The analog of Koopmans’ theorem for virtual Kohn–Sham orbital energies
Bibliographic record
Abstract
An analog of Koopmans’ theorem is formulated for the energies, ε a , of virtual Kohn–Sham (KS) molecular orbitals (MOs) from the requirement that the KS theory provides, in principle, not only the exact electron density, but also its exact response. The starting point is the Kohn–Sham analog of Koopmans’ theorem, relating the vertical ionization energies, I i , to the energies, ε i , of the occupied MOs ( Chong, D.P.; Gritsenko, O.V.; Baerends, E.J. J. Chem. Phys. 2002, 116, 1760 ). Combining this with the coupled-perturbed equations of time-dependent density functional theory (TDDFT), exact relations between the energies, ε a , of virtual KS MOs and the excitation energies, ω ia , and vertical ionization energies (VIPs), I i , are obtained. In the small matrix approximation for the coupling matrix K of TDDFT, two limiting cases of these relations are considered. In the limit of a negligible matrix element, K ia,ia , the energy, ε a , can be interpreted as (minus) the energy of ionization from the ? i → ? a excited state, ε a ≈ –I a , where –I a is defined from the relation I i = ω ia + I a . This relation breaks down in special cases, such as charge-transfer transitions and the HOMO–LUMO (highest occupied molecular orbital – lowest unoccupied molecular orbital) transition of a dissociating electron-pair bond (also of charge-transfer character). The present results highlight the important difference between virtual orbital energies in the Kohn–Sham model (ε a ≈ –I a ) and in the Hartree–Fock model (ε a ≈ –A a ). Kohn–Sham differences ε a – ε i approximate the excitation energy, ω ia , while Hartree–Fock differences [Formula: see text] do not approximate excitation energies but approximate the difference of an ionization energy and an electron affinity, I i – A a .
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".