Polyatomic molecules rotation

In most cases, for polyatomic molecules the nuclear partition function is again neglected, because it usually has a very small effect on the overall thermodynamic properties of polyatomic molecules. (Indeed, the only reason why we had to consider it for diatomic molecules is because it imposes an obvious, measurable effect on various observations, like spectra and thermodynamic properties to be considered in section 18.8.) In the high-temperature limit, a linear polyatomic molecule has the same rotational partition function as a homonuclear diatomic molecule  

A nonlinear polyatomic molecule can have up to three different moments of inertia, labeled 7a. 7b. and 7q. By convention, 7 is less than 7g, which is less than 7. Polyatomic molecules that have some symmetry may have some of their moments of inertia equal. If all three are equal, then the molecule is called a spherical top (see Chapter 14) and the rotational partition function can be written as 

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This integral has a known solution, and in terms of the variables in the above expression, the high-temperature limit for of a spherical top becomes 

If we define the rotational temperature 6 for a spherical-top polyatomic molecule as 

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In the case of a polyatomic molecule, rotation can occur in three dimensions about the molecular center of mass. Any possible mode of rotation can be expressed as projections on the three mutually perpendicular axes, x, y, and z hence, three moments of inertia are necessar y to give the resistance to angular acceleration by any torque (twisting force) in a , y, and z space. In the MM3 output file, they are denoted IX, lY, and IZ and are given in the nonstandard units of grams square centimeters. 

In a diatomic or linear polyatomic molecule rotational Raman scattering obeys the selection rule... 

Just as for diatomics, for a polyatomic molecule rotational levels are symmetric (5 ) or antisymmetric (a) to nuclear exchange which, when nuclear spins are taken into account, may result in an intensity alternation with J. These labels are given in Figure 6.24. 

The eigenfunctions of J2, Ja (or Jc) and Jz clearly play important roles in polyatomic molecule rotational motion they are the eigenstates for spherical-top and symmetric-top species, and they can be used as a basis in terms of which to expand the eigenstates of asymmetric-top molecules whose energy levels do not admit an analytical solution. These eigenfunctions IJ,M,K> are given in terms of the set of so-called "rotation matrices" which are denoted Dj m,K ... 

For diatomic and linear polyatomic molecules, rotational motion is restricted to an axis perpendicular to the molecular longitudinal axis. A corresponding diagram is shown in... 

For larger and nonlinear polyatomic molecules, rotational motion may occur with respect to three axes, resulting in a markedly raised number of rotational states. Moreover, the rotational constants determining the energy spacing of the rotational states (refer to Equation 2.39) diminish as a result of increased molecular moments of inertia. Consequently, both the spectral density and quantity of the rotational lines increase. The distance... 

There are two contributions to the polarizability of a molecule the distortion of the electronic wave function and the distortion of the nuclear framework. The major contribution is from the electrons, and can be considered to be the sum of contributions from the individual electrons. The contributions of the inner-shell electrons are nearly independent of orientation and these contributions can be ignored. The polarizability of electrons in a bond parallel to the bond direction is different from the polarizability perpendicular to that bond. As a diatomic molecule or linear polyatomic molecule rotates, the components of the polarizability in fixed directions are modulated (fluctuate periodically) as the ellipsoid of polarizability rotates. The rotation of a diatomic or linear polyatomic molecule will be Raman active (produce a Raman spectram). In a nonlinear polyatomic molecule, the polarizabilities of the individual bonds add vectorially to make up the total polarizability. If the molecule is a symmetric top, the total polarizability is the same in all directions and the ellipsoid of polarizability is a sphere. A spherical top molecule has no rotational Raman spectmm. Symmetric tops and asymmetric tops have anisotropic polarizabilities and produce rotational Raman spectra. 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

The above three sources are a classic and comprehensive treatment of rotation, vibration, and electronic spectra of diatomic and polyatomic molecules. 

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

In the experimental and theoretical study of energy transfer processes which involve some of the above mechanisms, one should distingiush processes in atoms and small molecules and in large polyatomic molecules. For small molecules a frill theoretical quantum treatment is possible and even computer program packages are available [, and ], with full state to state characterization. A good example are rotational energy transfer theory and experiments on Fie + CO [M] ... 

This Schrodinger equation relates to the rotation of diatomic and linear polyatomic molecules. It also arises when treating the angular motions of electrons in any spherically symmetric potential... 

D. Perturbative Treatment of Vibration-Rotation Coupling III. Rotation of Polyatomic Molecules... 

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in three-dimensions. 

The rotational motion of a linear polyatomic molecule can be treated as an extension of the diatomic molecule case. One obtains the Yj m (0,(1)) as rotational wavefunctions and, within the approximation in which the centrifugal potential is approximated at the equilibrium geometry of the molecule (Re), the energy levels are ... 

Treating the full internal nuclear-motion dynamics of a polyatomic molecule is complicated. It is conventional to examine the rotational movement of a hypothetical "rigid" molecule as well as the vibrational motion of a non-rotating molecule, and to then treat the rotation-vibration couplings using perturbation theory. 

The rotational kinetic energy operator for a rigid polyatomic molecule is shown in Appendix G to be... 

The first polyatomic molecule was detected in 1968 with use of a telescope having a dish 6.3 m in diameter at Hat Creek, California, USA, designed to operate in the millimetre wave region. Emission lines were found in the 1.25 cm wavelength region due to NH3. The transitions are not rotational but are between the very closely spaced 2 = 0 and 2 = 1 levels of the inversion vibration V2 (see Section 6.2.5.4). 

Rotational Raman spectra of diatomic and linear polyatomic molecules... 

As for diatomic molecules, there are stacks of rotational energy levels associated with all vibrational levels of a polyatomic molecule. The resulting term values S are given by the sum of the rotational and vibrational term values... 

The +, —, e, and/labels attached to the levels in Figure 7.25 have the same meaning as those in Figure 6.24 showing rotational levels associated with and Ig vibrational levels of a linear polyatomic molecule. Flowever, just as in that case, they can be ignored for a Z — I, type of electronic transition. 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

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