The rotational Hamiltonian

The simplest example of the way in which the various terms arise in the effective electronic Hamiltonian involves the rotational kinetic energy operator, 3Qot  

The operators B(R) and Nact only within each electronic state while the orbital angular momentum L acts both within and between such states. 

The first-order contribution to the effective Hamiltonian involves the diagonal matrix element of the operator in equation (7.80)  

to second order, we can write the effective rotational Hamiltonian as 

We see from the way in which the effective rotational Hamiltonian is constructed that it is naturally expressed in terms of the angular momentum operator N. In the scientific literature, however, it is frequently written in terms of the vector R (which represents the rotational angular momentum of the nuclei) rather than N. While R = N — L occurs in the fundamental Hamiltonian (7.71), its use in the effective Hamiltonian is not satisfactory because R has matrix elements (due to L) which connect different electronic states and so is not block diagonal in the electronic states. In practice, authors who claim to be using R in their formulations usually ignore any matrix elements which they find inconvenient such as those of Lx and Ly. We shall return to this point in more detail later in this chapter. 

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Moleeules for whieh all three prineipal moments of inertia (the li s) are equal are ealled spherieal tops. For these speeies, the rotational Hamiltonian ean be expressed in terms of the square of the total rotational angular momentum P ... 

The rotational Hamiltonian ean now be written in terms of and the eomponent of J along the unique moment of inertia s axis as ... 

The rotational eigenfunctions and energy levels of a molecule for which all three principal moments of inertia are distinct (a so-called asymmetric top) can not easily be expressed in terms of the angular momentum eigenstates and the J, M, and K quantum numbers. However, given the three principal moments of inertia la, Ib, and Ic, a matrix representation of each of the three contributions to the rotational Hamiltonian... 

The rotational Hamiltonian for a diatomic molecule as given in Chapter 3 is... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

Chapter 3. In the former case, the rotational Hamiltonian can be expressed in terms of J2 = Ja + Jb + Jc because all three moments of inertia are identical ... 

M-trans = a <%ivKRa " Ra,eq)l%fv> 8fl/8Ra > and derives its time dependence above from the rotational Hamiltonian ... 

The rotational Hamiltonian can then be written in terms of angular momenta and principal-axis moments of inertia as ... 

Now consider the eigenfunctions ip of the rotational Hamiltonian Hip = Eip. (This equation is more fully written as Hroliprot = Erotiprot, but since we are considering only rotation in this chapter, we omit the subscripts.) Because H commutes with P2 and Pz, the rotational eigenfunctions ip can be chosen as eigenfunctions of these two operators. 

The rotational Hamiltonian does not generally commute with Pc [(5.50)] ... 

Figure 7.7. A plot of the energy of E eigenstates of the rotational Hamiltonian as a function of mean angular momentum (mE) for different barrier heights, indicated in kcal/mol. (From Clough et al. [1982].)...

The characteristic features of coupled rotation show up for rotors that have identical frequencies and orientations. In this case, the rotation Hamiltonian assumes the form... 

Pendular state Superpositions of field-free rotational eigenstates in which the molecular axis librates about the field direction. Pendular states are eigenstates of the rotational Hamiltonian plus the dipole potential. 

We require only three nuclear coordinates to define the nuclear motion and we choose these to be R, the internuclear distance, and 0 the third Euler angle x is a redundant coordinate. In fact, because there are no nuclei lying off-axis in a diatomic molecule, X is undefineable it is, however, expedient to retain it because of simplification in the final form of the rotational Hamiltonian. We shall examine this point in more detail in... 

From (2.68) we see that we can add selected terms in 3/3/ to our expression for Pr in (2.51) and hence to the nuclear Hamiltonian, without altering the values of any of the physical observables. We choose these terms so that the rotational Hamiltonian has the same form as the rotational Hamiltonian of a spherical top molecule. We shall see later that with this choice for the rotational Hamiltonian, we can make use of the very powerful techniques of angular momentum theory, in particular, irreducible tensor methods, which would otherwise be denied to us. Accordingly, we modify equation (2.51) to be... 

In the present treatment, we retain essentially all the diagonal matrix elements of X these are the first-order contributions to the effective electronic Hamiltonian. There are many possible off-diagonal matrix elements but we shall consider only those due to the terms in Xrot and X o here since these are the largest and provide readily observable effects. The appropriate part of the rotational Hamiltonian is —2hcB(R)(NxLx + NyLy). The matrix elements of this operator are comparatively sparse because they are subject to the selection rules AA = 1, A,Y=0 and AF=0. The spin-orbit coupling term, on the other hand, has a much more extensive set of matrix elements allowed... 

The rotational Hamiltonian, including the effects of centrifugal distortion, is given by... 

The rotational Hamiltonian, and its expansion in the molecule-fixed axis system may be written... 

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... 

There is, then, a sequence of rotational levels for each fine-structure component, as illustrated in figure 11.26. The rotational Hamiltonian takes the conventional form... 

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