Hamiltonian centrifugal-distortion constants

S and A stand for the symmetric and asymmetric reductions of the rotational hamiltonian respectively see [53] for more details on the various possible representations of the centrifugal distortion constants. 

Tables in the present chapter contain in their upper part data on the spin-rotation interaction constants e q (q = Inertial axes a, b, c) and In their lower part data on the rotational constants A, B, and C. Methods of measurement, references, and remarks are the same for either constant. The quartic (A, 8 and A, 5) and sextic (O, (p) centrifugal distortion constants listed In the first table (for the electronic ground state of PHg) are defined by the A-reduced form (asymmetric reduction) of the respective Hamiltonian for spin-rotation Interaction and rotation [1 to 3]. Abbreviations used in the tables are MW for microwave absorption, EL AB for electronic absorption, LMR for laser magnetic resonance, FIR for far IR, IMF for Intermodulated fluorescence.

Rotational and centrifugal distortion constants for Watson s A-reduced Hamiltonian [96Kis2]. 

NH2 Radical. The NH2 radical Is an asymmetric top with the asymmetry parameter k = (2 B-A-C)/(A-C)= -0.38 (axes b C2, c molecular plane). An increase of the rotational quantum number N leads to a change from prolate- to oblate-top behavior. The rotational constants A, B, and C, the centrifugal distortion constants Ak, A k, A, 5k, and 5, and the spin-rotational coupling constants Ag, Bg, and Cg, for the vibrational ground and excited states are listed in Table 10, p. 182. The rotational Hamiltonian used for fitting the spectroscopic data is a combination of the A-reduced asymmetric rotor Hrot [1] and the spin-rotation Hamiltonian figR [2] ... 

Rotational and Centrifugal Distortion Constants. The rotational constants of gaseous NH2 were obtained by fitting the rovibrational bands of the high-resolution IR spectrum to a Watson-type S-reduced Hamiltonian. The results in cm for the ground state of the ion are Ao = 23.0508 0.0019, Bo = 13.0684 0.0015, Cq = 8.11463 0.00048, Dj = 0.001082 0.000022, Djk=-0.00381 0.00012, Dk = 0.02065 0.00013, Di =-0.000492 0.000014, and 2= -0.0000461 0.0000054. The corresponding values for the v l and the Va = 1 states and the estimated equilibrium rotational constants of the ground state are also listed. The analysis was restricted to quartic distortion terms, because the inclusion of sextic terms did not result in a better fit of the bands. The Hamiltonian used does not include the effects of the rotational interactions which are noticeable in some bands [2]. Rotational constants... 

Spectrum. If distortion is taken into account, terms in the angular momentum components of the fourth power, sixth power, and so forth are introduced into the Hamiltonian. Although the centrifugal distortion constant are very small relative to the rotational constants, they produce significant effects in the rotational spectrum, particularly for asymmetric tops. 

The Watson representation of the Hamiltonian for the near-symmetric asymmetric top was also used in order to obtain centrifugal distortion constants from FT-IR spectra [5 to 7], FT-MW spectra [8], and a continuous MW spectrum [2]. The resulting quartic constants are given in Table 9 see the cited papers for the values of the sextic and higher terms. 

The constant term B .Ia(.Ia + 1) is absorbed into the term value Tv( 2) and is not considered to be part of the rotational Hamiltonian. There are matrix elements off-diagonal in v but these are small and are taken into account as centrifugal distortion terms. The result of (10.145) is that the first-order rotational energies of the two electronic states (for Ja = 3 /2) are given by... 

Another complication arises from the fact that the rotational constants are usually obtained as effective fitting parameters of a reduced rotational Hamiltonian [14]. Not only do the numerical values of the rotational constants depend on the exact form of the reduced Hamiltonian, they also contain small contributions from quartic and higher order centrifugal distortion terms. Watson [14] has proposed to always determine the so-called determinable combinations of these constants. The values of these combinations are independent of the form of the reduction, although they still contain small contributions from the distortion terms. Up to the quartic centrifugal distortion terms, the determinable combinations of the rotational constants are... 

In Eq. (2.38), A, B, and C are the rotational constants in the PAS and H d the usual centrifugal distortion Hamiltonian. The main approximation made in the PAM is that the cross term -2FpP is considered as a perturbation which can be handled by successive Van Vleck transformations. The transformed Hamiltonian matrix can then be factored into smaller effective rotational matrices, one for... 

Well-agreeing sets of constants from three detailed analyses of the spectrum [4, 5, 6] are presented in Table 6, p. 162. Jhe distortion constants are defined by the effective rotational Hamiltonian for a nondegenerate vibrational state of a symmetric-top C y, molecule, including terms up to the eighth power in the angular momentum (according to [7]). An effective rotational Hamiltonian in form of a Pad operator was derived [8 to 11]. Optimum versions of a rational expansion of the effective rotational Hamiltonian for C3V molecules were developed and some of them critically discussed [12 to 17]. For an ab initio calculation of centrifugal distortion effects for phosphane, see [18]. 

The vibration-rotation spectra and/or the rotational spectra in excited vibrational states provide the af constants and, when all the a/ constants are determined, the equilibrium rotational constants can be obtained by extrapolation. This method has often been hampered by anharmonic or harmonic resonance interactions in excited vibrational states, such as Fermi resonances arising from cubic and higher anharmonic force constants in the vibrational potential, or by Coriolis resonances. Equihbrium rotational constants have so far been determined only for a limited number of simple molecules. To be even more precise, one has further to consider the contributions of electrons to the moments of inertia, and to correct for the small effects of centrifugal distortion which arise from transformation of the original Hamiltonian to eliminate indeterminacy terms [11]. Higher-order time-independent effects such as the breakdown of the Bom-Oppenheimer separation between the electronic and nuclear motions have been discussed so far only for diatomic molecules [12]. 

Spin-rotation coupling constants " ND2 and " NHD were calculated [28] using a reduced spin-rotation Hamiltonian [2]. Rotational, centrifugal distortion, and spin-rotation coupling constants of " ND2 and " NHD in the (0,0,0) state which were derived from an ab initio-calculated general valence force field of 4th order [22] agree well with experimentally derived constants. 

The analysis of the rotational spectrum of an asymmetric molecule in the vibrational state ui,... vj,... v u-6 normally allows the determination of the constants listed in this table. All rotating molecules show the influence of molecular deformation (centrifugal distortion, c.d.) in their spectra. The theory of centrifugal distortion was first developed by Kivelson and Wilson [52Kiv]. The rotational Hamiltonian in cylindrical tensor form has been given by Watson [77 Wat] in terms of the angular momentum operators J, J/and as follows ... 

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