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.


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==

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’.




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.

As usual, please rate/comment/share this post if you found it useful or if you think someone else might find it useful. Thanks for reading!


About joaquinbarroso

Computational and theoretical chemist in his early forties, in love with life, science, baseball, and literature. Science literacy makes us responsible citizens, it is therefore a scientific duty to talk, write, and engage with the general public; as Feynman said, if you find science boring, your learning from the wrong teacher. "Make like a molecule and react!"

Posted on July 27, 2012, in Computational Chemistry, Gaussian, Models, Quantum Mechanics, Science, Theoretical Chemistry, White papers and tagged , , , , , , , , . Bookmark the permalink. 33 Comments.

  1. Dr. Partha Sarathi Sengupta

    higly useful. Thanks

  2. Hello,

    could you explain the calculation of dynamic polarizabilities using 6-10 imaginary frequencies to be able to calculate the casimir-polder integral and to obtain a dispersion coefficient?

    • Hi,

      Could you be more specific please? I don’t think you can use Gaussian to do that, not even to obtain some of the parameters but it does seem interesting. If you have more information about what you are actually trying to accomplish, please share it.

      Have a nice day!

      • COuld you pleas , explain which value from the output file , form polarizability we take there are several values , like 10,87 , 11,22
        if i want to compute the polarizability for a molecule how i can get it from output file

  3. Dear Dr. Barroso,

    I tried the POLAR keyword in my system’s calculation, in order to to better understand the content of this post. In the output I got the following:


    According to the post, I guess those values for HyperPolar are the ones that you describe as “(…) an infinite series of dipole hyperpolarizabilities”. Would you mind to explain me the meaning of those values? Are they the components of a tensor or the beta, gamma, delta, etc terms of the expresion for the change in the dipole moment?

    I guess that for “dipole” those three values correspond to x,y & z values, but I am a bit confused with “Polar” and “HyperPolar”.

    Thanks a lot.

    Saludos cordiales

  4. Nice post. 1) Is there a way to visualize the anisotropy and the higher order polarizabilities since they are volumes? 2) What are the basis set effects/requirements for a good polarizability calculation using the PBC code, or is that not possible for periodic systems? Thank you!

    • Dear Sir,I hope to get a detail calculation for the final total value for each of hyperpolarizability and polarizibility of ,for example, H2O molecule
      Thank you very much.

      • Dear Sir,I hope to get a detail calculation for the final total value for each of hyperpolarizability and polarizibility of ,for example, H2O molecule
        Thank you very much.

  5. Hello muddy joaquin teacher wants to know that command line can determine the linear and nonlinear optical properties in the gaussian 09. and where in the output file show me this information thanks .. Doctor

  6. Hello, Dr. Joaquin, How can I calculate the imaginary part of Second Hyperpolarizibility using G09? Thanks in Advance.

  7. how to calculate the second hyperpolarizability using G09? how do I use keyword? I dont know which value I have to take from the output to get the second hyperpolarizability? Any suggestions will be helpful

    • Dear Kasa

      If you are using G09 D.01 you have to use polar=gamma. The output will put a sign pointing to these gamma values and I think it says also “second hyperpolarizabilities”, they are printed along and perpendicular to the dipole moment with their corresponding beta components.

      I hope this helps

      • Hi
        thanks for your answer
        But it did not work keywords and syntax error encountered
        #cam-b3lyp/6-31g* polar=gamma test
        QPErr — A syntax error was detected in the input line.
        -b3lyp/6-31g* polar=gamma test

        Last state=”Pol1″
        TCursr=57736 LCursr= 24
        Error termination via Lnk1e in d:\g09\l1.exe at Fri Oct 24 16:46:52 2014.
        Job cpu time: 0 days 0 hours 0 minutes 0.0 seconds.
        File lengths (MBytes): RWF= 1 Int= 0 D2E= 0 Chk= 1

      • The problem is your basis set. You cant use a * with 6-31g . Change it to 6-31g(d) and it should eork now unless you have atoms bigger than Kr for which this basis set isnt defined.

        I hope this helps

      • Dear Casa. The problem is in your bases set. Check it and re-define it.

  8. Dear Joaquin
    how to conversion Debye-Ang**3 to au?
    thanks alot

  9. Dear Joaquin,

    Currently I am using gaussian 09 to compute the polarizability of a water dimer system. The input keywords those I used for the calculation are following below

    # polar pop=chelpg mp2/aug-cc-pvdz density=current

    Water dimer 1

    0 1
    O -0.185814 -1.174947 0.766260
    H -0.128551 -0.898436 1.680861
    H -0.058278 -0.370255 0.263828
    O 0.174705 1.105000 -0.724443
    H -0.565084 1.313496 -1.294946
    H 0.928218 1.065299 -1.313403

    The input says that it is single point energy calculation at MP2 level, calculates atomic charge using CHELPG scheme and it also calculates the dipole moment, polarizability and higher order moment using the MP2 level electronic density.

    I got the outputs of those numbers from this calculation. The important section of this output is following below

    Sum of APT charges= -0.63408
    Electronic spatial extent (au): = 9203.7415
    Charge= 0.0000 electrons
    Dipole moment (field-independent basis, Debye):
    X= 2.3376 Y= 0.0000 Z= 0.1359 Tot= 2.3415
    Quadrupole moment (field-independent basis, Debye-Ang):
    XX= -8.6863 YY= -12.4505 ZZ= -13.0426
    XY= 0.0002 XZ= -34.5104 YZ= 0.0000
    Traceless Quadrupole moment (field-independent basis, Debye-Ang):
    XX= 2.7068 YY= -1.0574 ZZ= -1.6494
    XY= 0.0002 XZ= -34.5104 YZ= 0.0000
    Octapole moment (field-independent basis, Debye-Ang**2):
    XXX= 831.5434 YYY= 0.0000 ZZZ= 0.9460 XYY= 39.2300
    XXY= -0.0003 XXZ= 23.6280 XZZ= -14.7249 YZZ= 0.0000
    YYZ= -0.7517 XYZ= 0.0004
    Hexadecapole moment (field-independent basis, Debye-Ang**3):
    XXXX= -8717.5986 YYYY= -14.6198 ZZZZ= -15.4568 XXXY= 0.0345
    XXXZ= -4866.7421 YYYX= 0.0000 YYYZ= 0.0000 ZZZX= -27.0710
    ZZZY= 0.0000 XXYY= -1586.7247 XXZZ= -1665.2841 YYZZ= -5.2021
    XXYZ= 0.0004 YYXZ= -13.6156 ZZXY= -0.0001
    N-N= 2.072555395512D+01 E-N=-4.020567051105D+02 KE= 1.520892110237D+02
    Exact polarizability: 18.410 0.000 18.514 -0.524 0.000 18.189
    Approx polarizability: 13.876 0.000 13.654 -0.898 0.000 13.732
    Breneman (CHELPG) radii used.
    Generate Potential Derived Charges using the Breneman model, NDens= 1.
    Grid spacing= 0.300 Box extension= 2.800
    NStep X,Y,Z= 98 25 24 Total possible points= 58800
    Number of Points to Fit= 8009

    Now, my question is what are Exact and Approx polarizabilities? Are these three numbers correspond polarizabilities along three principal directions. How are those numbers calculated. What are units of Exact and Approx polarizabilities. Please reply me.


  10. Dear Sir,
    How calculate second order hyper polarizabilities of molecule from gaussian09 output.
    i was used this key word b3lyp/6-311++g(d,p) Polar=(DCSHG,Cubic) for calculation.

    Yours faithfully,

  11. Using Raman keyword we could get vibrational contribution of polarizability. However is there any reference of this calculation. I think g09 use double harmonic approximation to compute this. Anharmonic terms are not included. Any reference would make it clear

  12. How could I obtained polarizability and hyperpolarizability as numerical derivative of energy or dipole moment wrt applied electric field.HF DFT are implemented for analytical calculation.

  13. How calculate quadrupole moment in Gaussian 09?

  14. Thank you in advance for any help you can provide,
    Is it possible to calculate second hyperpolarizability values from the crystallographic data (cif file) ? Otherwise, it is possible to show me the method of gait of calculated from software (Gaussian 03W) that I have the chemical formula and the crystallographic data.
    Thank you for your cooperation.

  15. Dear Dr Barroso,

    Thanks for the info. What is the best type of first order hyperpolarizability to report? I’ve seen many papers talk about average or magnitude/total, but none specifically differentiates between them.

    Also, am I correct with the following descriptions of the beta ‘syntax’:

    Beta(0;0,0) – static first order hyperpolarizability
    Beta(-w;w,0) – frequency dependent electro-optic Pockels hyperpolarizability
    Beta(-2w;w,w) – frequency dependent single harmonic generation (SHG)

    I have pasted an excerpt of the input route line and from the output for clarity (dipole direction) below, sorry for the long message. I understand it is common to use the zzz, zxx and zyy values below due to them having the greatest contributions.

    #n B3LYP/6-311++G(d,p) POLAR=DCSHG int=UltraFine

    (Geom spec)

    0.0428 (Ha)

    First dipole hyperpolarizability, Beta (dipole orientation).
    ||, _|_ parallel and perpendicular components, (z) with respect to z axis,
    vector components x,y,z. Values do not include the 1/n! factor of 1/2.
    (esu units = statvolt**-1 cm**4 , SI units = C**3 m**3 J**-2)


    (au) (10**-30 esu) (10**-50 SI)
    || (z) -0.686327D+04 -0.592933D+02 -0.220061D+02
    _|_(z) -0.228776D+04 -0.197644D+02 -0.733537D+01
    x 0.000000D+00 0.000000D+00 0.000000D+00
    y 0.000000D+00 0.000000D+00 0.000000D+00
    z -0.343163D+05 -0.296466D+03 -0.110031D+03
    || 0.686327D+04 0.592933D+02 0.220061D+02
    xxx 0.000000D+00 0.000000D+00 0.000000D+00
    xxy 0.000000D+00 0.000000D+00 0.000000D+00
    yxy 0.000000D+00 0.000000D+00 0.000000D+00
    yyy 0.000000D+00 0.000000D+00 0.000000D+00
    xxz 0.224652D+01 0.194082D-01 0.720317D-02
    yxz 0.000000D+00 0.000000D+00 0.000000D+00
    yyz -0.102060D+05 -0.881722D+02 -0.327243D+02
    zxz 0.000000D+00 0.000000D+00 0.000000D+00
    zyz 0.000000D+00 0.000000D+00 0.000000D+00
    zzz -0.123498D+04 -0.106693D+02 -0.395979D+01

    Beta(-w;w,0) w= 1064.6nm:

    (au) (10**-30 esu) (10**-50 SI)
    || (z) -0.141076D+07 -0.121879D+05 -0.452342D+04
    _|_(z) -0.259084D+07 -0.223828D+05 -0.830716D+04
    x 0.000000D+00 0.000000D+00 0.000000D+00
    y 0.000000D+00 0.000000D+00 0.000000D+00
    z -0.705382D+07 -0.609396D+05 -0.226171D+05
    || 0.141076D+07 0.121879D+05 0.452342D+04
    xxx 0.000000D+00 0.000000D+00 0.000000D+00
    yxx 0.000000D+00 0.000000D+00 0.000000D+00
    yyx 0.000000D+00 0.000000D+00 0.000000D+00
    zxx 0.246027D+02 0.212548D+00 0.788852D-01
    zyx 0.000000D+00 0.000000D+00 0.000000D+00
    zzx 0.000000D+00 0.000000D+00 0.000000D+00
    xxy 0.000000D+00 0.000000D+00 0.000000D+00
    yxy 0.000000D+00 0.000000D+00 0.000000D+00
    yyy 0.000000D+00 0.000000D+00 0.000000D+00
    zxy 0.000000D+00 0.000000D+00 0.000000D+00
    zyy -0.228705D+06 -0.197583D+04 -0.733311D+03
    zzy 0.000000D+00 0.000000D+00 0.000000D+00
    xxz 0.188076D+02 0.162483D+00 0.603040D-01
    yxz 0.000000D+00 0.000000D+00 0.000000D+00
    yyz -0.659044D+07 -0.569363D+05 -0.211313D+05
    zxz 0.000000D+00 0.000000D+00 0.000000D+00
    zyz 0.000000D+00 0.000000D+00 0.000000D+00
    zzz -0.201391D+04 -0.173986D+02 -0.645733D+01

    Beta(-2w;w,w) w= 1064.6nm:

    (au) (10**-30 esu) (10**-50 SI)
    || (z) -0.818286D+06 -0.706935D+04 -0.262372D+04
    _|_(z) -0.176355D+07 -0.152357D+05 -0.565457D+04
    x 0.000000D+00 0.000000D+00 0.000000D+00
    y 0.000000D+00 0.000000D+00 0.000000D+00
    z -0.409143D+07 -0.353468D+05 -0.131186D+05
    || 0.818286D+06 0.706935D+04 0.262372D+04
    xxx 0.000000D+00 0.000000D+00 0.000000D+00
    yxx 0.000000D+00 0.000000D+00 0.000000D+00
    zxx 0.138515D+02 0.119667D+00 0.444130D-01
    xyx 0.000000D+00 0.000000D+00 0.000000D+00
    yyx 0.000000D+00 0.000000D+00 0.000000D+00
    zyx 0.000000D+00 0.000000D+00 0.000000D+00

    Best Regards,

  16. Hi,

    We can not calculate hyperpolarizabilities using Post-HF methods. Is it the same with DFT also? I was trying to calculate hyperpolarizabilities with B3LYP using Gaussian 09. But I could not get the values for hyperpolarizabilities.

    • Hi, yes you can !

      You have to use a line like this :

      #P B3LYP 6-31G* polar=enonly density=current

      The #P keyword to print the Beta tensor 10 values (assuming Kleinman symmetry for the static Betas) at the end of the job output file on the archive session.

      Polar calculations in Gaussian seem to have good linear parallel performance increase up to 3 processors so you won’t gain much by using 4 real cores on Xeon or i7 processors.

      Hope this helps


  17. Hi,
    How to calculate the polarizability & hyperpolarizability in presence of electric field ?

  18. What is the default applied electric field in Gaussian 09 when we use the “POLAR” keyword.

  19. Hi,

    The POLAR calculations are done applying finite fields along Cartesian vectors with default electric fields : w=0 and w=0.1 atomic units
    = 0.514220652×10ˆ11 volts / meter (kilogram . meters / ampere . second^3)
    = 5 volts / Angstrom.

    Not sure how changing this value will impact in the static results. I once tried with w=0.2 au and didn’t observe any meaningful differences for a set of molecules the size of Disperse Red 1

  20. Is it possible to get second hyperpolarizability (gamma) using the keyword polar=(dcshg, cubic) in the Gaussian program? If so please tell me where can I find the results in the output file. If not, please tell me how to calculate second hyperpolarizability (gamma) using Gaussian. And also please suggest me how to read the output file for these results. Any help would be grateful.

  21. Hi,
    To evaluate vibrational contribution to NLO susceptibilities, I have done the computation for molecule using Guassian-09 software, using the following keyword
    “#p B3LYP/6-311++G(d,p) freq=raman scrf=(solvent=chloroform)”.
    Plz suggest me how to extract the polarizability and hyperpolarizability values from the corresponding output file.
    My mail id: erandeyogesh@gmail.com

  22. Good morning Dear Sir
    Sir would you please explain me the finite field static beta and gamma calculations in gaussian. I also would like to calculate nlo properties as in the following paper. https://pubs.acs.org/doi/10.1021/jp046322x

  23. Hola, ¿existe alguna forma de calcular el tensor de la polarizabilidad para varias frecuencias?

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