How to align the atomic orbitals between pyscf and psi4

1

Dear Developers,
I found that the Hamiltonians obtained from pyscf and psi4 are different. For example,

from pyscf import gto, scf
import psi4

mol = gto.M(atom='O 0 0 0; H 0 1 0; H 0 0 1', basis='def2-svp')
rhf = scf.RHF(mol)
rhf.kernel()
H_pyscf = rhf.get_hcore()

geom = """
    O 0 0 0
    H 0 1 0
    H 0 0 1
    symmetry c1
    units angstrom
"""
mol_psi4 = psi4.geometry(geom)
psi4.set_options({'reference': 'rhf', 'basis': 'def2-SVP'})
scf_e, scf_wfn = psi4.energy('scf', molecule=mol_psi4, return_wfn=True)
H_psi4 = scf_wfn.H().np

Since the Hamiltonian is a (24, 24) matrix, I only print the first dimension here:

# H_pyscf[0]
array([-3.29856601e+01,  1.11013177e+01,  5.12600489e+00,  0.00000000e+00,
        1.99302401e-02,  1.99302401e-02,  0.00000000e+00,  2.67199490e-03,
        2.67199490e-03,  0.00000000e+00,  0.00000000e+00,  7.22474019e-04,
        0.00000000e+00, -1.25136171e-03,  1.37360210e+00,  1.96479169e+00,
        0.00000000e+00, -2.50325889e+00,  6.39602699e-04,  1.37360210e+00,
        1.96479169e+00,  0.00000000e+00,  6.39602699e-04, -2.50325889e+00])

# H_psi4[0]
array([-3.29856601e+01,  1.11013177e+01,  5.12600489e+00,  2.81856159e-02,
        0.00000000e+00,  0.00000000e+00,  3.77877143e-03,  0.00000000e+00,
        0.00000000e+00,  7.22474018e-04,  0.00000000e+00,  0.00000000e+00,
       -1.25136171e-03,  0.00000000e+00,  1.37360210e+00,  1.96479168e+00,
       -1.76961906e+00,  0.00000000e+00, -1.77052360e+00,  1.37360210e+00,
        1.96479168e+00, -1.76961906e+00,  0.00000000e+00,  1.77052360e+00])

The AOs in pyscf are

['0 O 1s    ', '0 O 2s    ', '0 O 3s    ', '0 O 2px   ', '0 O 2py   ', '0 O 2pz   ', '0 O 3px   ', '0 O 3py   ', '0 O 3pz   ', '0 O 3dxy  ', '0 O 3dyz  ', '0 O 3dz^2 ', '0 O 3dxz  ', '0 O 3dx2-y2', '1 H 1s    ', '1 H 2s    ', '1 H 2px   ', '1 H 2py   ', '1 H 2pz   ', '2 H 1s    ', '2 H 2s    ', '2 H 2px   ', '2 H 2py   ', '2 H 2pz   ']

The differences are occured when l>0 but it will not change the eigenvals. So what the difference between the definition of AOs in pyscf and psi4? And how to align the AOs between pyscf and psi4?

2

The molecule probably isn’t even aligned. Add

nocom
noreorient

to geom, just like you added symmetry c1. This will prevent Psi from moving the molecule in space.

3

Thank you for the suggestion. I tried with

geom = """
    O 0 0 0
    H 0 1 0
    H 0 0 1
    symmetry c1
    units angstrom
    nocom
    noreorient
"""

Then the H_psi4[0] is

array([-3.29856601e+01,  1.11013177e+01,  5.12600489e+00,  1.99302401e-02,
        0.00000000e+00,  1.99302401e-02,  2.67199490e-03,  0.00000000e+00,
        2.67199490e-03,  7.22474018e-04,  0.00000000e+00,  0.00000000e+00,
       -1.25136171e-03,  0.00000000e+00,  1.37360210e+00,  1.96479168e+00,
        6.39602697e-04,  0.00000000e+00, -2.50325888e+00,  1.37360210e+00,
        1.96479168e+00, -2.50325888e+00,  0.00000000e+00,  6.39602697e-04])

which are still different from that of pyscf.

4

PySCF and Psi4 use different AO orderings, and it looks like that’s the only remaining cause of disagreement. See here for details of AO ordering in Psi.

5

There’s also the question of whether PySCF is using the same molecular orientation as Psi4.

6

Thanks for your response. It is very helpful. I investigate the related information you provided, and find that the AO ordering in psi4 is 0, +1, -1, +2, -2. A quick check is to look at the Hamiltonian I provided above. The Hamiltonian under ‘O 2p’ AOs are

m -1 0 +1
H_pyscf 0.00000000e+00 1.99302401e-02 1.99302401e-02
H_psi4 1.99302401e-02 1.99302401e-02 0.00000000e+00

which seems have a permtation?
But for the Hamiltonian under ‘O 3d’ AOs

m -2 -1 0 +1 +2
H_pyscf 0.00000000e+00 0.00000000e+00 7.22474019e-04 0.00000000e+00 -1.25136171e-03
H_psi4 0.00000000e+00 0.00000000e+00 7.22474018e-04 0.00000000e+00 -1.25136171e-03

which are the same. I’m confuse about this.

7

Thank you for the response. I’m sorry that I don’t understand what the meaning of molecular orientation as the coordinates are the same.

8

Since the eigenvals of Hamiltonians from pyscf and psi4 are the same, these two Hamiltonians should be transformed by a unitary transformation. So what is this unitary transformation?

9

The transformation is not necessarily unitary due to differences in the way the basis functions are defined; see How to transform the Hamiltonians between pyscf and psi4? · Issue #2123 · pyscf/pyscf · GitHub for the PySCF side of the discussion.