Hyperpolarizability Explained

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

\alpha in isotropic media is defined as the ratio of the induced dipole moment p of an atom to the electric field E that produces this dipole moment.^[5]

Therefore, the dipole moment is:

p = \alpha E

In an isotropic medium

p is in the same direction as E , i.e. \alpha is a scalar. In an anisotropic medium p and E 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.:

P = \rho p = \rho \alpha E = \varepsilon _{0} \chi E ,

where

\rho is the concentration, \varepsilon _{0} is the vacuum permittivity, and \chi 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:P (t) = \varepsilon _{0} \left( \chi ^{(1)} E (t) + \chi ^{(2)} E ^{2(t)} + \chi ^{(3)} E ^{3(t)} + \ldots \right),

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

\chi ^{(n)} 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:p (t) = \alpha ^{(1)} E (t) + \alpha ^{(2)} E ^{2(t)} + \alpha ^{(3)} E ^{3(t)} + \ldots ,

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:

\alpha ^{(n)} = (\varepsilon _{0})/(\rho) \chi ^{(n)} .

With this definition

\alpha ^{(1)} is equal to \alpha defined above for the linear polarizability. Often \alpha ^{(2)} is given the symbol \beta and \alpha ^{(3)} is given the symbol \gamma . However, care is needed because some authors^[6] take out the factor \varepsilon _{0} from \alpha ^{(n)} , so that p = \varepsilon _{0\sum} ^{(n)} ^{n\alpha} E ^{n} and hence \alpha ^{(n)} = \chi ^{(n)} /\rho , 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

• Intrinsic hyperpolarizability

External links

• The Nonlinear Optics Web Site

Notes and References

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