Molecular Orbital Energy Calculator

Alpha is the Coulomb integral — the energy of an electron on an isolated p orbital (negative, approximately the ionization energy of carbon p). Beta is the resonance integral — the stabilization energy from p-orbital overlap between adjacent atoms (also negative for bonding).

The energy difference between the highest occupied and lowest unoccupied molecular orbitals. It determines the color of organic dyes (smaller gap = longer wavelength absorption = redder color) and the reactivity and conductivity of organic materials.

The HOMO-LUMO gap decreases as 1/(n+1) for long chains. More atoms means electrons are more delocalized, requiring less energy for a HOMO-LUMO transition, so absorption shifts to longer (redder) wavelengths.

The extra stabilization of a conjugated pi system compared to isolated double bonds. For benzene, the delocalization energy is about 36 kcal/mol (1.56 eV), explaining its unusual stability and aromatic character.

Yes, for cyclic conjugated systems (like benzene) the orbital energies are E_k = alpha + 2*beta*cos(2*pi*k/n), giving the famous 4n+2 Huckel aromaticity rule.

It ignores electron-electron repulsion, sigma-pi mixing, geometry optimization, and overlap integrals. It gives qualitatively correct results but quantitative accuracy requires ab initio or DFT calculations.

It explains aromaticity and antiaromaticity, predicts orbital symmetries for Woodward-Hoffmann rules (orbital symmetry control of pericyclic reactions), estimates relative stabilities of conjugated isomers, and guides dye design.

Conjugated organic molecules or polymers where pi electrons are delocalized across many atoms. The HOMO-LUMO gap determines conductivity. Polythiophene, pentacene, and PCBM are examples used in organic solar cells and transistors.

Common values for carbon: alpha = -7 to -10 eV (roughly the electron affinity of carbon), beta = -2.5 to -3.0 eV. These are fitting parameters. For other atoms (N, O) alpha is more negative. The ratio matters for gap calculations.