What is the convention of indices for the one and two-body integrals in qiskit

Question

I was able to calculate the one and two-body integrals for the H2 with the following code

from qiskit.chemistry.drivers import PySCFDriver, UnitsType

atom = 'H .0 .0 .0; H .0 .0 0.74' 
distance_unit = UnitsType.ANGSTROM
basis = 'sto3g'

driver = PySCFDriver(atom, unit=distance_unit, basis=basis)

molecule = driver.run()

h1 = molecule.one_body_integrals
h2 = molecule.two_body_integrals
print( h1 ) 
print( h2 ) 

and the results is

[[-1.25330979  0.          0.          0.        ]
 [ 0.         -0.47506885  0.          0.        ]
 [ 0.          0.         -1.25330979  0.        ]
 [ 0.          0.          0.         -0.47506885]]
[[[[-0.33737796  0.          0.          0.        ]
   [ 0.         -0.09060523  0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]]

  [[ 0.         -0.09060523  0.          0.        ]
   [-0.3318557   0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]]

  [[ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [-0.33737796  0.          0.          0.        ]
   [ 0.         -0.09060523  0.          0.        ]]

  [[ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.         -0.09060523  0.          0.        ]
   [-0.3318557   0.          0.          0.        ]]]


 [[[ 0.         -0.3318557   0.          0.        ]
   [-0.09060523  0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]]

  [[-0.09060523  0.          0.          0.        ]
   [ 0.         -0.34882575  0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]]

  [[ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.         -0.3318557   0.          0.        ]
   [-0.09060523  0.          0.          0.        ]]

  [[ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [-0.09060523  0.          0.          0.        ]
   [ 0.         -0.34882575  0.          0.        ]]]


 [[[ 0.          0.         -0.33737796  0.        ]
   [ 0.          0.          0.         -0.09060523]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]]

  [[ 0.          0.          0.         -0.09060523]
   [ 0.          0.         -0.3318557   0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]]

  [[ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.         -0.33737796  0.        ]
   [ 0.          0.          0.         -0.09060523]]

  [[ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.         -0.09060523]
   [ 0.          0.         -0.3318557   0.        ]]]


 [[[ 0.          0.          0.         -0.3318557 ]
   [ 0.          0.         -0.09060523  0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]]

  [[ 0.          0.         -0.09060523  0.        ]
   [ 0.          0.          0.         -0.34882575]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]]

  [[ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.          0.         -0.3318557 ]
   [ 0.          0.         -0.09060523  0.        ]]

  [[ 0.          0.          0.          0.        ]
   [ 0.          0.          0.          0.        ]
   [ 0.          0.         -0.09060523  0.        ]
   [ 0.          0.          0.         -0.34882575]]]]

This immediately raises the question of how qiskit labels the two-body integrals. If we look at the definition of second quantized Hamiltonian

$ H = \sum_{i,j}h_{ij}a^\dagger_i a_j + \frac{1}{2}\sum_{i,j,k,l}h_{ijkl}a^\dagger_ia^\dagger_ja_ka_l $

then we realize that $h_{0000}=0$, whereas qiskit gives -0.33737796. The answer for the integrals is given in the following table. Any helps are really appreciated.

Answer 1

Something to keep in mind is that integrals often hold two forms of symmetries - not only the symmetries enumerated in that table between the indices, but also the spins of the orbitals. I think this is explained well in Microsoft's documents. For example, if the input file notes $a^\dagger_0 a^\dagger_1 a_1 a_0$, we need to consider the potential permutations and the potential spins. So, we'd consider $a^\dagger_{0, \sigma} a^\dagger_{1, \rho} a_{1, \rho} a_{0, \sigma} $, and all of the permutations of the numbers, and all combinations of $\sigma, \rho \in \{ \uparrow, \downarrow \}$, where $\{ \uparrow, \downarrow \}$ are the spins.

So, while the $a^\dagger_0 a^\dagger_0 a_0 a_0$ may be invalid, $a^\dagger_{0, \uparrow}, a^\dagger_{0, \downarrow} a_{0, \downarrow} a_{0, \uparrow} $ is totally valid after indexing.

Admittedly, I have not dived into the PySCF documents, but my assumption is that programs like NWChem and PySCF all use the minimal integral representations and reconstruct from there.