Translation and rotation

MSS Molecule surface scattering [159-161] Translational and rotational energy distribution of a scattered molecular beam Quantum mechanics of scattering processes... 

Another distinction we make concerning synnnetry operations involves the active and passive pictures. Below we consider translational and rotational symmetry operations. We describe these operations in a space-fixed axis system (X,Y,Z) with axes parallel to the X, Y, Z) axes, but with the origin fixed in space. In the active picture, which we adopt here, a translational symmetry operation displaces all nuclei and electrons in the molecule along a vector, say. 

Variational RRKM theory is particularly important for imimolecular dissociation reactions, in which vibrational modes of the reactant molecule become translations and rotations in the products [22]. For CH —> CHg+H dissociation there are tlnee vibrational modes of this type, i.e. the C—H stretch which is the reaction coordinate and the two degenerate H—CH bends, which first transfomi from high-frequency to low-frequency vibrations and then hindered rotors as the H—C bond ruptures. These latter two degrees of freedom are called transitional modes [24,25]. C2Hg 2CH3 dissociation has five transitional modes, i.e. two pairs of degenerate CH rocking/rotational motions and the CH torsion. 

Thus the transfonnation matrix for the gradient is the inverse transpose of that for the coordinates. In the case of transfonnation from Cartesian displacement coordmates (Ax) to internal coordinates (Aq), the transfonnation is singular becanse the internal coordinates do not specify the six translational and rotational degrees of freedom. One conld angment the internal coordinate set by the latter bnt a simpler approach is to rise the generalized inverse [58]... 

Figure C3.3.9. A typical trajectory for a hard collision between a hot donor molecule and a CO2 bath molecule in which the CO 2 becomes translationally and rotationally excited.

Michaels C A, Lin Z, Mullin A S, Tapalian H C and Flynn G W 1997 Translational and rotational excitation of the C02(00°0) vibrationless state in the collisional quenching of highly vibrationally excited perfluorobenzene evidence for impulsive collisions accompanied by large energy transfers J. Chem. Phys. 106 7055-71... 

In the strictest meaning, the total wave function cannot be separated since there are many kinds of interactions between the nuclear and electronic degrees of freedom (see later). However, for practical purposes, one can separate the total wave function partially or completely, depending on considerations relative to the magnitude of the various interactions. Owing to the uniformity and isotropy of space, the translational and rotational degrees of freedom of an isolated molecule can be described by cyclic coordinates, and can in principle be separated. Note that the separation of the rotational degrees of freedom is not trivial [37]. 

Hermans, J., Wang, L. Inclusion of loss of translational and rotational freedom in theoretical estimates of free energies of binding. Application to a complex of benzene and mutant T4-lysozyme. J. Am. Chem. Soc. 119 (1997) 2707-2714... 

The length or dimension of the RDF code is independent of the number of atoms and the size of a molecule, unambiguous regarding the three-dimensional arrangement of the atoms, and invariant against translation and rotation of the entire molecule. 

Caution During a sininlation, solvent temperature may increase wh ile th e so In te cools. This is particii larly true of sm all solven t molecules, such as water, that can acquire high translational and rotational energies. In contrast, a macromolecule, such as a peptide, retains most of its kinetic energy in vibrational modes. This problem rem ains un solved, an d this n ote of cau tion is provided to advise you to give special care to simulations using solvent. 

An N-atom molecular system may he described by dX Cartesian coordinates. Six independent coordinates (five for linear molecules, three fora single atom) describe translation and rotation of the system as a whole. The remaining coordinates describe the nioleciiUir configuration and the internal structure. Whether you use molecular mechanics, quantum mechanics, or a specific computational method (AMBER, CXDO. etc.), yon can ask for the energy of the system at a specified configuration. This is called a single poin t calculation. ... 

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

Translation and rotation Translation, rotation, torsion Translation, rotation, torsion, vibration... 

In Table 6.6 the results for the point group are summarized and the translational and rotational degrees of freedom are subtracted to give, in the final column, the number of vibrations of each symmetry species. 

The classical values of each of drese components can be calculated by ascribing a contribution of R/2 for each degree of freedom. Thus tire U ansla-tional and tire rotational components are 3/27 each, for drree spatial components of translational and rotational movement, and (3 — 6)7 for die vibrational contribution in a non-linear polyatomic molecule containing n atoms and (3 — 5)7 for a linear molecule. For a diatomic molecule, the contributions ate 3/27 ti.a s -f 7 [.ot + 7 vib-... 

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

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