Transitions

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Some questions... Rotational transitions Are there any? How intense are they? What are the selection rules?

Vibrational transitions Are there any? How intense are they? What are the selection rules? Can the rotational state change in a vibrational transition?

Electronic transitions How intense are they? Can the rotational state change in an electronic transition? Can the vibrational state change in an electronic transition?

Recall that, for atoms, in the absence of an external fieldbecause |1> is a parity eigenstate, and d is odd

The expectation value of the electric dipole operator is zero => There are no permanent electric dipole moments

However, if |1> and |2> are of opposite parity

For molecules ... it’s exactly the same.

Some preliminary considerations

The intensity of an electric dipole transition is proportional to the square of the matrix element of the dipole operator between initial and final states:

In the Born-Oppenheimer approximation the total wavefunction is a product of electronic, vibrational and rotational parts:

The electronic wavefunction satisfies the electronic Schrodinger equation – the one you get by clamping the nuclei.

Usually referred to as “the dipole moment” or sometimes “the permanent dipole moment”. The name is a bit misleading, unless you are clear about what it means.

The expectation value of the electric dipole operator taken between these electronic wavefunctions

is usually not zero (unless there is a symmetry that makes it zero, e.g. homonuclear).

Some more preliminary considerations

Yet more preliminary considerations: vectors!Cartesian components of a vector: Vx, Vy, Vz

Often more convenient to work instead with the components V-1, V0, V+1

The two forms are related in a simple way:

Rank 1 spherical tensor

Scalar product between two vectors:

We care about

To make things simpler, let’s take the light to be plane polarized along z. Then, only need d0

The other polarization cases follow by analogy.

For a molecule, the electric dipole operator is

The intensity of an electric dipole transition is proportional to the square of the matrix element of the dipole operator between initial and final states:

Rotate the coordinate system so that the rotated z-axis lies along the internuclear axis. In this rotated system, let the electric dipole operator for the molecule be m. Its components are related to those of d in a simple way:

Component in the lab frame Angular

functions Components in the “molecule frame”

These are simply elements of the rotation matrix that rotates from one coordinate system to another. e.g. D00 = cos Q

In the Born-Oppenheimer approximation, we write the total molecular wavefunction as a product of electronic, vibrational and rotational functions. So, we need to evaluate:

With the help of this transformation, we have

Which separates into a part that depends only on , Q F, and a part independent of these:

Transitions within an electronic stateConsider rotational and vibrational transitions within an electronic state, i.e. n’=n

By symmetry, the only non-zero component of m is along the internuclear axis, i.e. m0

Define a rotational factor

Also define

The electric dipole moment function for electronic state n

cos Q

Angular momentumeigenstates

Its value at R0 is the dipole moment of the molecule in electronic state n

We are left with

The vibrational wavefunctions are only large close to R0. So, Taylor expand the dipole moment function around this point:

Transitions within an electronic state - RotationalTaking just the first term in this expansion (a constant), the integral is zero unless v’=v. So the first term gives us pure rotational transitions.

Pure rotation:

Recall that the angular factor Mrot is

This matrix element is straightforward to evaluate. We will do it when we study the Stark shift.For now, it is sufficient to know that it is zero unless DJ = J’-J = 0,±1 and DM = M’-M = 0

The selection rule on M is due to our choice of polarization. If we choose left/right circular light instead we get DM = -1/+1

If the dipole moment is zero (e.g. homonuclear ), there are no rotational transitions

Selection rules for rotational transitions: DJ = 0,±1 , DM = 0,±1

Have seen that the first term in our Taylor expansion cannot change the vibrational state

The second term can.

Vibrational transition:

If the dipole moment is zero (e.g. homonuclear ), there are no vibrational transitions

The vibrational integral is zero unless v’ = v ,±1 (see problem sheet)

Selection rule for vibrational transitions: Dv = ±1

The rotational state can also change in a vibrational transition, with selection rules DJ = 0,±1 , DM = 0,±1

The intensity of a vibrational transition is proportional to the gradient of the dipole moment function at the equilibrium internuclear separation

Transitions within an electronic state - Vibrational

Transitions that change an electronic state

Angular factor – same selection rules as before

Define a transition dipole moment function:

In the same way as before, Taylor expand about R0:

Then, to lowest order, we have the result:

Angular factor

Transition dipolemoment, n->n’

Vibrational overlap integral

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Franck-Condon factor

James Franck (1882-1964) Edward Condon (1902-1974)

The square of the overlap integral between vibrational wavefunctions in the two potential wells.N.B. They come from two different wells – not orthogonal (unless wells are identical).

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Some answers... Rotational transitions Are there any? How intense are they? What are the selection rules?

Vibrational transitions Are there any? How intense are they? What are the selection rules? Can the rotational state change in a vibrational transition?

Electronic transitions How intense are they? Can the rotational state change in an electronic transition? Can the vibrational state change in an electronic transition?

Yes, provided the dipole moment men is non-zero

Proportional to the dipole moment, men , squared

DJ = 0,±1 , DM = 0,±1

Yes, provided the dipole moment men is non-zero

Proportional to the gradient of the dipole moment, dmen/dR, squared

Dv = ±1

Yes, with DJ = 0,±1 , DM = 0,±1

Proportional to the transition dipole moment, men’,n , squared

Yes, with DJ = 0,±1 , DM = 0,±1

Yes, proportional to FC factor