Rigid linear rotators

Expressions of the partition function in the approximation of small quantum corrections are worked out using the following molecular models (1) rigid linear rotators (Section III) (2) rigid S5nn-metricaJ tops (Section IV). 

We will presently give the explicit expressions for the /> and the ( pj pX y for the linear momenta of the translational motion of the center of mass. The corresponding expressions for the rotational motion will be determined separately for the case of rigid linear rotators and rigid symmetric tops in the next sections. 

The correlation function for free rotation of a rigid linear molecule goes to zero at large times, since the axis of rotation is always perpendicular to the dipole, and is ven by... 

The geometrical meaning is that the Eckart subspace pf is perpendicular to the three-dimensional manifold of rigid-body rotation at the reference configuration z.si. The Eckart subspace is Euclidean since the conditions in Eq. (32) are linear. Therefore this space can be spanned by vectors , (p = 1,..., 3m — 6) that specify the 3m 6 directions of (vibrational) normal modes at the reference configuration zVI- in the (3m — 3(-dimensional configuration space. The vectors (m,m are orthonormal as Mi > = [Pg.107]

If we compare Eq. (4.78) with Eq. (4.73), it is clear that the algebraic three-dimensional model provides the correct rotational spectrum of a rigid linear rotor, where the (vibrational) angular momentum coefficient, ggg, is described by the algebraic parameters A 2 and A j2- The J-rotational band is obtained by recalling in Eq. (4.12), the branching law... 

Small. E W., and Isenbeig, I., 1977, Hydrodynamic operties of a rigid molecule Rotational and linear diflualon and fluonaeenoe anisotropy, Biopofypun 16 1907-1928. 

On the basis of plastic deformation visible at 2% extension, the original draw direction being contracted while the thickness of the sheet remained unchanged, the non-linearity at 0 = 90° was attributed to a slip mechanism involving rigid body rotation. This process occurred in conjunction with the linear mechanism associated with unordered r ons. Again c/c shear should be negligible. 

Since the components in each series can be varied in their detailed structure, and mixed between the two series, it seems that a valuable approach has been devised to prepare nanometer-scale lines. Although these lines can flex and rotate internally about single bonds, they are as rigidly linear as any molecules we have imagined. 

Electrokinetic Motion of Heterogeneous Particles, Fig. 1 Examples of unusual linear electrophoretic motion of heterogeneous particles, (a) A dumbhell consisting of two oppositely charged spheres connected by a rigid rod rotates to align as shown and moves in the direction of the electric field (positive mobility), even... 

A FIGURE 9.1 The rotational energy level diagram of a rigid linear molecule. 

Consider a rigid body rotating about a fixed axis with a constant angular velocity, cj, in radians per second. This rotation can be described by a vector CO with length cj and a direction parallel to the axis of rotation. A point P not on the rotation axis will then have a linear velocity given by... 

The rotation of a rigid, linear triatomic or polyatomic molecule is mechanically equivalent to the rotation of a rigid diatomic molecule. All are the rotations of an "infinitesimally thin rod" with two or more point masses attached. The basic analysis for the rotational spectroscopy of linear polyatomic molecules follows that of diatomic molecules however, it requires a generalization of the moment of inertia to more than two atoms. For a linear arrangement of point masses, the moment of inertia, I, about the center of mass is... 

A series of lines at 2B, 4B, 6B,... is thus expected for a rigid rotor. The energy levels, allowed transitions, and spectrum of a rigid linear rotor are illustrated in Fig. 6. The molecule OCS, which is commonly used as a standard for various purposes by microwave spectroscopists, has lines that occur at 12,162.97, 24,325.92, 36,488.80, 48,651.40 MHz,... for the most common isotope. For a light molecule such as CO, B = 57,635.97 MHz, and the rotational lines are spaced 115,271.94 MHz apart thus, high-frequency microwave techniques must be employed to measure even the 0 1 transition, which is at 115,271.94 MHz. The effect of centrifugal distortion is to produce a small shift to lower frequency in each transition. Illustrative rotational constants are collected in Table in. 

An anisotropic stress tensor means that there is non-zero dissipation if the entire fluid undergoes a rigid-body rotation, which is clearly unphysical. However, as emphasized in [28], this asymmetry is not a problem for most applications in the incompressible (or small Mach number) limit, since the form of the Navier-Stokes equation is not changed. This is in accordance with results obtained in SRD simulations of vortex shedding behind an obstacle [36], and vesicle [37] and polymer dynamics [14]. In particular, it has been shown that the linearized hydrodynamic modes are completely unaffected in two dimensions in three dimensions only the sound damping is slightly modified [28]. 

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. 

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. 

Within this "rigid rotor" model, the absorption speetrum of a rigid diatomie moleeule should display a series of peaks, eaeh of whieh eorresponds to a speeifie J ==> J + 1 transition. The energies at whieh these peaks oeeur should grow linearally with J. An example of sueh a progression of rotational lines is shown in the figure below. 

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 ... 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

If it is assumed that 2,2 -bipyridine is bonded to the catalyst by both nitrogen atoms, then the position of the chemisorbed molecule on the metal is rigidly fixed. Unless two molecules of this base can be adsorbed at the required distance from each other and in an arrangement which is close to linear, overlap of the uncoupled electrons at the a-position cannot occur. The failure to detect any quaterpyridine would then indicate that nickel atoms of the required orientation are rarely, if ever, available. Clearly the probability of carbon-carbon bond formation is greater between one chemisorbed molecule of 2,2 -bipyridine and one of pyridine, as the latter can correct its orientation relative to the fixed 2,2 -bipyridine by rotation around the nitrogen-nickel bond, at least within certain limits. 

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