The hyperpolarizability, a nonlinear-optical property of a molecule, is the second order electric susceptibility per unit volume.[1] The hyperpolarizability can be calculated using quantum chemical calculations developed in several software packages.[2][3][4] See nonlinear optics.

Definition and higher orders

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The linear electric polarizability   in isotropic media is defined as the ratio of the induced dipole moment   of an atom to the electric field   that produces this dipole moment.[5]

Therefore, the dipole moment is:

 

In an isotropic medium   is in the same direction as  , i.e.   is a scalar. In an anisotropic medium   and   can be in different directions and the polarisability is now a tensor.

The total density of induced polarization is the product of the number density of molecules multiplied by the dipole moment of each molecule, i.e.:

 

where   is the concentration,   is the vacuum permittivity, and   is the electric susceptibility.

In a nonlinear optical medium, the polarization density is written as a series expansion in powers of the applied electric field, and the coefficients are termed the non-linear susceptibility:

 

where the coefficients χ(n) are the n-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an n-th-order nonlinearity. In isotropic media   is zero for even n, and is a scalar for odd n. In general, χ(n) is an (n + 1)-th-rank tensor. It is natural to perform the same expansion for the non-linear molecular dipole moment:

 

i.e. the n-th-order susceptibility for an ensemble of molecules is simply related to the n-th-order hyperpolarizability for a single molecule by:

 

With this definition   is equal to   defined above for the linear polarizability. Often   is given the symbol   and   is given the symbol  . However, care is needed because some authors[6] take out the factor   from  , so that   and hence  , which is convenient because then the (hyper-)polarizability may be accurately called the (nonlinear-)susceptibility per molecule, but at the same time inconvenient because of the inconsistency with the usual linear polarisability definition above.

See also

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References

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  1. ^ "The Nonlinear Optics Home Page". www.nlosource.com. Retrieved 2019-12-29.
  2. ^ "GAMESS Input Documentation: TDHFX section". myweb.liu.edu. Retrieved 2019-12-29.
  3. ^ "Polar | Gaussian.com". gaussian.com. Retrieved 2019-12-29.
  4. ^ "The first calculation with DALTON". www.lct.jussieu.fr. Retrieved 2019-12-29.
  5. ^ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
  6. ^ Boyd, Robert. Nonlinear Optics (3rd ed.). Elsevier. ISBN 978-81-312-2292-8.
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