Motion degrees of freedom and

Conical intersections between electronically adiabatic potential energy surfaces are not only possible but actually quite frequent, if not prevalent, in polyatomic systems. Some examples are triatomic systems whose isolated atoms have 2S ground states [1,2] such as H3 and its isotopomers (DH2, HD2, HDT, etc.), LiH2 and its isotopomers, and tri-alkali systems such as Na3 and LiNaK. Many other kinds of polyatomic molecules also display such intersections. The reason is that they have three or more internal nuclear motion degrees of freedom, and only two independent relations between electronic Ham-... 

A joint is defined by its type, parameters, and auxiliary entities. The type of joint determines the degrees of freedom. Auxiliary entities are the reference points and lines needed for the definition of movements allowed by the degrees of freedom. Figure 5-20 introduces common joints from the everyday design of mechanisms by their motions, degrees of freedom, and auxiliary entities. 

Fig. 40. Attempt to schematically represent the ordered mesophase of PDES juxtaposed to the isotropic phase. The coUapsed coil structure assumed in the mesophase entails chain folds that provide additional motional degrees of freedom and is therefore thermodynamically stabilized in a certain temperature range (compare the statistical physics treatment in Ref. [124]). The shaded ellipses indicate somewhat closer packed, more ordered and less mobile areas in contrast to the more mobile, more distorted chain conformations in regions where many folds accumulate...

Because the degrees of freedom decouple in the linear approximation, it is easy to describe the dynamics in detail. There is the motion across a harmonic barrier in one degree of freedom and N — 1 harmonic oscillators. Phase-space plots of the dynamics are shown in Fig. 1. The transition from the reactant region at q <0 to the product region at q >0 is determined solely by the dynamics in (pi,qi), which in the traditional language of reaction dynamics is called the reactive mode. 

The internal degrees of freedom are associated with the rotation and vibration of the molecule. A linear molecule has 2 degrees of rotational motion and a non-linear molecule has three. The remaining (3N — 5) or (3N — 6) degrees of freedom describe the motion of the nuclei with respect to each other. For example, the linear CO2 has (3N — 5) = 4 vibrational degrees of freedom and the non-linear SO2 has three. The mode... 

The internal energy is, as indicated above, connected to the number of degrees of freedom of the molecule that is the number of squared terms in the Hamiltonian function or the number of independent coordinates needed to describe the motion of the system. Each degree of freedom contributes jRT to the molar internal energy in the classical limit, e.g. at sufficiently high temperatures. A monoatomic gas has three translational degrees of freedom and hence, as shown above, Um =3/2RT andCy m =3/2R. 

In this expression, N is the number of times a particular irreducible representation appears in the representation being reduced, h is the total number of operations in the group, is the character for a particular class of operation, jc, in the reducible representation, is the character of x in the irreducible representation, m is the number of operations in the class, and the summation is taken over all classes. The derivation of reducible representations will be covered in the next section. For now, we can illustrate use of the reduction formula by applying it to the following reducible representation, I-, for the motional degrees of freedom (translation, rotation, and vibration) in the water molecule ... 

A detailed analysis of the rotational degrees of freedom and of bending motion has been carried out by Freed and Band (2) and by Morse and Freed (52-54). The former authors considered the photodissociation of a linear triatomic molecule. In that work the rotational part of the initial wavefunction is written in the form... 

The number of coordinates needed to specify each type of motion is called the number of degrees of freedom e.g., non-linear molecules have three translational degrees of freedom, three rotational degrees of freedom and 3N — 6 vibrational degrees of freedom. 

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

A molecule composed of A atoms has in general 3N degrees of freedom, which include three each for translational and rotational motions, and (3N — 6) for the normal vibrations. During a normal vibration, all atoms execute simple harmonic motion at a characteristic frequency about their equilibrium positions. For a linear molecule, there are only two rotational degrees of freedom, and hence (3N — 5) vibrations. Note that normal vibrations that have the same symmetry and frequency constitute the equivalent components of a degenerate normal mode hence the number of normal modes is always equal to or less than the number of normal vibrations. In the following discussion, we shall demonstrate how to determine the symmetries and activities of the normal modes of a molecule, using NH3 as an example. 

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