Molecular polarizability

 

Perturbation theory: the Kramers-Heisenberg-Dirac equation

The treatment of light scattering involves the molecular polarizability. This is because light scattering is a two-photon process that involves coupling of the ground state to intermediate states. The Kramers-Heisenberg-Dirac equation for the polarizability expresses this coupling:

 

If we use the time-dependent wave functions that we have already derived we can obtain a first order correction to the transition moment.

The term mif(perm) is the transition moment evaluated with the zero-order states: mif(perm) = ái|m|fñ. The permanent transition moment is responsible for absorption and emission. The induced transition dipole is the molecular polarizability. The dipole and polarizability must be Hermitian: mif = m*fi and aif = a*fi. This means that the induced dipole will be of the form

The derivation will proceed as follows. We substitute in the first-order wave functions expand and compare to the above results. The comparison will give us an expression for aif.

The first order wave function for state f is

with a similar expression for state i. The coefficients are given by

The Kroenecker delta dnj = 1 if n = j and dnj = 0 if n ¹ j. The matrix elecment is

In this representation Vnj0 represents the magnitude of the pertubation

and mnj is the transition. The coefficients are found by direct integration

A similar expression holds for coefficient cni(t). The coefficients are now substituted into the expression for the perturbed wavefunction.

Again a similar expression holds for áyi(t)|. Physically we are interested in the induced dipole moment on a molecule at the frequency of an applied electric field. The wavefunction is perturbed at the field frequency e{±iwt}. The part of the transition frequency that depends on exp{iwnft} does not contribute to the induced transition moment. This is called the rotating wave approximation. We can rewrite the wavefunction as

Taking the complex conjugate and changing the index from f to i results in

 

The zero order wavefunctions ái| and |fñ combine to give the ordinary transition moment.

The induced dipole moment is

Substituting in the form of the matrix elements Vnf0 we have

Comparing this expression to our original hypothesis regarding the form of the induced dipole moment

gives the following expression for the transition polarizability:

 

It is crucial to keep in mind that each of the terms min and mnf is a transition moment connecting two states in the molecule. Thus, min and mnf are vectors and have components. It is customary to represent these by r and s where these refer to the x, y, and z directions in the molecular frame.

This is the Kramers-Heisenberg-Dirac (KHD) expression for the molecular polarizability. This expression gives the molecular polarizability as a sum of transition dipoles for transitions of the molecule. When the frequency of light is near that of an electronic transition there is a large enhancement (resonant enhancement). For resonant enhancement the first term (left term) can be dropped. When the states i and f are rotational or vibrational states within the same electronic state the KHD formula refers to Raman scattering. The KHD expression can also be used to determine the molecular polarizability of states (rather than transitions from i à f). For example, the ground state polarizability is

Lifetimes and dephasing contribute the imaginary component of polarizability

 The derivation above assumes that states are infinitely narrow in energy. As we discussed for the Fermi Golden Rule, in reality states have a finite width in energy due to their lifetime. In fact, they also have a finite energy width due to pure dephasing. Pure dephasing can be thought of as the loss of phase information (terms such as exp{iwijt} for states i and j) due to fluctuations in the electric field in the environment. We can define a rate G at which population decays or dephasing occurs. Thus, the probability of stationary state |n> decays exponentially in time:

To separate out contributions from population relaxation with time constant T1 and pure dephasing with time constant T2* we define the overall T2 time for a state as the inverse of the rate G.

The coefficient is

This term is incorporated into the expression for the coefficient

When propagated through the steps above the expression for the polarizability becomes

 

This expression suggests that polarizability has real and imaginary parts. These represent contributions to dispersion and absorption, respectively. This formalism is also related to higher order coupling in non-radiative transitions (so-called superexchange coupling) by replacing the dipole operator with the appropriate perturbation that connects two states.

 

Classical model of the frequency dependence of the molecular polarizability

The Lorentz model is given here. An electron in one dimension is subject to an electric field that results in a displacement from the center of charge. For a displacement in the x direction (this is the polarization of the electric field, i.e. the electric vector of radiation) the induced moment is given by

where e is the charge of an electron and a is the polarizability. The restoring force is

The force is balanced by the force due to the applied electric field

F = eE. The balance of forces is

Thus x is

and

The classical Lorentz polarizability is

The expression can be generalized by letting there be N electrons on a molecule. Each electron has an intrinsic harmonic oscillator frequency. The fraction of the electrons in each of the j modes is fj. The polarizability for all of the electrons is 

As above for the quantum mechanical treatment fj is the oscillator strength. The above treatment is for a static field. In the case where E(t) is a sinusoidal field the equation resembles that of a driven harmonic oscillator.

In addition to the electric force and the harmonic restoring force we have added a frictional force. The time dependent electric field must have the form E(t) = E0xexp[i(ky - wt)]. Therefore, we try a solution of the form

x(t) = x0exp[i(ky - wt)]

The equation becomes

Collecting terms in x(t) solving for x(t) we have

We can also define the polarizability a = -ex(t)/E(t) so

The polarizability has real and imaginary parts. These are obtained by multiplying both numerator and denominator by wj2 - w2 + iwG/m. 

Clearly the polarizability consists of real and imaginary parts.

We can write it as

The real and imaginary parts are related to one another through the Kramers-Kronig relations

Oscillator strength

The oscillator strength is a classical formalism. The oscillator strength fij is proportional to the intensity of a transition i à j. It is a number less than one and in fact the sum of all of the oscillator strengths in the molecule equals one. Thus,

Comparison of the quantum treatment with the Lorentz formulation gives

The factor of 1/3 arises because only one polarization of electromagnetic radiation is considered here.