Vibrational spectroscopy, 017 Uwe Burghaus, Fargo, ND, USA
CHEM761 Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states... by microwave spectroscopy or by far infrared spectroscopy https://en.wikipedia.org/wiki/rotational_spectroscopy Vibrational spectroscopy Radiation causing the vibration is absorbed Vibrational spectroscopy is a general term denoting two techniques: infrared and Raman spectroscopy. Raman, see next class
B: rotational constant I: moment of inertia l: angular moment also called J in rot.. Spectroscopy ν: frequency h E= ll ( + 1) I E = E E l+ 1 l h E = El+ 1 El = ( l+ 1) I E ( l 1) h ν = ( l+ 1)B = + I h B = E = hν 8π I Transmission Selection rule Please note that some textbooks use the wavenumber and not the frequency which leads to a different definition of B.
Figure 19.13/Engel
Rigid rotor-harmonic oscillator better approximation Eint = B hj( J + 1) + ( v + 1 ) hν + E e e el ν χ ( + ) h e e v 1 hα ( v + 1 ) J( J + 1) e Anharmonicity Rotation-vibration coupling hdj ( J + 1) Centrifugal distortion φ constant distance r 0 B e α e χ e Equilibrium rotational constant Equilibrium vib-rot coupling constant Equi. anharmonicity constant D: centrifugal distortion constant J, v: rot, vib. quantum numbers h: Planck s constant
energy Electronic excitations vibrations rotations of molecules. PChem364 quantum mechanics excited state rotations distance j=3 j= j=1 j=0 v=3 v= electronic energy levels ground state v=0 v=1 distance electronic energy level vibrations
HCl/DCl IR data from NDSU Pchem lab class 008 IR spectroscopy rotations & vibrations Vibration-rotation EXAMPLE HCl/DCl 90 80 inensity (a.u.) 70 P branch R branch 1 0 0 1 60 4 3 3 1 1 3 isotope splitting 600 800 3000 wave number in 1/cm inensity (a.u.) 90 80 70 60 900 90 940 960 980 3000 wave number in 1/cm v = 1 v = 0 j = 3 j = j = 1 j = 0 j = 3 j = j = 1 j = 0
Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, Chapter 7 J.M. Hollas, High resolution spectroscopy, Wiley, chapter 5 Every undergraduate/graduate Pchem book has typically a joint chapter about rotation-&- vibrations also remember/read harmonic oscillator chapter Physics of Atoms and Molecules, B.H. Bransden, C.J. Joachain, Wiley, Chapter 9, 10 Molecular Spectroscopy, J.M. Brown, Oxford Chemistry Primers, Vol. 55, Chapter 5, 6 https://en.wikipedia.org/wiki/infrared_spectroscopy# Number_of_vibrational_modes movie
Normal modes https://en.wikipedia.org/wiki/infrared_spect roscopy#number_of_vibrational_modes definition In a normal mode all oscillators oscillate at the same frequency and with a fixed phase lag. The frequencies are the systems natural frequencies / resonance frequencies. movie key point The modes are independently of each other meaning that an excitation of one mode will not cause excitation of a different mode. No coupling Small amplitudes Independent oscillators That decouples equation of motions making system solvable. http://en.wikipedia.org/wiki/normal_modes
vib-&-rot of diatomic Vibrational spectroscopy Theory Undergrad version Classical physics QM A little more beyond undergrad Polyatomic molecules Experimental Dispersion type spec. FT IR spec Spectro-fluorimetry, 017 Uwe Burghaus, Fargo, ND, USA
Harmonic oscillator classical physics point of view potential force V = 1 kx F = kx position speed x k = Asin( m t ); ν = 1 π v = x A k k m cos( m t) k m Total energy E tot = E + E = K + V kin pot 1 1 1 = mvx + kx = ka E = E E E pot tot kin kin 0 x max = A A x t
In-class homework Could we have a tunneling effect in classical mechanics? PChem Quantum mechanics Total energy E tot = E + E = K + V kin pot 1 1 1 = mvx + kx = ka E = E E E pot tot kin kin 0 x max = A RESULT RESULT Classically the potential energy cannot be larger than the total energy (see equation) since the kinetic energy cannot be negative. As a classical paradox: if the kinetic energy is assumed negative than E(pot) > E(tot), i.e., the amplitude A could then be larger than A(max) given by 1/kA_. In other words, the quantum tunneling would correspond to a negative kinetic energy in classical mechanics. [ see e.g. Levine ]
d ψ + Vψ = m dx V = 1 kx Eψ Solving the Schrödinger Eq. is complicated, so we omit it here See Levine, Physical Chemistry, p. 88 Chang, p. 58 Engel/Reid ch. 18.1 Atkins was/is actually more detailed
/ ) ( x n n n xe H A x α α ψ = Hermite polynominals n=0, 1,, 4 1/ ) (! 1 / π α µ α n A k n n = = n x n x n n dx e d e x H 1) ( ) ( =
Particle in box Harmonic oscillator 3D Rigid rotor n n E=0 l ZPE ZPE ZPE = 0 typical mistake
d ψ + Vψ = m dx V = 1 kx Eψ Solving the Schrödinger Eq. is complicated, so we omit it here See Levine, Physical Chemistry, p. 88 Chang, p. 58 Early editions (e.g. 3 rd ed.) of Atkins are actually more detailed, derivation is missing in newer editions (e.g. 7 th ) The other extreme Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, Chapter 7 90 pages
not 90 pages, but d ψ + Vψ = m dx 1 V = kx Eψ F = -kx harmonic force How to solve this problem?
By the way In-class homework Energy becomes continuous in classical limit. Aehhh how? n PChem Quantum mechanics n RESULT RESULT Calculate difference of adjacent energies. The difference will approach zero for large quantum numbers for box problem. cf., anharmonicity see e.g. Engel/Reid, p. 333 ( nd Ed.)
ψ ψ ψ E kx dx d m = + 1 α x y = Substitution ω ε 1 E = mk = α m k ω = 0 ) ( = + ψ ε ψ y dy d Asymptotic solution (y large) 0 = ψ ψ y dy d fe y = ψ with 0 1) ( = + f dy df y dy f d ε one obtains the Hermite diff Eq. / ) ( x n n n xe H A x α α ψ = Hermite polynominals n=0, 1,, 4 1/ ) (! 1 / π α µ α n A k n n = = n x n x n n dx e d e x H 1) ( ) ( = ) ω 1 ( + = n E n What is missing?
m d ψ dx m 1 1 d ψ + Vψ = dx Substitution/transformation (use center of mass coordinates) Eψ R = x 1 x m m x = + m m 1 x1 x Schrödinger Eq. for two masses bond distance center of mass When calculating the differentials in the new coordinates one obtains d ψ m dx d ψ + Vψ = µ dr Eψ Schrödinger Eq. in center of mass coordinates µ: reduced mass ψ =ψ translatio ψ Separation of variables ( x) ( R) n vibration d ψ m dx trans d ψ µ dr vib = E + Vψ trans vib ψ = trans E vib ψ vib That is the eq. we just solved
ψ ψ φ ψ ψ ψ µ E r V r r r r r r = + Θ + Θ Θ Θ Θ + ) ( )] ( sin 1 ) (sin sin 1 ) ( 1 [ ψ ψ ψ µ ψ µ E r V J r r r r r = + + ) ( ˆ 1 ) ( 1 ) ( ) ( ) (,, φ θ ψ Φ Θ = r R m l n Separation of variables Rotations: see last class Vibrations: select a potential Dunham potential: Taylor expansion Morse potential: common choice in spectroscopy, Schrödinger can be solved analytically G(v) see Pchem lab class manual (Birge-Sponer plots etc.) [ Bernath ]
What is the effect of anharmonicity? Eint = BehJ( J + 1) + ( v + 1 ) hν e + Eel hν e χ e ( + ) for diatomic Normal vibration modes are not independent of each other Energy transfer between modes v 1 Anharmonicity Resonance effects Fermi resonance - vibrations of same symmetry - & similar energy Energy transfer (weaker peak gets stronger) Shift in energy levels of the vibrations movie https://www.youtube.com/watch?v=tuz7if7n9rg (start at 5:50 min) Darling-Dennison resonance - vibration and overtone have same symmetry Fargo resonance
Einstein & Debye model of solid crystals
Q ϕ (0)/ kt 3N = e q with igh temperature limit ow temperature limit q Θ e = 1 e q q kt hν e T Θ T hν / kt ln( Q) ( ) T E = kt N, V C V Θ T e ( e Θ/ T = 3Nk( ) Θ/ T 1) Statistical thermodynamics describes the thermal excitation of vibrations Einstein temperature Θ = hν k
Lattice contributions to C v at low T: For T < 5 K, contribution of electron gas: C C C Einstein V V Debye V e T ( Θ = π T ( T Θ ) F Θ D E 3 ) / T C V T = 1 4 π 3 ( ) + ηr π 5 Θ T 3 + T D T ( Θ F ) C Θ T Θ V R η Θ Θ D F E D Heat capacity heat capacity Tempertaure Fermi temperature Gas constant number of valence electrons Einstein Temperature Debye temperature e.g. [Lau, ch. 13.6]
Normal modes definition In a normal mode all oscillators oscillate at the same frequency and with a fixed phase lag. The frequencies are the systems natural frequencies / resonance frequencies. key point The modes are independently of each other meaning that an excitation of one mode will not cause excitation of a different mode. No coupling Small amplitudes Independent oscillators That decouples equation of motions making system solvable. http://en.wikipedia.org/wiki/normal_modes
I discussed diatomic molecules (and solids/crystals) only. For that see e.g. Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, Chapter 7.
What does the potential look like? diatomic polyatomic CO n Q1 Q3 [animation] J.M. Hollas, High resolution spectroscopy, Wiley, chapter 5
What does the potential look like? ammonia NH 3 inversion vibration inversion vibration J.M. Hollas, High resolution spectroscopy, Wiley, chapter 5 animations http://chemtube3d.com/vibrationsnh3.htm http://www.ess.ucla.edu/~schauble/moleculehtml/ch4_h tml/ch4_page.html
What does the potential look like? ammonia NH 3 J.M. Hollas, High resolution spectroscopy, Wiley, chapter 5
n=1 Vibration 1 st excited state n=0 Vibration Ground state n j /n 0 = g j /g 0 e - E/RT gap J=0 Engel/Reid
HCl/DCl IR data from NDSU Pchem lab class 008 90 inensity (a.u.) 80 70 Vibration-rotation EXAMPLE HCl/DCl 60 000 500 3000 wave number in 1/cm 90 80 inensity (a.u.) 70 P branch R branch 1 0 0 1 60 4 3 3 1 1 3 90 isotope splitting 600 800 3000 wave number in 1/cm j = 3 inensity (a.u.) 80 70 60 900 90 940 960 980 3000 wave number in 1/cm v = 1 v = 0 j = j = 1 j = 0 j = 3 j = j = 1 j = 0
By the way PChem Quantum mechanics In-class homework What generates the larger peak splitting Cl 35 /Cl 37 or H/D? HCl 35, HCl 37, DCl 35, DCl 37 isotope mixture RESULT H/D due to effect on reduced mass. RESULT
How do vibrations affect the peak position of isotope labeled molecules? Deuterating has a larger effect than changing the Cl isotope.
textbook version Changing the Cl isotope leads to the small peak splitting.
v=3 v= v=1 v=0 vibrations fundamental first overtone nd overtone hot band electronic energy level
Infrared spectroscopy -- classification experimental hν hν hν e - Transmission Infrared spectroscopy hν RAIS Reflection-absorption Infrared spectroscopy e - HREELS High Resolution Electron Energy Loss Spectroscopy class 14
https://en.wikipedia.org/wiki/infrared_spectroscopy_correlation_table near-ir: 14000 4000 cm 1 (0.8.5 μm) - overtone or harmonics. mid-infrared: 4000 400 cm 1 (.5 5 μm) - fundamental vibrations, rotational-vibrational far-infrared: 400 10 cm 1 (5 1000 μm) - rotational spectroscopy
Signal detection techniques experimental 1) Direct detection, dispersion type spectrometers combined with lock-in technique ) Fourier transform techniques based on idea of Michelson interferometer
Infrared spectroscopy dispersion type system experimental Sometimes called dispersion IR spectrometer since it includes a prism or gratings to disperse the electro.mag. radiation Idea of operation, briefly: The chopper splits the beam that consecutive the reference cell or the sample cell signal will reach the detector The mirror acts as a diffraction gratings selects a partition of the spectra. If the sample does not adsorb at the selected frequency then the reference beam and sample beam have the same intensity and the amplifier will yield a zero signal Typically lock-in technique is use i.e. the amplifier is a little more sophisticated than shown in the scheme above.
AES Lock In technique experimental input AC amplifier band pass filters Phase sensitive amplifier Low pass filter output Phase shifter Modulator Reference Lock in idea: Input signal - modulated with ω and zero phase lag with respect to the reference. Output signal - constant voltage level. All other signals will be averaged out by the phase sensitive amplifier. A lock in amplifier is a filter with an extremely narrow band width. Side effects Nyquist equation: Signal-to-noise ratio ~ sqrt(band width) Background not modulated: it will be subtracted
Infrared spectroscopy setup FTIR experimental
FT IR idea - Michelson Interferometer experimental Figure 19.18 Engel/Reid
Infrared spectroscopy setup ---FTIR Fourier-transform IR spectrometer Idea of operation: experimental The source beam will be split. One part arrives at the detector directly (a,b,d,d,b,f,g). The other one will be delayed (a,c,e,e,c,f,g) by going a longer way. The difference in the traveling path length of these two beams will be modulated by a movable mirror. (That s the clue.) The interference of these two beams will be measured at the detector. (Intensity vs. position of the mirror) The spectra (intensity vs. wave number) is the Fourier transformation of the interference signal. Levine
More than one frequency of the source experimental Main idea: We measure an intensity as a function of time (i.e. as a function of the position of the nd mirror which is changing with time) that is the interference of the sample and reference beam. This intensity vs. time signal is converted in an intensity vs, frequency signal by means of a Fourier transformation. Engel/Reid
experimental
S 1 fluorescence intensity absorption frequency S 0
Spectroscopy Primer - Spectrofluorimeter Remember PChem364 quantum mechanics Excitation monochromator Excitation spectra excitation wavelength scanned emission wavelength fixed intensity (a. u.) Emission (Excit Mono 356 nm) Excitation (Emiss Mono 44 nm) 300 30 340 360 380 400 40 440 460 480 500 50 wavelength in nm light source Emission monochromator sample Emission (Excit Mono 356 nm) intensity (a. u.) 300 30 340 360 380 400 40 440 460 480 500 50 wavelength in nm detector Emission spectra emission wavelength scanned excitation wavelength fixed
Antracene data from NDSU Pchem lab class 006 Spectroscopy Primer - Spectrofluorimeter - Emission / Excitation spectra Emission (Excit Mono 356 nm) Excitation (Emiss Mono 44 nm) vibrations electronic states Antracene intensity (a. u.) 300 30 340 360 380 400 40 440 460 480 500 50 wavelength in nm Excitation spectra excitation wavelength scanned emission wavelength fixed Emission spectra emission wavelength scanned excitation wavelength fixed
E n 1 = ( n + ) ω Hermite polynominals n j /n 0 = g j /g 0 e - E/RT
How to use the center of mass system? How does the separation of variables work? What is a lock-in amplifier?
-) Why is the harmonic oscillator model important, i.e., what are the applications? -) Summarize the model of the classical harmonic oscillator. (Engel s book chapter 18.6) -) The energy spacing of the harmonic oscillator increases with increasing quantum number. Yes/No? -) The Q.M. harmonic oscillator shows the tunneling effect. Yes/No? -) What is the zero point energy of the Q.M. harmonic oscillator? -) The wave functions of the Q.M. harmonic oscillator are the so-called Hermite polynoms. -) The Q.M. harmonic oscillator has quantized energy eigenvalues. Yes/No? -) Again: what is actually an Eigenvalue? -) How to calculate the force constant of Q.M. harmonic oscillator if the vibrational frequency is known? -) What is the equation for the frequency of the Q.M. harmonic oscillator? -) What would you need to do in order to prove that the wave functions of the Q.M. harmonic oscillator are orthogonal? -) What would you need to do in order to prove that the wave functions of the Q.M. harmonic oscillator are normalized? -) What is it important that the wave functions of the Q.M. harmonic oscillator are normalized? -) What would you need to do in order to prove that a given wave function describes indeed correctly the Q.M. harmonic oscillator?
Pure rotation spectra, rotation and vibration spectra -) What are the selection rules for the particle-in-a box, harmonic oscillator, rigid rotor? -) What are typical energies for rotations and vibrations? -) What type of interaction do we consider? Magnetic? Electric? -) What are the energy eigenvalues for rotations in quantum mechanics? -) Calculate a rotational spectra, i.e., how large is the energy spacing between adjacent rotational levels? -) What is a wavenumber? -) How large is the degeneracy of states for rotations? Is that important for something? -) The R and P branches are a special office at US bank for private and business customers. Correct/wrong? -) Why is a gap in rotational spectra? -) Explain the intensity variations seen in rotational & vibrational spectra. -) Why can different definitions for the rotational constant be found in the literature?
https://en.wikipedia.org/wiki/vibration https://en.wikipedia.org/wiki/infrared_spectroscopy https://www.journals.elsevier.com/vibrational-spectroscopy/ Specialty journal https://en.wikipedia.org/wiki/rotational%e%80%93vibrational_spectroscopy
Raman and nanoscience Dresselhaus, Jorios, Saito, Annu.Rev.Condens.Matter Phys. 1 (010) 89 M.S. Dresselhaus, G. Dresselhaus, R. Saito, A. Jorio, Physics Reports 409 (005) 47-99 Raman Spectroscopy in Graphene Related Systsems, ISBN 978-3-57-40811-5 Rather didactic outline. Way too expensive, however, as a textbook for students. The basics E. Smith, G. Dent, Modern Raman Spectroscopy, A practical approach, Wiley, ISBN 0471497940 Undergraduate level description with many practical notes and an applied description of Raman scattering theory. A good starting book. Most undergrad books include a few pages long introduction. Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, Chapter 8 Physics of Atoms and Molecules, B.H. Bransden, C.J. Joachain, Wiley, Chapter 9, 10 Molecular Spectroscopy, J.M. Brown, Oxford Chemistry Primers, Vol. 55, Chapter 6
Self-study examples Undergrad level Engel / Reid
Microwave spectroscopy -- EXAMPLES Examples from Levine PChem Quantum mechanics
EXAMPLE: Rotation & vibration spectra Examples from Levine PChem Quantum mechanics
EXAMPLE: Rotation & vibration spectra Examples from Engel s book (1 st edition) PChem Quantum mechanics Q19.3) What is the difference between a permanent and a dynamic dipole moment? The permanent dipole moment arises from a difference in electronegativity between the bonded atoms. The dynamic dipole moment is the variation in the dipole moment as the molecule vibrates.
EXAMPLE: Rotation & vibration spectra Examples from Engel s book (1 st edition) PChem Quantum mechanics Q19.5) The number of molecules in a given energy level is proportional E to kt Where E e is the difference in energy between the level in question and the ground state. How is it possible that a higher lying rotational energy level can have a higher population than the ground state? Although the number of molecules in a given energy level is E proportional to kt e, it is also proportional to the degeneracy of the level, J+1. For small E values of the degeneracy of a level can increase faster with J than kt falls. Under this condition, a higher lying rotational energy level can have a higher population than the ground state.
Q19.7) What is the explanation for the absence of a peak in the rotationalvibrational spectrum near 3000 cm 1 in Figure 19.14? This energy would correspond to the J = 0 J = 0 transition, which is forbidden by the selection rule for rotational transitions in diatomic molecules.
P19.) Isotopic substitution is used to identify characteristic groups in an unknown compound using vibrational spectroscopy. Consider the C=C bond in ethene (C H 4 ). By what factor would the frequency change if deuterium were substituted for all the hydrogen atoms? Treat the H and D atoms as being rigidly attached to the carbon. ν µ D ν µ H CH -CH ( ) = = = CD -CD ( 16.0) 14.01 amu 8.0 0.935 3.04 amu
P19.11) An infrared absorption spectrum of an organic compound is shown in the following figure. Using the characteristic group frequencies listed in Section 19.6, decide whether this compound is more likely to be ethyl amine, pentanol, or acetone. The major peak near 1700 cm 1 is the C=O stretch and the peak near 100 cm 1 is a C C C stretch. These peaks are consistent with the compound being acetone. Ethyl amine should show a strong peak near 3350 cm 1 and pentanol should show a strong peak near 3400 cm 1. Because these peaks are absent, these compounds can be ruled out.
P19.14) The rotational constant for 17 I 79 Br determined from microwave spectroscopy is 0.1141619 cm 1. The atomic masses of 17 I and 79 Br are 16.904473 amu and 78.918336 amu, respectively. Calculate the bond length in 17 I 79 Br to the maximum number of significant figures consistent with this information. h h B= ; r = 8πµ 8πµ B r r 0 0 = 0 r0 ( 16.904473+78.918336) 34 6.660755 10 J s 16.904473 78.918336 amu 8π 1.660540 10 kg amu 0.1141619 cm.9979458 10 cm s 7 1 1 10 1 10 1.74035 10 m =
Microwave spectroscopy -- EXAMPLES PChem Quantum mechanics Standard example B 0 = B e from spectroscopy (see last slides) d = geometry of molecule
Microwave spectroscopy -- EXAMPLES PChem Quantum mechanics Practical example. Let s do that one qualitative, only. See Class (powerpoint)
Microwave spectroscopy -- EXAMPLES PChem Quantum mechanics
EXAMPLE: Rotation & vibration spectra Standard example Isotope effects (that s what we also look at in the Pchem lab class) PChem Quantum mechanics Eint = B hj( J + 1) + ( v + 1 ) hν + E ν χ ( + ) e e el h e e v 1 Anharmonicity hα ( v + 1 ) J( J + 1) Rotation-vibration coupling e hdj ( J + 1) Centrifugal distortion
EXAMPLE: Rotation & vibration spectra How do rotations affect the peak position of isotope labeled molecules? PChem Quantum mechanics