Molecular Orbital Energy Calculator

Calculator Inputs

Calculation model

Decimal precision

Energy unit

Two Orbital Secular Inputs

HAA atomic orbital energy

HBB atomic orbital energy

HAB resonance integral

Overlap integral S

Bonding occupancy

Antibonding occupancy

Linear Hückel Chain Inputs

Number of p orbitals

Alpha value

Beta value

Pi electrons

Example Data Table

Case	Model	Main Inputs	Expected Reading
Homonuclear pair	Two orbital	HAA = -11, HBB = -11, HAB = -2.5, S = 0	Bonding level is lower than -11.
Heteronuclear pair	Two orbital	HAA = -13, HBB = -10, HAB = -1.8, S = 0.1	Coefficients show unequal orbital mixing.
Butadiene estimate	Linear Hückel	N = 4, alpha = 0, beta = -1, electrons = 4	Two lowest pi levels are filled.

Formula Used

Two Orbital Secular Equation

The calculator solves this determinant:

| HAA - E      HAB - ES |
| HAB - ES     HBB - E | = 0

The expanded quadratic is:

(1 - S²)E² + (2HAB S - HAA - HBB)E + (HAAHBB - HAB²) = 0

The lower root is treated as the bonding molecular orbital. The higher root is treated as the antibonding molecular orbital.

Linear Hückel Chain Equation

E(k) = alpha + 2 beta cos(kπ / (N + 1))

Here, k is the level number. N is the number of connected p orbitals.

Extra Calculations

Total occupied energy = Σ(occupancy × orbital energy)
Bond order = (bonding electrons - antibonding electrons) / 2
Stabilization = total occupied energy - reference energy

How to Use This Calculator

Choose the calculation model first. Use the two orbital model for a pair of interacting atomic orbitals. Use the Hückel model for a simple linear pi system.

Enter all energy values in the same unit. Do not mix electron volts with kilojoules per mole. Keep overlap S between -1 and 1.

Press Calculate to show the result above the form. Review the summary table and energy level table. Use the CSV or PDF option to save the output.

Molecular Orbital Energy in Chemistry

Molecular orbital energy describes how atomic orbitals combine and split. A lower level is usually bonding. A higher level is usually antibonding. The gap depends on atomic orbital energies, interaction strength, and overlap. This calculator gives a practical estimate from those values.

Why the Calculation Matters

Chemists use orbital energy to compare stability, reactivity, and electron placement. A strong interaction makes a larger split. A small overlap makes weaker splitting. Electron occupancy then decides total electronic energy. When more electrons occupy bonding levels, the molecule is more stable. When antibonding levels fill, the bond order falls.

Two Orbital Secular Model

The two orbital model is useful for diatomic fragments and simple mixtures. It solves the secular determinant. The inputs are HAA, HBB, HAB, and S. HAA and HBB are atomic orbital energy terms. HAB is the resonance or coupling term. S is the overlap integral. The calculator solves the quadratic equation, then labels the lower root as bonding and the upper root as antibonding.

Hückel Chain Option

The Hückel option estimates pi orbital levels for a linear conjugated chain. It uses alpha, beta, and the number of atoms. Levels are filled by the electron count. This is helpful for quick comparisons of butadiene, hexatriene, and related systems.

Reading the Results

The result table shows each level, energy, occupancy, and contribution. The contribution equals occupancy times level energy. Bond order is also shown for the two orbital method. Stabilization is compared with a simple atomic reference. Negative stabilization means the chosen occupancy is lower in energy than the reference.

Good Input Practice

Use one energy unit throughout the form. Electron volts are common, but kilojoules per mole can also work if every input uses that unit. Keep overlap between negative one and positive one. Use negative beta for typical bonding Hückel work. Round results only after checking the final table. Export the table when you need a record for a lab note, worksheet, or report.

The tool is an educational estimator. It does not replace full quantum chemistry. Real molecules may need basis sets, electron repulsion terms, and symmetry checks. Still, these simple models reveal trends quickly and help students test assumptions before detailed software work.

FAQs

What is molecular orbital energy?

It is the energy assigned to a molecular orbital after atomic orbitals combine. Lower values usually indicate stabilizing bonding orbitals. Higher values often indicate antibonding orbitals.

What does HAA mean?

HAA is the energy term for atomic orbital A. In simple models, it represents the diagonal Hamiltonian value for that orbital.

What does HAB mean?

HAB is the interaction or resonance integral between two atomic orbitals. A stronger interaction usually creates a larger split between bonding and antibonding levels.

What is the overlap integral?

The overlap integral S measures how much two orbitals overlap in space. It should stay between -1 and 1 in this calculator.

Can I use kilojoules per mole?

Yes. You may use any consistent energy unit. Every entered energy value must use the same unit for the result to be meaningful.

What is bond order?

Bond order estimates net bonding. This calculator uses bonding electrons minus antibonding electrons, then divides by two.

What is the Hückel model used for?

It estimates pi molecular orbital energies in simple conjugated systems. It is useful for fast classroom and worksheet calculations.

Why are my roots not real?

Complex roots can appear when input values are inconsistent. Check the overlap value, resonance integral, and atomic orbital energies.