Answers of different complexity are possible. One of the simplest (but not exhaustive) is that in a single atom we have bandgap equal to the distance between ground state and first excited state, while in the bulk both levels are broadened. This broadening leads to narrowing of the bandgap. In a nanoparticle, broadening must be less than in the bulk, and narrower bandgap is expected. This explanation is simplified and does not take a lot of details into account....

Answers of different complexity are possible. One of the simplest (but not exhaustive) is that in a single atom we have bandgap equal to the distance between ground state and first excited state, while in the bulk both levels are broadened. This broadening leads to narrowing of the bandgap. In a nanoparticle, broadening must be less than in the bulk, and narrower bandgap is expected. This explanation is simplified and does not take a lot of details into account....

I like Alexander's answer.

Nanomaterials are nothing but group of few atoms, where the interaction between atoms is less as compared to atoms in bulk material. Thus the energy levels are similar to that of isolated atom. And thereby as the number of atoms are decreasing (i.e. size of nanomaterial decrease ) the energy levels are becoming closely similar to isolated atom.

BTW,  "This broadening leads to narrowing of the bandgap. In a nanoparticle, broadening must be less than in the bulk, and narrower bandgap is expected. " is this sentence is correct, or you mean BROAD bandgap is expected?

Yes, in nanoparticle the bangap is expected to be narrower than in single atom but broader then in the bulk

"In a nanoparticle, broadening must be less than in the bulk, and narrower bandgap is expected." (Aleks) 

I would say: a broader bandgap is expected 

When the particle size is decreased the band gap in between ground and excited state are increased normally due to the larger surface area occurred inside the nano particle. Also  that broader band gap increases the energy levels of the nanomaterials, step wise energy levels are easily used for the transaction.

I think a better answer to your questions is simply that you have more confinement in a nano particle.  Quantum wells and quantum dots also have higher optical bandgaps for the same reason.  This is a semi-classical answer and can be traced to simple quantum mechanics.  If you take a square well potential (with a barrier potential higher than the potential in the well), the ground state potential will be higher than the potential for the material in the well.  The smaller the well, the closer the ground state potential (of the bound state or states) will be to the ground state energy of the bulk barrier material (outside the well).  With a nanoparticle, the energy state for electrons outside the particle would normally be higher than the energy for electrons (or holes) inside the nanoparticle (otherwise the electrons would flow out of the nanoparticle).

As mentioned above, below a certain size nano materials behave more like single atoms than a continuous material. The electrons become more localized and energy gaps increase. This in turn increases the optical gap.  Also, there is a tendency for indirect gaps (which are not iptically active) to transform into direct gaps (optically active) as momentum dependence is reduced in the electronic band structure from localization. This creates different and usually larger optical gaps in indirect semiconductors - as in MoS2. Metallic systems are more complex as the optical properties are related to plasmons typically rather than a true energy gap opening except perhaps in the smallest of nanoparticles.

Dear Thomas Varghese,

This is a result of quantum confinement; when the size of materials goes to nanometer scale/ comparable to particle wave length, energy spectrum clearly turn to discrete and this result widened Eg, which is mostly size dependent.  some times also the band transition between valance band maximum (VBM) and conduction band minimum (CBM) changes from indirect to direct, and other fundamental properties also changes (electrical, Magnetic, Mechanical,...)         

The most important factor is described in Aleksandr's answer.  To get a quantitative answer, one can use the simple 1-D Huckel model of a chain of atoms with one orbital per site. If the atomic energy corresponding to that orbital is E0, and the Hamiltonian matrix element between neighbouring orbitals is V (i.e. V=, where |n> is the orbital corresponding to site n), then the energy Ek of the chain is Ek=E0-2|V|cos(πk/(N+1)), k=1,2,...,N. Note that V

I do agree with short and easy to follow answer by Aleksandr.

The simple explanation given by Aleksandre is easy to follow and to understand  the change in the band gap. You keep on reducing the size of a nano particle; that means you are reducing the number of atoms which are forming the nano-particle. However, the limit on the largest band gap in the atomic energy separation between the ground state and the first excited state and the lowest band gap will be for the bulk sample.

This blueshift behavior (The increase of band gap enerhy) can in principle be explained by the Moss–Burstein band filling effect ]. Based on the Moss–Burstein theory, nanoparticles are an n-type semiconductor, the Fermi level will be inside of the conduction band. Since electrons occupy the states below the Fermi level in the conduction band, so the absorption edge should shift to the higher energy or blue-shift.

You can also think about it, at least heuristically, form the uncertainty principle point of view that after all entitles all wave phenomena - Confinement to smaller volume means larger spreading in the k-space (momentum). If the dispersion is a monotonic function, as in free space, this also means larger energy gaps of whatever band structure a particle may posses.

I agree with aleksandr

Due to quantum confinement

Nano particles are larger than individual atoms and molecules but are smaller than bulk solid. Hence they obey neither absolute quantum chemistry nor laws of classical physics and have properties that differ markedly from those expected.

There are two major phenomenons that are responsible for these differences:

First is the high dispersity of nanocrystalline systems. As the size of a crystal is reduced, the number of atoms at the surface of the crystal compared to the number of atoms in the crystal itself, increases. & The second phenomenon occurs noticeably only in metals and semiconductors. It is called size quantisation and arises because the size of a nanoparticle is comparable to the de Broglie wavelength of its charge carriers (i.e. electrons and holes). Due to the spatial confinement of the charge carriers, the edge of the valance and conduction bands split into discrete, quantized, electronic levels. These electronic levels are similar to those in atoms and molecules.

The spacing of the electronic levels and the bandgap increases with decreasing particle size. This is because the electron hole pairs are now much closer together and the Coulombic interaction between them can no longer be neglected giving an overall higher kinetic energy.