I'd like to be able to recover the maximum value that any of the basis functions can take, as well as their maximum derivative. How can I do this?

For context, I need it because I want to calculate the values of the variable listed in Eq. 66 from the paper by Babbush et al. [1].


  1. Babbush, R.; Berry, D. W.; Kivlichan, I. D.; Wei, A. Y.; Love, P. J.; Aspuru-Guzik, A. Exponentially More Precise Quantum Simulation of Fermions in Second Quantization. New J. Phys. 2016, 18 (3), 033032. DOI: 10.1088/1367-2630/18/3/033032. (Open Access)
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    $\begingroup$ +1. Welcome to the site! In the AO basis it is probably not too hard to figure out $\phi_{\textrm{max}}$ using the exponents of the Gaussians, but you need the software to do the AO to MO transformation, then you can figure out what you want based on the LCAO coefficients. You might have to modify the code of PySCF to print out what you want, since these are not typical values that interest most people. Codes often print out the LCAO coefficients, integrals, SCF energies, etc. but not the maximum value that a spin-orbital can obtain. $\endgroup$ – Nike Dattani Sep 16 '20 at 0:32
  • $\begingroup$ Thanks @NikeDattani for the warm welcome. I'm quite new to PySCF and Psi4, so I am unsure what should I be looking for in this case. I've seen for instance that there files such as github.com/sunqm/pyscf/blob/master/pyscf/gto/basis/cc-pvdz.dat but these indicate the energies, not the coefficients or exponents. Do you know where can I look for? $\endgroup$ – Pablo Sep 17 '20 at 7:41

The two-electron integrals $(\alpha \beta | \gamma \delta )$ are typically evaluated in terms of the atomic orbital (AO) basis functions, since in a Gaussian basis set these can be done analytically. Figuring out the maximal value of the AO and its derivatives is trivial, since you know the analytic form of the function, which is highly monotonic.

However, for calculations including electron correlation i.e. quantum computing applications and post-Hartree-Fock methods you actually need the integrals in the molecular orbital (MO) basis, which are obtained from the AO integrals with the transform $(ij|kl)=\sum_{\alpha\beta\gamma\delta} C_{\alpha i} C_{\beta j} C_{\gamma k} C_{\delta l} (\alpha \beta | \gamma \delta )$, where ${\bf C}$ are the MO coefficients. (In practice, the transform is carried out as four consecutive one-index transforms that can be expressed as matrix-matrix products, since this scales as $N^5$ instead of $N^8$.)

Now, the question is: how to determine the maximal value of a MO $\phi_i({\bf r}) = \sum_\alpha C_{\alpha i} \chi_{\alpha} ({\bf r})$ and its derivative, where $\chi_\alpha$ is the AO. This is a global optimization problem, for which there is no simple solution. You essentially need to sample over all of ${\bf r}$ to find where the minima lie, and you still don't have a guarantee of finding the real minimum.

There are clever schemes, though, to directly estimate the value of the AO two-electron integral; this allows avoiding explicit computation of AO small integrals.

  • $\begingroup$ Actually you're right, Susy. I was thinking that the $\varphi_{max}$ was referring to AO but it is actually MO. I'm ok with only achieving a local maximum, since I am more interested in the order of magnitude. Getting more into depth, I've consulted (Atomic orbital basis sets) and I believe I understand that $C_{\alpha,i}$ are determined via Hartree-Fock (the basis change matrix). Therefore my question would be how to get that matrix in PySCF or similar, and if the numbers in ie basissetexchange.org/basis/cc-pvdz/format/psi4/… mean the contraction coefficients. $\endgroup$ – Pablo Sep 18 '20 at 14:54
  • $\begingroup$ For instance, in the first box in psicode.org/psi4manual/1.3.2/… what are those numbers exactly? From [F. Jensen, Atomic orbital basis sets], I believe they are the coefficients $d_{ij}$ from its equation 3, that says $\kappa_j = \sum_i d_{ij} \chi_i$, and $\phi = \sum_j c_j \kappa_j$. Is that right? On the other hand I assume that via Hartree-Fock one is determining the $c_j$ coefficients from the second equation. $\endgroup$ – Pablo Sep 18 '20 at 15:01
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    $\begingroup$ Hello, yes, the contracted GTO basis functions have a set of coefficients defined by the basis set; the molecular orbitals are then expanded in terms of the contracted GTOs. However, you can't really estimate the global maximum with a local maximum! $\endgroup$ – Susi Lehtola Sep 20 '20 at 11:43

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