H₆ Ring (RHF)

The H₆ ring is the simplest EDF test case: 6 hydrogen atoms at equal spacing, RHF wavefunction, one QTAIM basin per atom, 6 fragments.

Files

test/h6.edfinp   — input
test/h6.wfn      — RHF wavefunction (GAMESS/aimpac format)
test/h6.aom      — AOM from PROMOLDEN
test/h6.edfout   — reference output

Input

0
h6.aom
h6.wfn
ngroup 6
1 1
1 2
1 3
1 4
1 5
1 6
end

Line	Content	Meaning
0	ioverlap	AOM from PROMOLDEN
h6.aom	filedat	AOM filename
h6.wfn	wfnfile	WFN filename
ngroup 6	Record 4	6 fragments
1 1 … 1 6	Records 5.1–5.6	Each fragment contains 1 atom
end	Terminator	End of input

Running

cd test/
OMP_NUM_THREADS=4 ../edf-omp < h6.edfinp > h6_new.edfout

Key output

Probability distribution (spinless)

H₆ has 6 electrons, each atom ideally contributing 1. The dominant spinless configuration:

#     Probability            n1    n2    n3    n4    n5    n6
#     0.9994921682  <-- SUM,      56 PROBABILITIES

The dominant configuration is 1 1 1 1 1 1 — one electron per hydrogen — confirming the delocalized ring character. Configurations with 2 electrons on one H and 0 on an adjacent one appear at probabilities ~0.002.

Average populations

<n(  1)_alpha>  =  0.499746
<n(  1)_beta>   =  0.499746

Each H carries exactly 0.5 alpha + 0.5 beta = 1.0 electron on average, as expected by symmetry. Small numerical deviations (< 10⁻⁴) reflect the finite integration grid used by PROMOLDEN.

Delocalization indices

# delta_(  2  1)  =  0.443349     <- adjacent H atoms (1 bond apart)
# delta_(  3  1)  =  0.060120     <- 2 bonds apart
# delta_(  4  1)  =  0.112734     <- 3 bonds apart (para)

Adjacent atoms share ~0.44 electrons — strong delocalization consistent with the aromatic π system. The alternating pattern (adjacent > para > ortho') reflects the ring topology.

Timer

#     _binedf       0.014 s
#     _linearsys    0.001 s
#     _splitedf     0.000 s

For this small system the entire calculation completes in ~20 ms. The parallelized _linearsys and _splitedf regions are fast because H₆ has only 3 alpha and 3 beta electrons (small configuration space).

Physical interpretation

The EDF of H₆ reveals that the "covalent" picture (1 electron per H) is dominant (~99.9% summed probability), while ionic configurations (2 or 0 electrons on one H) carry the remaining ~0.1%. The large delocalization index between adjacent H atoms (~0.44) is characteristic of a 6π Hückel aromatic system, where each bond has bond-order contributions from all ring positions.