There are six components in exact polarizability value (431.627 8.057 198.981 6.048 29.377 198.648) and each component represents αxx, αxy, αyy, αxz, αyz, αzz respectively.  For hyperpolarizability in the output you will get 10 HyperPolar components and each represents βxxx, βxxy, βxyy, βyyy, βxxz, βxyz, βyyz, βxzz, βyzz, βzzz respectively. In your example these components are not present.

hellow 

to calculate first hyperpolarizability under Gaussian you can use in the output for your calculation (you will get 10 HyperPolar components and each represents βxxx, βxxy, βxyy, βyyy, βxxz, βxyz, βyyz, βxzz, βyzz, βzzz respectively

that is: βx, βy and βz

To calculate (hyper)polarizability in Gaussian, you should use "polar" keyword rather than "freq" keyword. The outputted (hyper)polarizability information is somewhat complicated and difficult to understand. I suggest you use Multiwfn (freely availble at http://sobereva.com/multiwfn) to convert these information to a much more readable format.

For example, after boot up Multiwfn, input below commands:

polar.out  //A Gaussian output file generated with "polar" keyword

200

7  //Parse (hyper)polarizability information from Gaussian output file

1  //Assume that this is a DFT task

Then you will see information like below, which are very clear

Static polarizability:

XX=       21.386200

XY=       -0.565213

YY=       23.988300

XZ=        0.000000

YZ=        0.000000

ZZ=       20.782800

Isotropic average polarizability:       22.052433

Polarizability anisotropy (definition 1):        3.108620

Polarizability anisotropy (definition 2):        2.950445

Eigenvalues of polarizability tensor:     20.78280     21.26873     24.10577

Polarizability anisotropy (definition 3):        3.080004

Static first hyperpolarizability:

XXX=       18.742800

XXY=      -10.139100

XYY=       11.244000

YYY=      -39.824800

XXZ=        0.000000

XYZ=        0.000000

YYZ=        0.000000

XZZ=      -11.834200

YZZ=       -1.150830

ZZZ=        0.000000

Beta_X=       18.15260  Beta_Y=      -51.11473  Beta_Z=        0.00000

Magnitude of first hyperpolarizability:       54.242350

Projection of beta on dipole moment:      -32.754248

Beta ||     :      -19.652549

Beta ||(z)  :        0.000000

Beta _|_(z) :        0.000000

For more information, please check Section 3.200.7 of Multiwfn manual.

The procedure explained by Professor Lu is very simple, straightforward and much helpful. The "Multiwfn" code is  very user friendly and is able to perform too many tasks for the chemical as well as physical purposes.

Thank u all 

The calculation of 'beta' depending on the symmetry and in static electric field is done using the shown XXX, YYY, etc., values. Their description is in terms of permanent moment and variety of polarizabilities. The equations depends on the symmetry of the molecules/crystals.

The number of constants specifying the tensor 'beta' for C1 symmetry are 10. So, the data in the shown output are computed for C1 molecular symmetry. For Cs symmetry the number is 6; C2 - 4; C2h - 0; C2v - 3; D4h - 0; C3v - 3; etc.

When the computations by Gaussian is carried out using the frequency-dependent option, the interpretation of the data is in term of periodic electric field and of angular frequency omega. In this case the interactions molecular-field is described with modified polarizabilities which are function of the omega. The molecular multi-poles are given by the equations shown as attachment.

(For the static case, the corresponding equations are shown, here:

https://www.researchgate.net/post/How_to_calculate_the_polarisability_using_DFT)

This si towards the comments by Mr. Deb.

Hello everyone

How can I get second order hyper-polarizability factor (i.e. gamma) from Gaussian output?

Dear Prabhjot Kaur,

Simply use polar=gamma keyword, you will find summary of gamma at the end of output file. Note that this keyword is only available since later version of G09.

Thanks a lot Tian Lu.

Are frequency calculation required along with this keyword or simply I can compute it after optimization process?

Dear Prabhjot Kaur,

freq task is completely irrelvant to the gamma calculation. You simply need to optimize the geometry first, and then use polar=gamma. Of course, if you want to confirm whether the geometry is a minimum point, you can do freq task and then examine frquencies.

Hi everyone, thank you for the explanation. I have one question: results of hyperpolarizability comes in Debye-Ang**2, how do I convert to esu?

hello everyone my assignment is that how can we calculate beta vector using guassian 09 programme?

when i search the results display for second hyperpolarizability. these both words are used for same term?