Molecular vibration polyatomic molecules

Absorption of electromagnetic radiation in the NIR region is caused by overtone and combination vibrations. Polyatomic molecules exhibit many overtone and combination vibrations, their spectral bands overlap and make typical NIR bands look very broad and featureless. Nevertheless, NIR spectra contain molecular information about the sample, and this information can be extracted by means of chemo-metric methods (cf Chapter 13). A prerequisite for chemometric evaluations is high quahty of the collected spectral data. Therefore, wavelength precision, resolution, photometric precision and signal-to-noise ratio are important criteria for the selection of an NIR spectrometer. 

Even with these complications due to anliannonicity, tlie vibrating diatomic molecule is a relatively simple mechanical system. In polyatomics, the problem is fiindamentally more complicated with the presence of more than two atoms. The anliannonicity leads to many extremely interestmg effects in tlie internal molecular motion, including the possibility of chaotic dynamics. 

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

The starting points for many conventions in spectroscopy are the paper by R. S. Mulliken in the Journal of Chemical Physics (23, 1997, 1955) and the books of G. Herzberg. Apart from straightforward recommendations of symbols for physical quantities, which are generally adhered to, there are rather more contentious recommendations. These include the labelling of cartesian axes in discussions of molecular symmetry and the numbering of vibrations in a polyatomic molecule, which are often, but not always, used. In such cases it is important that any author make it clear what convention is being used. 

Another area of research ia laser photochemistry is the dissociation of molecular species by absorption of many photons (105). The dissociation energy of many molecules is around 4.8 x 10 J (3 eV). If one uses an iafrared laser with a photon energy around 1.6 x 10 ° J (0.1 eV), about 30 photons would have to be absorbed to produce dissociation (Eig. 17). The curve shows the molecular binding energy for a polyatomic molecule as a function of interatomic distance. The horizontal lines iadicate bound excited states of the molecule. These are the vibrational states of the molecule. Eor... 

I. N. Levine (1975) Molecular Spectroscopy (John Wiley Sons, New York). A survey of the theory of rotational, vibrational, and electronic spectroscopy of diatomic and polyatomic molecules and of nuclear magnetic resonance spectroscopy. 

The last step in the calculation of the frequencies of molecular vibrations, as observed in the infrared spectra, is carried out by combining Eqs. (54) and (55). The vibrational energy of a polyatomic molecule is then given in this, the harmonic approximation, by... 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

A combination of both methods was realized by Uehara et al. 85,88) They investigated the Stark spectrum of polyatomic molecules in strong electric fields by probing the different Stark components with the Zeeman-tuned laser line. Since the molecular constants of the vibrational ground state are often known from microwave investiga-... 

The Section on Molecular Rotation and Vibration provides an introduction to how vibrational and rotational energy levels and wavefunctions are expressed for diatomic, linear polyatomic, and non-linear polyatomic molecules whose electronic energies are described by a single potential energy surface. Rotations of "rigid" molecules and harmonic vibrations of uncoupled normal modes constitute the starting point of such treatments. 

In this section, we focus our attention on applications of the CDF protocol to control of population transfer between vibrational levels of a nonrotating polyatomic molecule. The vibrational spectrum of a polyatomic molecule is rich, and if one wishes to transfer population between two states in a subset of selected states that is embedded in the complete manifold of molecular vibrational states, it is... 

In a polyatomic molecule witli many vibrations, we simplify the vibrational partition function much as the original molecular partition function was simplified we assume that the total vibrational energy can be expressed as a sum of individual energies associated with each mode, in which case, for a non-linear molecule, we have... 

In this contribution we demonstrate vibrationally induced molecular dissociation in two polyatomic molecules, namely chromium hexacarbonyl and diazomethane (Fig. 1), employing fs mid-infrared laser sources [1,2]. As will be shown below, in both cases the initially excited mode does not lead directly to reaction but provides the doorway to access the right combination of modes. Thus both reactions have three steps (1) Stepwise multiphoton excitation of the doorway mode, (2) sharing of energy with other modes, and (3) formation of... 

The fundamental frequencies 9t (t = 1, 2,... 3tf—6) are related to and since Xt are the roots of det B—XE) — 0, r, are related to the matrix B and to the molecular force constants Bif. Hence the vibrational energy levels for a non-linear polyatomic molecule in the harmonic oscillator approximation are given by... 

G. Herzberg, Molecular Spectra and Molecular Structure. I. Diatomic Molecule, Prentice-Hall, New York, 1939 Infrared and Raman Spectra of Polyatomic Molecules, D. Van Nostrand Co., New York, 1945 E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations The Theory of Infrared and Raman Vibrational Spectrat McGraw-Hill Book Co., New York, 1955. 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

From a molecular viewpoint, we know that heat capacity is closely connected to internal modes of molecular vibration. According to the classical equipartition theorem (Sidebar 3.8), a nonlinear polyatomic molecule of Aat atoms has ftmodes = 3Aat — 6 independent internal modes of vibration, each of which would contribute equally to heat capacity... 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

The major changes in the new edition are as follows There are three new chapters. Chapter 1 is a review and summary of aspects of quantum mechanics and electronic structure relevant to molecular spectroscopy. This chapter replaces the chapter on electronic structure of polyatomic molecules that was repeated from Volume I of Quantum Chemistry. Chapter 2 is a substantially expanded presentation of matrices. Previously, matrices were covered in the last chapter. The placement of matrices early in the book allows their use throughout the book in particular, the very tedious and involved treatment of normal vibrations has been replaced by a simpler and clearer treatment using matrices. Chapter 7 covers molecular electronic spectroscopy, and contains two new sections, one on electronic spectra of polyatomic molecules, and one on photoelectron spectroscopy, together with the section on electronic spectra of diatomic molecules from the previous edition. In addition to the new material on matrices, electronic spectra of polyatomic molecules, and photoelectron... 

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

Analysis of the rotational fine structure of IR bands yields the moments of inertia 7°, 7°, and 7 . From these, the molecular structure can be fitted. (It may be necessary to assign spectra of isotopically substituted species in order to have sufficient data for a structural determination.) Such structures are subject to the usual errors due to zero-point vibrations. Values of moments of inertia determined from IR work are less accurate than those obtained from microwave work. However, the pure-rotation spectra of many polyatomic molecules cannot be observed because the molecules have no permanent electric dipole moment in contrast, all polyatomic molecules have IR-active vibration-rotation bands, from which the rotational constants and structure can be determined. For example, the structure of the nonpolar molecule ethylene, CH2=CH2, was determined from IR study of the normal species and of CD2=CD2 to be8... 

Raman spectroscopy (Section 4.10) aids the study of the vibrations of polyatomic molecules. For a vibration to be Raman active, it must give a change in the molecular polarizability. For many molecules with some symmetry, one or more of the normal modes correspond to no change in... 

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