Vibrational wave function normal

Group theory can be applied to several different areas of molecular quantum mechanics, including the symmetry of electronic and vibrational wave functions, normal modes... 

The symmetry properties of a fundamental vibrational wave function are the same as those of the corresponding normal coordinate Q. For example, when the C3 operation is carried out on Qi, the normal coordinate for Vj, it is transformed into Q[, where... 

Figure 3 UMP2/6-311G potential energy profile (a) and hyperfine coupling constants (b) of CH3 versus s. Vibrational wave functions are normalized to 5.

We now allow nuclear motion and seek vibrational wave functions corresponding to states i i and tjfg. We assume throughout that the subunits have the same point group symmetry in both oxidation states (M and N), and then it is only necessary to consider explicitly totally symmetric normal coordinates of the two subunits (4, 5). Let us assume that there are two on each... 

A normal mode of vibration is said to be infrared active if the fundamental transition, in which the mode is excited by one quantum of vibrational energy, is allowed. Initial and final states are described by vibrational wave functions, of which the ground state wave function has Ai symmetry and the excited state has the same symmetry as the normal mode. Thus the fundamental transition... 

The fundamental vibrational energy levels lie at an energy kvp above the ground state. Each level has associated with it a certain fundamental frequency vp and therefore a certain X value Xp and each Xp has associated with it np normal coordinates Qp(m) (tn = l,2,...n ). Coupled with each of these normal coordinates there is a vibrational wave-function ppm which is proportional to (see e(ln (9-3.11)). Thus... 

Just as the electronic wave functions of molecules can be classified as g or u, so can the vibrational wave functions of such molecules. This classification of pvib refers to its behavior on inversion of the normal coordinates with respect to molecule-fixed axes. The ground vib of a molecule is always a function, since the polynomial factor in the ground-state Vlb is a constant. 

Since each pk is normalized, the vibrational wave function is normalized ... 

The vibrational wave function is a product of harmonic-oscillator functions, one for each normal coordinate hence... 

The integral < vib vib) maY vanish because of symmetry considerations. For example, the C02 normal mode v3 in Fig. 6.2 has eigenvalue — 1 for the inversion operation. Hence (Section 6.4), the v3 factor in the vibrational wave function is an even or odd function of the normal coordinate Q3, depending on whether v3 is even or odd. For a change of 1,3,5,... in the vibrational quantum number v3, the functions p vib and p"ib have different parities and their product is an odd function, so that ( ibl vib) vanishes. Thus we have the selection rule Ac3 = 0,2,4,... for electronic transitions in... 

The direct product enables one to find the symmetry of a wave function when the symmetries of its factors are known. For example, consider In the harmonic-oscillator approximation, the vibrational wave function is the product of 3N—6 harmonic-oscillator functions, one for each normal mode. To find the symmetry of we first examine the symmetries of its factors. Let the distinct vibrational frequencies of the molecule be vx>v2,..., vk,...,vn, and let vk be <4-fold degenerate let the harmonic-oscillator... 

This phenomenon of vibronic coupling can be treated very effectively by using group theoretical methods. As will be shown in Chapter 10, the vibrational wave function of a molecule can be written as the product of wave functions for individual modes of vibration called normal modes, of which there will be 3n - 6 for a nonlinear, /i-atomic molecule. That is, we can... 

It is further shown in Chapter 10 that, when each of the normal modes is in its ground state, each of the y/j is totally symmetric and hence y/v is totally symmetric. If one of the normal modes is excited by one quantum number, the corresponding it may then belong to one of the irreducible representations other than the totally symmetric one, say T, and thus the entire vibrational wave function f/Y will belong to the representation T,. Simple methods for finding the representations to which the first excited states of the normal modes belong are explained in Chapter 10. In this section we will quote without proof results obtained by these methods. 

As shown in Section 5.1, the wave functions must form bases for irreducible representations of the symmetry group of the molecule, and the same holds, of course, for all kinds of wave functions, vibrational, rotational, electronic, and so on. Let us now see what representations are generated by the vibrational wave functions of the normal modes. Inserting Hn(Va4,) into 10.6-1, we obtain... 

Here i//0 is the ground vibrational wave function and ij/ is the wavefunction corresponding to the first excited vibrational state of the th normal mode /< is the electric dipole moment operator Qj is the normal coordinate for the /th vibrational mode the subscript 0 at derivative indicates that the term is evaluated at the equilibrium geometry. The related rotational strength or VCD intensity is determined by the dot product between the electric dipole and magnetic dipole transition moment vectors, as given in (2) ... 

The vibrational wave function of the ground state belongs to the totally symmetric irreducible representation of the point group of the molecule. The wave function of the first excited state will belong to the irreducible representation to which the normal mode undergoing the particular transition belongs. 

Here F is the total vibrational wave function for the ground state, 4, is the total vibrational wave function for the first excited state referring to the z th normal mode and x, y and z are Cartesian coordinates. 

The approximation involved in factorization of the total wave function of a molecule into electronic, vibrational and rotational parts is known as the Bom-Oppenheimer approximation. Furthermore, the Schrodinger equation for the vibrational wave function (which is the only part considered here), transformed to the normal coordinates Qi (which are linear functions of the "infinitesimal displacements q yields equations of the harmonic oscillator t5q>e. For these reasons Lifson and Warshel have stressed that the force-field calculations should not be considered as classical-me-... 

The function y/mx is the solution of the rotational problem. The vibrational part, is a function of the normal coordinates and is the vibrational wave function. Substituting Eq. (4.24) in Eq. (4.23) and ignoring the rotational and translational contributions, the Schrddinger equation for the vibrational wave function will be ... 

For a more complete description of the vibronic levels associated with the Tig(A2 + E) electronic excited state, the wave functions Tj [Eq. f32)] must be augmented to include vibrational wave functions associated with all of the normal modes other than a = 2,5a, and 5e. Here we shall restrict our attention to just those vibronic levels derived from the vibrational modes a = 2,5a, and 5e. Assuming no vibronic couplings involving the ground electronic state of our model system, the rotatory strength of a transition between the lowest vibrational level of the ground electronic state and the j-th vibronic level of the Tjg (t2g + eg) coupled state may be written as... 

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