Quantum Mechanics of Rotational Motion

Home , Energy operator, Good quantum number, Rigid rotor, Rotational transition

Chapter 17 Quantum Mechanics of Rotational Motion

P. J. Grandinetti

Chem. 4300

Would be useful to review Chapter 4 on Classical Rotational Motion.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum of a System of Particles ⃗ System of particles: Ltotal divided into orbital and spin contributions ⃗ ⃗ ⃗ Ltotal = Lorbital + Lspin ⃗ ⃗ Imagine earth orbiting sun at origin with Lorbital and spinning about its center of mass with Lspin. For molecules, relabel contributions as translational and rotational ⃗ ⃗ ⃗ , Ltotal = Ltrans + Jrot ▶ ⃗ Ltrans is associated with center of mass of rigid body translating relative to some ﬁxed origin, ▶ ⃗ Jrot is associated with rigid body rotating about its center of mass. ⃗ ⃗ With no external torques, to good approximation, Ltrans and Jrot contributions are separately conserved. Approximate molecule as rigid body, and describe motion with ▶ 3 translational coordinates to follow center of mass ▶ 3 rotational coordinates to follow orientation of its moment of inertia tensor PAS, , i.e., Euler angles, 휙, 휃, and 휒

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators in the Body-Fixed Frame

Angular momentum vector components for rigid body about center of mass, J⃗, are given in terms of a space-fixed frame and a body-fixed frame. ̂ ̂ ̂ ⃗ Ja, Jb, and Jc represent J components in body-fixed frame, i.e., PAS. Body-fixed frame components are given by ( ) 휕 휕 휕 ̂ ℏ 휙 휙 휃 휙 휃 Ja = i − sin 휕휃 − cos cot 휕휙 + cos csc 휕휒 ( ) 휕 휕 휕 ̂ ℏ 휙 휙 휃 휙 휃 Jb = i cos 휕휃 − sin cot 휕휙 + sin csc 휕휒

휕 ̂ ℏ Jc = i 휕휙

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators in the Space-Fixed Frame

Angular momentum vector components for rigid body about center of mass, J⃗, are given in terms of a space-fixed frame and a body-fixed frame. ̂ ̂ ̂ ⃗ Jx, Jy, and Jz represent J components in space-fixed frame. Space-fixed frame components are given by ( ) 휕 휕 휕 ̂ ℏ 휙 휙 휃 휙 휃 Jx = −i − sin 휕휃 − cos cot 휕휙 + cos csc 휕휒 ( ) 휕 휕 휕 ̂ ℏ 휙 휙 휃 휙 휃 Jy = −i cos 휕휃 − sin cot 휕휙 + sin csc 휕휒

휕 ̂ ℏ Jz = −i 휕휙

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators Body-ﬁxed frame are related to space-ﬁxed frame components

̂2 ̂2 ̂2 ̂2 ̂2 ̂2 ̂2 J = Jx + Jy + Jz = Ja + Jb + Jc [ ] 휕2 휕2 휕2 휕2 휕 Ĵ2 = −ℏ2 csc2 휃 + csc2 휃 − 2 cot 휃 csc 휃 + + cot 휃 휕휙2 휕휒2 휕휙휕휒 휕휃2 휕휃

̂ , ̂ ℏ̂ ̂ , ̂ ℏ̂ ̂ , ̂ ℏ̂ Commutators in space-fixed frame: [Jx Jy] = i Jz [Jy Jz] = i Jx [Jz Jx] = i Jy ̂ , ̂ ℏ̂ ̂ , ̂ ℏ̂ ̂ , ̂ ℏ̂ Commutators in body-fixed frame: [Ja Jb] = −i Jc [Jb Jc] = −i Ja [Jc Ja] = −i Jb ̂ , ̂ Space fixed components commutes with body-fixed components, e.g., [Jx Jb] = 0 { i = x, y, z [Ĵ2, Ĵ ] = 0 [Ĵ2, Ĵ ] = 0 and [Ĵ , Ĵ ] = 0 where 훼 i i 훼 훼 = a, b, c

Uncertainty principle says you can simultaneously know angular momentum vector length, 1 body-ﬁxed frame component, and 1 space-ﬁxed frame component.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators Eigenvalues of rotational angular momentum vector component operators: ▶ Space-ﬁxed frame:

̂ 휓 ℏ휓 , , , , , , , Ji = MJ where MJ = −J −J + 1 … J − 1 J i = x y or z

▶ Body-ﬁxed frame:

̂ 휓 ℏ휓 , , , , , 훼 , , J훼 = KJ where KJ = −J −J + 1 … J − 1 J = a b or c Eigenvalues of rotational angular momentum vector squared operator:

Ĵ2휓 = J(J + 1)ℏ2휓.

Rotational angular momentum quantum number, J can only take on integer values of

J = 0, 1, 2, … .

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Kinetic Energy Operator

Classical mechanics of rotational motion: kinetic energy in terms of principal components of moments of inertia tensor J2 J2 J2 K = a + b + c 2Ia 2Ib 2Ic

Converting to QM, need solutions of Schrödinger equation [ ] Ĵ2 Ĵ2 Ĵ2 ̂ 휓(휃, 휙, 휒) = a + b + c 휓(휃, 휙, 휒) = E휓(휃, 휙, 휒) 2Ia 2Ib 2Ic

휓(휃, 휙, 휒), is function of molecule orientation.

Euler angles, (휃, 휙, 휒), deﬁne orientation of molecule’s moment of inertia tensor PAS relative to some inertial space-ﬁxed frame.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Kinetic Energy Operator Common to re-express Hamiltonian as

̂2 ̂2 ̂2 J J J 2휋 [ ] ̂ = a + b + c = AĴ2 + BĴ2 + CĴ2 ℏ a b c 2Ia 2Ib 2Ic where ℏ ℏ ℏ ≡ ≡ ≡ A 휋 B 휋 C 휋 4 Ia 4 Ib 4 Ic are called rotational constants where A ≥ B ≥ C, and have dimensionality of frequency. Also common to re-express Hamiltonian as 2휋c [ ] ̂ = 0 Ã Ĵ2 + B̃ Ĵ2 + C̃ Ĵ2 ℏ a b c where ℏ ℏ ℏ ̃ ≡ A ̃ ≡ B ̃ ≡ C A = 휋 B = 휋 C = 휋 c0 4 c0Ia c0 4 c0Ib c0 4 c0Ic are called wavenumber rotational constants, and have dimensionality of wave numbers.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Classiﬁcation of molecules Molecules grouped into 5 classes based on principal components. Name Diagonal values Examples

Spherical Ia = Ib = Ic = I CH4

< Prolate Symmetric I|| = Ia Ib = Ic = I⟂ CH3F

< Oblate Symmetric I⟂ = Ia = Ib Ic = I|| CHF3

< < Asymmetric Ia Ib Ic CH2Cl2, CH2CHCl

Linear Ia = 0, Ib = Ic = I OCS, CO2 Molecules with two or more 3-fold or higher rotational symmetry axes are spherical tops. All molecules with one 3-fold or higher rotational symmetry axis are symmetric tops because principal moments about two axes normal to n-fold rotational symmetry axis (n ≥ 3) are equal.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Spherically Symmetric Molecules

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Spherically Symmetric Molecule Rotational Energy Eigenvalues

For spherical symmetric molecule we have Ia = Ib = Ic = I ̂2 ̂2 ̂2 ̂2 ̂2 ̂2 ̂2 J J J Ja + J + Jc J ̂ = a + b + c = b = 2Ia 2Ib 2Ic 2I 2I Schrödinger equation becomes Ĵ2 ̂ 휓(휃, 휙, 휒) = 휓(휃, 휙, 휒) = E휓(휃, 휙, 휒) 2I Eigenvalue of Ĵ2 is J(J + 1)ℏ2 and J = 0, 1, 2, …. Rotational energy is J(J + 1)ℏ2 E = = hc BJ̃ (J + 1), J = 0, 1, 2, … J 2I 0 B̃ is wavenumber rotational constant ℏ ̃ ≡ B 휋 4 c0I No zero point energy. Why? Free rotation, like free translation has no zero point energy. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Spherically Symmetric Molecule Rotational Energy Eigenstates Solutions to Schrödinger equation are [ ] 1∕2 휓 휙, 휃, 휒 2J + 1 (J) 휙, 휃, 휒 J,M ,K ( ) = D , ( ) J J 8휋2 −MJ −KJ (J) D , (휙, 휃, 휒) are called Wigner D-functions, m1 m2 (J) im 휙 (J) im 휒 D , (휙, 휃, 휒) = e 1 d , (휃) e 2 m1 m2 m1 m2 (J) where d , (휃) are called reduced Wigner-d functions and are tabulated in various texts. m1 m2 J = 0, 1, 2, … , , , , , , KJ = −J −J + 1 … 0 … J − 1 J body-ﬁxed frame component , , , , , , MJ = −J −J + 1 … 0 … J − 1 J space-ﬁxed frame component

ℏ ⃗ KJ is quantum number for rotation about c in PAS. KJ is projection of J onto c of PAS frame. ℏ ⃗ MJ is quantum number for rotation about z in space-ﬁxed frame. MJ is projection of J onto z of space-ﬁxed frame.

Energy is independent of KJ and MJ. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Energy levels and degeneracies of spherical rotor molecule energy degeneracy ̃ Rotational state energy divided by hc0 gives quantity, F(J), in wave numbers, E J = F̃ (J) = BJ̃ (J + 1) hc0

Each energy level, EJ, has gJ = (2J + 1)(2J + 1) from range of both MJ and KJ values. ▶ , , , , KJ = −J −J + 1 … J − 1 J ▶ , , , , MJ = −J −J + 1 … J − 1 J ΔE in wave numbers between adjacent levels is

EJ+1 − EJ ̃ ̃ = FJ+1 − FJ hc0 = B̃ (J + 1)(J + 2) − BJ̃ (J + 1) = 2B̃ (J + 1)

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Symmetric Molecules

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Prolate Symmetric Molecule think “cigar shaped” < In prolate symmetric molecule: Ia Ib = Ic. Deﬁne I|| = Ia & I⟂ = Ib = Ic Hamiltonian is ̂ ̂ ̂ ( ) Ĵ2 J2 Ĵ2 Ĵ2 J2 + J2 ̂2 ̂ a b c a b c J 1 1 ̂2 = + + = + = + − Ja 2Ia 2Ib 2Ic 2I|| 2I⟂ 2I⟂ 2I|| 2I⟂

̂2 ̂ ̂2, ̂ Operators appearing in last Hamiltonian are J and Ja, which commute, [J Ja] = 0. ̂2 ̂ Wigner-D functions are also eigenfunctions of J and Ja: Prolate and spherically symmetric eigenstates have same form.

KJ is quantum number for rotation about a axis in PAS frame. ℏ ⃗ KJ is projection of J onto a axis of PAS frame.

| | If KJ = J prolate molecule is found rotating about axis nearly parallel to a axis. | | If KJ is small or zero prolate molecule is found rotating about b or c axes, primarily. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Prolate Symmetric Molecules think “cigar shaped” Rotational energy of a prolate symmetric molecule is ( ) ℏ2 J(J + 1) 1 1 2ℏ2 EJ,K = + − KJ J 2I⟂ 2I|| 2I⟂

or with some rearranging becomes

E , J KJ ̃ ̃ ̃ ̃ 2 = F , = BJ(J + 1) + (A − B)K J KJ J hc0 Ã and B̃ are rotational constants deﬁned as ℏ ℏ ̃ ≡ and ̃ ≡ B 휋 A 휋 4 c0I⟂ 4 c0I|| Since rotational energy is independent of orientation of J⃗ in isotropic space—space-ﬁxed frame—rotational energy of prolate symmetric molecule is independent of MJ. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Prolate Symmetric Molecule Rotational Energy Levels

E , J KJ ̃ ̃ ̃ ̃ 2 = F , = BJ(J + 1) + (A − B)K J KJ J J hc0 6 For prolate symmetric molecules 5 (A − B) > 0 so rotational energy levels of prolate symmetric molecule increase with 4 increasing KJ |KJ|=3 3 K = 0 energy levels are (2J + 1)-fold

Energy J 2 |K |=2 1 J degenerate 0 |KJ|=1 | | ≠ |KJ|=0 KJ 0 energy levels are 2(2J + 1)-fold Prolate degenerate

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Oblate Symmetric Molecules think “frisbee shaped” > In oblate symmetric molecule: Ic Ia = Ib. Deﬁne I|| = Ic & I⟂ = Ia = Ib Hamiltonian is ̂ ( ) Ĵ2 J2 Ĵ2 ̂2 ̂ a b c J 1 1 ̂2 = + + = + − Jc 2Ia 2Ib 2Ic 2I⟂ 2I|| 2I⟂

̂2 ̂ ̂2, ̂ Operators appearing in last Hamiltonian are J and Jc, which commute, [J Jc] = 0. ̂2 ̂ Wigner-D functions are also eigenfunctions of J and Jc: Oblate and spherically symmetric eigenstates have same form.

KJ is quantum number for rotation about c axis in PAS frame. ℏ ⃗ KJ is projection of J onto c axis of PAS frame.

| | If KJ = J oblate molecule is found rotating about axis nearly parallel to c axis. | | If KJ is small or zero oblate molecule is found rotating about a or b axes, primarily.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Oblate Symmetric Molecules think “frisbee shaped”

Rotational energy of a oblate symmetric molecule is

̃ ̃ ̃ ̃ 2 FJ,K = BJ(J + 1) + (C − B)KJ

Rotational constants deﬁned as ℏ ℏ ̃ ≡ and ̃ ≡ B 휋 C 휋 4 c0I⟂ 4 c0I||

Again, since rotational energy is independent of orientation of J⃗ in isotropic space—space-ﬁxed frame—rotational energy of oblate symmetric molecule is independent of MJ.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Oblate Symmetric Molecule Rotational Energy Levels

̃ ̃ ̃ ̃ 2 FJ,K = BJ(J + 1) + (C − B)KJ

6 KJ = 0 energy levels are (2J + 1)-fold degenerate 5 | | ≠ KJ 0 energy levels are 2(2J + 1)-fold degenerate 4 For oblate symmetric molecules 3 (C − B) < 0 so rotational energy levels of Energy 2 prolate symmetric molecule decrease with |KJ|=3 1 |KJ|=2 0 increasing K . |KJ|=1 |K |=0 J J Opposite behavior of prolate symmetric Oblate molecules.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Symmetric Molecules

J 6

4 |K |=3 3 J

Energy 2 |KJ|=2 |KJ|=3 1 |KJ|=2 0 |KJ|=1 |KJ|=1 |KJ|=0 |KJ|=0 Oblate Prolate

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Asymmetric Molecules

caﬀeine P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion ≠ ≠ Asymmetric Molecules - Ia Ib Ic Most molecules become asymmetric as number of atoms increase. Ĵ2 Ĵ2 Ĵ2 ̂ = a + b + c 2Ia 2Ib 2Ic

Hamiltonian eigenstate is NOT same as Ĵ2 and one angular momentum vector component. ⃗ In absence of external torques, J is conserved, thus, J and MJ are good quantum numbers. ⃗ Projection of J onto any molecule axis is not constant, thus, KJ is NOT good quantum number. One approach: treat asymmetric molecule as deviating from prolate symmetry, ( ) [ ( )] ( ) 2휋 B + C 2휋 B + C 2휋 B − C ( ) ̂ = Ĵ2 + A − Ĵ2 + Ĵ2 − Ĵ2 ℏ ℏ a ℏ b c ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟2 2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟2 ̂ (0) ̂ (1) ̂ (0) is prolate symmetry Hamiltonian and ̂ (1) is perturbation due to deviation. When ||̂ (1)|| ≪ ||̂ (0)|| employ static perturbation theory to obtain energy eigenvalues. Can similarly treat asymmetric molecule as deviating from oblate symmetry.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Linear Molecules

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Linear Molecules - Classical Expression for Kinetic Energy

For linear molecule we have Ia = 0 and Ib = Ic = I. Because Ia = 0 there can be no rotational motion around the line of atoms. All rotation occurs about an axis perpendicular to the a axis. 휔 Since Ia = 0 and a = 0 we modify 1 ( ) K = 휔2I + 휔2I + 휔2I 2 a a b b c c to be 1 ( ) 1 K = 휔2 + 휔2 I = 휔2I 2 b c 2 2 ⃗ ⋅ ⃗ 2휔2 2휔2 2휔2 Similarly we see that J = J J = Ib b + Ic c = I and we have J2 K = 2I as kinetic energy of linear rigid rotor.

Warning: looks identical to spherical case, but not the same. Linear case derived with Ia = 0. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Linear Molecules

For linear molecule we have Ia = 0 and Ib = Ic = I. Hamiltonian is

Ĵ2 ̂ = , with energies F̃ = BJ̃ (J + 1) 2I J

Appears to be same as Spherical molecule but diﬀerent in a key way.

Spherical molecule has 3 non-zero and equal components, i.e., Ia = Ib = Ic

Linear molecule has 2 non-zero and equal components, Ib = Ic, but 1 zero component, Ia = 0

There can be no rotational angular momentum along a axis, so only KJ = 0 is allowed for linear molecule.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Linear Molecules Rotational Wave Function & Energy

Linear molecule rotational energy is energy KJ degeneracy 0 ̃ ̃ FJ = BJ(J + 1)

When KJ = 0, Wigner D-functions reduce to spherical harmonics, [ ] 휋 1∕2 0 (J) 휙, 휃, 휒 4 휃, 휙 D , ( ) = YJ,M ( ) −MJ 0 2J + 1 J and rotational eigenstates for linear molecule are

0 1 휓 , (휙, 휃, 휒) = √ Y , (휃, 휙) J MJ J MJ 2휋

0 Remember, only KJ = 0 allowed 0 MJ has 2J + 1 values, so each energy level degeneracy is 2J + 1

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Pure∗ Rotational Motion Transition Selection Rules

∗No changes in the molecule’s vibrational energy state

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Pure Rotational Motion Transition Selection Rules Transitions between rotational energy states through absorption and emission of light require non-zero electric dipole transition moment, 2휋 2휋 휋 ⟨ ⃗휇⟩ 휙 휒 휃 휃 휓∗ 휙, 휃, 휒 ⃗̂휇휓 휙, 휃, 휒 mn = ∫ d ∫ d ∫ sin d m( ) n( ) 0 0 0

Only polar molecules can have a pure rotational spectrum. Orientation of permanent electric dipole, ⃗휇, is constant in body-ﬁxed frame. In the space-ﬁxed frame, the dipole moment vector is given by [ ] ⃗휇 휇 휃 휙⃗ 휃 휙⃗ 휃⃗ = sin cos ex + sin sin ey + cos ex For spherical, prolate, oblate symmetric and linear molecules, transition moment integral is 2휋 2휋 휋 ∗ ⃗ ⟨ ⃗휇⟩ , , , ′, ′ , ′ = d휙 d휒 sin 휃 d휃 휓 (휙, 휃, 휒) ̂휇휓 ′, ′ , ′ (휙, 휃, 휒) J MJ KJ J M K ∫ ∫ ∫ J,M ,K J M K J J 0 0 0 J J J J and pure rotational transition selection rules are , , ̂휇 , ̂휇 ̂휇 ΔJ = ±1 ΔKJ = 0 ΔMJ = 0 for z ΔMJ = ±1 for x and y

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Transition Selection Rules Linear, Spherical, and Prolate and Oblate Symmetric Molecules

, , ̂휇 , ̂휇 ̂휇 ΔJ = ±1 ΔKJ = 0 ΔMJ = 0 for z ΔMJ = ±1 for x and y

By symmetry ⃗휇 must be parallel to ﬁgure axis (principal axis with highest rotational symmetry) in spherical, linear, and prolate or oblate molecules.

KJ is associated with rotation about ﬁgure axis.

Changes in KJ in spherical, linear, and prolate or oblate molecules cannot change ⃗휇 direction, so transition moment is not inﬂuenced by changes in KJ, thus ΔKJ = 0.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Breakdown of Rigid Molecule Approximation

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Breakdown of Rigid Molecule Approximation Experimentally, energy levels and spacing decrease from predicted values with increasing J. Fictitious forces in rotating frame lead to breakdowns in rigid body approximation: ▶ Under centrifugal force, bonds lengths increase with increasing J causing principal moments of inertia increase and in turn causing E to decrease from values expected for rigid molecule. ▶ Coriolis force aﬀects the atoms as they vibrate about their equilibrium positions. Energy correction terms added. In linear molecule case, energy expression becomes ̃휈 ̃휈 2 ⋯ , Evib∕(hc0) = e(n + 1∕2) − exe(n + 1∕2) + ̃ ̃ ̃ 2 2 ⋯ , Erot∕(hc0) = FJ,n = BnJ(J + 1) − DnJ (J + 1) + where ̃ ̃ 훼 ⋯ , ̃ ̃ ⋯ . Bn = Be + e(n + 1∕2) + and Dn = De + ̃ ̃ Both Bn and Dn depend on molecule’s vibrational state. ̃ De constant is always positive, and always leads to decrease in rotational energy levels. ̃ ≡ ̃ ≡ ̃휈 ≡ Be equilibrium rotational constant, De centrifugal distortion constant, and e vibrational frequency of bond in wave numbers.

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Centrifugal Distortion Constants

̃휈 −1 ̃휈 −1 ̃ −1 훼 −1 ̃ −6 −1 Molecule e/cm exe/cm Be/cm e/cm De/10 cm H2 4401.21 121.43 60.853 3.062 47100 D2 3115.50 61.82 30.444 1.0786 11410 H19F 4138.39 89.94 20.953712 0.7933704 2150 H35Cl 2990.92 52.80 10.5933002 0.3069985 531.94 H81Br 2648.97 45.22 8.46488 0.23328 345.8 H127I 2309.01 39.64 6.4262650 0.1689 206.9 19F 916.93 11.32 0.889294 0.0125952 3.3 35 Cl2 559.75 2.69 0.24415 000152 0.186 127 I2 214.50 0.61 0.03737 0.000114 0.0043 14 N2 2358.56 14.32 1.998236 0.017310 5.737 12C16O 2169.81 13.29 1.931280985 0.01750439 6.1216 16 O2 1580.19 11.98 1.445622 0.015933 4.839 14N16O 1904.20 14.07 1.67195 0.0171 0.5

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Example Calculate energy diﬀerence between J = 10 and J = 11 rotational energy levels of HCl molecule in n = 1 vibrational state assuming (a) pure rigid molecule behavior and (b) centrifugal distortion.

̃ ̃ 훼 . −1 . −1 . −1 Bn = Be + e(n + 1∕2) = 10 5933002 cm + 0 3069985 cm (1 + 1∕2) = 11 05379795 cm

(a) Using the rigid molecule approximation, ̃ ̃ ̃ ̃ . −1 . −1 ΔFJ = FJ+1 − FJ = 2B1(J + 1) = 2(11 05379795 cm )(10 + 1) = 243 2 cm (b) Including the centrifugal distortion with n = 1, [ ] [ ] ̃ ̃ ̃ ̃ ̃ 2 2 ̃ ̃ 2 2 ΔFJ,1 = FJ+1,1 − FJ,1 = B1(J + 1)(J + 2) − D1(J + 1) (J + 2) − B1J(J + 1) − D1J (J + 1)

̃ ̃ ̃ 3 which simpliﬁes to ΔFJ,n = 2B1(J + 1) − 4D1(J + 1)

̃ . −1 . −6 −1 3 . −1 and evaluates to ΔFJ = 2(11 05379795 cm )(10+1)−4(531 94×10 cm )(10+1) = 240 35 cm 243.2 cm−1 − 240.35 cm−1 Corresponds to × 100% = 1.171875% decrease 243.2 cm−1

P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion