Symmetry and normal coordinates

In Section 1.6, we have shown the relationship between Cartesian and normal coordinates (Eq. (1-44)). Similar relationships exist between internal (symmetry) and normal coordinates ... 

From Eq. (22) (Section VII), it follows, assuming that molecular symmetry and normal coordinates can be equated, that... 

No attempt will be made here to compute further force constants instead the reader is referred to the papers of Crawford and Miller cited above. However, as an example of the calculation of frequencies from a set of known constants, the Eiu. factor (infrared active) will be worked out, using Crawford and Miller s constants. Although a prolilcm of this size (three frequencies) can be carried through by direct expansion to the polynomial form, we shall not use such a method, but shall instead illustrate some of the matrix numerical methods which were described in Sec. 9-7. This method will also yield the transformation coefficients between symmetry and normal coordinates. 

Outside the treatment in the space of purely vibrational coordinates, such as internal, symmetry and normal coordinates, vibrational motion can be described on die... 

The allyl radical [115] trapped in an argon matrix can be photolytically (A = 410 nm) converted into the cyclopropyl radical [116] (Holtzhauer er a/., 1990). Dicyclopropane and cyclopropane were formed when the photolysed matrix was warmed from 18 to 35 K. The intermediate [116] was shown to be a cr-type (Cs symmetry) and not a rr-type symmetry) radical. Normal coordinate analysis of the radical [116] has been carried out and the IR band at 3118 cm has been assigned to the stretching vibration of the C—H bond at the radical centre. 

Here q and Q symbolize the sets of all electronic and nuclear coordinates q (i = 1,. . ., 3 ) and Qt (i = 1,2,.. . , 3AT — 6), respectively. The derivatives are taken at the coordinate values Qf and the summation runs over all nuclear coordinates of independent vibrations. The expansion may be carried out with respect to different types of nuclear coordinates, i.e. symmetry coordinates and normal coordinates of the ground or the excited states. If the Q, s are normal coordinates and the Q s are taken at the potential minimum of an electronic state E the coordinate values are by definition Q = 0 for all i. In this case the matrix elements of the electron dependent part in the second term of Eq. (1) should vanish due to the minimal condition, i.e. 

Thus, the force constants of the bonds, the masses of the atoms, and the molecular geometry determine the frequencies and the relative motions of the atoms. Fig. 2.1-3 shows the three normal vibrations of the water molecule, the symmetric and the antisymmetric stretching vibration of the OH bonds, and Va, and the deformation vibration 6. The normal frequencies and normal coordinates, even of crystals and macromolecules, may be calculated as described in Sec. 5.2. In a symmetric molecule, the motion of symmetrically equivalent atoms is either symmetric or antisymmetric with respect to the symmetry operations (see Section 2.7). Since in the case of normal vibrations the center of gravity and the orientation of the molecular axes remain stationary, equivalent atoms move with the same amplitude. 

Fig. 7 Summary of the JT effect in cobaltocene. Symmetries of the corresponding geometries, electronic states and normal coordinates, numbering of C atoms in the cyclopentadienyl rings, as well as the relative energies of the different structures is given...

For [XeF5]+ of C4v symmetry nine fundamental vibrations spanning the irreducible representations 3at + 2bj + b2 + 3e should be observed. All nine modes should be Raman active, but only the a, and e modes infrared active. On the basis of these spectra all nine fundamentals for [XeF,]+ were assigned and normal coordinate analysis carried out. The assignments for the fundamentals are given in Table 16. 

The total deformation space of ethylene and hence of any olefin can be described in a systematic manner by group theory. This approach has competently been elaborated by Ermer (11). From the irreducible representation of ethylene (1) with its Dlh symmetry, twelve normal coordinates are obtained, three of which are related to out-of-plane and nine to the in-plane... 

I.r. and Raman spectra of H3P,BCl3 and D3P,BCl3 at —196 °C have been reported and assigned, on the basis of symmetry. A normal-coordinate analysis was carried out, indicating a force constant of 1.96 mdyn A for the P— B stretch. Similar experiments and calculations on Mc3P,BH3 and Mc3P,BD3 show that v(PB) and (PC) are extensively mixed, while the P—B stretching force constant in this case is 2.37 mdyn A . ... 

Once the reference force field F is properly derived, diagonalization of the GF° matrix [see Eq. (8)] yields the reference frequencies normal coordinates Q, . We use capital Q s to distinguish normal coordinates from the generic nuclear coordinates q in Eq. (9). This is a partial solution, since we still have to account for linear e-ph coupling. The expansion in Eq. (9) is conveniently carried out on the basis of the reference normal coordinates. The simplest case arises when, at least for a given symmetry subspace, only one electronic operator, 0, is coupled to the phonons. 

U is an orthogonal matiix. The vibrational problem is solved separately for vibrations belonging to different symmetry species of the molecular point group. In this way the number of independent force constants defining the potential field is considerably reduced. The relation between symmeby and normal coordinates is given by die expression... 

The cyclopentadienyl radical and the cyclopentadienyl cation are two well-known Jahn-Teller problems The traditional Jahn-Teller heatment starts at the D k symmetry, and looks for the normal modes that reduce the symmetry by first-01 second-order vibronic coupling. A Longuet-Higgins treatment will search for anchors that may be used to form the proper loop. The coordinates relevant to this approach are reaction coordinates. 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

Summarizing, in the crystal there are 36 Raman active internal modes (symmetry species Ug, hig, 2g> and 26 infrared active internal modes (biw b2w hsu) as well as 12 Raman active and 7 infrared active external vibrations (librations and translations). Vibrations of the type are inactive because there appears no dipole moment along the normal coordinates in these vibrations of the crystal. 

After the first unsuccessful attempts to record a matrix IR spectrum of the methyl radical, reliable data were obtained by the use of the vacuum pyrolysis method. IR spectra of the radicals CH3 and CD3 frozen in neon matrices were measured among the products of dissociation of CH3I, (CH3)2Hg and CD3I (Snelson, 1970a). The spectra contained three absorptions at 3162 (1 3), 1396 V2) and 617 cm (I l) belonging to the radical CH3 and three bands 2381, 1026 and 463 cm assigned to the radical CD3. Normal coordinate analysis of these intermediates was performed and a valence force field calculated. In accordance with the calculations, methyl radical is a planar species having symmetry >31,. 

In the example considered above, Arj - A/s is the only symmetry coordinate of species B2. Thus, it results in a factor of degree one in the completely reduced secular determinant It is therefore a normal coordinate. On the other hand, the two normal coordinates of species Ai are linear combinations of the symmetry coordinates Acr and Arj + Ar2. They can only be found by solution of the secular equations. 

Fig. 18. Top transition coordinates (with symmetry species) of conformational transition states of cyclohexane (top and side views). Hydrogen displacements are omitted. The displacement amplitudes given are towards the C2v-symmetric boat form, and towards >2-symmetric twist forms (from left), respectively. Inversion of these displacements leads to the chair and an equivalent T>2-form, respectively. Displacements of obscured atoms are given as open arrows, obscured displacements as an additional top. See Fig. 17 for perspective conformational drawings. Bottom pseudorotational normal coordinates (with symmetry species) of the Cs- and C2-symmetric transition states. The phases of the displacement amplitudes are chosen such that a mutual interconversion of both forms results. The two conformations are viewed down the CC-bonds around which the ring torsion angles - 7.3 and - 13.1° are calculated (Fig. 17). The displacement components perpendicular to the drawing plane are comparatively small. - See text for further details.

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