Rotational constant operator

In other words, each of the parameters is the sum of a first-order and a second-order contribution. We have met equation (7.126) for the effective rotational constant operator before, in an earlier section, where we pointed out that the second-order contribution Ba> is very much smaller than BiV) and that these two contributions have a different reducedmass dependence. It is importantto realise thatthis is not generally true. Indeed, except for molecules with very light atoms such as H2, the second-order contribution to the spin-rotation parameter is usually very much larger in magnitude than the first-order contribution. The same is also often true for the spin-spin coupling parameter /.. The reduced mass dependences of the two contributions to the spin-rotation parameter y are different from each other and quite complicated. However, Brown and Watson [17] were able to show the rather remarkable result that when one takes the first- and second-order contributions together as in equation (7.127), the reduced mass dependence of the resultant parameter y(R) is simply /u-1. 

The experimental evidence for such a contribution to the spin-rotation interaction in the effective Hamiltonian was somewhat elusive in the early days although there are now well documented cases of its involvement, for example for CH in its 4E state [21]. Equation (7.166) suggests one reason why this parameter is not as important in practice as might be expected. The last factor on the right-hand side of (7.166) is just the difference of the rotational constant operators for the upper and lower states. This causes a considerable degree of cancellation in a typical situation because the B value is not expected to vary markedly between the electronic states. 

The rotational constant operator, B+(R), contains small contributions from rotation-electronic interactions with remote states (see Watson, 1999, and Section 4.2). The pair of terms in in Eq. (3.2.22) give rise to off-diagonal matrix elements (AJ = AN+ = Al = 0, AA+ = 1) between members of different Rydberg series built on different 25++1A+ ion-core states. The case (d) l complex states are denoted n2Z(2S++1 A+). When Hel is evaluated in a case (d) basis set, Hel generates both energy differences and nonzero off-diagonal mar-trix elements between components of an l complex (Jungen and Miescher, 1969 Watson, 1999) or supercomplex (Miescher, 1976 Jakubek and Field, 1994). 

In other words, each of the parameters is the sum of a first-order and a second-order contribution. We have met equation (7.126) for the effective rotational constant operator before, in an earher section, where we pointed out that die second-order contribution... 

Characteristic length [Eq. (121)] L Impeller diameter also characteristic distance from the interface where the concentration remains constant at cL Li Impeller blade length N Impeller rotational speed also number of bubbles [Eq, (246)]. N Ratio of absorption rate in presence of chemical reaction to rate of physical absorption when tank contains no dissolved gas Na Instantaneous mass-transfer rate per unit bubble-surface area Na Local rate of mass-transfer per unit bubble-surface area Na..Average mass-transfer rate per unit bubble-surface area Nb Number of bubbles in the vessel at any instant at constant operating conditions N Number of bubbles per unit volume of dispersion [Eq. (24)] Nb Defined in Eq. (134)... 

Similar operational definitions have to be taken into account for every experimental tool of structural chemistry to define the meaning of the observables that it provides6. In microwave spectroscopy, for example, structural information is obtained from the rotational constants... 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

B is called the rotational constant, and will be discussed in detail later the operator form of the rotational energy in (6.143) has profound consequences, as we shall see. 

The effects of the off-diagonal terms when folded-in by perturbation theory are of two types. They can either produce operators of the same form as those which already exist in the Hamiltonian constructed from the Azl = 0 matrix elements (the zeroth-order Hamiltonian), or they can have a completely novel form. A good example of the former type is the second-order contribution to the rotational constant which arises from admixture of excited and A states,... 

In HD the nuclei are different and so distinguishable, the Pauli principle no longer operates and transitions between any two quantum states, J, are allowed without restriction to odd or even number and no spin trapping occurs. The rotational energies are still given by Eq. (6.9) and the rotational constant for HD is, = 44.7 cm [1] We shall see that the use of HD in INS spectroscopy has benefits. 

The rc and p-Kr coordinates are determined by adjusting coordinates to fit moments of inertia and differences in moments of inertia, respectively. This is again an operational definition, so the only uncertainty is the result of experimental uncertainty in the rotational constants, and this may be treated by standard methods.18 However, the standard deviation between observed rotational constants and those computed from the final structure should not be used in such a calculation. In the comparison shown above the structural parameters of NF2CN are more reliable than those of PF2CN, but the deviation between observed and calculated moments is greater for NF2CN. The standard deviations of the rotational constants are best obtained from the fit of the rotational spectra. These values should be used to estimate the experimental uncertainties in rQ and p-Kr coordinates. The uncertainties in r0 or p-Kr coordinates as estimates of the equilibrium coordinates are very difficult to compute. The Costain rule is probably satisfactory for p-Kr coordinates, but less so for rQ coordinates. 

As molecular applications of the extended DK approach, we have calculated the spectroscopic constants for At2 equilibrium bond lengths (RJ, harmonic frequencies (rotational constants (B ), and dissociation energies (Dg). A strong spin-orbit effect is expected for these properties because the outer p orbital participates in their molecular bonds. Electron correlation effects were treated by the hybrid DFT approach with the B3LYP functional. Since several approximations to both the one-electron and two-electron parts of the DK Hamiltonian are available, we dehne that the DKnl -f DKn2 Hamiltonian ( 1, 2= 1-3) denotes the DK Hamiltonian with DKnl and DKn2 transformations for the one-electron and two-electron parts, respectively. The DKwl -I- DKl Hamiltonian is equivalent to the no-pair DKwl Hamiltonian. For the two-electron part the electron-electron Coulomb operator in the non-relativistic form can also be adopted. The DKwl Hamiltonian with the non-relativistic Coulomb operator is denoted by the DKwl - - NR Hamiltonian. 

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... 

Although centrifugal distortion is not a perturbation effect, a derivation of the form of the centrifugal distortion terms in Heff provides an excellent illustration of the Van Vleck transformation. If the vibrational eigenfunctions of the nonrotating molecular potential, V(R) rather than [V(R) + J(J + 1)H2/2/j,R2, are chosen as the vibrational basis set, then the rotational constant becomes an operator,... 

Neither of the above results include the effects of the instantaneous distortion of the molecule, as the p, u are taken to be constants. In both these calculations we neglect the coordinate dependence of the inverse moment of inertia tensor in Eq. (5) and evaluate p, in the equilibrium configuration. This definition was the one of several definitions considered by Frederick et al. (84) that leads to the smallest standard deviation. If we were to include the coordinate dependence of p in our definition of the rotational energy operators, then both A v and AEA would be substantially larger. 

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