Dunham coefficients

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... 

Apart from the distance variable x that Dunham used in his function V(x) for potential energy, other variables are amenable to production of term coefficients in symbolic form as functions of the corresponding coefficients in a power series of exactly the same form as in formula 16. Through any method to derive algebraic expressions for Dunham coefficients l j, the hamiltonian might have x as its distance variable, but after those expressions are produced they are convertible to contain coefficients of other variables possessing more convenient properties. To replace x, two defined variables are y [38],... 

According to apphcation of Dunham s formalism to analysis of molecular spectra, as for GaH and H2, these radial coefficients of seven types represent many Dunham coefficients Ym and their auxiliary coefficients Zki of various types that collectively allow wave numbers of observed transitions to be reproduced almost within their uncertainty of measurement through formula 54. Mostly because of inconsistency between reported values of frequencies of pure rotational transitions [118,119], the reduced standard deviation of the fit reported in table 3 is 1.25, slightly greater than unity that would be applicable with consistent data for which uncertainty of each measurement were carefully assigned. 

Use of equation (7.205) shows that A4 is proportional to /i 1. We thus see that there is a contribution 6hA4 to the Dunham coefficient Uo (or coexe) since it is the coefficient of (v + 1/2)2 in the expansion (7.198) this term has the expected reduced mass dependence. However, there is also a small contribution 3hA4/2 to the coefficient Too with the wrong reduced mass dependence (/i ). This is an example of the higher-order corrections revealed by Dunham s calculation. 

Table 10.5. Dunham coefficients determined for four isotopomers of LiH [100], The parameters are given in MHz, except for the A parameters which are dimensionless...

As already discusf d for the single-oscillator case in Section II.C.2, it is possible to obtain Dunham-like form for the local-mode eigenspectrum (4.33). Given the anharmonic character of the algebraic formulation, we expect to find a certain number of relations between the Dunham coefficients similar to those of the x-K relations of Lehmann [73, 74]. In fact, we can compare the traditional Dunham series for a triatomic molecule... 

The choice of signs in front of the different parameters in eqs. (2. la) and (2.6a) has historical reasons. Usefiil relations hold in good approximation between tiie Dunham coefficients and the standard spectroscopic parameters, and read as follows ... 

Dunham [32Dun] gave equations for several Yfj (Dunham coefficients) in terms of the potential coefficients Oj which hold within the WKB approximation. This was extended to additional T j and... 

The Dunham coefficients Yy are related to the spectroscopical parameters as follows 7io = cOe to the fundamental vibrational frequency, Y20 = cOeXe to the anharmonicity constant, Y02 = D to the centrifugal distortion constant, Yn = oie to the vibrational-rotational interaction constant, and Ym = / to the rotational constant. These coefficients can be expressed in terms of different derivatives of U R) at the equilibrium point, r=Re. The derivatives can be either calculated analytically or by using numerical differentiation applied to the PEC points. The numerical differentiation of the total energy of the system, Ecasccsd, point by point is the simplest way to obtain the parameters. In our works we have used the standard five-point numerical differentiation formula. In the comparison of the calculated values with the experimental results we utilize the experimental PECs obtained with the Rydberg-Klein-Rees (RKR) approach [58-60] and with the inverted perturbation approach (IPA) [61,62]. The IPA is method originally intended to improve the RKR potentials. 

The Dunham coefficients of Equation 16.11 are determined by fitting the measured spectra. The sensitivity coefficients Ki can be found by making use of Equations 16.12 and 16.13. Some rovibrational levels of different electronic excited states lie very close to each other. Eor such levels, additional nonadiabatic corrections can be included within the two-level approximation [57]. 

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