we calculated the number of states and simply extrapolated this number into the continuum. We believe that this is the best what can be done, provided a global potential eneigy surface and full dimensionality dynamics calculations for this potential are available. Because of the much smaller number of states, for HCO this procedure is less well defined. In our final analysis (Ref. 33 of our paper in this volume) we tested the extrapolation from the bound to the continuum region and an estimation of the density of states based on a Dunham expansion of the term energies and found that both recipes give essentially the same result. 

Comparing the results obtained from the Morse potential with those from the Dunham expansion, we see that the coefficients are related as follows,... 

This quantity is the leading term in the Dunham expansion which is obtained by higher-order quantisation than that given in equation (6.358). In the semi-classical treatment therefore,... 

Watson s treatment applies to vibration-rotation effects in diatomic molecules in singlet electronic states it has not yet been extended to include spin-dependent phenomena. It is based on the Dunham expansion [23] which we met earlier in the previous chapter,... 

The Fermi resonance Hamiltonian consists of two terms. The first one, Ho, is the Dunham expansion, which characterizes the uncoupled system, while the second term, Hp, is the Fermi resonance coupling, which describes the energy flow between the reactive mode and one perpendicular mode. For the three systems, HCP CPH, HOCl HO - - Cl and HOBr HO + Br, the reactive degree of freedom is the slow component of the Fermi pair and will therefore be labeled s, while the fast component will be labeled /. Thus, the resonance condition writes co/ w 2c0s. More explicitly, for HCP the slow reactive mode is the bend (mode 2) and the fast one is the CP stretch (mode 3), while for HOCl and HOBr the slow mode is the OX stretch (X = Cl,Br) (mode 3) and the fast one is the bend (mode 2). The third, uncoupled mode— that is, the CH stretch (mode 1) for HCP and the OH stretch (mode 1) for HOCl and HOBr—will be labeled u. With these notations, the Dunham expansion writes in the form... 

Let us first neglect the Fermi resonance and analyze the dynamics of the uncoupled systems described by the Dunham expansion alone [Eq. (24)]. Because of the resonance condition co/ 2c0s, quantum states are organized in clumps, or polyads. Each polyad is defined by two quantum numbers, namely the number v of quanta in the uncoupled degree of freedom and the so-called polyad number P ... 

Figure 10. Density probability in the ( 3, 2) plane for the eight states of uncoupled HOBr belonging to polyad [v ,P] = [0,14], The Hamiltonian is the Dunham expansion of Eq. (24) with parameters from Table 1 of Ref. 41. (OBr stretch) ranges from to —6.5 to 6.5, and qz (bend) ranges from -5.0 to 5.0. The energy (in cm ) above the quantum mechanical ground state, as well as the good quantum numbers (vj,vy) = (v3,V2), are indicated for each state.

Numerically, this is most easily analyzed by rewriting the Dunham expansion of Eq. (24) in terms of coordinates, which are adapted to the polyad structure of the spectmm. One defines new sets (/ , / ), Jp, //>), and Jo, v[/o) of conjugate action-angle-like coordinates, according to... 

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