Methodology
How the calculator turns a molecular structure into a LUMO energy: the engine it uses, the methods it offers, how input is prepared, and how every number in a result is defined.
Overview
Each calculation follows the same four stages. Results always report which settings were used at each stage.
Computational engine
- Engine
- Molecule LUMO Calculation Extended Hückel solver, written for this site
- Version
- 1.0.0 (reported with every result)
- License
- Site code. Parameters are Hoffmann's published values as tabulated in YAeHMOP.
- Where it runs
- Entirely in your web browser, in a background Web Worker. No structure is sent to a server.
- Cheminformatics toolkit
- OpenChemLib JavaScript 9.25.1 (BSD-3-Clause) for SMILES and MOL parsing, 2D depiction, conformer generation and the MMFF94s force field
The solver builds a valence-only Slater-type basis for each atom, evaluates every overlap integral numerically in prolate spheroidal coordinates, forms the Extended Hückel Hamiltonian and solves the generalized eigenvalue problem HC = SCE by Löwdin orthogonalization followed by Householder tridiagonalization and QL iteration. To cite a result, name the method variant, the geometry source and the engine version shown in the results panel, and link to this page.
Available methods
Only methods listed here appear in the calculator's method selector.
| Method | Level of theory | Basis set | Best for |
|---|---|---|---|
| Extended Hückel (EHT), standard | Semi-empirical, non-iterative; Hij = K Sij(Hii + Hjj)/2, K = 1.75 | Minimal valence Slater-type orbitals | Fast screening of trends; the classic Hoffmann formulation |
| Extended Hückel (EHT), weighted formula | As above with the weighted Wolfsberg-Helmholz formula of Ammeter et al. | Minimal valence Slater-type orbitals | Molecules with very different atom types, where the weighted formula reduces artificial mixing |
In the weighted formula, K is replaced by K' = K + Δ2 + Δ4(1 − K), with Δ = (Hii − Hjj)/(Hii + Hjj). For hydrocarbons the two variants give nearly identical frontier orbitals; they differ most for bonds between atoms of very different electronegativity.
Atomic parameters
Valence orbital ionization energies (Hii) and Slater exponents (ζ) are Hoffmann's standard values, as distributed with the YAeHMOP package.
| Element | Orbitals | Hii s (eV) | ζ s | Hii p (eV) | ζ p |
|---|---|---|---|---|---|
| H | 1s | −13.60 | 1.300 | n/a | n/a |
| B | 2s, 2p | −15.20 | 1.300 | −8.50 | 1.300 |
| C | 2s, 2p | −21.40 | 1.625 | −11.40 | 1.625 |
| N | 2s, 2p | −26.00 | 1.950 | −13.40 | 1.950 |
| O | 2s, 2p | −32.30 | 2.275 | −14.80 | 2.275 |
| F | 2s, 2p | −40.00 | 2.425 | −18.10 | 2.425 |
| Si | 3s, 3p | −17.30 | 1.383 | −9.20 | 1.383 |
| P | 3s, 3p | −18.60 | 1.750 | −14.00 | 1.300 |
| S | 3s, 3p | −20.00 | 2.122 | −11.00 | 1.827 |
| Cl | 3s, 3p | −26.30 | 2.183 | −14.20 | 1.733 |
| Br | 4s, 4p | −22.07 | 2.588 | −13.10 | 2.131 |
| I | 5s, 5p | −18.00 | 2.679 | −12.70 | 2.322 |
No d orbitals are included for any element, which is the usual minimal choice for main-group molecules.
Molecular input handling
Accepted formats
SMILES strings for a single molecule, and structure files in MOL or SDF format (V2000 or V3000, first record only) and XYZ format. Files are read in the browser and never uploaded.
Preparation steps
- Implicit hydrogens are added before geometry generation.
- For SMILES, and for MOL or SDF files without complete 3D coordinates, a conformer is generated with the OpenChemLib conformer generator (fixed random seed 42, so results are reproducible) and then minimized with the MMFF94s force field. If MMFF94s has no parameters for an atom type, the unminimized conformer is used and the result says so.
- MOL or SDF files that already contain 3D coordinates and all hydrogens are used as supplied. XYZ coordinates are always used as supplied.
- Total charge is taken from the formal charges in the structure. XYZ files carry no charge, so it is set in the calculator's advanced options.
- Spin multiplicity is singlet. Structures with an odd number of electrons, or with radical centers, are refused because the method as implemented handles closed shells only.
- Structures with more than 150 atoms, counting hydrogens, are not calculated. Mixtures and salts entered as several fragments are refused.
- Supported elements are H, B, C, N, O, F, Si, P, S, Cl, Br and I.
Energy units and output definitions
- LUMO energy
- Eigenvalue of the lowest unoccupied orbital, with electrons placed two per orbital from the bottom up.
- HOMO energy
- Eigenvalue of the highest occupied orbital.
- HOMO-LUMO gap
- E(LUMO) minus E(HOMO), in the same unit.
- LUMO+1 energy
- Eigenvalue of the next orbital above the LUMO. A degenerate LUMO is reported as such.
- Hartree to eV
- 1 Eh = 27.211386245988 eV
- eV to kJ/mol
- 1 eV = 96.485332 kJ/mol
- Displayed precision
- 3 decimals in eV, 5 in Hartree, 1 in kJ/mol. The JSON download keeps full double precision, every orbital energy and the 3D coordinates used.
Conversion factors are CODATA 2018 recommended values published by NIST. Extended Hückel energies are native in eV, so Hartree and kJ/mol values are conversions, not separate calculations.
Validation
The engine was checked in three ways before release:
- Overlap integrals were compared with closed-form expressions for 1s-1s, 2s-2s, 2pσ-2pσ and 2pπ-2pπ Slater overlaps at several distances; agreement is better than 10−12.
- Ethylene gives a π (HOMO) energy of −13.23 eV and a π* (LUMO) energy of −8.21 eV, matching the values of about −13.2 and −8.2 eV that are standard for Extended Hückel calculations on ethylene.
- Chemical trends follow expectations: the LUMO falls from benzene (−8.31 eV) to pyridine (−9.24 eV) to 1,4-benzoquinone (−11.39 eV), and the gap shrinks from ethylene to benzene to naphthalene.
Assumptions and limitations
- Molecules are treated as isolated and in the gas phase. No solvent model is applied.
- One conformer is calculated. Flexible molecules may give different values for other conformers.
- Orbital energies are model quantities and change with method and basis set. Extended Hückel LUMO energies are several eV lower than Hartree-Fock or DFT values and should be compared only with other Extended Hückel results.
- Extended Hückel theory is not self-consistent, so charge redistribution is not captured and ions are described less reliably than neutral molecules.
The full discussion is on the Accuracy and Limitations page.
Scientific references
- Hoffmann, R. An Extended Hückel Theory. I. Hydrocarbons. J. Chem. Phys. 1963, 39, 1397-1412.
- Wolfsberg, M.; Helmholz, L. J. Chem. Phys. 1952, 20, 837-843.
- Ammeter, J. H.; Bürgi, H.-B.; Thibeault, J. C.; Hoffmann, R. Counterintuitive Orbital Mixing in Semiempirical and ab Initio Molecular Orbital Calculations. J. Am. Chem. Soc. 1978, 100, 3686-3692.
- Mulliken, R. S.; Rieke, C. A.; Orloff, D.; Orloff, H. Formulas and Numerical Tables for Overlap Integrals. J. Chem. Phys. 1949, 17, 1248-1267.
- Landrum, G. A.; Glassey, W. V. YAeHMOP: Yet Another extended Hückel Molecular Orbital Package. github.com/greglandrum/yaehmop
- Halgren, T. A. Merck Molecular Force Field. I. Basis, Form, Scope, Parameterization, and Performance of MMFF94. J. Comput. Chem. 1996, 17, 490-519.
- Halgren, T. A. MMFF VI. MMFF94s Option for Energy Minimization Studies. J. Comput. Chem. 1999, 20, 720-729.
- OpenChemLib and openchemlib-js. github.com/Actelion/openchemlib and github.com/cheminfo/openchemlib-js
- Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry. Dover, 1996.
- Koopmans, T. Physica 1934, 1, 104-113.
- Tiesinga, E. et al. CODATA Recommended Values of the Fundamental Physical Constants: 2018. Rev. Mod. Phys. 2021, 93, 025010.