I was trying to calculate the hyperpolarizability $\beta$ of a cluster of two water molecules in Gaussian16 using the route card:

  1. #t cam-b3lyp/daug-cc-pvtz Polar nosymm scf=tight int=grid=superfine

  2. #t cam-b3lyp/daug-cc-pvtz Polar=doubleNumer nosymm scf=tight int=grid=superfine

The files which can be used for testing are attached in these links [1], [2].

In case (1) I got a value of 83.44 au for the $\beta_{xxx}$. For (2) I got a value of 23.66 au for $\beta_{xxx}$. I also altered the distance between two water molecules by $\pm$ 0.5 angstroms and I got an average $\beta_{xxx}$ value of 23.44 au .

I am not sure about the capabilities of other electronic structure codes (I know ORCA does not have an option to calculate hyperpolarizability; Dalton Turbomole does, but I don't have access to it), but is this error reproducible in other codes or this is just a problem of G16?

From Gaussian16 support, I got the following response:

We have reviewed this further and it is a numerical error which is caused by near linear dependencies in the basis set which the analytic second derivative for beta is not handling well. You can see this if you use aug-cc-pvtz in place of daug-cc-pvtz.

It is also possible to get a better analytic value using Polar=(Cubic,Fourpoint) which is more numerically stable but also significantly more time consuming computationally. With that method the results are:

  1. #p cam-b3lyp/daug-cc-pvtz Polar=(Cubic,fourpoint) nosymm scf=conver=10 int=grid=superfine


  1. #p cam-b3lyp/daug-cc-pvtz Polar=DoubleNumer nosymm scf=Conver=10 int=grid=superfine


where you see the comparison with numerical is about 5 significant figures.

  • $\begingroup$ Okay, it's looking better now. $\endgroup$
    – Cody Aldaz
    Jul 30, 2020 at 14:21
  • 1
    $\begingroup$ +1. Welcome to the site, thanks for asking here and we hope to see much more of you !!! I have one question: Why is it that you don't have access to Dalton? Dalton is free and you can download it from Git. $\endgroup$ Jul 30, 2020 at 14:23
  • $\begingroup$ @NikeDattani My apologies. I wanted to write Turbomole instead of Dalton. Edited $\endgroup$
    – mykd
    Jul 30, 2020 at 15:39
  • 2
    $\begingroup$ Dalton should have beta as well. $\endgroup$
    – MSwart
    Aug 1, 2020 at 7:19

1 Answer 1


Since the basis set seems to play a big role, from my paper (1999):

"A lot of effort has been put into constructing basis sets specially designed for accurate calculations of the polarizability [18–22]."

[18] H.-J. Werner, W. Meyer, Mol. Phys. 31 (1976) 855–872.
[19] A.J. Sadlej, Theor. Chim. Acta 81 (1991) 329.
[20] A.J. Sadlej, Theor. Chim. Acta 81 (1991) 45.
[21] A.J. Sadlej, M. Urban, J. Mol. Struct. (THEOCHEM) 80 (1991) 234.
[22] A.J. Sadlej, Theor. Chim. Acta 79 (1991) 123.

Have you tried the Sadlej basis sets? Or the Frank Jensen ones?

We used Jensen's pcS-n for NMR properties, and I know that for polarizability it is not the nuclear region but the extended region that needs to be described well, but I was just wondering how good they are. At least in our study they converged much better with basis-set cardinal number than Dunning's ones.

  • $\begingroup$ Thanks for your response! We initially tested the beta values for Dunning's basis aug-cc-pVXZ (X=2-6) and realized that an additional diffused function was required for the basis for the beta values to be close to the MP2 benchmark. Also Sadlej basis and aug-cc-pvdz produces data at a similar accuracy. Also from Jensen's paper ( it looks like that the comparative standard was cc-pVXZ without any augmentations or diffusion. $\endgroup$
    – mykd
    Aug 1, 2020 at 10:19
  • $\begingroup$ We were also concerned by the fact that for a second order property based on the electron density, beta values should be highly sensitive to the quality of the electronic wavefunction as compared to dipole moment or polarizability. That's why we were looking for the largest feasible basis which could provide beta value with high accuracy. $\endgroup$
    – mykd
    Aug 1, 2020 at 10:22

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