Hamiltonian rotations vibrations

The Hamiltonian operator for molecular rotational-vibrational spectra, which in the uncoupled form is written as in Eq. (2.17), can now be written in the coupled form as... 

We have discussed up to now vibrational spectra of linear and bent triatomic molecules. We address here the problem of rotational spectra and rotation-vibration interactions.3 At the level of Hamiltonians discussed up to this point we only have two contributions to rotational energies, coming from the operators C(0(3]2)) and IC(0(412))I2. The eigenvalues of these operators are... 

Jensen, P. (1983), The Nonrigid Bender Hamiltonian for Calculating the Rotation-Vibration Energy Levels of a Triatomic Molecule, Comp. Phys. Rep. 1,1. 

Nearly all kinetic isotope effects (KIE) have their origin in the difference of isotopic mass due to the explicit occurrence of nuclear mass in the Schrodinger equation. In the nonrelativistic Bom-Oppenheimer approximation, isotopic substitution affects only the nuclear part of the Hamiltonian and causes shifts in the rotational, vibrational, and translational eigenvalues and eigenfunctions. In general, reasonable predictions of the effects of these shifts on various kinetic processes can be made from fairly elementary considerations using simple dynamical models. 

Within the Born-Oppenheimer approximation, we assume the nuclei are held fixed while the electrons move really fast around them, (note Mp/Me 1840.) In this case, nuclear motion and electronic motion are seperated. The last two terms can be removed from the total hamiltonian to give the electronic hamiltonian, He, since Vnn = K, and = 0. The nuclear motion is handled in a rotational/vibrational analysis. We will be working within the B-0 approximation, so realizing... 

Figure 16. Scattering resonances of the full rotational-vibrational Hamiltonian describing the dissociation of CO2 on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with 7 = 0,..., 10 are given by dots. Their close vicinity explains the formation of hyphens , i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with V2 = 0,. .. 5. The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

From the fact that r Ncf 3% is the isometric group of the NC (4.1) it follows by the same reasoning as for SRMs that e is the symmetry group of the rotation-vibration hamiltonian. Though the representation (4.11) is commonly used in vibrational spectroscopy55-S7 it only seldom has been characterized as a group of isometric transformations57. ... 

Contact Transformation for the Effective Hamiltonian.—The vibration-rotation hamiltonian of a polyatomic molecule, expressed in terms of normal co-ordinates, has been discussed in particular by Wilson, Decius, and Cross,24 and by Watson.27- 28 It is given by the following expression for a non-linearf polyatomic molecule, to be compared with equation (17) for a diatomic molecule ... 

Equation (1-13) or its body-fixed equivalent is of little use for Van der Waals complexes, as it discriminates one nuclear coordinate, e.g. y = 1. Specific mathematical forms of Hamiltonians describing the nuclear motions in Van der Waals dimers have been developed (7). This point will be discussed in more details in Section 12.4. Here we only want to stress that whatever the mathematical form of the Hamiltonian is used to solve the problem of nuclear motions, the results will be the same, if the Schrodinger equation is solved exactly. However, in weakly bound complexes there is a hierarchy of motions due to the strong intramolecular forces which determine the internal vibrations of the molecules, and to much weaker intermolecular forces which determine their relative translations and rotations. This hierarchy allows to make a separation between the intramolecular vibrations with high frequencies and the intermolecular modes with much lower frequencies. Such a separation of the fast intramolecular vibrations and slow rotation-vibration-tunneling motions can be performed if a suitable form of the Hamiltonian for the nuclear motions in Van der Waals molecules is used. 

The Bom-Oppenheimer Hamiltonian is not dilatation analytic if only the electronic co-ordinates are scaled since relS — R vanishes for a continuous range of 0 3 r = R and r.R = cos(0) /80/. The conceptual and numerical difficulties associated with complex scaling of both the nuclear and electronic coordinates and solution of the combined problem with a plethora of rotational, vibrational and electronic thresholds have prompted the formulation of (see ref. 22) two different schemes by McCurdy and Rescigno /81 / and Moiseyev and Corcoran... 

K. L. Mardis and E. L. Sibert III, Derivation of rotation-vibration Hamiltonians that satisfy the Casimir condition. J. Chem. Phys. 106, 6618-6621 (1997). 

In order to proceed further one must write down the Hamiltonian for the molecule, that is, the expression for its kinetic and potential energy. The former contains up to five contributions, namely, for its translational, rotational, vibrational, electronic, and nuclear energies. Since the molecules do not interact with one another the potential energy is zero. Thus, one may write... 

In this connection an extended Hamiltonian has been developed that includes an additional vibrational degree of freedom. This extended model was tested with three molecules. It was possible to fit the splittings in the rotational spectra of the lowest three internal rotation/vibration states with one set of coefficients. There is an indication that for higher states an even more extended Hamiltonian is necessary. 

The first high resolution spectroscopic measurements of FAD with fully resolved rotational-vibrational-tunneling transitions were reported by Madeja and Havenith in 2002 [39]. They measured the C-O vibrational band of (DC00H)2 which could be analyzed in terms of an asymmetric top rigid rotor Hamiltonian. The vibrational frequency of the C-O stretch in (DCOOH)2 was determined to be 1244.8461(2) cm i. This deviated considerably from values for the band center as given by Wachs et al. [55] (1231.85 cm i). Previous measurements include the Raman transition as reported by Bertie et al. (1230 2 cm ) [54] and the value obtained by Millikan and Pitzer (1239 cm ) [56]. 

Chapters 2, 3, and 5 form the core of this book. Perturbations are defined and simple procedures for evaluating matrix elements of angular momentum operators are presented in Chapter 2. Chapter 3 deals with the troublesome terms in the molecular Hamiltonian that are responsible for perturbations. Particular attention is devoted to the reduction of matrix elements to separately evaluable rotational, vibrational, and electronic factors. Whenever possible the electronic factor is reduced to one- and two-electron orbital matrix elements. The magnitudes and physical interpretations of matrix elements are discussed in Chapter 5. In Chapter 4 the process of reducing spectra to molecular constants and the difficulty of relating empirical-parameters to terms in the exact molecular Hamiltonian are described. Transition intensities, especially quantum mechanical interference effects, are discussed in Chapter 6. Also included in Chapter 6 are examples of experiments that illustrate, sample, or utilize perturbation effects. The phenomena of predissociation and autoionization are forms of perturbation and are discussed in Chapters 7 and 8. 

Figure 1 Schematic picture of the relationship between Born-Oppenheimer potential energy surfaces, the rotation-vibration Hamiltonian, and the observed spectroscopy and dynamics.

The quantum mechanical rotation-vibration Hamiltonian (63) takes the form... 

---