Ψ(PR,Pr) 'ψel(R,Pr)ψnuc(PR)[Tnuc% E(R)]ψnuc(PR) ' Eψnuc(PR)V-1CHAPTER V. MOLECULAR SPECTROSCOPYA. Energies of MoleculesFrom our previous discussion of the quantum treatment of molecules within the Born-Oppenheimer approximation, recall that the 3n coordinates of the nuclei were separated from theelectron coordinates such that the total wavefunction could be represented by an electronic partdescribing the motion of the electrons, and a nuclear part describing the motion of the nuclei; .In the case of a diatomic molecule, the Schrödinger equation for the relative motion of the nucleiis simply where E is the total energy and E(R) is the molecular energy obtained by solving the electronicSchrödinger equation for different values of R. (Recall that the electronic energy, E (R), iselE(R)!Z Z e /R). 1 22The kinetic energy operator can be expressed in terms of the translational, rotational, andvibrational kinetic energies of the molecule. In the absence of forces that couple the degrees offreedom of the nuclei, the energies associated with the three kinds of nuclear motion may betreated as being independent. Hence, we may writeH = H + H + H .nuc tr rot vibThis is always true for translation, but only approximately so for rotation and vibration becausethe state of vibration affects the moment of inertia. The translational and rotational Hamiltonianscontain only kinetic energy terms, while the vibrational Hamiltonian contains the E(R) term. Inthis context, E(R) is the potential energy of the nuclei in the field of the electrons.Since we have separated the nuclear Hamiltonian into three parts, the wavefunction maybe expressed as the productV-2Ψ = ψ ψ ψ.nuc tr rot vibThis leads to the separated Schrödinger equationsH ψ = E ψtr tr tr trH ψ = E ψrot rot rot rotH ψ = E ψ.vib vib vib vibThe first equation above is the free particle Schrödinger equation for which the energyeigenvalues are continuous. For a diatomic molecule, the second equation is the same as theSchrödinger equation for rotation of a particle on a sphere. As we will see later, the E(R) term inthe third equation may be approximated by a quadratic function, in which case it reduces to theSchrödinger equation for a harmonic oscillator. Therefore, we may use the results obtained for3-D rotation and for the harmonic oscillator to discuss the vibrational and rotational energy levelsof diatomic molecules.The total energy of a molecule is thenE = E + E ,el nucwhereE . E + E + E .nuc tr rot vibSince The electronic, vibrational, and rotational energies of a molecule are quantized, oneexpects to be able to observe discrete transitions between the different energy levels. When a molecule absorbs electromagnetic radiation, its energy increases and when itemits electromagnetic radiation, its energy decreases. In spectroscopy, the transition itself maybe specified by stating its energy E = hν, wavelength, frequency ν = c/λ, or wavenumber ν& = ν/c= 1/λ. The regions of the electromagnetic spectrum in which the various types of moleculartransitions occur are as follows:PE 'PE0cos2πνtPE0V-3Electronic:E - 5 eV λ - 250 nm} Visible to Ultra-violet ν& - 40,330 cm -1Vibrational:E - 0.2 eV λ - 6.2 µm } Infrared ν& - 1613 cm-1Rotational:E - 4x10 eV-4 λ - 3.1 mm } Microwave ν& - 3.2 cm-1A schematic diagram showing the relative ordering of the three types of molecular energy levelsis shown in Fig. V-1 and the electromagnetic spectrum is shown in Fig. V-2.B. Transition Rates and Selection RulesConsider a molecule in a time varying electromagnetic field, such as that generated by abeam of electromagnetic radiation.Both the electric and magnetic field components interact with the charge distribution of themolecule. However, for atoms and molecules, the magnetic interaction is usually so muchsmaller than the electric interaction that it can be neglected. The time dependent electric fieldmay be written as ,where is the electric field vector and ν is the frequency of the radiation.The interaction of an electric field with an arbitrary charge distribution can be expressedas a multipole expansion, whereby the potential energy may be written in terms of multipolemoments that characterize the spatial dependence of the charge distribution;J111091v’=0v’=1vibrationalenergy levelsrotationalenergy levelsv’=2excited electronic energy levelJ111091v=0v=1v=2ground state electronic energy levelrotationalenergy levelsvibrationalenergy levelsROY G BIVwavelength (m)radiowavesmicro-wavesinfraredvisibleultra-violetx rayswavenumber (103cm- 1)103110- 310- 610- 91214161820 22 24 26V-4Figure V-1. Schematic diagram showing the relative ordering of electronic, vibrational, androtational energy levels.Figure V-2. The electromagnetic spectrum. The visible region is shown in greater detail.V(Pr)V(Pr,t) ' ! Pµ @PE0cos2πνPµ 'jj!ePrj(the dipole moment)V-5 = f(monopole moment) + f(dipole moment) + f(quadrupole moment) +... Usually, for atoms and molecules, the magnitudes are such that only the dipole term of thisexpansion needs to be considered. Hence, the potential energy associated with the interaction ofthe electric field with a molecule is given byt,where .The interaction of a time varying electric field with an atom or molecule can induce atransition from a given energy level to a higher energy level (stimulated absorption) or to alower energy level (stimulated emission). The emitted radiation stimulated by photons iscoherent, which means it has the same phase and propagation direction as the incident photons. This property is utilized in lasers, as will be discussed further in section G4. The probability perunit time (λ) for a transition may be obtained from time-dependent perturbation theory. Theresult for a stimulated absorption or emission transition from state m to state n is ,where ρ(ν) = time averaged energy density of the field at the transition frequency,mnand .The quantity µ is referred to as the transition dipole moment integral.mn A third process, called spontaneous emission, also can contribute to the total emissionrate. As the name implies, this mechanism does not require the presence of an electromagneticfield. However, because the transition probability expression for spontaneous emission dependsAnm'8πhν3nmc3BnmV-6on the transition frequency raised to the third power, it is relatively unimportant for lowfrequency transitions, such as those for rotation and