COOP and COHP: bonding analysis of a periodic system

A density of states counts states per energy; a projected DOS says which atoms they live on; neither says whether a state holds two atoms together or pushes them apart. The crystal orbital overlap population (COOP, Hughbanks, Timothy and Hoffmann, Roald (1983). *Chains of trans-edge-sharing molybdenum octahedra: metal-metal bonding in extended systems.* Journal of the American Chemical Society, **105**(11), 3528–3537. doi:[10.1021/ja00349a027](https://doi.org/10.1021/ja00349a027)) and the crystal orbital Hamilton population (COHP, Dronskowski, Richard and Blöchl, Peter E. (1993). *Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations.* Journal of Physical Chemistry, **97**(33), 8617–8624. doi:[10.1021/j100135a014](https://doi.org/10.1021/j100135a014)) answer that question. Both resolve the DOS by atom pair and weight every Bloch state by the sign of its overlap (COOP) or Hamiltonian (COHP) matrix element between the two atoms: bonding states come out with one sign, antibonding states with the other, and the integral up to the Fermi level, ICOOP or ICOHP, is a single number that ranks bonds in a crystal the way a bond order ranks them in a molecule. The periodic Mayer bond order (Mayer, I. (1983). *Charge, bond order and valence in the ab initio SCF theory.* Chemical Physics Letters, **97**(3), 270–274. doi:[10.1016/0009-2614(83)80005-0](https://doi.org/10.1016/0009-2614(83)80005-0)) is that molecular index carried over to a k mesh.

vibe-qc computes all three after a periodic SCF and writes them into the QVF archive as dos.coop, dos.cohp and bond_orders sections. On the GDF routes the analysis is production: it projects the accepted SCF eigenpairs, the COHP weights use the analysed spin Fock matrix (the operator that defines the bands), a request for any other operator is refused rather than mislabelled, and the integrated values contract the accepted density so that fractional occupations are kept. On the BIPOLE, AICCM and external-XC routes the request fails closed before the SCF, with a message naming the missing converged-Hamiltonian projection. See Methods: theory and status for the label vocabulary. This page runs the analysis on a dimerised hydrogen chain, reads the archive three ways, stretches the bond, and shows the low-level API that the runner wraps.

Setup

The system is a chain of H\(_2\) molecules along \(x\): a 1.4 bohr bond in a 6 bohr cell, so that the intra-cell H-H distance is 1.4 bohr and the inter-cell one 4.6 bohr. The GDF route needs a three-dimensional cell (a strictly one-dimensional dim=1 system is refused because its bare nuclear lattice sums have no consistent Coulomb gauge), so the chain sits in 10 bohr of vacuum in \(y\) and \(z\). Eight k points along the chain are plenty for an insulator.

import json
import zipfile

import numpy as np
import vibeqc as vq

a, vac = 6.0, 10.0
system = vq.PeriodicSystem(3, [[a, 0.0, 0.0], [0.0, vac, 0.0], [0.0, 0.0, vac]],
                           [vq.Atom(1, [0.0, 0.0, 0.0]), vq.Atom(1, [1.4, 0.0, 0.0])])
basis = vq.BasisSet(system.unit_cell_molecule(), "sto-3g")

result = vq.run_periodic_job(system, basis, method="RHF", jk_method="gdf", kpoints=[8, 1, 1],
                             coop_cohp=True, output_qvf=True, output="h2_chain")
print(f"E = {result.energy:.6f} Ha per cell, converged = {result.converged}")

def read_section(kind):
    """Metadata and the binary arrays of one QVF section."""
    with zipfile.ZipFile("h2_chain.qvf") as qvf:
        manifest = json.loads(qvf.read("manifest.json"))
        sec = next(s for s in manifest["sections"] if s["kind"] == kind)
        arrays = {name: np.frombuffer(qvf.read(m["path"]), dtype=m.get("dtype", "float64"))
                  .reshape(m["shape"])
                  for name, m in sec["members"].items() if not m["path"].endswith(".json")}
    return {k: v for k, v in sec.items() if k != "members"}, arrays

meta, coop = read_section("dos.coop")
_, cohp = read_section("dos.cohp")
print("pair:", meta["pairs"][0], " sigma =", meta["sigma_ev"], "eV")

e = coop["energies"]                                   # eV, relative to the Fermi level
proj = coop["projections"][0]                          # COOP(E) of the first (only) pair
i_b, i_a = np.argmax(proj), np.argmin(proj)
print(f"ICOOP  = {coop['integrated'][0]:+.4f} electrons")
print(f"-ICOHP = {cohp['integrated'][0]:+.3f} eV")
print(f"COOP peaks: bonding {proj[i_b]:+.3f} at {e[i_b]:.2f} eV, antibonding {proj[i_a]:+.3f} at {e[i_a]:.2f} eV")
print("occupied COOP >= 0:", bool((proj[e <= 0] >= -1e-9).all()),
      " empty COOP <= 0:", bool((proj[e > 0] <= 1e-9).all()))

with zipfile.ZipFile("h2_chain.qvf") as qvf:
    manifest = json.loads(qvf.read("manifest.json"))
    sec = next(s for s in manifest["sections"] if s["kind"] == "bond_orders")
    bonds = json.loads(qvf.read(sec["members"]["bond_orders"]["path"]))
print("Mayer:", [(f"{p['symbol_i']}{p['i']}-{p['symbol_j']}{p['j']}", round(p["order"], 4))
                 for p in bonds["pairs"]])

On a vibe-qc 0.17.10 development build (Apple M5 Max) the run takes about 40 seconds on an idle machine, nearly all of it the GDF lattice sums in the vacuum box; the analysis itself is a fraction of a second.

E = -1.116339 Ha per cell, converged = True
pair: {'i': 0, 'j': 1, 'symbol_i': 'H', 'symbol_j': 'H', 'distance_ang': 0.7408480952641999}  sigma = 0.05 eV
ICOOP  = +0.3972 electrons
-ICOHP = +9.602 eV
COOP peaks: bonding +0.817 at -16.32 eV, antibonding -4.400 at 19.98 eV
occupied COOP >= 0: True  empty COOP <= 0: True
Mayer: [('H0-H1', 1.0002)]

The same archive can be read without Python:

vibeqc coop h2_chain.qvf            # pair table with ICOOP and -ICOHP
vibeqc coop --json h2_chain.qvf     # machine-readable
vibeqc mayer h2_chain.qvf           # periodic Mayer bond orders above 0.05
vibe-view open h2_chain.qvf         # the curves in the viewer; terminal mode charts dos.coop and dos.cohp

# COOP  (E_F = 0.495 eV,  σ = 0.050 eV)
#         pair  distance/Å         ICOOP
# ------------------------------------------
       H0-H1        0.74       +0.3972

# -COHP  (E_F = 0.495 eV,  σ = 0.050 eV)
#         pair  distance/Å        -ICOHP
# ------------------------------------------
       H0-H1        0.74       +9.6020
# Mayer bond orders (method=mayer, threshold=0.05)
#         pair  distance/Å       order
# --------------------------------------
       H0-H1        0.74      1.0002

What just happened

After the SCF converged, the runner took the accepted eigenpairs \(\varepsilon_{n\mathbf k}\), \(\mathbf C(\mathbf k)\), the overlap \(\mathbf S(\mathbf k)\), the Fock matrix \(\mathbf F(\mathbf k)\) and the density \(\mathbf P(\mathbf k)\) at each of the eight k points, formed the list of atom pairs in the home cell within 8 bohr of each other (here exactly one, H0-H1 at 1.4 bohr), and for every band and k point computed the pair-resolved overlap population and Hamilton population defined in the theory section. Broadening each with a 0.05 eV Gaussian and summing with the k weights gives the two curves stored in dos.coop and dos.cohp, on a common energy axis in eV measured from the Fermi level. The .system manifest records the provenance of the operator: property_hamiltonian = "accepted-scf" and property_kpoints = 8, and the .references file gains the four COOP/COHP papers and Mayer’s.

The output is the textbook picture of H\(_2\). The one occupied band, the \(\sigma_g\) combination, has a positive COOP (bonding overlap population) and lies 16 eV below the Fermi level; the empty \(\sigma_u^*\) band, 20 eV above it, has a negative COOP, larger in magnitude because the antibonding combination has the larger absolute overlap population (\(-S/(1-S)\) against \(+S/(1+S)\) per electron for a two-centre bond). The sign checks in the script confirm that every occupied state is bonding and every empty one antibonding. Integrating the COOP up to the Fermi level gives the overlap population of the bond, \(\mathrm{ICOOP} = 0.397\) electrons, the Mulliken overlap population of an H\(_2\) molecule in a minimal basis. The Hamilton population carries the energy weight: the runner reports \(-\mathrm{ICOHP} = 9.60\) eV, the bonding energy the pair contributes to the band-structure energy in the sense of Dronskowski and Blochl, positive because the Lobster sign convention is used (the returned curve is \(-\mathrm{COHP}\), so bonding is positive in both panels; see the theory). The periodic Mayer bond order, 1.0002, says the pair shares one electron pair, as it should for a two-electron bond that has no partner within the cutoff.

What the runner did not do matters as much. It did not enumerate the inter-cell H1-H0\('\) pair at 4.6 bohr: pairs are formed from raw home-cell coordinates, so a bond that crosses the cell boundary, or a bond from an atom to its own periodic image, does not appear (the user guide states this limitation; choose the cell so that the bonds you care about lie inside it). It did not interpolate the Fock matrix to a finer mesh: on the GDF route the spectrum is built on the accepted SCF k mesh, and a different dos_kmesh is refused (with a warning and no property payload) rather than served from an interpolated operator. And it did not use any operator other than the SCF Fock matrix for the COHP; the low-level API below shows the refusal.

Variation: stretch the bond

The chemistry test of any bond descriptor is what it does when the bond weakens. Repeat the run with the intra-cell distance at 1.8 and 2.2 bohr (the inter-cell gap shrinks from 4.6 to 3.8 bohr, so the chain is still clearly dimerised).

for bond in (1.4, 1.8, 2.2):
    sys_b = vq.PeriodicSystem(3, [[6.0, 0.0, 0.0], [0.0, 10.0, 0.0], [0.0, 0.0, 10.0]],
                              [vq.Atom(1, [0.0, 0.0, 0.0]), vq.Atom(1, [bond, 0.0, 0.0])])
    res_b = vq.run_periodic_job(sys_b, vq.BasisSet(sys_b.unit_cell_molecule(), "sto-3g"),
                                method="RHF", jk_method="gdf", kpoints=[8, 1, 1],
                                coop_cohp=True, output_qvf=True,
                                output=f"h2_chain_{bond:.1f}", verbose=0)
    with zipfile.ZipFile(f"h2_chain_{bond:.1f}.qvf") as qvf:
        manifest = json.loads(qvf.read("manifest.json"))
        def integrated(kind):
            m = next(s for s in manifest["sections"] if s["kind"] == kind)["members"]["integrated"]
            return float(np.frombuffer(qvf.read(m["path"]), dtype=m.get("dtype", "float64"))[0])
        m = next(s for s in manifest["sections"] if s["kind"] == "bond_orders")["members"]["bond_orders"]
        mayer = json.loads(qvf.read(m["path"]))["pairs"][0]["order"]
        print(f"d = {bond:.1f} bohr  E = {res_b.energy:.6f} Ha  ICOOP = {integrated('dos.coop'):.4f}"
              f"  -ICOHP = {integrated('dos.cohp'):.3f} eV  Mayer = {mayer:.4f}")

d = 1.4 bohr  E = -1.116339 Ha  ICOOP = 0.3972  -ICOHP = 9.602 eV  Mayer = 1.0002
d = 1.8 bohr  E = -1.076705 Ha  ICOOP = 0.3441  -ICOHP = 8.728 eV  Mayer = 1.0004
d = 2.2 bohr  E = -1.011717 Ha  ICOOP = 0.2886  -ICOHP = 7.828 eV  Mayer = 1.0010

The two integrated populations fall monotonically as the bond stretches, by 27 percent (ICOOP) and 18 percent (\(-\)ICOHP) between 1.4 and 2.2 bohr, tracking the loss of overlap and of bonding energy. The Mayer bond order does not move: it counts the electron pair shared between the two atoms, and a stretched H\(_2\) still shares exactly one pair at the RHF level (it would only drop once the RHF determinant is allowed to break symmetry or correlation is included). This is the practical division of labour: the Mayer index classifies a bond (single, double, none), the integrated populations grade its strength, and only the energy-resolved curves say which states are responsible.

Variation: the low-level API on an explicit operator

compute_coop_cohp and periodic_mayer_bond_orders take real-space lattice blocks (LatticeMatrixSet objects) and a k mesh directly. The runner’s GDF path does not hand its Fock blocks back, so the low-level route is for workflows that already own an operator. The shipped example examples/periodic/coop_cohp_h2_chain.py runs the same GDF SCF in a 3-D vacuum box as this tutorial and then, in its last block, applies the low-level API to the one-electron Hamiltonian \(\mathbf T + \mathbf V\) on a strictly one-dimensional chain for exactly that reason; that block is a Huckel-like teaching operator, not the SCF Hamiltonian, and its absolute energies and ICOHP are not comparable with the runner’s.

import numpy as np
import vibeqc as vq
from vibeqc.coop_cohp import compute_coop_cohp, periodic_mayer_bond_orders

EV = 27.211386245988
a, vac = 6.0, 30.0
chain = vq.PeriodicSystem(1, [[a, 0.0, 0.0], [0.0, vac, 0.0], [0.0, 0.0, vac]],
                          [vq.Atom(1, [0.0, vac / 2, vac / 2]), vq.Atom(1, [1.4, vac / 2, vac / 2])])
basis_1d = vq.BasisSet(chain.unit_cell_molecule(), "sto-3g")

opts = vq.LatticeSumOptions()
S = vq.compute_overlap_lattice(basis_1d, chain, opts)
T = vq.compute_kinetic_lattice(basis_1d, chain, opts)
V = vq.compute_nuclear_lattice(basis_1d, chain, opts)
kmesh = vq.monkhorst_pack(chain, [32, 1, 1])

cc = compute_coop_cohp([T, V], S, chain, basis_1d, kmesh, include_cohp=True,
                       sigma=0.01, n_grid=401, n_electrons_per_cell=2)
print("pairs:", [(p["symbol_i"], p["i"], p["symbol_j"], p["j"]) for p in cc.pairs])
print(f"E_F = {cc.fermi_energy:.4f} Ha, grid {cc.energies[0]:.3f} .. {cc.energies[-1]:.3f} Ha, "
      f"coop shape {cc.coop.shape}")
print(f"ICOOP = {cc.integrated_coop[0]:.4f} e, -ICOHP = {cc.integrated_cohp[0]:.4f} Ha "
      f"= {cc.integrated_cohp[0] * EV:.2f} eV")
bo = periodic_mayer_bond_orders([T, V], S, chain, basis_1d, kmesh, n_electrons_per_cell=2)
print(f"Mayer(H0-H1) = {bo[0, 1]:.4f}")

try:                                   # a different operator is refused, not relabelled
    compute_coop_cohp([T, V], S, chain, basis_1d, kmesh, H_terms=[T], n_electrons_per_cell=2)
except ValueError as exc:
    print("refused:", exc)

pairs: [('H', 0, 'H', 1)]
E_F = -2.6542 Ha, grid -2.716 .. -1.784 Ha, coop shape (1, 401)
ICOOP = 0.1341 e, -ICOHP = 0.3828 Ha = 10.42 eV
Mayer(H0-H1) = 1.0004
refused: COHP requires the analyzed spin Hamiltonian; H_terms describes a different operator. Use include_cohp=True for standard COHP or cohp_weights_k for explicit operator projections.

Three differences from the runner path are instructive. The energies are in Hartree and absolute (the low-level result is not shifted to the Fermi level; cc.fermi_energy is the highest occupied eigenvalue for the electron count you pass). The Mayer index is again 1.00, because the one-electron operator also fills the bonding band with one pair. But the ICOOP is 0.134 instead of 0.397: the H_core bands are a different operator with different Bloch coefficients, which is the whole point of insisting on the SCF Fock matrix for a quantitative COHP. include_cohp=True projects the operator that generated the eigenpairs; the H_terms argument is accepted only when its Bloch sum equals that operator, and the cohp_weights_k kernel exists for explicitly labelled projections of anything else. For an unrestricted operator pass the beta blocks as F_terms_beta= and the arrays gain a leading spin axis.

Theory

Expand the Bloch orbitals of band \(n\) at wave vector \(\mathbf k\) in the atom-centred basis, \(\psi_{n\mathbf k} = \sum_\mu C_{\mu n}(\mathbf k)\, \chi_{\mu\mathbf k}\), with \(\mathbf S(\mathbf k)\) and \(\mathbf H(\mathbf k)\) the Bloch-summed overlap and Hamiltonian. The Mulliken population of that state, \(\sum_{\mu\nu} C^*_{\mu n} S_{\mu\nu} C_{\nu n} = 1\), can be split into on-site and pair terms, and the pair term between the functions on atom \(A\) and those on atom \(B\) is the overlap population of the state. Weighting it with a broadened delta function and summing over states gives the COOP,

\[ \mathrm{COOP}_{AB}(E) = \sum_{\mathbf k} w_{\mathbf k} \sum_n \Big[\operatorname{Re}\sum_{\mu\in A}\sum_{\nu\in B} C^*_{\mu n}(\mathbf k)\,S_{\mu\nu}(\mathbf k)\,C_{\nu n}(\mathbf k)\Big] \,\delta_\sigma(E - \varepsilon_{n\mathbf k}), \]

and replacing the overlap matrix element by the Hamiltonian matrix element gives the COHP,

\[ \mathrm{COHP}_{AB}(E) = \sum_{\mathbf k} w_{\mathbf k} \sum_n \Big[\operatorname{Re}\sum_{\mu\in A}\sum_{\nu\in B} C^*_{\mu n}(\mathbf k)\,H_{\mu\nu}(\mathbf k)\,C_{\nu n}(\mathbf k)\Big] \,\delta_\sigma(E - \varepsilon_{n\mathbf k}). \]

The restricted spin degeneracy multiplies both by two. The sign conventions follow from the two-centre case: a bonding combination \(\chi_A + \chi_B\) has a positive overlap population and, because the off-diagonal Hamiltonian element \(H_{AB}\) is negative for a bonding interaction, a negative Hamilton population; antibonding states reverse both. COOP is therefore positive for bonding and COHP negative, and because plots are easier to read when bonding points up, the Lobster convention (Maintz, Stefan, Deringer, Volker L., et al. (2016). *LOBSTER: A tool to extract chemical bonding from plane-wave based DFT.* Journal of Computational Chemistry, **37**(11), 1030–1035. doi:[10.1002/jcc.24300](https://doi.org/10.1002/jcc.24300)) is to plot and report \(-\mathrm{COHP}\); vibe-qc’s cohp array and its integrated_cohp already carry the minus sign, so both curves and both integrals are positive for bonding. Always say which convention a number uses, because some programs print raw COHP.

The integrals up to the Fermi level, \(\mathrm{ICOOP}_{AB} = \int^{E_F}\mathrm{COOP}_{AB}(E)\,dE\) and likewise \(\mathrm{ICOHP}_{AB}\), sum over the occupied states. ICOOP is the Mulliken overlap population of the pair, a dimensionless electron count that correlates with bond order but is basis-dependent (it shrinks toward zero as the basis becomes more diffuse or more complete, the usual Mulliken pathology). ICOHP has the dimension of an energy and is the pair’s share of the band-structure energy \(\sum_{n\mathbf k} f_{n\mathbf k}\varepsilon_{n\mathbf k}\), which Dronskowski and Blochl showed by writing that energy as \(\sum_{AB}\mathrm{ICOHP}_{AB}\) plus on-site terms. It is more robust to the basis than ICOOP (Deringer, Tchougreeff and Dronskowski project even plane-wave states onto local orbitals to obtain it, Deringer, Volker L., Tchougréeff, Andrei L., and Dronskowski, Richard (2011). *Crystal orbital Hamilton population (COHP) analysis as projected from plane-wave basis sets.* Journal of Physical Chemistry A, **115**(21), 5461–5466. doi:[10.1021/jp202489s](https://doi.org/10.1021/jp202489s)), but it has a gauge: adding a constant \(c\) to the Hamiltonian, $\mathbf H \to \mathbf H

• c,\mathbf S\(, shifts \)\mathrm{COHP}{AB}\( by \)c,\mathrm{COOP}{AB}\(. In a periodic calculation the average electrostatic potential is not absolutely defined, so \)-\mathrm{ICOHP}\( values are comparable within one code, one route and one basis, as the stretch series above, and not across programs or against a molecular bond energy. On the GDF route the runner integrates with the accepted density matrix rather than by refilling integer occupations, so smeared or fractional occupations are honoured, and the Gaussian width \)\sigma$ (0.05 eV in the archive, sigma in Hartree in the low-level call) is a presentation choice that does not change the integrals.

The periodic Mayer bond order generalises Mayer’s molecular index (Mayer, I. (1983). *Charge, bond order and valence in the ab initio SCF theory.* Chemical Physics Letters, **97**(3), 270–274. doi:[10.1016/0009-2614(83)80005-0](https://doi.org/10.1016/0009-2614(83)80005-0)), \(B_{AB} = \sum_{\mu\in A} \sum_{\nu\in B} (\mathbf{PS})_{\mu\nu}(\mathbf{PS})_{\nu\mu}\), by averaging over the mesh:

\[ B_{AB} = \sum_{\mathbf k} w_{\mathbf k} \sum_{\mu\in A}\sum_{\nu\in B} \big[\mathbf P(\mathbf k)\mathbf S(\mathbf k)\big]_{\mu\nu}\, \big[\mathbf P(\mathbf k)\mathbf S(\mathbf k)\big]_{\nu\mu}, \]

with \(\mathbf P(\mathbf k) = 2\,\mathbf C_\text{occ}(\mathbf k)\mathbf C_\text{occ}^\dagger(\mathbf k)\) for a restricted state and the spin-summed form (\(2\sum_\sigma\) over the spin densities) for an unrestricted one; the k average is exact for the lattice-summed products by the inverse Bloch relation. Like its molecular parent it counts shared electron pairs, is exactly 1 for an isolated two-electron bond, and does not measure bond energy, as the stretch series shows.

Where the analysis helps: assigning bands to bonds and antibonds, locating the antibonding states that a Peierls distortion or a dopant would empty, ranking chemically similar contacts within one structure, and following a bond through a series of geometries or compositions. Where it fails: absolute comparisons across bases, codes or the potential gauge; systems whose bonds cross the cell boundary or whose atoms bond to their own images, which the home-cell pair list does not enumerate; and any diffuse basis, in which Mulliken-type projections become meaningless. Converge the k mesh and the SCF thresholds before the broadening, and keep the QVF with its provenance so that the operator, mesh and convention travel with the numbers.

Resources

• COOP and COHP user guide: result shapes, pair selection, the plotters coop_figure and cohp_figure, and the convergence checklist.

• Bond analysis for the molecular Mayer and Wiberg indices; Projected DOS and Band structure for the spectra the populations resolve.

• Peierls distortion: the dimerised chain of this page as a physical phenomenon.

• Smearing for metals, where the Fermi level and the occupations must be converged together.

• examples/periodic/coop_cohp_h2_chain.py: this tutorial’s SCF in a vacuum box, plus the low-level example on the one-dimensional chain.

• QVF file format and vibe-view terminal mode, which charts dos.coop and dos.cohp.

• Feature matrix and Methods: theory and status.

Cost: the analysis diagonalises nothing new on the GDF route (it reuses the accepted eigenpairs) and scales as \(N_{\mathbf k}\,N_\text{pair}\, N_\text{bf}^2\) for the pair weights plus \(N_\text{pair}\,N_E\) for the curves; it reserves its spectral storage under a 128 MiB cap before allocating. The 40 seconds of the example are the GDF SCF in a vacuum box, not the analysis.

References

• COOP. T. Hughbanks and R. Hoffmann, “Chains of trans-edge-sharing molybdenum octahedra: metal-metal bonding in extended systems,” J. Am. Chem. Soc. 105, 3528 (1983), Hughbanks, Timothy and Hoffmann, Roald (1983). *Chains of trans-edge-sharing molybdenum octahedra: metal-metal bonding in extended systems.* Journal of the American Chemical Society, **105**(11), 3528–3537. doi:[10.1021/ja00349a027](https://doi.org/10.1021/ja00349a027). The crystal orbital overlap population.

• COHP. R. Dronskowski and P. E. Blochl, “Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations,” J. Phys. Chem. 97, 8617 (1993), Dronskowski, Richard and Blöchl, Peter E. (1993). *Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations.* Journal of Physical Chemistry, **97**(33), 8617–8624. doi:[10.1021/j100135a014](https://doi.org/10.1021/j100135a014). The Hamilton population and the band-energy partition.

• Projected COHP. V. L. Deringer, A. L. Tchougreeff and R. Dronskowski, “Crystal orbital Hamilton population (COHP) analysis as projected from plane-wave basis sets,” J. Phys. Chem. A 115, 5461 (2011), Deringer, Volker L., Tchougréeff, Andrei L., and Dronskowski, Richard (2011). *Crystal orbital Hamilton population (COHP) analysis as projected from plane-wave basis sets.* Journal of Physical Chemistry A, **115**(21), 5461–5466. doi:[10.1021/jp202489s](https://doi.org/10.1021/jp202489s).

• Lobster and the sign convention. S. Maintz, V. L. Deringer, A. L. Tchougreeff and R. Dronskowski, “LOBSTER: A tool to extract chemical bonding from plane-wave based DFT,” J. Comput. Chem. 37, 1030 (2016), Maintz, Stefan, Deringer, Volker L., et al. (2016). *LOBSTER: A tool to extract chemical bonding from plane-wave based DFT.* Journal of Computational Chemistry, **37**(11), 1030–1035. doi:[10.1002/jcc.24300](https://doi.org/10.1002/jcc.24300).

• Mayer bond order. I. Mayer, “Charge, bond order and valence in the ab initio SCF theory,” Chem. Phys. Lett. 97, 270 (1983), Mayer, I. (1983). *Charge, bond order and valence in the ab initio SCF theory.* Chemical Physics Letters, **97**(3), 270–274. doi:[10.1016/0009-2614(83)80005-0](https://doi.org/10.1016/0009-2614(83)80005-0).

Next

• Projected DOS resolves the same bands by atom and angular momentum instead of by pair.

• Peierls distortion follows the dimerisation of a hydrogen chain and is where the antibonding states above the Fermi level start to matter.

• Periodic HF and Periodic methods compared explain the GDF route this page ran on and the ones that refuse the analysis.