Water, normal modes

Normal modes of water. Experimental and (calculated) frequencies are shown. Theoretical frequencies "ed using a 6-310 basis set. 

A nonlinear molecule consisting of N atoms can vibrate in 3N — 6 different ways, and a linear molecule can vibrate in 3N — 5 different ways. The number of ways in which a molecule can vibrate increases rapidly with the number of atoms a water molecule, with N = 3, can vibrate in 3 ways, but a benzene molecule, with N = 12, can vibrate in 30 different ways. Some of the vibrations of benzene correspond to expansion and contraction of the ring, others to its elongation, and still others to flexing and bending. Each way in which a molecule can vibrate is called a normal mode, and so we say that benzene has 30 normal modes of vibration. Each normal mode has a frequency that depends in a complicated way on the masses of the atoms that move during the vibration and the force constants associated with the motions involved (Fig. 2). 

Assuming that a reasonable force field is known, the solution of the above equations to obtain the vibrational frequencies of water is not difficult However, in more complicated molecules it becomes very rapidly a formidable one. If there are N atoms in the molecule, there are 3N total degrees of freedom and 3N-6 for the vibrational frequencies. The molecular symmetry can often aid in simplifying the calculations, although in large molecules there may be no true symmetry. In some cases the notion of local symmetry can be introduced to simplify the calculation of vibrational frequencies and the corresponding forms of the normal modes of vibration. 

For die example of the water molecule it is of interest to calculate the forms of the vibrational modes, as obtained from the evaluation of the matrix L = UL. The results can be presented most simply as shown in Fig. 4. The calculation of the specific form of the normal modes is complicated, although with the aid of current computer programs it becomes routine - at least for relatively simple molecules. 

It is apparent from Fig. 4 that the normal modes of vibration of the water molecule, as calculated from the eigenvectors, can be described approximately as a symmetrical stretching vibration (Mj) and a symmetrical bending vibration... 

In water since we have 3N-6 = 3 allowed vibrational modes, these are referred as normal mode... 

Figure 4.19 Normal-mode vibrational quantum numbers for a bent triatomic molecule. Contrast the results for water, which is (cf. Table 4.6) near the local-mode limit with that for S02, which is near the normal-mode limit.

How one obtains the three normal mode vibrational frequencies of the water molecule corresponding to the three vibrational degrees of freedom of the water molecule will be the subject of the following section. The H20 molecule has three normal vibrational frequencies which can be determined by vibrational spectroscopy. There are four force constants in the harmonic force field that are not known (see Equation 3.6). The values of four force constants cannot be determined from three observed frequencies. One needs additional information about the potential function in order to determine all four force constants. Here comes one of the first applications of isotope effects. If one has frequencies for both H20 and D20, one knows that these frequencies result from different atomic masses vibrating on the same potential function within the Born-Oppenheimer approximation. Thus, we... 

The vibrational frequencies of isotopic isotopomers obey certain combining rules (such as the Teller-Redlich product rule which states that the ratio of the products of the frequencies of two isotopic isotopomers depends only on molecular geometry and atomic masses). As a consequence not all of the 2(3N — 6) normal mode frequencies in a given isotopomer pair provide independent information. Even for a simple case like water with only three frequencies and four force constants, it is better to know the frequencies for three or more isotopic isotopomers in order to deduce values of the harmonic force constants. One of the difficulties is that the exact normal mode (harmonic) frequencies need to be determined from spectroscopic information and this process involves some uncertainty. Thus, in the end, there is no isotope independent force field that leads to perfect agreement with experimental results. One looks for a best fit of all the data. At the end of this chapter reference will be made to the extensive literature on the use of vibrational isotope effects to deduce isotope independent harmonic force constants from spectroscopic measurements. 

Odelius and co-workers reported some time ago an important study involving a combined quantum chemistry and molecular dynamics (MD) simulation of the ZFS fluctuations in aqueous Ni(II) (128). The ab initio calculations for hexa-aquo Ni(II) complex were used to generate an expression for the ZFS as a function of the distortions of the idealized 7), symmetry of the complex along the normal modes of Eg and T2s symmetries. An MD simulation provided a 200 ps trajectory of motion of a system consisting of a Ni(II) ion and 255 water molecules, which was analyzed in terms of the structure and dynamics of the first solvation shell of the ion. The fluctuations of the structure could be converted in the time variation of the ZFS. The distribution of eigenvalues of ZFS tensor was found to be consistent with the rhombic, rather than axial, symmetry of the tensor, which prompted the development of the analytical theory mentioned above (89). The time-correlation... 

The analysis of vibration spectra proceeds by the use of normal modes. For instance, the vibration of a nonlinear water molecule has three degrees of freedom, which can be represented as three normal modes. The first mode is a symmetric stretch at 3586 cm , where the O atom moves up and the two H atoms move away from the O atom the second is an asymmetric stretch at 3725 cm where one H atom draws closer to the O atom but the other H atom pulls away and the third is a bending moment at 1595 cm , where the O atom moves down and the two H atoms move up and away diagonally. The linear CO2 molecule has four normal modes of vibration. The first is a symmetric stretch, which is inactive in the infrared, where the two O atoms move away from the central C atom the second is an asymmetric stretch at 2335 cm where both O atoms move right while the C atom moves left and the third and fourth together constitute a doubly degenerate bending motion at 663 cm where both O atoms move forward and the C atom moves backward, or both O atoms move upward and the C atom moves downward. 

By the symmetry of a normal mode of vibration, we mean tbe symmetry of the nuclear framework under the distortion introduced by the vibration. Pictorially, the symmetry of the normal mode is equal to the symmetry of the pattern of arrows drawn to indicate the directions of the nuclear displacements. The normal modes of vibration of water are the symmetric and antisymmetric stretches, and the angle bend, shown in Figure 6-1. 

Figure 6.1. Normal modes of vibration of the water molecule.

Another fact suggested by the examples is that not every subgroup of the original group need have an associated normal mode or electronic state. Thus there were two normal modes of water with full C2v symmetry and none having C2 symmetry. It will be useful to have computational tools for predicting which symmetry species will correspond to normal modes or electronic states, and how many and which symmetry species will not occur. 

Problem 7-18. Set up and solve the secular equation for the normal modes of water in internal coordinates. 

Problem 11-26. From the vector of general displacements of the three atoms of the water molecule, project out the unique normal mode of B2 symmetry. 

Normalization, 6 Normal modes, 240-244 of benzene, 438-439 of boron trifluoride, 281, 290 of carbon dioxide, 242, 248, 262, 265 of ethylene, 291 and group frequencies, 266-268 IR active, 457 Raman active, 457 and symmetry, 246-249,430-439 of water, 431-437 Normal operator, 108 Nuclear g factor, 3 24 Nuclear magnetic moments, 323-325 Nuclear magnetic resonance, 129-130, 323-366... 

Consider the symmetry behavior of the normal modes of H20 and COz. For water, the symmetry operations are E, C2 b), ov(ab), and av(bc), where the a and b principal axes lie in the molecular plane the bond angle in water is relatively obtuse, so that the C2 axis is the axis of intermediate moment of inertia. Each of these operations converts the displacement vectors of p, in Fig. 6.1 to an indistinguishable configuration the same is true for v2. Normal modes that are symmetric with respect to all the molecular symmetry operations are called totally symmetric. For H20, the modes and v2 are totally symmetric, but v3 is not. Figure 6.3 shows that v3 is antisymmetric with respect to C2(b). We can tabulate the behavior of the H20 normal modes with respect to the molecular symmetry operations ... 

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