# Polarizability and Hyperpolarizability Calculation

Units for the series expansion "charge - dipole - alpha - beta - gamma" are as follows:

Charge: positrons (Charge on a fluoride ion is "-1", on a sodium cation is "+1")

Dipole: Debye

Polarizability (alpha): Cubic atomic units and cubic Angstroms.  1 au3 = (0.529)3 Angstroms3

First hyperpolarizability (beta): ESU or atomic units (atomic units)5 and (Angstroms)5.  The description given by D. Kanis, M. Ratner and T. Marks, Chem. Rev. vol 94, pg 239 (1994), is as follows:

"The units for molecular hyperpolarizabilities can be somewhat complicated. Most contributions quote b in either atomic or esu units.  Dimensional analysis of the quadratic hyperpolarizability indicates that is possesses the units of dipole moment3/energy2, or in atomic units:

b = e3ao3/Eh2

where ao is the Bohr radius and Eh is the ionization energy of hydrogen (a Hartree). Substituting Eh (Hartree) =e2/ao

b = e3ao3/(e2/ao)2 = ao5/e

in esu units; the second-order response is given as cm5statcoulomb-1 which is identical to cm5esu-1.  Sometimes these units are simply referred to as esu.  The conversion factor between atomic units and esu is as follows:

1 atomic unit = (ao5/e) (0.529 x 10-8cm/ao)5(e/4.803 x 10-10esu) or

1 atomic unit =8.641 x 10-33 cm5.esu-1

in SI units, 1 atomic unit =3.206 x 10-53 C3m3J-2 and 1  = 2.693 x 1020 cm5esu-1."

(In earlier MOPACs, the quantity "4.803" was not used, therefore until December 2003 the value in cm5esu-1 was too large by a factor of 4.803).

Second hyperpolarizability (gamma): Esu.