Degrees of freedom for a molecule

The number of degrees of freedom for a molecule is the number of independent coordinates needed to specify its position and configuration. Hence a molecule of Adatoms has 3A/ degrees of freedom. These could be taken as the three Cartesian coordinates of the N individual atoms, but it is more convenient to classify them as follows. 

We will restrict the further considerations to the case, where only one product in the expansion of the total wave function is relevant. Instead of the MCTDSCF approximation the solution is approximated by a single product function wherein these functions are determined in a self consistent way (time dependent SCF approximation, TDSCF). The situation is similar to that where there are several electronic degrees of freedom for a molecule, but where it has been demonstrated that the a batic Bom-Oppenheimer approximation works substantially well for the description of most spectroscopic and other properties of molecules. 

To calculate the values of partition functions of the molecule each of the motions (electronic, translational, rotational and vibrational) should be coinsidered keeping in mind that a molecule of n atoms has 3 n degrees of freedom. For a molecule there are three degrees of translational... 

Point 4 is related to the molecular motion discussed in Section 19.3. In general, the number of degrees of freedom for a molecule increases with increasing number of... 

Point 4 is related to the molecular motion discussed in Section 19.3. In general, the number of degrees of freedom for a molecule increases with increasing number of atoms, and thus the number of possible microstates also increases. Figure 19.13 compares the standard molar entropies of three hydrocarbons in the gas phase. Notice how the entropy increases as the number of atoms in the molecule increases. 

In Chapter 1, the number of degrees of freedom for a molecule was calculated by considering the motion of each atom in the X,Y and Z directions. Each atom can move in three dimensions, and so a molecule containing N atoms has 3N degrees of freedom. The discussion of the guitar vibrations above shows that the actual motion of the atoms in a... 

Figure 10.2 Translational, rotational, and vibrational degrees of freedom for a simple diatomic molecule.

Letter from G. N. Lewis to Paul Ehrenfest, undated but probably 1925, G. N. Lewis Correspondence, BL.UCB. G. N. Lewis and D. F. Smith promised in their paper, "The Theory of Reaction Rate," JACS 47 (1925) 15081520, to publish a demonstration that a range of frequencies of radiation affecting degrees of freedom in a molecule is responsible for chemical reaction. This paper was the subject of the letter, with anonymous referee s report, from Arthur B. Lamb to G. N. Lewis, 28 February 1925, G. N. Lewis Papers, BL.UCB. The referee said "No real unimolecular reaction has actually been observed they have been shown to be merely catalytic the idea that a unimolecular reaction is due to collision between a quantum and a molecule is not original with Lewis."... 

For a molecule with N atoms, its 3iV degrees of freedom would be split into three translational degrees of freedom (corresponding to x-, y-, and z-directions), and three rotational degrees of freedom for nonlinear molecules and two for linear ones. Therefore, 3N—6 and 3N— 5 vibrational degrees of freedom exist for nonlinear and linear molecules, respectively. Vibrational frequencies can be obtained from convenient tabulations (see, for example, Shimanouchi, 1972 Chase et al., 1985). 

The utility of Eq. (9.49) depends on the ease with which the Hessian matrix may be constructed. Methods that allow for the analytic calculation of second derivatives are obviously the most efficient, but if analytic first derivatives are available, it may still be worth the time required to determine the second derivatives from finite differences in the first derivatives (where such a calculation requires that the first derivatives be evaluated at a number of perturbed geometries at least equal to the number of independent degrees of freedom for tlie molecule). If analytic first derivatives are not available, it is rarely practical to attempt to construct the Hessian matrix. 

The number of angles required to specify a molecule s orientation depends on whether it is linear or nonlinear. It takes only two angles, 0 and , to specify the orientation for a linear molecule, as illustrated in Fig. 8.2. Thus there are two rotational degrees of freedom for a linear molecule. It takes three angles, 6, 0, and nonlinear molecule in space, so a nonlinear molecule has three rotational degrees of freedom. 

The statistical mechanical form of transition theory makes considerable use of the concepts of translational, rotational and vibrational degrees of freedom. For a system made up of N atoms, 3N coordinates are required to completely specify the positions of all the atoms. Three coordinates are required to specify translational motion in space. For a linear molecule, two coordinates are required to specify rotation of the molecule as a whole, while for a non-linear molecule three coordinates are required. These correspond to two and three rotational modes respectively. This leaves 3N — 5 (linear) or 3N — 6 (non-linear) coordinates to specify vibrations in the molecule. If the vibrations can be approximated to harmonic oscillators, these will correspond to normal modes of vibration. 

Question. Given that the total number of degrees of freedom for a polyatomic molecule is 3N, calculate the number of vibrational modes open to (a) an atom, (b) a diatomic molecule, (c) a non-linear polyatomic molecule with N atoms and (d) a non-linear activated complex with N atoms. 

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