Rotational spectroscopy, 2017 Uwe Burghaus, Fargo, ND, USA
Atomic spectroscopy (part I) Absorption spectroscopy Bohr model QM of H atom (review) Atomic spectroscopy (part II) Visualization of wave functions Atomic spectroscopy (part III) Angular momentum (details) Orbital angular momentum Spin Spin-Orbit coupling Zeeman effect Many-electron atoms, part I Pauli Principle Singlet vs. triplet Term symbols LS vs. jj coupling Many-electron atoms, part II Hund s rules Selection rules Hyperfine structure Stark effect, 2017 Uwe Burghaus, Fargo, ND, USA
Hund s rules PChem Quantum mechanics 1) Hund s maximum multiplicity rule: An atom in its ground state adopts a configuration with the greatest number of unpaired electrons. Or For a set of terms arising from a given electron configuration, the lowest-lying term is generally the one with the maximum spin multiplicity. 2) For terms that have the same spin multiplicity, the term with the greatest orbital angular momentum lies lowest in energy. 3) If the unfilled subshell is exactly or more than half full, the level with the highest J value less than half full, the level with the lowest J values has the lowest energy Friedrich Hund, Göttingen, in the 1920s [ from Wikipedia ]
https://en.wikipedia.org/wiki/selection_rule mathematically complex topic f I = Intensity of transition i I M = ψ µψ dτ * 2 fi i j transition dipole moment Symmetry-forbidden transitions µ: odd for dipole transitions ψ i ψ j :must have different symmetry (product will then be even) Spin-forbidden transitions S=0 singlet triplet forbidden Atoms (electric dipole transitions) J = 0, ± 1 but J=0 0 L = 0, ± 1 but L=0 0 l = ± 1 g u, u g (magnetic dipole transitions) (electric quadrupole transitions) Molecules * * * See e.g. P.W. Atkins Quanta for a complete list
Feinstructure same term same energy same j same energy Zeeman effect
F = J + I oupling of nuclear spin, I, and J (electronic spin) Bernath, p. 147
Stark effect Zeeman effect Haken, Wolf, p. 228 What is the difference?
Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, Chapter 6 Every undergraduate/graduate Pchem book has typically a joint chapter about rotation-&- vibrations also remember/read ridged rotor chapter https://en.wikipedia.org/wiki/rotational_spectroscopy
Rotational spectroscopy Examples Classical mechanics Quantum mechanics Better approximations, 2017 Uwe Burghaus, Fargo, ND, USA
microwave spectroscopy T = (2.735+-0.06) K John Cromwell Mather Nobel price for measuring cosmic background radiation I could not find a copyright free photo of George Smoot who shared the Nobel price with J. Mather https://en.wikipedia.org/wiki/cosmic_background_explorer movie https://www.youtube.com/watch?v=orxq7zrhdbs See Physics Today, December 2006
John Cromwell Mather Proposed in 1900 by Max Planck http://en.wikipedia.org/wiki/planck%27s_law#history George Smoot Big bang experiment movie (60 min) watch at home https://www.youtube.co m/watch?v=orxq7zrh Dbs http://en.wikipedia.org/wiki/john_c._mather http://en.wikipedia.org/wiki/george_smoot
Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states... microwave spectroscopy or by far infrared spectroscopy https://en.wikipedia.org/wiki/rotational_spectroscopy some comments Pure rotations: high resolution microwave spectroscopy Rotational Raman spectroscopy Usually described together with vibrations. pure rotational spectroscopy CHEM761 rotational-vibrational spectroscopy (rotational, vibrational energy changes) ro-vibronic spectroscopy (rotational, vibrational and electronic energy changes)
Rotation spectra of HCl text book example Transmission
Etot = Evib + Etrans + Erot + Eelectronic
CHEM761 constant distance no potential no zero-point energy
CHEM761 Determine moments of inertia bond lengths bond angles Stark splitting in electrical field determine electric dipole moments radio telescopes chemical composition of the interstellar medium
rotations in classical physics CHEM761 Translations v = v0 s + v a at distance velocity acceleration Rotations Θ= s r ω = Θ t dω α = dt ω = ω0 + αt
rotations in classical physics Translations Rotations CHEM761 m p = mv mass momentum I = mr 2 moment of inertia L = Iω z l = rxp ω l Rule for ω Ekin = 1 v 2 m 2 kinetic energy Ekin E kin 1 2 = Iω 2 = E = rot 2 l 2I
Math primer
rotations in classical physics Translations Rotations CHEM761 F F = ma = dp dt force Newton s law T torque T = Iα = T = r x F dl dt
rotations in classical physics CHEM761 z l = rxp ω l single particle Rule for ω angular moment l l l x y z vector I = I I moment of inertia xx xy xz I I I xy yy yz tensor L = I ω I I I xz yz zz angular velocity ωx ω y ωz vector extended object
CHEM761 Why am I telling you this? Why is the moment of inertia important? Classical mechanics: E rot = E rot (moment of inertia) E kin = E = rot 2 l 2I We have our rotational spectrum
classical ridged rotor Classical mechanics: E rot = E rot (moment of inertia) Idea in Q.M. basically identical with classical concept. Look at the principal axes of symmetry of the molecule. Those determine the moment of inertia and hence E rot. Example: Principal axes of symmetry Use symmetry of problem E rot 1 I + 1 I + 1 I a b c
Motivation - molecular rotation E = E ( moment of inertia) 1 + 1 + 1 rot rot I I I Use energy quantization and a b c A symmetric top is a molecule in which two moments of inertia are the same.
Classical ridged rotor [ classification ] CHEM761 spherical top [ try this link ] linear top symmetric top asymmetric top
CHEM761 = z y x zz yz xz yz yy xy xz xy xx z y x I I I I I I I I I l l l ω ω ω = z y x z y x z y x I I I l l l ω ω ω 0 0 0 0 0 0 diagonalizable: principal axes system See, Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, Chapter 6.1 E rot classical ridged rotor
classical ridged rotor CHEM761 E rot principal axes system
Etot = Evib + Etrans + Erot + Eelectronic CHEM761
This image cannot currently be displayed. Separation of variables non interacting particles Coordinates of the system x,y,z or sets of coordinates of different particles Assume H = H H H... H 1 + 2 + 3+ + r then ψ = f ( q ) f ( q ) f ( q )... f ( q ) E = E + E + E +... + E with H f H f... H f 1 1 2 2 3 3 = 1 2 3 = = E f 1 1 1 1 E f 2 2 2 2 E f r r r r r r r
Separation of variables same deal here 1) Center of mass (CM) + internal coordinates (int) 2) translations, vibrations, rotations Hˆ total = Hˆ translations ( CM ) + Hˆ vibrations (int) + Hˆ rotations ( Θ ψ total = ψ translations ( CM )* ψ vibrations (int)* ψ rotations ( ΘCM, φcm ) CM, φ CM ) 2D version: rotations 3D: H atom φ constant distance r 0 2 1 2µ 2 r o 2 d Φ( φ) 2 dφ Solutions = EΦ( φ) Φ + ( φ) = A + e Schrödinger Eq. in polar coordinates imφ Φ ( φ) = A e imφ
2D rotations in quantum mechanics 2D rotations are analog to a particle moving on a ring Solutions Φ + ( φ) = Φ ( φ) = A + e A e imφ imφ Engel/Reid
2D rotations in quantum mechanics Boundary condition again leads to the quantization: m must be integer. I( φ) = I( φ+ 2 π) im( φ+ 2 π) imφ e = e m must be integer m = 0, ± 1, ± 2, ± 3,...(= m ) l Engel/Reid
2D rotations QM summary E m l = hm 2 2 l 2I ; I= µ r 0 m = 0, ± 1, ± 2, ± 3,... l Φ ( φ) = Ae ± ± ± imφ 2 angular momentum lˆ zφ ( φ) =± mh l Φ( φ) l = l= ± mh l E m l = 2 l 2I same as in classical mechanics No zero point energy (V=0) Degeneracy of states
spectroscopy
spectroscopy Transmission
Selection rule for pure rotations (classical point of view) Pure rotation excitation ( V = 0) is only allowed if µ(r) is not zero, i.e. the molecule must have a permanent dipole moment. Rotational inactive: Symmetrical linear molecules CO 2 Homonuclear diatomic molecules Exception: Distortion by rotation may be large enough causing a dipole moment
Selection rules another point of view J = +1, -1 Conservation of momentum rule.
Two more details one should know
population of states n 2 J j( j + 1)/(2IkT ) n 0 = (2J + 1) e Figure 19.15 Engel/Reid
Rigid rotor-harmonic oscillator better approximation Eint = B hj( J + 1) + ( v + 1 2 ) hν + E e e el 2 2 hdj ( J + 1) Centrifugal distortion D: centrifugal distortion constant J, v rot, vib. quantum numbers h: Planck s constant
Vibrations in detail in next class energy Electronic excitations vibrations rotations of molecules. PChem364 quantum mechanics excited state rotations distance j=3 j=2 j=1 j=0 v=3 v=2 electronic energy levels ground state v=0 v=1 distance electronic energy level vibrations
Rigid rotor-harmonic oscillator better approximation Eint = B hj( J + 1) + ( v + 1 2 ) hν + E e e el ν χ ( + ) h e e v 1 2 hα ( v + 1 2) J( J + 1) e 2 Anharmonicity Rotation-vibration coupling 2 2 hdj ( J + 1) Centrifugal distortion Ok many parameters here B Equilibrium rotational constant e α e χ e Equilibrium vib-rot coupling constant Equi. Anharmonicity constant D: centrifugal distortion constant J, v rot, vib. quantum numbers h: Planck s constant
CHEM761 coupling of nuclear spin angular momentum with rotational angular momentum causes splitting of the rotational energy levels In the presence of a static external electric field the 2J+1 degeneracy of each rotational state is partly removed Fourier transform microwave (FTMW) spectroscopy Vibrational angular momentum
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https://en.wikipedia.org/wiki/rotational_spectroscopy http://www.chemtube3d.com/spectrorotcd0-ce-test-rotate-all.html http://web.mit.edu/5.33/www/lec/spec5.pdf http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/rotrig.html https://www.kent.edu/chemistry/profile/michael-tubergen Summary draft Good one Research group working on high resolution microwave spectroscopy http://www.colorado.edu/chemistry/volkamer/teaching/index.html#lecturenotes https://en.wikipedia.org/wiki/microwave https://chem.libretexts.org/core/physical_and_theoretical_chemistry/spectrosc opy/rotational_spectroscopy/microwave_rotational_spectroscopy
Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, Chapter 7 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
Self-study examples Undergrad level Engel / Reid
Microwave spectroscopy -- EXAMPLES 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, 2J+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.
P19.2) Isotopic substitution is used to identify characteristic groups in an unknown compound using vibrational spectroscopy. Consider the C=C bond in ethene (C 2 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 ( ) 2 2 = = = CD -CD 2 2 ( 16.02) 2 14.01 amu 28.02 0.935 32.04 2 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 1200 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 127 I 79 Br determined from microwave spectroscopy is 0.1141619 cm 1. The atomic masses of 127 I and 79 Br are 126.904473 amu and 78.918336 amu, respectively. Calculate the bond length in 127 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 = 2 2 0 2 r0 ( 126.904473+78.918336) 34 6.6260755 10 J s 126.904473 78.918336 amu 8π 1.6605402 10 kg amu 0.1141619 cm 2.99792458 10 cm s 2 27 1 1 10 1 10 1.742035 10 m =