Azulene is the most famous example (S2-S0 transition) which does not obey Kasha's rule. There are few recent examples where emission is monitored from S2 state.

http://pubs.acs.org/doi/10.1021/jp9906785

http://pubs.acs.org/doi/abs/10.1021/acs.jpclett.5b00882

Vavilov rule says that the quantum yield is independent of excitation wavelength. An implication of Kasha's rule is that the emission wavelength is independent of excitation wavelength. A violation of this rule is red-edge excitation shift. This is observed in polar molecule in viscous media where the solvent relaxation is slow and is in the order of decay rate. This phenomenon is more common in biological systems.

http://pubs.acs.org/doi/abs/10.1021/bi00308a026

Here is an example in the field of QD

https://newscenter.lbl.gov/2011/07/01/breaking-kashas-rule/

Thank you very much dear Tuhin Khan. The last link you mentioned is very useful. However, I want to know if there is a QD which can change emission frequency between several states, not only two states?

Bashir Footuhi,

Carbon (organic) dots usually do not obey the Kasha rule. The reasons may be different, most often because of the presence of independent fluorophores (emmitters) in the nanoparticle. Such c-dots can be bi- or multimodal. As an example, see enclosed papers.

Best regards

Article Fluorescent carbon nanomaterials: "quantum dots" or nanoclusters?

Article Facile fabrication of luminescent organic dots by the thermo...

Article Molecular Fluorescence in Citric Acid Based Carbon Dots

@A. B. Shcherbakov

If the emission in carbon dots are due to presence of multiple fluorophore (which I believe is) can we call the excitation dependent emission a deviation from Kasha's rule? Because the emission is still from the lowest excited state (from the individual fluorophore). For example, if we have a mixture of rhodamine dye (ground state heterogeneity) we will observe an excitation dependent emission. Can we call this a violation of Kasha's rule?

Actually some references say that this is not a violation of Kasha's rule

"The observed ground–state heterogeneity leading to huge excitation dependent shifting of emission spectra of CND does not violate the classical Kasha–Vavilov rule"

http://pubs.acs.org/doi/10.1021/acs.jpclett.6b01791

"This interpretation contrasts with studies where fluorescence tunability was rather interpreted as a violation of Kasha’s rule due to an unusually slow relaxation of polar solvents around photoexcited CDs"

http://pubs.acs.org/doi/10.1021/acs.jpclett.6b01590

Research in carbon dots is still in it's early stage (regarding its origin of emission). The first reference ("Fluorescent carbon nanomaterials: "quantum dots" or nanoclusters?") actually the best work I have came across.

Tuhin Khan,

Yes, I agree, every individual fluorophore of the c-dot is subject to the Kasha rule. Although many researchers consider the c-dot as a single object (emitting unit) that does not obey the rule - it is probably wrong.

Dear A. B. Shcherbakov 

Thank you for your kind and in-detail response. It is very interesting to have such a multi-modal emission. But I have one more question regarding your suggestion:

1. Is the non-Kasha behavior related to individual C-based QDs? Or it is a cluster specification?

Dear Sirs

According to your conversations, now I need to understand what type of molecule or QD is not subject to Kasha rule, individually?

Dear Bashir Footuhi,

It seems to me the statement of Mary Dekaliuk (first cited paper) is quite right. The authors write that the C-dot is not ‘‘quantum dot’’ and are not even ‘‘dot’’: its fluorescence response is not collective and represents a composition of individual emitters. In the next cited paper, we studied one of the possible mechanisms for the formation of such a cluster by the joining of individual fluorophores into a polymer matrix of nanoparticle. Pay attention to Scheme 5 in the third marked paper: the multimodal emission of similar citrazinic-based dot is shown as a single electronic system on the Jablonski diagram. In any cases, You are absolutely right: from a practical point of view, it is very interesting to have such a multi-modal emission nanoobject – for example, for specific staining of desired objects in the system of plenty luminescent dyes, which obey the Kasha rule (see Figure 7 in the second cited paper).

Best reards