The Energy levels

The Energy Levels.—Introducing for X its value as given in Equation 18-27, and solving for W, it is found that Equation 18-39 leads to the energy expression 

This expression is identical with that of the old quantum theory (Eq. 7-24), even to the inclusion of the reduced mass p. It is seen that the energy of a hydrogenlike atom in the state represented by the quantum numbers n, l, and m does not depend on their individual values but only on the value of the total quantum number n = n + l + 1. Inasmuch as both n and l by their nature can assume the values 0, 1, 2, , we see that the allowed values of n are 1, 2, 3, 4, , as assumed in the old quantum theory and verified by experiment (discussed in Sec. 76). 

Except for n = 1, each energy level is degenerate, being represented by more than one independent solution of the wave equation. If we introduce the quantum numbers n, l, and m as subscripts (using n in preference to n ), the wave functions we have found as acceptable solutions of the wave equation may be written as 

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The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

The energy level spectrum of the hamionic oscillator is completely regular. The ground state energy is given... 

The variation of tlie frequency can be approximated by a series in the number of quanta, so the energy levels are given by... 

Often, it is a fair approximation to tnmcate the series at the quadratic tenn with The energy levels are then approximated as... 

Wliat does one actually observe in the experunental spectrum, when the levels are characterized by the set of quantum numbers n. Mj ) for the nonnal modes The most obvious spectral observation is simply the set of energies of the levels another important observable quantity is the intensities. The latter depend very sensitively on the type of probe of the molecule used to obtain the spectmm for example, the intensities in absorption spectroscopy are in general far different from those in Raman spectroscopy. From now on we will focus on the energy levels of the spectmm, although the intensities most certainly carry much additional infonnation about the molecule, and are extremely interesting from the point of view of theoretical dynamics. 

The irreducible representations of a symmetry group of a molecule are used to label its energy levels. The way we label the energy levels follows from an examination of the effect of a synnnetry operation on the molecular Sclnodinger equation. 

The energy level of an an /-fold degenerate eigenstate can be labelled according to an /-fold degenerate irreducible representation of the synmietry group, as we now show. 

Note that the diagonal elements of the matrix, ap and hp, correspond to the populations in the energy levels, a and b, and contain no time dependence, while the off-diagonal elements, called the coherences, contain all the time dependence. 

Equation (A2.1.8) turns out to be consistent with die changes of the energy levels measured spectroscopically, so the energy produced by work defined this way is frequently called the spectroscopic energy . Note that the electric and magnetic parts of the equations are now synnnetrical. 

Figure A3.12.10. Schematic diagram of the one-dimensional reaction coordinate and the energy levels perpendicular to it in the region of the transition state. As the molecule s energy is increased, the number of states perpendicular to the reaction coordinate increases, thereby increasing the rate of reaction. (Adapted from [4].)...

There is one special class of reaction systems in which a simplification occurs. If collisional energy redistribution of some reactant occurs by collisions with an excess of heat bath atoms or molecules that are considered kinetically structureless, and if fiirthennore the reaction is either unimolecular or occurs again with a reaction partner M having an excess concentration, dien one will have generalized first-order kinetics for populations Pj of the energy levels of the reactant, i.e. with... 

Before presenting the quantum mechanical description of a hannonic oscillator and selection rules, it is worthwhile presenting the energy level expressions that the reader is probably already familiar with. A vibrational mode v, witii an equilibrium frequency of (in wavenumbers) has energy levels (also in... 

Of course, real molecules are not hamronic oscillators, and the energy level expression can be expanded in powers of (v + 1/2). For a single mode we have... 

Figure Bl.2.3. Comparison of the hannonic oscillator potential energy curve and energy levels (dashed lines) with those for an anliannonic oscillator. The hannonic oscillator is a fair representation of the tnie potential energy curve at the bottom of the well. Note that the energy levels become closer together with increasing vibrational energy for the anliannonic oscillator. The aidiannonicity has been greatly exaggerated.

Upon solving the Schrodinger equation, the energy levels are , = + 1/2), where is related to the force... 

Av = 1 hannonic oscillator selection mle. Furthennore, the overtone intensities for an anhannonic oscillator are obtained in a straightforward maimer by detennining the eigenfiinctions of the energy levels in a hannonic oscillator basis set, and then simnning the weighted contributions from the hannonic oscillator integrals. 

Of the NMR-active nuclei around tluee-quarters have / > 1 so that the quadnipole interaction can affect their spectra. The quadnipole inter action can be significant relative to the Zeeman splitting. The splitting of the energy levels by the quadnipole interaction alone gives rise to pure nuclear quadnipole resonance (NQR) spectroscopy. This chapter will only deal with the case when the quadnipole interaction can be regarded as simply a perturbation of the Zeeman levels. 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

Figure Bl.15.8. (A) Left side energy levels for an electron spin coupled to one nuclear spin in a magnetic field, S= I =, gj >0, a<0, and a l 2h)<(a. Right side schematic representation of the four energy levels with )= Mg= , Mj= ). +-)=1, ++)=2, -)=3 and -+)=4. The possible relaxation paths are characterized by the respective relaxation rates W. The energy levels are separated horizontally to distinguish between the two electron spin transitions. Bottom ENDOR spectra shown when a /(21j)< ca (B) and when co < a /(2fj) (C).

Figure Bl.26.18. Schematic diagram of the energy levels in a solid.

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