Approximate: yes. Accurate: not that I know of.

The only method I know is: weight your sample (at least around 10 mg to reduce imprecision on measurement). Then prepare a solution of known concentration using weight and molecular weight. Repeat 2 other times. For your three solutions, prepare a serial dilution (generally in the micromolar range) and measure absorbance. Plot absorbance vs concentration making a mean of your 3 samples and take the slope in the linear (low concentrations) domain.

There might be new methods involving cross-section measurements that I am not aware of, but to the best of my knowledge this is the only method for measuring epsilon. It's not exactly accurate due to weight and volume measurement errors, which is the reason why most extinction coefficients in the litterature and provider's website are displayed as multiple of 1000.

Thank you Julien. I asked for a theoretical way using density functional theory.

Sorry for the misunderstanding. I hope that someone will have a more suitable answer than.

All I can tell as far as this problem is concerned is that there is apparently a way to evluate two photon cross-section using density functional theory, but don't know if such a method exist for one-photon cross section.

I have no experience of calculating absorption coefficients by DFT or other theories. The (exp. vs. theory) problem is that the epsilon(max) values are sensitive to the solvent, as is the bandwith (the excitation energies/wavenumbers also evidently depend on the medium). A simple epsilon(max) value usually found in the literature cannot be compared unless you consider the integrated value: in polar solvents the bandwith is usually much larger than in weakly polar solvents.

I am not an expert in theroretical calculation. However, I believe experimental method is the best way to judge the extinction coefficient since in theory the solvent may not be taken into account. 

There are methods for modelling the solvent effects (for example the polarizable continuum model = PCM). I know that for thermochemical calculations of basicity parameters in the gas phase and in solution. I don't know if the models are good enough for spectroscopy, but I have seen a lot of recent papers on "time dependent" DFT and electronic excitation, see for example:

J. Chem. Phys. 140, 034109 (2014); 

J. Phys. Chem. B, Article ASAP - DOI: 10.1021/jp507962n

Computational and Theoretical Chemistry - Volumes 1040–1041, 15 July 2014, Pages 328–337 (Coumarin C153 and N-methyl-6-Quinolinium betaine dyes)

Molecular SimulationVolume 40, Issue 1-3, 2014, pages 169-175; (Simulations of UV–visible spectra for analytical applications: phenothiazines ...)

and so on... There are about 200-400 answers between 2014-2015 if you type in Google scholar :

solvent effect PCM electronic spectrum

It is also possible to add water molecule(s) (discrete solvation) to simulate the effect od Hydrogen bonding.

Good luck

J-F

I don't believe that DFT is the only, nor always the best, modeling method. My personal interest is in carbon-rich molecules. For these, I mostly use the PM3 semi-empirical method implemented in the Hyperchem package (Hypercube, Inc.), which also includes the other main semi-empirical methods. PM3 is MNDO parametrized by J.P. Stewart especially for such molecules by comparison with corresponding experimental data (J. Comp. Chem. 10 , 209, 221). It was shown to perform relatively satisfactorily (Hehre et al. 1989, Experiments in Computational Organic Chemistry, 1993, Facsimile (714)955-2118 from Wavefunction, Inc.). The integrated absorption coefficients are automatically computed.

I have used this for molecules up to ~500 atoms in size. Other similar packages are SPARTAN and MOPAC.

Best

Renaud

Renaud, Do know about a study comparing the exp. and calc. (PM3 or other method)absorption coefficients (as well as absorption maximums) ?

Thanks,

J-F

The problem is that TDDFT as simplified description of reality gives a kind of relative values for the oscillator strengths. If you plot them against experimental one you should get a line (slope/intercept) and then you can re-calculate back (improve) your theoretical values. In most of the DFT functionals oscillator strength is overestimated depending on the functional (in my own experience, I might be wrong in general). Look for example the attached paper.

However I do not understand how/why do you compare the theoretically estimated oscillator strength with experimental molar absorptivity? First one represents an UV-Vis absorption band area, and second one - single point from the spectrum. Just imagine that you have low intensive, but wide band - this might mean large oscillator strength. So please clarify what do you mean under predicted "molar absorptivity".

Article Prediction of the color of dyes by using time-dependent dens...

I think Liudmil gave a precise answer on the relationship between oscillator strength and molar abosption coefficient. I will be interested to know if he can complete with the solvent effects on oscillator strengths.

It is not easy question. There is empirical relation between gas phase (vapor) and solition oscillator strength (o.s.), where the solvent participates with its dielectric constant. See for instance: Myers,A.B.and Birge,R.R.(1980)J.Chem.Phys.,73,5314–5321. This is for the experiement. I have never tried to recalculate the solution o.s. taking theoretical one as gas phase.

In respect of the theory - nowdays TDDFT does not make much difference (at least for my systems) between gas phase and solution o.s. if you do not model specific interactions and use just the solvent as a medium.

To conclude - in practice o.s. depends on the environment by its dielectric constant and specifically (it is related on how given solvent stabilize corresponding ground and excited state).

It may be possible to simulate the solvation by adding explicit solvent molecules (water...) to charged sites when possible.

If extinction coefficient in any of the solvent known, you can make a stock solution in that solvent. Dilute this stock to the same extent one by using the same solvent and the other by the solvent of interest. This way you will get the  extinction coefficient in the mixed solvent. Now reverse the experiment making the stock solution in the solvent of interest and then diluting the stock solution to the same extent as earlier, one by using the solvent of interest and the other by using the reference solvent. Using the earlier estimate of the extinction coefficient in the mixed solvent, one can now estimate the extinction coefficient in the solvent of interest. 

I have got a TDDFT spectrum of my organic molecule which when plotted using gaussview gave me higher epsilon values for peaks with lower oscillatory strength. What does that mean??which should band should be taken as the most intense band in that case?? The one with higher oscillotory strength or the one with higher epsilon??

In TDDFT calculations, oscillator strength is always used for representing transition strength in UV-Vis spectrum, however the precise relationship with molar absorptivity hasn’t been built. Program such as Gaussview or Multiwfn use a known relation between theoretical data and experimental spectra: If the unit eV and L/mol/cm are used for X-axis and Y-axis respectively, then the area under the curve that broadened from per unit oscillator strength should be 1/4.32/8.0655*106=28700. By this relation, UV-Vis spectrum can be emulated by theoretical data and you get your epsilon. See Multiwfn manual for details. For my systems, I usually use the oscillator strength for analyzing my results since epsilon still an experimental coefficient difficult to be predicted from theoretical calculations.

In this context it may be of interest, that the extinction coefficient itself is, even in the absence of any interactions, dependent on concentration (and absorbance is depending only approximately linearly on concentration, albeit the deviations are very small for low concentrations and oscillator strengths). See e.g. Article Beer's Law – Why Absorbance Depends (Almost) Linearly on Concentration