Harmonic oscillator vibrational energy levels

We expect, therefore, that a reasonable approximation to the vibrational energy levels vib of a diatomic molecule would be the harmonic-oscillator vibrational energy levels Eqs. (4.47) and (4.25) give... 

In order to evaluate the vibrational partition function we consider a single harmonic oscillator. The energy levels, with the zero of energy as the zero-point level, are En = hi/sn, with n = 0,1,..., and degeneracy ojn = 1. The partition function takes the form... 

In the electronic ground state, acetylene is a linear molecule. By using the harmonic approximation, vibrational energy levels for small amplitude oscillations can be labeled as (Uj, V2, v, v, v ) where is the quantum... 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

We have seen in Section 1.3.6 how the vibrational energy levels of a diatomic molecule, treated in the harmonic oscillator approximation, are given by... 

The simplest model for vibrational energy levels is the harmonic oscillator, which has allowed levels with energy... 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

Molecules also possess internal degrees of freedom, namely vibration and rotation. The vibrational energy levels in the harmonic oscillator approximation of a vibration with frequency hv are given by... 

This result is the same as for the harmonic oscillator and the allowed vibrational energy levels follow immediately as... 

Without derivation, we may consider the bonds between atoms as a spring connecting two atoms in a harmonic oscillator. The energy difference between two vibrational energy levels is... 

When the electron transfer process is coupled to classical reorientation modes and to only one harmonic oscillator whose energy quantum h( is high enough for only the ground vibrational level to be populated, the expression of the electron transfer rate is given by [4, 9] ... 

The general form of the energy of the harmonic oscillator indicates that the vibrational energy levels are equally spaced. Due to the vector character of the dipole transition operator, the transition between vibronic energy levels is allowed only if the following selection rule is satisfied ... 

Substitution of the potential energy for this harmonic oscillator into the Schrodinger wave equation gives the allowed vibrational energy levels, which are quantified and have energies Ev given by... 

However, as seen in Fig. 3.2, this idealized harmonic oscillator (Fig. 3.2b) is satisfactory only for low vibrational energy levels. For real molecules, the potential energy rises sharply at small values of r, when the atoms approach each other closely and experience significant charge repulsion furthermore, as the atoms move apart to large values of r, the bond stretches until it ultimately breaks and dissociation occurs (Fig. 3.2c). 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

Figure 9.7 Vibrational energy levels determined from solution of the one-dimensional Schrodinger equation for some arbitrary variable 6 (some higher levels not shown). In addition to the energy levels (horizontal lines across the potential curve), the vibrational wave functions are shown for levels 0 and 3. Conventionally, the wave functions are plotted in units of (probability) with the same abscissa as the potential curve and an individual ordinate having its zero at the same height as the location of the vibrational level on the energy ordinate - those coordinate systems are explicitly represented here. Note that the absorption frequency typically measured by infrared spectroscopy is associated with the 0 —> 1 transition, as indicated on the plot. For the harmonic oscillator potential, all energy levels are separated by the same amount, but this is not necessarily the case for a more general potential...

The fundamental frequencies 9t (t = 1, 2,... 3tf—6) are related to and since Xt are the roots of det B—XE) — 0, r, are related to the matrix B and to the molecular force constants Bif. Hence the vibrational energy levels for a non-linear polyatomic molecule in the harmonic oscillator approximation are given by... 

The quantum mechanical vibrational energy levels for a harmonic oscillator are... 

Theory predicts that for a harmonic oscillator only a change from one vibrational energy level to the next higher is allowed, but for anharmonic oscillators weaker transitions to higher vibrational energy levels can occur. The resulting "overtones" are found at approximate multiples of the frequency of the fundamental. Combination frequencies representing sums... 

Vibrational energy levels can be approximated by the harmonic oscillator. For a diatomic molecule the relationship is... 

If the rotational quantum number J is zero, the molecule possesses no angular momentum arising from the motion of the nuclei nuclear motion is purely vibrational. The vibrational energy levels depend on the shape of the potential function 1/(7 ), most often of the well-known diatomic form. Near the minimum the potential function approximates to a parabola. The eigenvalues and functions are thus approximately those appropriate for a harmonic oscillator,... 

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