Vibrational-rotational motions

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

Many elements of ehemists pietures of moleeular stmeture hinge on the point of view that separates the eleetronie motions from the vibrational/rotational motions and treats eouplings between these (approximately) separated motions as perturbations. It is essential to understand the origins and limitations of this separated-motions pieture. 

The BO picture is what gives rise to the concept of a manifold of potential energy surfaces on which vibrational/rotational motions occur. 

Even within the BO approximation, motion of the nuelei on the various eleetronie energy surfaees is different beeause the nature of the ehemieal bonding differs from surfaee to surfaee. That is, the vibrational/rotational motion on the ground-state surfaee is eertainly... 

The treatment of electronic motion is treated in detail in Sections 2, 3, and 6 where molecular orbitals and configurations and their computer evaluation is covered. The vibration/rotation motion of molecules on BO surfaces is introduced above, but should be treated in more detail in a subsequent course in molecular spectroscopy. 

Improving on the semi-classical treatment of the vibration-rotation motion only slightly allows Rt to be recast in a form... 

More sensitive to the level of theory is the vibrational component of the interaction energy. In the first place, the harmonic frequencies typically require rather high levels of theory for accurate evaluation. It has become part of conventional wisdom, for example, that these frequencies are routinely overestimated by 10% or so at the Hartree-Fock level, even with excellent basis sets. A second consideration arises from the weak nature of the H-bond-ing interaction itself. Whereas the harmonic approximation may be quite reasonable for the individual monomers, the high-amplitude intermolecular modes are subject to significant anharmonic effects. On the other hand, some of the errors made in the computation of vibrational frequencies in the separate monomers are likely to be canceled by errors of like magnitude in the complex. Errors of up to 1 kcal/mol might be expected in the combination of zero-point vibrational and thermal population energies under normal circumstances. The most effective means to reduce this error would be a more detailed analysis of the vibration-rotational motion of the complex that includes anharmonicity. 

Theoretical chemistry is the discipline that uses quantum mechanics, classical mechanics, and statistical mechanics to explain the structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of (1) electronic structure, potential energy surfaces, and force fields (2) vibrational-rotational motion and (3) equilibrium properties of condensed-phase systems and macromolecules. Chemical dynamics includes (1) bimolecular kinetics and the collision theory of reactions and energy transfer (2) unimolecular rate theory and metastable states and (3) condensed-phase and macromolecular aspects of dynamics. 

At this point the first-principles perturbative (FP) approach becomes valuable. The same kinds of perturbative models are used to describe the vibrational-rotational motions as in the SP approach. However, data from electronic structure theory computations or potential energy functions are used to parameterize the formulas instead of spectroscopically obtained data. The FP approach has for example, been pursued by Martin et al. [16-18] and by Isaacson, Truhlar, and co-workers [19-25]. This avenue is especially valuable when spectroscopic data are not available for a molecule of interest. Codes are available that can carry out vibrational perturbation theory computations, using a grid of ab initio data as input SURVIBTM... 

The first step in a unimolecular reaction is the excitation of the reactant molecule s energy levels. Thus, a complete description of the unimolecular reaction requires an understanding of such levels. In this chapter molecular vibrational/rotational levels are considered. The chapter begins with a discussion of the Bom-Oppenheimer principle (Eyring, Walter, and Kimball, 1944), which separates electronic motion from vibrational/rotational motion. This is followed by a discussion of classical molecular Hamiltonians, Hamilton s equations of motion, and coordinate systems. Hamiltonians for vibrational, rotational, and vibrational/rotational motion are then discussed. The chapter ends with analyses of energy levels for vibrational/rotational motion. 

The first term in brackets represents the vibrational/rotational motion of the molecule, while the second term represents the molecule s translational motion i.e., H = +... 

The vibrational/rotational motion of a diatomic molecule can be confined to one plane (e.g., the x,y-plane) and the relative coordinates and momenta in Eq. (2.10) can be transformed to polar coordinates and momenta by the relations... 

The separation of the reaction coordinate x from the other coordinates of the reacting system allows us to treat independently the relative translation or vibration of reactants and its non-reac-tive motions from the point of view of statistical physics, too. Classical statistics may be used in most cases for the motion along the reaction coordinate however, quantum statistics is usually necessary for the non-reactive vibration-rotation motions of reactants. 

Ill) but does involve an energy exchange between the translation and vibration-rotation motions as far as the reaction coordinate in the transition region can be treated as a separable one Cosequently, then and only then for an electronically adiabatic reaction = 1 ... 

The Born-Oppenheimer (B.O.) approximation is the cornerstone of most theories dealing with the effect of isotopic substitution on molecular properties. Within the framework of this approximation, the potential energy surface for the vibrational-rotational motions of a molecular system depends on the nuclear charges of the substituent atoms and on the number of electrons in the system but is independent of the masses of the nuclei. Thus isotope effects arise from the fact that nuclei of different mass move differently on the same potential surface. 

An important simplification is possible in the cases of vibrational-rotational motions. Throughout this review we have seen that the relevant operators Y are functions of position operators (i.e., vibrational displacements and molecular Euler angles), and that they do not explicitly involve the conjugate momenta. Presently, is quadratic in the conjugate momenta, and as a consequence... 

The center of mass translational energy may be removed from in eqn (20.26) to give a Hamiltonian for vibrational/rotational motion, which may be expressed in Cartesian, internal or normal mode coordinates. The... 

Molecular vibrations Rotational motion Nuclear spin transitions... 

A Fourier transform of the appropriate intensity function (see eqs. 3.9 or 3.12) over the band will directly yield a time-correlation function (Cmit), for example), but one whose time-dependence reflects only the motions giving rise to the particular band of Interest. Consequently, we can calculate (and compare with experiment ) time-correlation functions for pure rotational motion, for vibration-rotation motion for the 4 th normal mode or, in favorable cases, for the translation-rotation motions that give rise to an induced spectrum. 

The relationship (1.24) can be further rewritten by considering the translation-vibration-rotation motion of the nuclei system, with 3N-6 cardinal as the associate degree of freedom, so making the derivatives ... 

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