Hello Debashree,

in the following I describe my simple point of view for this relation, maybe it helps.

in an absorption experiment there is an incoming radiation power, P0, and an outgoing one, P, where both rays are considered in one line. The absorbance A is defined as A=lg(P0/P). The absorption process in the sample may be interpreted in the following simple picture: the sample is cut into many very thin layers (perpendicular to the ray); in each layer, the probability of ray being absorbed is proportional to the density of the particles in this layer (particles per area); the proportionality constant is the cross section (of each particle), sigma. Integrating over all layers and converting particle densities into molar concentration with Avogadro constant NA yields the Lambert-Beer law A=eps * b * c where eps is the molar extinction coefficient, b is the sample thickness, and c is the molar concentration.

During this derivation it turns out that eps=lg(e)*sigma*NA (or something similar, please check that in the literature). The great thing now is: The absorption cross section can be defined in terms of the isotropic polarizability alpha_iso (isotropic means that the trace of the polarizability tensor alpha is taken because we want to have the cross section averaged for all orientations of a molecule):

sigma = 4*pi*omega/c * Im(alpha_iso(omega))

where omega is the angular frequency of the light and c is speed of light and

alpha_iso = 1/3 Trace[alpha(omega)] .

And finally: Deriving alpha_iso from linear response theory, it can also be defined in terms of oscillator strengths f_n:

alpha_iso(omega) = limit(eta->0+) { \sum_{n=1}^\infty \frac{f_n}{(\hbar \omega + i\hbar\eta)^2 -\Omega_n^2}}

where eta is a scalar which is taken to the limit zero from the positive and

f_n=2/3 * \Omega_n \sum_{\mu=x,y,z} |M_\mu^{0n}|^2

and the absolute square of the electric transition dipole moment is

|M_\mu^{0n}|^2 = *

with r_\mu being the \mu-th component of the electric dipole vector operator r=\sum_i^electrons (x_i,y_i,z_i)=(r_x,r_y,r_z).

SO YOU SEE: OSCILLATOR STRENGTHS AND MOLAR EXTINCTION COEFFICIENTS ARE RELATED VIA THE CROSS SECTION AND THE (DYNAMIC ELECTRIC DIPOLE-ELECTRIC DIPOLE) POLARIZABILITY TENSOR ALPHA.

An excellent starting point for reading about these relations is the following book:

"Time-Dependent Density-Functional Theory - Concepts and Applications" from Carsten Ullrich published in Oxford University Press 2012.

I apologize in advance for any typos and inaccuracies in the description but I wanted to show the main track. Best regards. Asbjoern.

Hello Debashree,

in the following I describe my simple point of view for this relation, maybe it helps.

in an absorption experiment there is an incoming radiation power, P0, and an outgoing one, P, where both rays are considered in one line. The absorbance A is defined as A=lg(P0/P). The absorption process in the sample may be interpreted in the following simple picture: the sample is cut into many very thin layers (perpendicular to the ray); in each layer, the probability of ray being absorbed is proportional to the density of the particles in this layer (particles per area); the proportionality constant is the cross section (of each particle), sigma. Integrating over all layers and converting particle densities into molar concentration with Avogadro constant NA yields the Lambert-Beer law A=eps * b * c where eps is the molar extinction coefficient, b is the sample thickness, and c is the molar concentration.

During this derivation it turns out that eps=lg(e)*sigma*NA (or something similar, please check that in the literature). The great thing now is: The absorption cross section can be defined in terms of the isotropic polarizability alpha_iso (isotropic means that the trace of the polarizability tensor alpha is taken because we want to have the cross section averaged for all orientations of a molecule):

sigma = 4*pi*omega/c * Im(alpha_iso(omega))

where omega is the angular frequency of the light and c is speed of light and

alpha_iso = 1/3 Trace[alpha(omega)] .

And finally: Deriving alpha_iso from linear response theory, it can also be defined in terms of oscillator strengths f_n:

alpha_iso(omega) = limit(eta->0+) { \sum_{n=1}^\infty \frac{f_n}{(\hbar \omega + i\hbar\eta)^2 -\Omega_n^2}}

where eta is a scalar which is taken to the limit zero from the positive and

f_n=2/3 * \Omega_n \sum_{\mu=x,y,z} |M_\mu^{0n}|^2

and the absolute square of the electric transition dipole moment is

|M_\mu^{0n}|^2 = *

with r_\mu being the \mu-th component of the electric dipole vector operator r=\sum_i^electrons (x_i,y_i,z_i)=(r_x,r_y,r_z).

SO YOU SEE: OSCILLATOR STRENGTHS AND MOLAR EXTINCTION COEFFICIENTS ARE RELATED VIA THE CROSS SECTION AND THE (DYNAMIC ELECTRIC DIPOLE-ELECTRIC DIPOLE) POLARIZABILITY TENSOR ALPHA.

An excellent starting point for reading about these relations is the following book:

"Time-Dependent Density-Functional Theory - Concepts and Applications" from Carsten Ullrich published in Oxford University Press 2012.

I apologize in advance for any typos and inaccuracies in the description but I wanted to show the main track. Best regards. Asbjoern.

Hello Asbjoern,

Thanks a lot for your answer.

Your reply has cleared my doubt whether there exist any direct correlation between molar extinction coefficient and oscillator strength. Now I understand that there is some direct correlation.

From your discussion it is also clear to me that the molar extinction coefficient is measured in terms of absorption cross-section and absorption cross-section is defined in terms of polarizability tensor (alpha_iso) to include average orientation of the molecule in all possible direction.

I have little confusion about inclusion of alpha_iso (polarizability tensor) in oscillator strength. Oscillator strength in the above equation includes omega_n and square of electric transition dipole moment. Where alpha_iso will fit into oscillator strength?

Debashree

Hello Debashree,

The scalar value alpha_iso is 1/3 of the trace of the polarizability tensor alpha (basically a 3x3 matrix in Cartesian space), i.e., alpha_iso=1/3*sum_{i=x,y,z}alpha_{ii}. The expression of alpha_iso in terms of the oscillator strengths f_n is easier to write down because because f_n contains also the sum over x,y,z.

Therefore, alpha_iso may be represented in terms of the oscillator strengths.

Thanks Asbjorn.

So, conceptually is it something like this? Alpha_iso is a function of oscillator strength. On the other hand molar extinction coefficient is a function of alpha_iso. Hence, molar extinction coefficient is a function of oscillator strength, as well.

Thanks for the reference also. I shall collect the reference to get started with the details.