Coefficients of the density fitting

1

Hello,

I have been trying to do density fittings. I would like to obtain the coefficients of the expansion of the density in the auxiliary basis.

Is there a way to output the coefficients when running a Psi4 calculation with scf_type=df?

Thanks in advance,
Liss

2

To my knowledge, there is no way to print more details about the density fitting process currently in the code. The coefficients you want are:
\sum_{D} (D | P )^{-1/2}
where D and P are auxiliary basis functions, right ?

I think it would not be difficult to modify the code to print them, if it is for a SCF procedure. I can tell you where to look.

3

Thank you for your reply!

I actually want the coefficients
D_mna = \Sum_a (mn | a)[J^{-1}]_ab
where J is the repulsion integral between the auxiliary basis functions |a> and |b>, |m> and |n> are from the primary basis.
But now thinking again, maybe the codes do not compute those coefficients but only the ones you mention to get (kl|mn)?. In that case, maybe I can get one from the other. I appreciate if you could tell me where to look in the code or if there is a way to have access to the integrals (mn | a) and [J^{-1}]_ab.

4

Probably the easiest way to do this is through the psi4numpy project: https://github.com/dgasmith/psi4numpy

An example Psi4 input code is shown below:

import numpy as np

molecule mol {
O
H 1 1.1
H 1 1.1 2 108
symmetry c1
}

set {
basis sto-3g
df_basis_scf cc-pVDZ-ri
}

energy('SCF')
wfn = wavefunction()
df = DFTensor(wfn, "DF_BASIS_SCF")

Qso =  np.array(df.Qso())
print Qso

df_ao_eri = np.einsum('pqQ, rsQ->pqrs', Qso, Qso)
print df_ao_eri

mints = MintsHelper()
exact_ao_eri = mints.ao_eri()

This will print out | Q | ao , ao> and then the reformed two electron integrals. We are undergoing some infrastructure changes so this piece of code is in a bit of flux. If it breaks, let me know and I can get you an updated copy.

The inverse coulomb metric is not exposed python side, all 3index tensors are pre-contracted with this. The metric code is in the following file /src/lib/lib3index/fittingmetric.cc.

5

Oh, okay! Yes, the code is forming the symmetric density-fitted tensors
b_{\mu \nu}^{D} = \sum_{D} (\mu \nu | D) (D | P )^{-1/2}

So we don’t directly have the coefficients you want. You can use what dgasmith indicated with Psi4NumPy to get the b_{\mu \nu}^{D} tensor (the Qso NumPy array in his example).
The inverse Coulomb matrix is not explicitly computed, but in addition to looking into /src/lib/lib3index/fittingmetric.cc as suggested, you can also check out /src/lib/libfock/DFJK.cc, in particular the initialize_JK_core subroutine. Around line 550 (if it did not change too much) is where (D|P)^{-1/2} is built (this part of the code may not be executed for your particular case, though, because we also have initialize_JK_disk, but it is more difficult to understand).

6

Thank you all!
I’m looking at the code where you indicated.
The input runs perfectly. I’m running it now and leaning how it works. Very helpful! Thanks!