Rotations and vibrations of polyatomic molecules

When the potential energy surface V(~R1, ~R2, . . . , ~RN) is known we can com-pute the energy levels of the molecule. These levels can be an effect of:

• Rotation of a molecule as a whole (end-over-end rotation)

• Small vibrations around equilibrium configuration of the nuclei

• Internal rotation - free or hindered

• Tunneling

• Large amplitude vibrations (van der Waals vibrations)

Transitions between these levels are are observed in spectroscopy:

• Infrared/Raman – vibrations

• Terahertz (submillimeter) – rotations, large amplitude vibrations

•Microwave – rotations, tunneling

These transitions occur between the eigenvalues of the Hamiltonian for thenuclear motion:

H = −1

2

N∑i=1

1

mi

∆~Ri+ V(~R1, ~R2, . . . , ~RN)

where mi is the mass of the i-th nucleus.The Born-Oppenheimer approximation is assumed here .

Rotations and vibrations of polyatomic molecules, continued

For polyatomic molecules rotations and vibrations cannot be separated

The separation is possible only approximately(often with good accuracy) forthe so called rigid moleucles such as:

• water • carbon dioxide • methane • ethylene • benzene

Rotation is then described as a rotation of a rigid body (a rotating top of theright kind).

Vibrations are described as small amplitude oscillations around the (rotating)equilibrium positions of the nuclei.

The total energy is a sum of the rotation energy and the vibration energy

For nonrigid molecules (floppy molecules), such as:

• dimethylacetylene • propane • ammonia • ethanol • water dimer

such an approximation does not work and each case must be treated separately.

Theory of nuclear motion in floppy molecules is still in development and willnot be presented in these lectures since:

There are no commercial programs offering a possibility of computingrovibrational spectra of nonrigid molecules.

Rotations of polyatomic molecules

Classically, the rotation of a rigid body is described by the vector of angularvelocity ~ω such that the velocity ~vi of i-th particle (atomic nucleus) is givenby:

~vi = ~ω × ~Ri

The angular momentum of the rigid body is then given by

~L =N∑i=1

~Ri × (mi ~vi) =N∑i=1

mi~Ri × (~ω × ~Ri) = II~ω,

where II is the matrix (tensor) of the moment of inertia:

II =

∑

imi (Y2i + Z2

i ) −∑

imiXi Yi −∑

imiXi , Zi

−∑

imi YiXi

∑imi (X

2i + Z2

i ) −∑

imi YiZi

−∑

imiZiXi −∑

imiZi Yi∑

imi (X2i + Y 2

i )

The matrix II is symemtric, e.g., IIxy=IIyx. In the coordinate system fixed tothe rotating molecule the matrix II does not depend on time.

Rotations of polyatomic molecules, continued

As each symmetric matrix the matrix II can be diagonalized. In the newcoordinate frame (also fixed to the molecule) this matrix has the form:

II =

IA 0 0

0 IB 0

0 0 IC

where IA, IB, and IC are the so-called principal moments of inertia of themolecule. The normalized eigenvectors vectors ~A, ~B, and ~C, corresponding tothe eigenvalues IA, IB, and IC determines the so-called principal exes of thetop (a molecule). If the molecule has a symmetry then ~A, ~B, i ~C s ↪a coincidewith the symmetry axes of the molecule. The principal moments of inertiaare given by the formulas:

IA =∑i

mi (B2i + C2

i ) IB =∑i

mi (A2i + C2

i ) IC =∑i

mi (A2i +B2

i )

where Ai, Bi and Ci are the coordinates of the i-th nucleus in the coordinateframe determined by the principal axes ~A, ~B, and ~C, for instance B2

i + C2i is

the distance of the i-th nucleus form the ~A axis.

Rotations of polyatomic molecules, continued

In the coordinate frame ~A, ~B, and ~C the matrix II is diagonal so the angularmomentum vector ~L = II~ω has the following coordinates in this frame:

LA = IA ωA, LB = IB ωB, LC = IC ωC,

One can easily show that the kinetic energy of rotations is given by

Erot =1

2

N∑i=1

mi ~vi · ~vi =1

2

N∑i=1

mi ~vi · (~ω × ~Ri) =1

2

N∑i=1

mi ~ω · (~Ri × ~vi) =1

2~ω · ~L

In the frame of principal axes ~ω · ~L = ωALA + ωBLB + ωCLC, hence

Erot =1

2IA ω

2A +

1

2IB ω

2B +

1

2IC ω

2C

or

Erot =1

2IAL2A +

1

2IBL2B +

1

2ICL2C

To quantize and obtain the quantum mechanical Hamiltonian we add hats:

Hrot =1

2IAL2A +

1

2IBL2B +

1

2ICL2C

Rotations of polyatomic molecules, continued

Eigenvalues and eigenfunctions of Hrot depend significantly on the relativevalues of the principal moments of inertia. We distinguish four kinds of tops:

• spherical tops, IA = IB = IC (methane, hexafluoride of uranium)

• prolate symmetric tops, IA < IB = IC (ammonia, CH3Cl)

• oblate symmetric tops, IA = IB < IC (benzene, SO3, CHCl3)

• asymmetric tops, IA 6= IB 6= IC (water, ethylene, naphthalene)

For the oblate symmetric top IA = IB so we can write:

Hrot =1

2IAL2A+

1

2IBL2B +

1

2ICL2C =

1

2IAL2A+

1

2IAL2B +

1

2IAL2C +

1

2ICL2C−

1

2IAL2C

henceHrot =

1

2IAL2 +

(1

2IC−

1

2IA

)L2C

Since the eigenvalues of L2 and LC are given by J(J + 1)~2 i K~, where−J ≤ K ≤ J , the eigenvalues of Hrot must be given by (in atomic units)

EJK =1

2IAJ(J + 1) +

(1

2IC−

1

2IA

)K2

Rotations of polyatomic molecules, continued

For the prolate symmetric top we obtain a similar formula only the smallestand largest moments of inertia IA and IC are interchanged:

EJK =1

2ICJ(J + 1) +

(1

2IA−

1

2IC

)K2

For symmetric tops the degeneracy of the level EJK is 2(2J+1), 2 with respectto K K and 2J + 1 with respect to M . For the spherical top IA=IB=IC thus:

EJ =1

2IAJ(J + 1)

The degeneracy in this case is (2J + 1)2.

For the asymetric top (water, ethylene,) there is no analytical formula for therotational levels. The degeneracy with respect to K is lifted. K ceases ti bea god quantum number.

Linear molecules in states with Λ 6= 0, i.e. Π, ∆, etc. states are also symmetrictops (prolate ones) with K = Λ where Λ = 1 for the Π, Λ = 2 for the ∆,states, etc. The rotational energy of this states is given by:

EJΛ =1

2IC[J(J + 1)− Λ2] +

1

2IAΛ2

where IC=µR2e and Λ2/(2IA) is a constant included in the electronic energy.

Rotational energies pattern for symmetric rotors

Rotational energies pattern for asymmetric rotors

Vibrations of polyatomic molecules

Consider N atomic nuclei. Position of the i-th nucleus will be specified by theCartesian coordinates xi, yi and zi of its displacement from the equilibriumposition (assumed fixed in space). The Hamiltonian in this coordinates is:

Hvib = −3N∑i=1

1

2mi

∂2

∂u2i

+ V (u1, u2, . . . , u3N)

where ui, i=1,...,3N denote collectively all Cartesian coordinates x1, y1, z1,x2, y2, z2,... and masses are defined such that m1=m2=m3, m4= m5=m6,...

The potential V (u1, u2, . . . , u3N) can be expanded in the Taylor series aroundu1=u2=u3=· · ·=0

V (u1, u2, . . . , u3N) = V0 +3N∑i=1

(∂V

∂ui

)ui +

1

2

3N∑i=1

3N∑j=1

(∂2V

∂ui∂uj

)uiuj + · · ·

The first derivatives vanish at the minimum, the third ones are neglected(harmonic approximations). The constant V0 is included in the electronicenergy, hence:

Hvib = −3N∑i=1

1

2mi

∂2

∂u2i

+ +1

2

3N∑i=1

3N∑j=1

Vij uiuj

where Vij is the matrix of second derivatives ( the Hessian) at the minimum.

Vibrations of polyatomic molecules, continued

To find eigenvalues of this Hamiltonian it is convenient to introducethe so-called mass scaled coordinates :

wi =√mi ui

In these coordinates the Hamiltonian has the form :

Hvib = −1

2

3N∑i=1

∂2

∂w2i

+1

2

3N∑i=1

3N∑j=1

Vij√mimj

wiwj

The matrix Vij/√mimj is symmetric so it can be diagonalized with a matrix

Uij and one can introduce the normal coordinates qk using this Uij matrix:

qk =3N∑i=1

Uikwi henceczyli ui =1√mi

3N∑k=1

Uikqk e.g., u(k)i =

1√mi

Uikqk

In this normal coordinates the Hamiltonian has a very simple from:

Hvib =3N∑k=1

(−

1

2

∂2

∂q2k

+1

2λk q

2k

)where λk are eigenvalues of the Vij/

√mimj matrix. The eigenvalules of Hvib

are:

Ev1,v2,... =3N∑k=1

~ωk(vk +

1

2

)gdzie ωk =

√λk

Translations, rotations, and vibrations of a triatomic molecule

Harmonic vibration spectrum of water

Resonance tunneling in the vibration spectrum of ammonia

Fermie resonance of the vibrational excitations

~ω1 = 1333 cm−1 and 2~ω2 = 1338 cm−1 in the CO2 molecule

Coriolis force effects on the normal vibrations of the CO2 molecule