Dear Thomas Mayerhöfer ,

The absorbance is proprotional with E2 if you consider weak fields and molecules only in gas phase and neglect the induced dipole components. In this case the transition probability is proportional with E2.  If you study condensed matter you can not neglect the different component of the dipole and the transition probability will not be proportional with E2.

I recommend you the following book (mainly chapter 5 and 9):

George C. Schatz and Mark A. Ratner

Quantum Mechanics in Chemistry

Moreover I attached two articles, see Eq. 2-5 in Guillot's paper.

Dear Peter, thanks, but my problem is actually how the macroscopic property -lg T can be linked to the microscopic property "transition probability". Actually, Transmittance and Reflectance are defined through intensities or irradiances which are proportional to E2. But how can then the negative decadic logarithm of T be also proportional to E2? 

As Guillot (and also others) shows the absorption coefficient is also proportional with the square of the transition dipole which is some kind of function of the the electric field (in most of the cases it depends on E2) . If the Lambert-Beer law is valid then the absorption coefficient is proportional with the absorbance. This formula (see the attached pic) is valid for the macroscopic absorption coefficients, it was deduced from the statistical mechanical extension of the microscopic quantum mechanical transition probabilities.

In general case the cross-section of the absorption can be calculated

                     σabs = rate of the absorption / incident photon flux

here the rate of the absorption (wabs) can be calculated from Fermi's golden rule. This cross section can be considered not only for one molecule, but also for an ensemble of molecules (eg. in a cuvette). The cross section is directly connected to the  absorbance or to the absorption coefficient. This is the first point of connection to the macroscopic picture.

The other part of the macroscopic point of view comes into the story through the rate of the absorption:

                                                            wabs  ~  |µ|2

here

                                              µ =

is the quantum mechanical expectation value of the transition dipole between the initial ground state and the final excited states. If you consider not only one molecule in a closed system, rather molecules in an open thermal ensemble, then you need to replace the quantum mechanical expectation value with the Boltzmann average over the thermally available molecular quantum states. The obtained formula (see the attached picture) is valid for a macroscopic (equilibrium) system, because it considers all of the thermally available molecular quantum states. If you want to be more exact, you can take into account the intermolecular interactions. In this case µ has to contain more extra terms  that describes the collision (or other intra- or intermolecular...surface enhanced) induced dipoles.

Dear Peter, thanks again!

If I understand you correctly, then the absorbance depends on the electric field squared (or, at least, somehow from some functional dependence involving E), because the absorption coefficient depends on E2 according to literature.

If I look into the Beer-Lambert law (assuming a pure substance) then the absorption coefficient alpha = lg e*4 pi*(1/lamda)*k, wherein k is the absorption index and lamda the wavelength. Therefore, obviously, k would depend on the electric field intensity.

The absorption index, however, is part of the complex index of refraction, which does not depend on the electric field intensity according to the textbooks of optics (except for very high field strengths, where non-linear effects start to occur).

So, always provided I did not misunderstand you, we moved from a comparably unimportant problem (dependence of absorbance on electric field intensity) to a much more challenging one...

Just a short update concerning this question: Searching through literature we found hints in literature that show that it is not absorbance, but absorptance (1-reflectance-transmittance), which is proportional to the electric field intensity. Indeed, this is also what we found in the systems we investigated.

Dear Thomas Mayerhöfer ,

Have you found a satisfied answer for this question. I am in trouble in understanding EM mechanism of the surface-enhanced vibration spectroscopy. It seems that EM mechanism can be explained by the increased probability of the Raman scattering process or IR absorption process, which i think is the same meaning of "transition probability" in Peter's answer. As far as i understanding the "transition probability", it means that for a given bond, a flux of photon interact with this bond while only one photon is absorbed. And the "transition probability" described the ratio between this absorbed photon and the total number of photon in the flux (probability of a photon is absorbed by this bond). But, the given bond is only capable of absorbing one photon by jumping from ground state to excited state. Thus, if the measurement is complete within the time scale of exciting and relaxing process, the surface-enhanced phenomenon do not exit. Am I right?

Dear Lie Wu,

yes, I have found an answer. Absorbance and SEIRA are two concepts that do not match. To derive absorbance, the basic assumption is that E2 is just influenced by the light source and any local increase of E2 is a contradiction to the derivation. If you like you can read the details in the draft of my book project directly in the introduction: https://www.researchgate.net/project/Book-Project-Wave-Optics-in-Infrared-Spectroscopy (see the pdf file). The correct quantity that has to be invoked is the absorptance (1-R-T). However, these are the macroscopic quantities, and like in Thermodynamics you get no information about what happens on the level of bonds. I have to admit that I have had no time to see into measurements where the dynamics are probed. Generally things seem to be even more complicated, because one theory states that in a SEIRA spectrum it is actually not the molecular excitation that you see, but that of the coupled system plasmon-molecule. Therefore, I would naively assume that there are additional relaxation pathways through the interaction with the electrons in the metal...

As a chemist I may not follow all the details of your reasonings, but I may give my understanding of the problem of absorbance. Intensity is proportional to the square of the electric field but this not relevant to the Beer Lambert law nor to the A = -lg T relation. Intensity is proportional to the photons flux (number per second). The number of molecules on the photon trajectory is proportional to the optical path length and concentration (or pressure...). The number of interactions (intensity decrease) is proportional to the number of photons and the number of molecule, with a probability governed by a cross section or transition probability. This intensity decrease is valid for a infinitesimal small path dL, dI/I = -kc(dL ), and it gives the log relation when integrated from I0 to I over the total path L. Transmittance is (I/I0) and log(I0/I) is the absorbance of the sample. Note that in condensed phase studies (mostly dilute solutions in cuvettes) the reflection is automatically corrected by using a reference cuvette, and absorptance is not relevant, only transmittance or absorbance are used (note that absorptance is the complement to 1 of transmittance, but absorbance is on log scale of transmittance).

I am a chemist, too, but I had to learn the hard way that Beer-Lambert is often just a rough approximation and in many situations not compatible to the all-governing Maxwell-equations. The situation you mention that the reflection is automatically corrected by the reference cuvette is correct for solution measurements, but just to a certain degree and only in the UV/Vis range. For the thin cuvettes in infrared spectroscopy things are completely different due to the electric field standing wave effect: Article The Electric Field Standing Wave Effect in Infrared Transmis...

Article The electric field standing wave effect in infrared transfle...

If you want to understand the underlying approximations in the UV/Vis range, see Article Employing Theories Far beyond Their Limits-The Case of the (...