Harmonic vibrational energy levels

The electronic energy of a molecule, ion, or radical at geometries near a stable structure can be expanded in a Taylor series in powers of displacement coordinates as was done in the preceding section of this Chapter. This expansion leads to a picture of uncoupled harmonic vibrational energy levels... 

For harmonic vibrational energy levels, eq. (36) reduces to the well-known closed-form result, which we write as... 

Figure 7-8. The low lying harmonic vibrational energy levels for a gas phase water molecule are shown. Note that as the energy increases, the density of states increases.

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... 

Just as the electrical behaviour of a real diatomic molecule is not accurately harmonic, neither is its mechanical behaviour. The potential function, vibrational energy levels and wave functions shown in Figure f.i3 were derived by assuming that vibrational motion obeys Hooke s law, as expressed by Equation (1.63), but this assumption is reasonable only... 

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

Vibrational Energy Levels A diatomic molecule has a single set of vibrational energy levels resulting from the vibration of the two atoms around the center of mass of the molecule. A vibrating molecule is usually approximated by a harmonic oscillatord for which... 

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... 

Figure 8.1 The harmonic potential and the Morse potential, together with vibrational energy levels. The harmonic potential is an acceptable approximation for molecular separations close to the equilibrium distance and vibrations up to the first excited level, but fails for higher excitations. The Morse potential is more realistic. Note that the separation between the vibrational levels decreases with increasing quantum number, implying, for example, that the second overtone occurs at a frequency slightly less than twice that of the fundamental vibration.

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... 

The spacing between vibrational energy levels cannot be ignored for calculations of qvib at room temperature. For a single harmonic vibration, the energy levels are given by... 

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 ... 

At larger bond lengths, the true potential is "softer" than the harmonic potential, and eventually reaches its asymptote which lies at the dissociation energy De above its minimum. This negative deviation of the hue V(R) from 1/2 k(R-Re)2 causes the true vibrational energy levels to lie below the harmonic predictions. 

Explain how the conclusion is "obvious", how for J = 0, k = R, and A = 0, we obtain the usual harmonic oscillator energy levels. Describe how the energy levels would be expected to vary as J increases from zero and explain how these changes arise from changes in k and re. Explain in terms of physical forces involved in the rotating-vibrating molecule why re and k are changed by rotation. 

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...

Figure 10.4—Vibrational energy levels of a bond, a) For isolated molecules b) For molecules in the condensed phase. The transition from V — 0 to V = 2 corresponds to a weak harmonic band. Because of the photon energy involved in the mid IR, it can be calculated that the first excited state (V = 1) is 106 times less populated than the ground state. Harmonic transitions are exploited in the near IR.

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... 

For the c = 0 and v= 1 vibrational levels of CO, calculate the maximum departure of each nucleus from its equilibrium position in the principal-axis coordinate system if it is assumed the nuclei move classically. Assume harmonic-oscillator energy levels. 

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... 

When the function V(r) is used in the vibrational Hamiltonian instead of the simple harmonic V(x), the quantised vibrational energy levels are... 

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