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)

.

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’.

Thank u 4r  this ...

Mr. Bhattacharje,

Let us first start with the fact that you do not need frequency-dependent computations by Gaussian in order to obtain the permanent properties of the molecular systems in external electric field, just because of there are valid equations (1)-(4) (Attachment). This is towards the comments by Mr. Gour.

Using the keyword 'Polar' you have obtained all there N-pole moments (at the end for the output-file).

Example of an output:

Electronic spatial extent (au): = 17.7841

Charge= 0.0000 electrons

Dipole moment (field-independent basis, Debye):

X= 0.0000 Y= -1.6455 Z= 0.0001 Tot= 1.6455

Quadrupole moment (field-independent basis, Debye-Ang):

XX= -4.5174 YY= -5.4344 ZZ= -6.0949

XY= 0.0000 XZ= 0.0000 YZ= 0.0000

Traceless Quadrupole moment (field-independent basis, Debye-Ang):

XX= 0.8315 YY= -0.0855 ZZ= -0.7460

XY= 0.0000 XZ= 0.0000 YZ= 0.0000

Octapole moment (field-independent basis, Debye-Ang**2):

XXX= 0.0000 YYY= -0.1213 ZZZ= -0.0001 XYY= 0.0000

XXY= -0.4379 XXZ= -0.0001 XZZ= 0.0000 YZZ= 0.0105

YYZ= -0.0001 XYZ= 0.0000

Hexadecapole moment (field-independent basis, Debye-Ang**3):

XXXX= -6.6056 YYYY= -4.9187 ZZZZ= -3.2403 XXXY= 0.0000

XXXZ= 0.0000 YYYX= 0.0000 YYYZ= 0.0000 ZZZX= 0.0000

ZZZY= 0.0000 XXYY= -1.7372 XXZZ= -1.7664 YYZZ= -1.3919

XXYZ= 0.0000 YYXZ= 0.0000 ZZXY= 0.0000

To compute dipole polarizability (equation 5) there are needed to take into consideration the corresponding Xii components among all N-pole moments (The components in the Gaussian's output-file are given in Cartesian coordinates). Depending on the symmetry of your molecule, equation (5) can be simplified, yielding to equation (6), where you should just substitute azz and axx with the values of the quadrupole moments XX and ZZ from the field independent basis.

Second, the polarizability tensor, in general, consists of two components: electronic and vibronic parts. Furthermore the latter part consists of real and imaginary components. In this context, the data in your output file represent the vibronic contribution to the dipole polarizability.

Thank u mam 4r ur valuable rply ..