# Blog Archives

## Polarizability and Hyperpolarizability in Gaussian

Calculating both Polarizability and the Hyperpolarizability in Gaussian is actually very easy and straightforward. However, interpreting the results requires a deeper understanding of the underlying physics of such phenomena. Herein I will try to describe the most common procedures for calculating both quantities in Gaussian09 and the way to interpret the results; if possible I will also try to address some of the most usual problems associated with their calculation.

**Introduction**

The dipole moment of a molecule changes when is placed under a static electric field, and this change can be calculated as

*pe = pe,0 + α:E + (1/2) β:EE + …* (**1**)

where pe,0 is the dipole moment in the absence of an electric field; α is a second rank tensor called the polarizability tensor and β is the first in an infinite series of dipole hiperpolarizabilities. The molecular potential energy changes as well with the influence of an external field in the following way

*U = U0 – pe.E – (1/2) α:EE – (1/6) β:EEE –* … (**2**)

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Route Section Keyword: **Polar**

This keyword requests calculation of the polarizability and, if available, hyperpolarizability for the molecule under study. This keyword is both available for DFT and HF methods. Hyperpolarizabilities are NOT available for methods that lack analytic derivatives, for example CCSD(T), QCISD, MP4 and other post Hartree-Fock methods.

Frequency dependent polarizabilities may be calculated by including CPHF=RdFreq in the route section and then specifying the frequency (**expressed in Hartrees**!!!) to which the calculation should be performed, after the molecule specification preceded by a blank line. Example:

#HF/6-31G(d) Polar CPHF=RdFreq Title Section Charge Multiplicity Molecular coordinates==blank line==0.15

In this example 0.15 is the frequency in Hartrees to which the calculation is to be performed. By default the output file will also include the static calculation, that is, ω = 0.0. Below you can find an example of the output when the CPHF=RdFreq is employed (taken from Gaussian’s website) Notice that the second section is performed at ω = 0.1 Ha

SCF Polarizability for W= 0.000000: 1 2 3 1 0.482729D+01 2 0.000000D+00 0.112001D+02 3 0.000000D+00 0.000000D+00 0.165696D+02 Isotropic polarizability for W= 0.000000 10.87 Bohr**3. SCF Polarizability for W= 0.100000: 1 2 3 1 0.491893D+01 2 0.000000D+00 0.115663D+02 3 0.000000D+00 0.000000D+00 0.171826D+02 Isotropic polarizability for W= 0.100000 11.22 Bohr**3.

You may have noticed now that the polarizabilities are expressed in volume units (Bohr^3) and the reason is the following:

Consider the simplest case of an atom with nuclear charge *Q*, radius *r*, and subjected to an electric field, *E*, which creates a force *QE*, and displaces the nucleus by a distance* d*. According to Gauss’ law this latter force is given by:

(dQ^2)/(4πεr^3) = QE (Hey! WordPress! I could really use an equation editor in here!)

if the polarizability is defined by Qd/E then we can rearrange the previous equation and yield

α = 4πεr^3 which in atomic units yields volume units, r^3, since 4πε = 1. This is why polarizabilities are usually referred to as ‘polarizability volumes’.

******THIS POST IS STILL IN PROGRESS. WILL COMPLETE IT IN SHORT. SORRY FOR ANY INCONVENIENCE******

The use of double zeta quality basis sets is paramount but it also makes these calculations more time consuming. Polarization functions on the basis set functions are a requirement for good results.

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