1Vibrational Spectroscopy Harmonic Oscillator Potential Energy ( )2e1VxkxxRR2==− where Re – equilibrium bond length Selection Rules The dipole moment of a molecule can be expanded as a function of x = R – Re. ( ) ( )23232300011x0xxxx2x6x∂µ∂µ∂µµ=µ++++∂∂∂L ()00µ=µ – dipole momentum at equilibrium, permanent dipole moment. 0x∂µ∂ – change in dipole moment along vibrational coordinate – effect of stretching, compression, bending, etc… on dipole moment. Recall that transition is possible only when transition dipole matrix element is nonzero, that is ˆfi0µ≠ For the dipole operator in the transition dipole matrix element, substitute the series approximation above. 23230023000011ˆfifxxxifxix2x6xx∂µ∂µ∂µ∂µµ=µ++++≈µ+∂∂∂∂L 000000fxififxififxixxx∂µ∂µ∂µµ+=µ+=µ+∂∂∂ A transition implies that the final state must be different from the initial statefi0= Thus fxi0≠ for a transition to occur. Also, 00x∂µ≠∂ for a transition to occur. This requirement means that the vibration that is being excited must cause the dipole moment of the molecule to change. This requirement is true for transitions in infrared vibrational spectroscopy.2Recall that ( )22NHeξ−ννν=ξ where 142kxµξ=h Careful: µ is reduced mass. ( ) ( )2211224422ffii0012200fifxiNHeNHedxxkkfifixkxk∞ ξξ−−−∞∂µ∂µµ==ξξξξ∂∂µµ∂µ∂µω=ξ=ξ∂µ∂∫hhhh Recall that 1111HHH1122νν−ν+ξ=ν+⇒ξν=νν−+ν+ 0001fififii1fi1xkxk21ifi1fi1xk2∂µω∂µωµ=ξ=−++∂∂∂µω=−++∂hhh Or perhaps stated a little more clearly. i01fifi1fi1xk2∂µωµ=ν−++∂h The first integral is nonzero only when fi1−ν=ν ⇒ 1∆ν=− The second integral is nonzero only when fi1+ν=ν ⇒ 1∆ν=+ Therefore the selection rule for a harmonic oscillator in vibrational spectroscopy is 1∆ν=± Anharmonic Oscillator General Potential Energy The actual potential energy (not idealized) can be written as a Taylor series about the point R = Re (i.e., x = 0). ( ) ( )2342342340000V1V1V1VVxV0xxxxx2x6x24x∂∂∂∂=+++++∂∂∂∂L The origin of the potential energy can be set arbitrarily (and sensibly) to zero. That is, V(0) = 0 by definition. Also, since the slope of the potential energy curve at Re is zero 0V0x∂=∂3Thus the first nonzero coefficient of the potential energy series above is 2201V2x∂∂ This coefficient can be related to the force constant. 2222001V1Vkk2x2x∂∂=⇒=∂∂ The next nonzero coefficient of the potential energy series causes the oscillator to be anharmonic. The nonzero coefficients can be manipulated to yield to yield the cubic anharmonicity constant, χeνe, the quartic anharmonicity constant, yeνe, etc, … Energy levels of the anharmonic oscillator. 23eeeee111Ehy222ν=νν+−χνν++νν++L Note: The energy levels for the harmonic oscillator are evenly spaced. (eEh∆=ν) However the energy levels for the anharmonic oscillator are not evenly spaced. (eEh∆<ν). The energy levels for the anharmonic oscillator get closer together as the quantum number ν increases. Morse Potential Electronic potential energy curve is not purely quadratic, i.e., ( )21Vxkx2≠ A better approximation of an actual potential energy curve is the Morse potential. ( )()2axeVxD1e−=− where 122ea2Dµω= and De is the potential well depth. V(x)xharmonic (quadratic)anharmonic (Morse)D0De D0 – dissociation energy e01DD2−=ωh – zero point energy4Using the Morse potential, the spectroscopic term becomes ( )2eee11G22ν=νν+−χνν+%% where eeeh4Dνχ= This expression is an exact solution for the Morse potential. Worth emphasizing is that the Morse potential is only an approximation (albeit, a good one). If the above spectroscopic term, cannot reproduce the experimental spectra, then a more accurate potential energy function is needed. (Usually based on Taylor series expansion.) When the polynomial found from the Taylor expansion is used, a more general spectroscopic term can be written. ( )234eeeeeee1111Gyz2222ν=νν+−χνν++νν++νν++%%%%L Selection Rules of the Anharmonic Oscillator Nonzero transition dipole matrix elements of the anharmonic oscillator have two origins 1.) Anharmonic wavefunctions are combinations of HO wavefunctions. 2.) Dipole can have quadratic (cubic, etc…) dependence i.e., 2200x∂µ≠∂ thus fiµ and 2fxi, 3fxi, 4fxi, etc… The resultant selection rules are complicated, but transitions where ∆ν is a single digit are all allowed. That is, 1,2,3,4,5,∆ν=±±±±±K Consequences of Anharmonicity 1.) Transitions where 1∆ν> are allowed - Such transitions are called overtones. - 1 ← 0 transition is called the fundamental. 2.) More than one vibration can be excited with a single transition. - Such transitions are combination bands. - Consider water with its three modes of vibration. OHHOHHsymmetric stretch - νs OHHantisymmetric stretch - νa OHHbending - νb OHH OHH5 - The total vibrational wavefunction is product of individual vibrational wavefunctions. sabsabννν=ννν 000 - vibrational ground state 100 ← 000 - fundamental for symmetric stretching mode 010 ← 000 - fundamental for antisymmetric stretching mode 001 ← 000 - fundamental for bending mode - Examples of combinations bands would be 201 ← 000 110 ← 000 312 ← 221 001 ← 211 Miscellaneous A hot band is an absorptive transition from an excited state. Vibration – Rotation Spectra Analysis for