Calculator Inputs
Formula Used
Hydrogen 1s atomic orbital:
ψ1s(r) = Z3/2 / √π × e-Zr
Overlap integral:
S = e-ZR × [1 + ZR + (ZR)2 / 3]
Molecular orbital:
ψ = N × [cAψA + cB eiθψB]
Normalization:
N = 1 / √(cA² + cB² + 2cAcB cosθ S)
Probability density:
|ψ|² = real(ψ)² + imaginary(ψ)²
Energy estimate:
E = [α(cA² + cB²) + 2cAcB cosθ β] / [cA² + cB² + 2cAcB cosθ S]
How to Use This Calculator
Enter rA and rB in Bohr radius units. Enter R as the distance between both hydrogen nuclei. Choose bonding for a constructive orbital. Choose antibonding for destructive mixing. Use custom mode when you want direct control over both coefficients. Press calculate to view the wave amplitude, density, overlap, normalization, population values, and estimated energy.
Example Data Table
| Case | rA | rB | R | Z | Type | Expected Use |
|---|---|---|---|---|---|---|
| Balanced bond point | 1.00 | 1.00 | 1.40 | 1.00 | Bonding | Checks a common symmetric estimate. |
| Near nucleus A | 0.40 | 1.60 | 1.40 | 1.00 | Bonding | Shows density shift toward A. |
| Antibonding trial | 1.00 | 1.00 | 1.40 | 1.00 | Antibonding | Tests cancellation and node behavior. |
Hydrogen Molecule Wave Function Guide
What This Model Means
The hydrogen molecule is a central example in quantum chemistry. It shows how two atomic orbitals can form a molecular orbital. This calculator uses a linear combination of atomic orbitals. The method is often called an LCAO approach. It is not a full many electron solution. It is a compact model for learning, checking, and comparing orbital behavior.
Bonding and Antibonding Views
A bonding wave function adds the two 1s orbitals. This raises electron density between the nuclei. That extra density helps explain attraction in a stable bond. An antibonding wave function subtracts one orbital from the other. This can create a node between the nuclei. A node is a region where the wave amplitude becomes very small.
Why Overlap Matters
The overlap integral shows how much the two orbitals share space. A short internuclear distance usually gives a stronger overlap. A long distance gives a weaker overlap. The calculator uses a simple hydrogen 1s overlap expression. Effective nuclear charge is included, so you can test tighter or looser orbital shapes.
Normalization and Density
The wave function must be normalized before density is interpreted. Normalization keeps the probability scale consistent. The calculator finds a normalization constant from the coefficients, phase, and overlap. It then computes the real part, imaginary part, and probability density. Density is the square magnitude of the wave function.
Energy Estimate
The energy value uses Coulomb and resonance integral inputs. Alpha represents the average atomic orbital energy. Beta represents interaction between the two orbital centers. These values are simplified, but they are useful for comparison. Change beta to see how stronger orbital mixing changes the result. Change phase or coefficients to study constructive and destructive combinations.
Best Use
Use this tool for study notes, quick reports, and classroom demonstrations. It helps compare bonding cases without complex software. For research grade molecular calculations, use advanced quantum chemistry packages. For concept checks, this calculator gives clear and direct output.
FAQs
1. What does this calculator find?
It estimates a hydrogen molecule molecular orbital value. It also gives overlap, normalization, probability density, population shares, and an energy estimate.
2. Is this a full quantum solution?
No. It is a simplified LCAO model. It is best for learning, comparison, and quick checks, not high accuracy research.
3. What units should I use for distance?
Use Bohr radius units for rA, rB, and R. Keeping the same unit system makes the overlap and orbital values consistent.
4. What is the overlap integral?
The overlap integral measures shared space between two atomic orbitals. Larger overlap usually means stronger orbital interaction.
5. What is a bonding orbital?
A bonding orbital combines atomic orbitals constructively. It usually increases electron density between the two hydrogen nuclei.
6. What is an antibonding orbital?
An antibonding orbital combines orbitals destructively. It can create a node and reduce electron density between the nuclei.
7. Why is normalization needed?
Normalization places the wave function on a consistent probability scale. Without it, density comparisons can be misleading.
8. What are alpha and beta?
Alpha is the Coulomb integral estimate. Beta is the resonance integral estimate. Together they create a simple orbital energy value.