Harmonic oscillator wavefunctions

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

If we further assume that the vibrational wavefunctions associated with normal mode i are the usual harmonic oscillator ones, and r = u + 1, then the integrated intensity of the infrared absorption band becomes... 

Finally, it is a weU-known result of quantum mechanics" that the wavefunctions of harmonic oscillators extend outside of the bounds dictated by classical energy barriers, as shown schematically in Figure 10.1. Thus, in situations with narrow barriers it can... 

The selection rules illustrated above are general, as they depend only on the symmetry properties of the functions involved. However, more limiting, selection rules depend on the form of the wavefunctions involved. A relatively simple example of the development of specific selection rules is provided by the harmonic oscillator. The solution of this problem in quantum mechanics,... 

In this approximation the nuclear wavefunctions are a product of N harmonic oscillator functions, one for each normal mode ... 

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2,..., n/v), and those for the final state by m = (mi,m2,..., tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore... 

Now it becomes apparent why it was useful to replace the delta function by its Fourier transform. The wavefUnctions Xin are products of harmonic oscillator functions, the Hamiltonians Hi and H/ are sums of harmonic oscillator terms. Therefore the terms in the brackets factorize in the form ... 

In a first model, these motions are represented by harmonic vibrations, and the functions (Q) and Xbw (Q) are then replaced by products of harmonic oscillator-like wavefunctions. The solutions of Eqs. (9) take this particular form when the T jJ are negligible and when and H b can be expanded in terms of normal coordinates ... 

The absorption (emission) band shape at 0 K can be estimated from Equation (5.30) if the square of the overlap integral of harmonic functions, l(Xm(2)l/o(2)>P. is known for each excited (terminal) m level. Using the wavefunctions of the harmonic oscillator, these overlap functions can be expressed as a function of S ... 

The low-lying excited states of the hydrogen molecule conhned in the harmonic potential were studied using the configuration interaction method and large basis sets. Axially symmetric harmonic oscillator potentials were used. The effect of the confinement on the geometry and spectroscopic constants was analyzed. Detailed analysis of the effect of confinement on the composition of the wavefunction was performed. 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

The harmonic oscillator energies and wavefunctions comprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomic molecule are often characterized in terms of individual bond-stretching and angle-bending motions each of which is, in turn, approximated harmonically. This results in a total vibrational wavefunction that is written as a product of functions one for each of the vibrational coordinates. 

Here, De is the bond dissociation energy, re is the equilibrium bond length, and a is a constant that characterizes the steepness of the potential and determines the vibrational frequencies. The advantage of using the Morse potential to improve upon harmonic-oscillator-level predictions is that its energy levels and wavefunctions are also known exactly. The energies are given in terms of the parameters of the potential as follows ... 

Calculate the expectation value of the x2 operator for the first two states of the harmonic oscillator. Use the v=0 and v=l harmonic oscillator wavefunctions given below... 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

The molecular vibrational wavefunctions are now approximated in terms of products of harmonic-oscillator wavefunctions Xmu(vmu) and Xnv vnv), where vmu and vnv correspond to the vibrational quantum numbers ... 

The vibrational overlap integrals play a key role in electron transfer. A region of vibrational overlap defines values of the normal coordinate where a finite probability exists for finding coordinates appropriate for both reactants and products. The greater the overlap, the greater the transition rate. The vibrational overlap integrals can be evaluated explicitly for harmonic oscillator wavefunctions. An example is shown in equation (26) for the overlap between an initial level with vibrational quantum number v = 0 to a level v = v where the frequency (and force constant) are taken to be the same before and after electron transfer. 

For the predissociative C2N2 (C- TIU) state in the collinear approximation, the nuclear wavefunction is approximated by the product of three harmonic oscillator functions describing the normal modes vibrations. The frequencies and normal coordinates of the three linear stretching vibrations were obtained from ab initio MCHF calculations. The validity of the harmonic approximations is supported from experimental data (8) where absorption spectra of C2N2 is found to give a set of equidistant bands. 

The wavefunction and its square are known as gaussian or bell curves they occur in probability theory as the normal distribution. This function, together with three higher-energy solutions for the harmonic oscillator, is shown in Fig. 3.5. 

Here, instead of the usual delta function d(Eg, — Eg — ftv), which supplies only discontinuous spectral lines, a Lorentz function with a line width 29(normalized to one) is introduced for all vibrational levels labeled by n and m of electronic states j and k which contribute to the transition rate of Eq. (9) [58]. M is the transition operator, in general, pertinent to an electric dipole transition. The nuclear wavefunctions are approximated by products of harmonic oscillator functions... 

The Gj(t) functions of Eq. (15) have been calculated by Lin [60] when summing over Franck-Condon factors obtained from all possible (infinite) wavefunctions in the harmonic oscillator approximation. These Gj(t) are rather complicated functions of the frequencies arf, co and reduced masses M j, M which are attributed to the corresponding normal coordinates Qf and Q j. They are collected in parameters describing the frequency relation ft2 and the potential minimum shift Aj of the excited state with respect to the ground state... 

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

In order to elucidate the general behavior we also show in Figure 10.3 what we will call the harmonic oscillator (HO) approximation, i.e., expression (10.7) without the sinusoidal factor. It represents the momentum distribution of the harmonic oscillator in the fcth bending vibrational state. Suppression of the fast oscillations has the advantage of elucidating more clearly the wide oscillations for k > 0, which reflect the nodal structure of the excited bending wavefunctions. The superimposed fast oscillations, on the other hand, reflect the shift of the equilibrium angle 7e away from zero. They are absent for a linear molecule and most pronounced for 7e 7t/2, as for H2S and H2O, for example. 

The wavefunction of the parent molecule in the electronic ground state is assumed to be a product of two harmonic oscillator wavefunctions with m and n quanta of excitation along R and r, respectively. In Figure 13.2(b) only the vibrational mode of BC is excited, n = 3, while the dissociation mode is in its lowest state, m = 0. The corresponding spectrum is smooth without any reflection structures. Conversely, the wavefunction in Figure 13.2(a) shows excitation in the dissociation mode, m = 3, while the vibrational mode of BC is unexcited. The resulting spectrum displays very clear reflection structures in the same way as in the one-dimensional case. Thus, we conclude that, in general ... 

Fig. 13.2. Illustration of the origin of reflection structures for polyatomic molecules. The potential energy surface is of the form (6.35) with e = 0. The wave-function of the parent molecule is simply the product of two harmonic oscillator wavefunctions. The heavy arrows illustrate the dissociation path.

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