Your question is incorrect.

1. What is "a polarizability of a bond"? The polarizability tensor is the coefficients, corresponding to the quadratic terms in the Taylor expansion of the energy of the molecule with respect to the electric field. So it is the second derivative of the energy with respect to the field at zero field strength. Or, if you wish, a first derivative of a molecule's dipole moment with respect to the field.

2. Bond is not a physical observable, so you are free to define the object which you call "a bond".

3. What is "polarizability in the presence and absence of external field for a molecule"? You want to see how the polarizability changes with field? Then you need to calculate at least the first hyperpolarizability.

Your question is incorrect.

1. What is "a polarizability of a bond"? The polarizability tensor is the coefficients, corresponding to the quadratic terms in the Taylor expansion of the energy of the molecule with respect to the electric field. So it is the second derivative of the energy with respect to the field at zero field strength. Or, if you wish, a first derivative of a molecule's dipole moment with respect to the field.

2. Bond is not a physical observable, so you are free to define the object which you call "a bond".

3. What is "polarizability in the presence and absence of external field for a molecule"? You want to see how the polarizability changes with field? Then you need to calculate at least the first hyperpolarizability.

We recently developed a technique to calculate distributed atomic polarizabilities and gave a definition of bond polariabilities. See

http://www.amercrystalassn.org/documents/2011%20Transactions/Macchi.pdf

I agree with Denis that the bond polarizability is not unambiguous first of all because bond is not a physical observable, then because the contribution of a bond to the molecular polarizability is not easy to define.

Thank you for reply to both Denis and Macchi. I understand what Denis has said but is it possible to capture the different polarizability of a sp bond compare to a sp2 bond in a molecule.

I have seen the link sent by Macchi. It is interesting that you have calculated the atomic polarizability in a mlecule but i have to spend little time to understand it.

Dear Professor Macchi,

thanks for your very instructive reference,

and also,

I think analysis of wfn files (from G09, Gamess or any other ....) by using QTAIM method to determine charge distribution and polarizability of each atom will be helpful,

Dear Denis Usvyat,

Kindly check with NBO analysis... It will generate polarisation coefficient of every individual bonds in a molecular system....

You can calculte the polarizability by taking the difference in electronegativity of both atoms the higher the difference the more polar the bond and the more ionic

Dear Hatem, Bond polarizability is NOT bond polarity. So it does not measure the bond dipole moment (or better the bond contribution to a molecular dipole moment). Instead, bond polarizability should measure the change of this bond dipole as a function of the applied electric field, which could somewhat inform us on how a bond polarity may change for example along a chemical reaction path.

Unfortunately, bond polarizability is not unequivocally defined in the literature.

Dear Dr Piero

Thank you very much for your notice but from the question any one should expect what i did think of it , but your analysis is really correct .Thank you again It is a pleasure for this discussion

Atom-atom, atom-bond, bond-bond "polarizabilities" can be defined within the Hueckel model, see e.g. Quantum Chemistry, 3rd ed. by J.P.Lowe and K.A.Peterson, pp. 406-408. The atom-atom polarizability lk shows how the electron density at atom l will change if the Coulomb integral at atom k changes etc.