I have used GAMESS before for TDHF (RPA) / TDDFT level first hyperpolarizability ($\beta$) calculations. However, I cannot figure out how to calculate $\beta$ with MP2 and CCSD for comparative purposes. Do I have to use the FFIELD keyword for the finite field approach? If that is the case, what is the general workflow necessary to calculate the time-varying hyperpolarizability?

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1 Answer 1


As you guessed, to calculate $\beta$ within GAMESS for MP2 and CCSD, you would need to perform first- and second-order finite difference, respectively, as there is no coupled cluster gradient needed, so no true CCSD dipole moment.

However, there is a more fundamental problem: it is impossible to calculate time-dependent response using a finite field approach. Take the general form of 1-dimensional 2nd-order central difference, $$ f''(x) \approx \lim_{h \to 0} \frac{f(x+h)-2f(x)+f(x-h)}{h^2}, $$

which can directly translate to a diagonal element of the polarizability tensor formed from energies at finite electric fields:

$$ \alpha_{zz}(x) \approx \lim_{h \to 0} \frac{E(x+h)-2E(x)+E(x-h)}{h^2} $$

where $h$ is a finite electric dipole field of some strength applied along the $z$-direction. What is $x$? Is it as simple as converting the desired frequency $\omega$ to units of electric field strength? No; that still corresponds to a static field rather than a time-varying field. How are the different perturbation directions handled? $x$ must be set to 0.

From a molecular properties review by Jurgen Gauss, page 5:

(...) disadvantages of the numerical differentiation scheme are

a) that there is no straightforward extension to the computation of frequency-dependent properties (...)

Page 37:

While analytic derivative theory is sufficient for the theoretical treatment of time-independent (static) properties, the underlying theory needs to be extended for the calculation of time-dependent (dynamical) properties. In particular, the fact that there is -- unlike for the static case -- in the time-dependent case no well-defined energy explains why the simple derivative theory discussed so far is not applicable.

Page 39 (with some manipulation by me):

It can be shown that the linear, quadratic, etc. response functions $\left<\left<A;B\right>\right>_{w_{b}}$, $\left<\left<A;B,C\right>\right>_{w_{b},w_{c}}$, etc. and thus the frequency-dependent properties of interest can be determined as derivatives of the so-called time-averaged quasi energy

$$ Q(t) = \left<\tilde{\psi}\left|\left(H - i\frac{\partial}{\partial t}\right)\right|\tilde{\psi}\right> $$

with the phase-isolated wavefunction

$$ \tilde{\psi}(t) = e^{iF(t)} \psi(t) $$

where $F$ is a time-varying external field, and $\psi(t)$ is the usual wavefunction where MO coefficients and thus the density are explicitly time-dependent.

The only numerical (rather than analytic) method for calculating dynamic properties I am aware of is direct time-propagation of the wavefunction. If you want to calculate frequency-dependent CCSD hyperpolarizabilities, your best (free) bet is DALTON. Note that their coupled cluster module is closed-shell only.

  • $\begingroup$ Thank you for your response; it was very informative and I am grateful for the resource you have provided. I believe analytic gradients for CCSD and MP2 for the first hyperpolarizability have previously been published, is it just the case that they have not been implemented into GAMESS? Or am I mistaken about their derivation? $\endgroup$ Jul 26, 2017 at 13:35
  • $\begingroup$ The point is that they are not implemented in GAMESS. Analytic gradients for MP2 and CCSD are in most packages. Very few programs (DALTON, Gaussian) can calculate correlated hyperpolarizabilities. $\endgroup$ Jul 26, 2017 at 13:57
  • $\begingroup$ @ComputationalNovice If my answer was useful, please consider accepting it. $\endgroup$ Jul 28, 2017 at 19:47
  • $\begingroup$ NWChem has had CCSD quadratic response hyperpolarizabilities with non-zero frequencies since 2008. You can do RHF, UHF or ROHF, but I strongly recommend ROHF for open shells, for reasons discussed in the paper linked last. nwchemgit.github.io/TCE.html#input pubs.aip.org/aip/jcp/article-abstract/130/19/194108/296267/… pubs.aip.org/aip/jcp/article-abstract/128/22/224102/924805/… $\endgroup$ Jan 10 at 11:59
  • $\begingroup$ github.com/nwchemgit/nwchem/tree/… is an example input file, since it seems the user manual is not complete (my fault) $\endgroup$ Jan 10 at 13:55

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