Green’s-function ionisation potentials and electron affinities (GF2 and OVGF)

Koopmans’ theorem reads an ionisation potential off an orbital energy and is wrong by one to two electronvolts. The one-particle electron propagator corrects every orbital energy of one reference calculation at once through a frequency-dependent self-energy and attaches to each ionisation a pole strength that says whether the one-hole picture still holds. run_job(method="ovgf") runs a Hartree-Fock reference (RHF for singlets, UHF otherwise) and then solves the quasiparticle equation with the diagonal second-order self-energy (GF2) for the outer-valence orbitals. It is production for that scope, including open-shell references through a spin-orbital kernel. The result list result.ovgf holds the GF2 quasiparticle energies and pole strengths. For small closed-shell systems the .out additionally prints one renormalised HOMO value that includes the full diagonal third-order self-energy. What is not implemented is the literal outer-valence Green’s function (OVGF) of von Niessen, Schirmer and Cederbaum with its A, B and C screening recipes; the name ovgf records the lineage (Cederbaum, L. S. (1975). *One-body Green’s function for atoms and molecules: theory and application.* Journal of Physics B: Atomic and Molecular Physics, **8**(2), 290–303. doi:[10.1088/0022-3700/8/2/018](https://doi.org/10.1088/0022-3700/8/2/018), von Niessen, W., Schirmer, J., and Cederbaum, L. S. (1984). *Computational methods for the one-particle Green’s function.* Computer Physics Reports, **1**(2), 57–125. doi:[10.1016/0167-7977(84)90002-9](https://doi.org/10.1016/0167-7977(84)90002-9)), as the vibeqc.propagator module docstring states. The label vocabulary is defined in Methods: theory and status. The tutorial Ionisation potentials from the electron propagator works water through in detail, with a Delta-SCF comparison and the third-order theory; this page is the reference.

Quick start

Water at RHF/6-31G*:

from vibeqc import Atom, Molecule, run_job

ang = 1.8897259886
EV = 27.211386245988
mol = Molecule([
    Atom(8, [0.0, 0.0, 0.1173 * ang]),
    Atom(1, [0.0, 0.7572 * ang, -0.4692 * ang]),
    Atom(1, [0.0, -0.7572 * ang, -0.4692 * ang]),
], charge=0, multiplicity=1)

result = run_job(mol, basis="6-31g*", method="ovgf", output="h2o_ovgf")
print(result)                                            # RHFResult+OVGF(4 quasiparticle orbitals)
n_occ = mol.n_electrons() // 2
for q in result.ovgf:                                    # one QuasiparticleResult per orbital
    tag = "HOMO" if q.orbital == n_occ - 1 else "LUMO" if q.orbital == n_occ else ""
    print(f"MO {q.orbital} {tag:>4}  Koopmans {-q.eps_scf * EV:8.3f} eV  "
          f"GF2 {-q.eps_qp * EV:8.3f} eV  Z = {q.pole_strength:.3f}  "
          f"({q.iterations} Dyson iterations, converged={q.converged})")

On a vibe-qc 0.17.10 development build (Apple M5 Max) the job takes under two seconds. The list holds the three highest occupied orbitals and the LUMO. The orbital-energy columns of the .out are energies in eV, negative for bound orbitals; an ionisation potential is the negative of the corrected energy.

MO 2       Koopmans   19.221 eV  GF2   17.977 eV  Z = 0.930  (3 Dyson iterations, converged=True)
MO 3       Koopmans   15.536 eV  GF2   13.448 eV  Z = 0.914  (3 Dyson iterations, converged=True)
MO 4 HOMO  Koopmans   13.539 eV  GF2   11.004 eV  Z = 0.907  (4 Dyson iterations, converged=True)
MO 5 LUMO  Koopmans   -5.839 eV  GF2   -5.292 eV  Z = 0.980  (3 Dyson iterations, converged=True)

and the .out prints, under OVGF / GF2 quasiparticle energies (diagonal second-order self-energy),

  MO  label  Koopmans(eV)  OVGF (eV)  pole Z
   4   HOMO      -13.5394   -11.0040  0.9067
   5   LUMO        5.8389     5.2917  0.9805
  OVGF ionization potential (HOMO) = 11.0040 eV
  OVGF electron affinity   (LUMO) = -5.2917 eV   (small-basis EA -- qualitative only)
  GF2(renorm) ionization potential (HOMO) = 11.6628 eV   (renormalized; pole Z=0.925)

(the table also lists MOs 2 and 3). Against the experimental vertical IP of 12.62 eV: Koopmans 13.54, GF2 11.00, renormalised 11.66, and 10.85 for a Delta-SCF cation (from the tutorial). The GF2 value undershoots because second order treats the two-hole-one-particle states as non-interacting and overestimates relaxation; the renormalised value pulls it back. The negative “electron affinity” for water is correct in sign but is the energy of a 6-31G* virtual orbital without diffuse functions and should not be quoted.

An open-shell molecule takes the UHF spin-orbital kernel automatically. The OH radical at RHF/6-31G*, with a 0.97 Å bond:

from vibeqc import Atom, Molecule, run_job

ang = 1.8897259886
EV = 27.211386245988
oh = Molecule([Atom(8, [0.0, 0.0, 0.0]), Atom(1, [0.0, 0.0, 0.97 * ang])], 0, 2)
r = run_job(oh, basis="6-31g*", method="ovgf", output="oh_ovgf")
for spin, q in r.ovgf:                                   # (spin, QuasiparticleResult)
    print(f"{spin:5s} MO {q.orbital}  Koopmans {q.eps_scf * EV:9.4f}  "
          f"GF2 {q.eps_qp * EV:9.4f} eV  Z = {q.pole_strength:.3f}")

alpha MO 4  Koopmans  -14.9808  GF2  -12.7864 eV  Z = 0.922
alpha MO 5  Koopmans    5.9828  GF2    5.5918 eV  Z = 0.984
beta  MO 3  Koopmans  -13.7108  GF2  -11.6509 eV  Z = 0.925
beta  MO 4  Koopmans    3.6091  GF2    0.2722 eV  Z = 0.939

The .out reports OVGF first ionization potential = 11.6509 eV: the smallest ionisation potential over the alpha and beta channels. For each spin it corrects the spin-HOMO and the spin-LUMO. The job takes about 3 s. The MSINDO analogue is msindo_ovgf([8, 1, 1], xyz), described below.

Theory

The one-particle Green’s function has poles at the exact ionisation potentials and electron affinities, and the residues are the pole strengths. For a Hartree-Fock reference the zeroth-order function has poles at the orbital energies, which is Koopmans’ theorem, and the exact function follows from the Dyson equation with a self-energy \(\Sigma(\omega)\) that carries relaxation and correlation. In the diagonal approximation each ionisation \(p\) solves the quasiparticle equation

\[ \omega_p=\varepsilon_p+\Sigma_{pp}(\omega_p),\qquad Z_p=\Big(1-\frac{\partial\Sigma_{pp}}{\partial\omega}\Big|_{\omega_p}\Big)^{-1}, \]

by Newton iteration from \(\omega=\varepsilon_p\) (default: tol=1e-8, at most 50 iterations), and \(Z_p\in(0,1]\) is the weight of the one-hole configuration. With the non-iterative choice iterate=False one gets \(\varepsilon_p+\Sigma_{pp}(\varepsilon_p)\) instead, which differed from the converged value by 0.3 eV for the water HOMO.

The second-order self-energy for a closed-shell reference, with \(i,j\) occupied and \(a,b\) virtual spatial orbitals and chemist’s notation, is

\[ \Sigma^{(2)}_{pp}(\omega)=\sum_{iaj}\frac{\big[2(pi|aj)-(pj|ai)\big](pi|aj)}{\omega+\varepsilon_a-\varepsilon_i-\varepsilon_j} +\sum_{aib}\frac{\big[2(pa|ib)-(pb|ia)\big](pa|ib)}{\omega+\varepsilon_i-\varepsilon_a-\varepsilon_b}, \]

the two-hole-one-particle (\(2h1p\)) channel that lowers ionisation potentials and the two-particle-one-hole (\(2p1h\)) channel that matters for electron attachment. For a UHF reference the same quantity is written over spin orbitals with antisymmetrised integrals \(\langle pq\|rs\rangle\), \(\Sigma_{pp}(\omega)=\tfrac12\sum_{iab}|\langle pi\|ab\rangle|^2/(\omega+\varepsilon_i-\varepsilon_a-\varepsilon_b)+\tfrac12\sum_{aij}|\langle pa\|ij\rangle|^2/(\omega+\varepsilon_a-\varepsilon_i-\varepsilon_j)\), the electron-propagator analogue of UMP2. Both need the full MO two-electron tensor, which is why the routes are size-guarded. The approximations are the diagonal truncation (the ionised state keeps the shape of one canonical orbital), the second-order truncation of the interaction and the Hartree-Fock reference. The second-order truncation makes the relaxation part too large, so GF2 ionisation potentials sit below experiment.

The renormalised value adds the diagonal third-order self-energy, \(\Sigma^{(3)}_c\) per channel \(c\in\{2h1p,2p1h\}\), with its static (\(W\)) and dynamic (\(U\)) terms, and resums the ratio geometrically per channel,

\[ \Sigma_c(\omega)\approx\frac{\Sigma^{(2)}_c(\omega)}{1-\Sigma^{(3)}_c(\omega)/\Sigma^{(2)}_c(\omega)}, \]

falling back to the bare \(\Sigma^{(2)}_c+\Sigma^{(3)}_c\) when \(|\Sigma^{(3)}_c/\Sigma^{(2)}_c|\) exceeds screen_cap (0.5) or \(\Sigma^{(2)}_c\approx0\). This is the outer-valence Green’s-function idea of Cederbaum (Cederbaum, L. S. (1975). *One-body Green’s function for atoms and molecules: theory and application.* Journal of Physics B: Atomic and Molecular Physics, **8**(2), 290–303. doi:[10.1088/0022-3700/8/2/018](https://doi.org/10.1088/0022-3700/8/2/018)) and of von Niessen, Schirmer and Cederbaum (von Niessen, W., Schirmer, J., and Cederbaum, L. S. (1984). *Computational methods for the one-particle Green’s function.* Computer Physics Reports, **1**(2), 57–125. doi:[10.1016/0167-7977(84)90002-9](https://doi.org/10.1016/0167-7977(84)90002-9)), applied to the full diagonal third-order self-energy, and it is deliberately not their A/B/C screening partition, which cannot be validated here against an external code. Semiempirical NDDO references work the same way (Danovich, Danovich, David (1997). *NDDO semiempirical approximations coupled with Green’s function technique—a reliable approach for calculating ionization potentials.* Journal of Molecular Structure: THEOCHEM, **401**(3), 235–252. doi:[10.1016/S0166-1280(97)00026-2](https://doi.org/10.1016/S0166-1280(97)00026-2)).

Where it helps: outer-valence ionisations of closed-shell molecules with \(Z>0.85\), all corrected from one reference calculation at MP2-like cost, to a few tenths of an electronvolt in a good basis. Where it fails, and says so through a small \(Z\): inner-valence and core ionisations and satellite structure (the water \(2a_1\) has \(Z=0.30\) at second order), strongly correlated references, and electron affinities in a basis without diffuse functions.

Options

There are no OVGF-specific run_job keywords: method="ovgf" selects the route, basis sets the basis, and the reference is chosen by the multiplicity.

What	Behaviour
Reference	RHF for multiplicity == 1, UHF otherwise. ROHF and Kohn-Sham references are not used by this route.
Orbitals corrected, closed shell	The three highest occupied orbitals and the LUMO (the whole outer valence for small molecules).
Orbitals corrected, open shell	The spin-HOMO and spin-LUMO of each spin channel.
Solver	Newton iteration on the Dyson equation, always iterated.
result	An object that forwards every attribute of the SCF result and adds .ovgf and .method.
result.ovgf	Closed shell: a list of QuasiparticleResult. Open shell: a list of (spin, QuasiparticleResult), spin being "alpha" or "beta".

QuasiparticleResult (frozen dataclass, energies in Hartree): orbital (0-based spatial MO index), eps_scf (Koopmans), eps_qp, sigma (the self-energy at the solution), pole_strength, iterations, converged, and the property correction (eps_qp - eps_scf).

The kernels are importable from vibeqc.propagator for your own orbital windows and references (see the tutorial for a complete example):

Function	Purpose
quasiparticle_energies(eps, n_occ, eri_mo, orbitals, *, iterate=True, max_iter=50, tol=1e-8)	Closed-shell GF2 for a list of MO indices; quasiparticle_energy does one.
diagonal_self_energy(eps, n_occ, eri_mo, p, omega)	\(\Sigma^{(2)}_{pp}(\omega)\) in Hartree.
unrestricted_quasiparticle_energies(eri_ao, Ca, Cb, eps_a, eps_b, n_a, n_b, orbitals, *, iterate=True, max_iter=50, tol=1e-8, max_spinorbitals=80)	Spin-orbital GF2; orbitals are (spin, index) pairs.
renormalized_quasiparticle_energies(eri_ao, Ca, Cb, eps_a, eps_b, n_a, n_b, orbitals, *, iterate=True, max_iter=50, tol=1e-7, screen_cap=0.5, max_spinorbitals=60)	Third-order renormalised GF2; a closed shell is Ca == Cb, n_a == n_b.

The kernels are reference-agnostic: any set of orbital energies and MO integrals (Hartree-Fock, Kohn-Sham, semiempirical) can be fed in, though the runner itself uses Hartree-Fock.

MSINDO and other references

vibeqc.semiempirical.methods.msindo.msindo_ovgf(atomic_numbers, coords_angstrom, *, charge=0, multiplicity=None, orbitals=None, renormalize=False, iterate=True, max_iter=200, conv_tol=1e-10, output=None) runs the MSINDO SCF and the same self-energy over the rigorous INDO integral tensor. It returns MsindoOVGFResult for a closed shell or MsindoUOVGFResult for an open shell (odd valence-electron count, or multiplicity > 1; s and p elements only). Verified for water:

from vibeqc.semiempirical.methods.msindo import msindo_ovgf

xyz = [[0, 0, 0], [0, 0.757, 0.587], [0, -0.757, 0.587]]     # Angstrom
bare = msindo_ovgf([8, 1, 1], xyz)
print(f"{bare.ip_homo_ev:.3f} eV  EA {bare.ea_lumo_ev:.3f} eV  Z {[round(float(z), 3) for z in bare.pole_strength]}")
print(f"{msindo_ovgf([8, 1, 1], xyz, renormalize=True).ip_homo_ev:.3f} eV (renormalised)")

13.129 eV  EA -7.083 eV  Z [0.98, 0.988]
12.874 eV (renormalised)

The MSINDO propagator is a Python reference implementation; see MSINDO for scope and the comparison with the reference program’s factorised-integral approximation. Its output= writes the Danovich, Cederbaum and von Niessen citations. Kohn-Sham (DFT) orbitals can be passed to the vibeqc.propagator kernels by hand, but there is no run_job route for a DFT-referenced propagator.

Output

• .out: a header OVGF / GF2 quasiparticle energies (diagonal second-order self-energy) (open shell: (open-shell, spin-resolved second-order self-energy)) with a table of MO index, label, Koopmans energy, corrected energy and pole strength, the summary lines shown above, and for a small closed-shell system the GF2(renorm) line.

• Citations (.bibtex, .references): Cederbaum 1975 and von Niessen 1984 with the route, next to the basis and SCF entries; Danovich 1997 joins for msindo_ovgf.

• Python: result.ovgf, as above. A job with optimize=True or hessian=True relaxes or differentiates the Hartree-Fock reference, not the propagator surface, and tags the surface accordingly.

Limits and gates

• Size: the closed-shell route builds the \(n_\text{bf}^4\) MO tensor and refuses more than 100 basis functions with NotImplementedError. The open-shell route builds a \((2n_\text{bf})^4\) spin-orbital tensor and refuses more than 40 basis functions (2 n_bf <= 80); verified with the OH radical in cc-pVTZ (2.n_bf=88), which raises after the SCF has converged. The renormalised third-order HOMO line is printed only when \(2n_\text{bf}\le60\) (at most 30 functions), and renormalized_quasiparticle_energies raises NotImplementedError beyond max_spinorbitals (60). Use a smaller basis or the vibeqc.propagator API on a truncated window. The size guards fire after the SCF.

• Electron affinities in a basis without diffuse functions are qualitative: the virtual orbitals are not variational and a bound anion needs diffuse functions. The .out flags them; a positive \(-\varepsilon^\text{qp}\) does not establish a stable anion.

• Approximation: diagonal self-energy only; no off-diagonal (ADC(3)) treatment. Pole strengths below about 0.8 mark a breakdown of the orbital picture; do not quote such a line as an ionisation potential.

• ECPs: a heavy-element reference with an effective core potential runs (HI at RHF/def2-SVP, valence orbitals, first ionisation 10.09 eV, \(Z\) = 0.95) and the self-energy is built over the valence orbitals the SCF filled. The feature matrix and the methods page still say that OVGF refuses an ECP reference; that statement is out of date for this build, and there is no scalar-relativistic spin-orbit treatment behind the number.

• Solvent: solvent= with method="ovgf" is refused (no validated post-SCF composition in solvent).

• Frozen core: none; all occupied orbitals enter the sums.

• Unconverged SCF: no propagator block is produced.

See also

• Ionisation potentials from the electron propagator: water with a Delta-SCF comparison, all valence orbitals and the theory of the third-order terms.

• MSINDO: msindo_ovgf and semiempirical ionisation potentials.

• MP2 and double hybrids and CCSD: the correlated energies that use the same MO integral seam.

• TDDFT, TDA and CIS excited states: neutral excitations from the same response-theory family, and Properties.

• Citations, Methods: theory and status and the feature matrix.