Molecular transitions and vibrations

Molecular spectra arise from Electronic, vibrational, rotational transtitions

Erot < Evib < Eelec hirarchy

Powerfull:shapes sizes of molecules

strenght and stiffness of bonds Information needed to account for chemical reactions

Gross selection rulesstatements about the properties that a molecule must

possess to perform a specific transition

Specific selection ruleschanges in quantum number

http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/molspecon.html#c1

Absorption and emissionTransitions are induced by the interaction of the electric component of the electromagnetic field with the electric dipole associated with the transition

= Electric dipol moment operator

Physical interpretation: measure of dipolar migration of charge that accompanies the transition.

When is calculated it can be used for the Rates of transitions:

Stimulated: W = Brad(E) Spontaneous: W = A

iffi

Bc

hA fi

3

38

20

2

6 fi

B

Raman processesInelastic scattering of a photon when it is incident on a moleculeSelection rules for Raman transitions are based on aspects of the polarizability of a molecule, the measure of its responce to an electric field.

Classical argumentConsider time-variation of magnitude of the dipole moment induced in a molecule by an electromagnetic field E(t):

(t) = (t)E(t) (t) = polarizability,

Incident radiation with frequency

of molecule changes between min and max at frequency int as a result of its rotation or vibrationtEtt cos)cos

2

1()( 0int

with a range of variation = min-max , the product expands to:

ttEtEt )cos()cos(4

1cos)( intint00

Rayleigh Stokes Anti-Stokes

Molecular rotations

Rotational energy levels

q qq

qq

qqq I

JIT

22

12

2

q is the angular frequency about the axis

The classical kinetic energy of a freely rotating molecule can be expressed as the kinetic energy of rotation of a body of moment of inertia Iqq about an axis q

R

m2

m1

cm

q

Rotational Spectra

Assume that diatomic molecules rotate as rigid rotors.The energies can be modeled in a manner parallel to the classical description of the rotational kinetik energy of a rigid object.

From these descriptions, structural information can be obtained (bond lengths and angles).

Rotational spectrum of diatomic molecules

R

m2

m1

cm

Since the rotational kinetic energy of the rigid rotor can be expressed in terms of the angular momentum, we can imply the form for the Hamiltonian associated with the rotation around a single principal axis.

For this limited case of rotation about a single axis, the Schrödinger equation can be formulated in terms of the total angular momentum and the form of the energy eigenvalues implied.

Determining the rotational constant B enables you to calculate the bond length R.

Centrifugal distortion

As the degree of rotational excitation increases the bonds are stressed.

A diatomic molecule with reduced mass rotating at an angular velocity will experience a centrifugal force.

Tends to stretch the bond acting like a spring with restoring force obeying Hook’s law proportional to the displacement from equilibrium R0 with

k(R – R0) k = force constant.

The increase in moment of inertia that accompanies this centrifugal distortion results in a lowering of the rotational constant the energy levels are less far apart at high J than expected on the basis of the rigid rotor assumption.

Pure rotational selection rulesuse Born-Oppenheimer approximationvibrations are much faster than rotations can be separated too.The overall wavefunctionof the molecule can be written

The transition matrix factorizes into:

JMJ ,,

jJJJJJ MJMJMJMJMJMJ ,,,,,,,, ''''''

= permanent electric dipole moment of the molecule in the state .

The transition element is the matrix element of the permanent electric dipole moment between the two states connected by the transition.

Only polar molecules ( 0) can have a pure rotational spectrum.

Z

Jz

MJ

K

The specific selection rules governing rotational transitions can be established by investigating the eigenvalues of J’ and M’J for given eigenvalues of J and MJ for which the matrix element

0,,' ' JJ MJMJ

Linear molecule: rotational wavefunctions are eigenfunctions of the operators J2 and Jz (z = laboratory axis).In connection with orbital angular momentum the eigenfunctions are the spherical harmonics YJMJ

(,).

ddYYMJMJJJ

JMQMJJQJ sin,,0

2

0

''''

To evaluate the matrix elements we need to evaluate

ddYYIJJ

JMQMJM sin0

2

0

''

With M = 0, 1

Ideal for group theoretical arguments and the joint selection rules are

J = 1 MJ = 0, 1

For a polar linear rotor.

Symmetric rotors can invole changes in quantum number K.

Any permanent electric dipole moment must lie parallel to Cn axis, not perpendicular.

The electromagnetic field cannot couple to any transitions that correspond to chages in the component of angular momentum around the principal axis and to changes in K.

There is no handle perpendicular to the principal axis on which an electric field can exert a torque. The selection rules become

J = 1 MJ = 0, 1 K = 0

Spherical rotors do not have permanent dipole moments by symmetry. They do not show pure rotational transitions.

Rotational Raman selection rulesMolecules with anisotropic electric polarizabilites can show pure rotational Raman lines. The selection rules are

J = 2, 1 K = 0 but K = 0 0 is forbidden for J = 1

Rules out J = 1 for linear molecules.

Why a 2 for J? Raman effect depends on polarizability of molecule changing with time, with an internal frequency.)cos

2

1()( inttt

For a rotation the polarizability returns to its original value twice per revolution int = 2rot.

Molecule seams to be rotating twice as fast as its mechanical motion.

Idealized depiction of a Raman line produced by interaction of a photon with a diatomic molecule for which the rotational energy levels depend upon one moment of inertia

Establishing selection rules:

Recognize that the anisotropy of the polarizability has components that vary with time with angle Y2M(,).

Consider diatomic molecule with polarizabilities and an electric field E applied in the laboratory z direction.

The induced dipole is parallel to z so z = zzE.

In the molecular frame the components of the dipole moment

will be x y and z

z = xsincos + xsinsin + z cos

Ex = Esincos EY = Esinsin Ez = Ecos

The molecular component of the induced electric dipole moment is related to the molecular component of the electric field by q = qqE

z = xxExsincos + yyEysinsin + zzEzcos

= Esin2cos2 + Esin2sin2 + ||Ezcos2

=Esin2 + ||Ezcos2

With = xx = yy and || = zz the mean polarizability is = 1/3( || + 2) and

EYz

,

53

420

21

The first term does not contribute to off-diagonal elements but the second gives a contribution to the transition dipole moment

JJJZJ MJYMJEMJMJ ,,'53

4,, 20

'2

1

''

The integral that determins wether or not this matrix element vanishes is

ddYYYIJJ

JMMJsin,,, 20

0

2

0

''

The integral is zero unless J’ = J 2.

Raman lines can be expected at the following wavenumbers:

Stokes lines (J = + 2 ): J = 0 – 4B(J + 3/2) J = 0,1,2,….Anti-Stokes lines (J = - 2 ): J = 0 – 4B(J - 3/2) J = 2,3,….

Where 0 is the wavenumber of the incident radiation.

Nuclear statistics

Certain molecules show a peculiar alternation in intensity of the rotational Raman spectra.

A linear molecule shows an alternation in intensitydue to the Pauli principle and the fact that the rotation of a molecule may interchange identical nuclei having spin I (analogue of s for electrons).

Spinn of nuclei can be integral or half integral depending on specific nuclide. According to the Pauli principle the interchange of identical fermions (fractional spin particles, such as protons or carbon-13 nuclei or) or bosons ( integral spin particles like carbon-12 or oxygen-16 nuclei) must obey:

fermions

bosons

)2,1(

)2,1()1,2(

These symmetries are obeyed when a molecule rotates through or some other angle for symmetric rotors.

Diatomic molecules have only one degree of vibrational freedom, namely the stretching of the bond.

The molecular energy of a diatomic molecule increases if the nuclei are displaced from their equilibrium positions.

For small displacements (x = R - Re) the potential energy can be expressed as the first few terms of a Taylor series where the interesting term is

V(x) = ½ kx2 k=(d2V/dx2)0

The potential energy close to equilibrium is parabolic. The hamiltonian for the two atoms of masses m1 and m2 is

221

2

2

2

21

2

1

2

2

1

22kx

dx

d

mdx

d

mH

Molecular vibration

Vibrational Spectra of Diatomic Molecules The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations.

Sampling of transition frequencies from the n=0 to n=1 vibrational level for diatomic molecules and the calculated force constants.

When the potential energy depends only on the separation of the particles, the hamiltonian can be expressed as a sum, one term referring to the motion of the center of mass of the system and the other to the relative motion. The former is of no concern and the latter is

22

22

2

1

2kx

dx

dH

With being the effective mass

1/ = 1/m1 + 1/m2

The motion is dominated by the lighter atom, when m1>>m2 m2,

A hamiltonian with a parabolic potential energy is characteristic to a harmonic oscillator with:

2

1vEv

21

k

With = 0, 1, 2,…. Uniform ladder with separation .

The corresponding wavefunctions are bell-shaped gaussian functions multiplied by an Hermite polynomial.

Anharmonic oscillationSolve Schrödinger equation with a potential energy term that matches the true potential energy the Morse potential

V(x) = hcDe{1-e-ax}2 a = (k/2hcDe)1/2

The parameter De is the depth of the minimum of the curve. The Schrödinger equation becomes:

ev xvvE 2

2

1

2

1

2

2axe

21

k

Xe is the anharmonicity constant as v becomes large second term becomes imporant, at high excitations the energy converges.

Vibrational selection rulesThe transition matrix element

vvvv

'' = dipole moment of the molecule in electronic state , with bond lenght R. depends on R since the electronic wavefunction depends parametrically on the internuclear separation.The transition matrix element is

...'2

1' 2

0

2

2

0

'

vxv

dx

dvxv

dx

dvv

The gross selection rule for the vibrational transitions of diatomic molecules is that they must have a dipole moment that varies with extension homonuclear diatomic molecules do not undergo electric dipole vibrational transitions

of a molecule can vary linearly with the extention of the bond for small displacements; true for a heteronuclear molecule in which the partial charges on the two atoms are independent of the internuclear distance, then the quadratic and higher terms in the expansion can be ignored and

vxvdx

dvv

'0

'

When is the matrix element not zero?Use the following property of Hermite polynomials:

2yHv(y) = Hv+1(y) + 2vHv-1(y)

The only nonzero contributions to v’v will be obtained when v’= v 1The selection rule for the electronic dipole transition within the harmonic approximation is v = 1

The wavenumbers of the transitions that can be observed by electric dipole transitions in a harmonic oscillator are

chchc

EEv vv

2~ 1

The spectrum would consist of a single line regardless of the initial vibrational states. In real life anharmonicities cause different transitions to occur with different wavenumbers.Large displacements adjust the partial charges as the internuclear distance changes the electrical anharmonicities permit transitions with v = 2 which are the first overtones or second harmonics of the vibrational spectrum.

Vibration-rotation spectra of diatomic molecules

The vibrational transition of a diatomic molecule is accompanied by a simultaneous rotational transition with J = 1 The total energy changes and the frequency of the transition depends on the rotational constant, B, of the molecule and the initial value of J. The energy is: ...)1()1(

2

1

2

1),( 22

2

JJhcDJJhcBxvvJvE vve

The transition v= +1 and J = -1 give rise to P-branch of the vibrational spectrum. The wavenumbers of the transitions are

....)()(...~)1(2~),(~ 211 JBBJBBxvvvJvv vvvve

P

A series of lines is obtained since many initial rotational states are occupied

Transitions with J = 0 give rise to the Q-branch of the vibrational spectrum. This is only allowed when the molecule possesses angular momentum parallel to the internuclear axis a diatomic molecule can possess a Q-branch only if the total orbital angular momentum for the electrons around the internuclear axis is nonzero.The wavenumbers of this branch are:

....)()(...~)1(2~),(~ 211 JBBJBBxvvvJvv vvvve

Q

The transition with J = 1 give rise to the R branch of the vibrational spectrum with the wavenumbers:

...)()3(2...~)1(2~

/),()1,1(),(~

2111

JBBJBBBxvvv

hcJvEJvEJvv

vvvvve

R

Vibrational Raman transitions of diatomic molecules

The gross selection rule for the observation of vibrational Raman spectra of diatomic molecules is that the molecular polarizability should vary with internuclear separation.

That is generally the case with diatomic molecules regardless of their polarity, so all diatomic molecules are vibrationally Raman active.

The electronic and vibrational wavefunctions can be separated in the Born-Oppenheimer approximation and evaluatedfor a series of selected displacements, x, from equilibrium. Expand the polarizability as a Taylor function in the displacement

The origin of the gross selection rule,and the derivation of the particular:Consider the the transition dipole moment without troubling about the orientation dependence of the interaction between the electromagnetic field and the molecule:

Evvvvvv ,',,','

...')0('...)0('0

'

Evxv

dx

dEvvEvx

dx

dvvv

The first matrix element is zero on account of the orthogonality of the vibrational states when v’v:

Evxvdx

dvv '

0'

The selection rule is v = 1

Stokes lines v = +1Anti-Stokes lines v = -1

Only Stokes lines are normaly observed since initially most molecules have v = 0

In the gas phase the Stokes and anti-Stokes lines show branch structure with the selection for diatomic molecules.The selection rules are J = 0, 2.In addition to the Q-branch, there are also O- and S-branches for J = -2 and J = +2 respectively. A Q- branch is observed for all diatomic molecules regardless of their orbital angular momentum

Summary Ineraction of electromagnetic field with electric dipoleErot

Diatomic moleculerigid rotor rotating around single axisSchrödinger equation in terms of total angular momentumIs good for calculating bond lengthsSelection rule: only polar molecules can have pure rotational spectrumchange in quantum number k for symmetriesMolecules with anisotropic electric polarizabilities can show pure rotational Raman lines. Certain molecules show alternations of intensity

Evib

Diatomic molecules have only one degree of vibrational freedom, the stretching of bonds (two beads on a spring) approximate quantum mechanical harmonic oscillatorGross selection rules Dipole moment must vary with extensionhomonuclear diatomic molecules do not undergo electric dipole vibrational transitionsVibrational Raman transitionsthe molecular polarizability varies with internuclear separationtrue for all diatomic molecules regardless of polarity