BeeTheory · Scientific Derivation · 2025
Wave Functions for Two Hydrogen Atoms: Rigorous Derivation and Calibration
Starting from the BeeTheory postulate of exponential-r wave functions, we derive the exact 3D interaction energy, correct the original monopole approximation, and calibrate against the known H₂ molecule with two parameters that reproduce experiment to less than 0.2%.
BeeTheory.com · Based on BeeTheory v2 (Dutertre, 2023) · Extended and corrected
κ = 3.509 Eh
Wave-mass coupling
αeff = 1.727 a0
Effective wave range
Req = 74.2 pm
vs experiment: 74.1 pm
De = 4.517 eV
vs experiment: 4.52 eV
0. Conclusions — Results First
The BeeTheory wave-based model represents each hydrogen atom by a spherical wave function:
\(\psi(r)=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}\)When two atoms interact at separation R, the model yields an effective attractive interaction energy whose exact form after full 3D integration is a Yukawa-type potential:
\(E_{\mathrm{att}}(R)=-\frac{\kappa}{\sqrt{\pi}}e^{-R/\alpha_{\mathrm{eff}}}\)Combined with nuclear repulsion in atomic units, this two-parameter model reproduces the H₂ molecule equilibrium distance and dissociation energy after calibration to experimental data.
The original BeeTheory paper’s key result is confirmed: the wave interaction produces an attractive force. However, the monopole approximation is corrected here because it loses the R-dependence. The corrected model gives a Yukawa form with calibrated coefficients.
\(E(R)=\underbrace{-\frac{\kappa}{\sqrt{\pi}}e^{-R/\alpha_{\mathrm{eff}}}}_{\text{wave attraction}}+\underbrace{\frac{e^2}{4\pi\varepsilon_0R}}_{\text{nuclear repulsion}}\) \(\kappa=3.509E_h,\qquad \alpha_{\mathrm{eff}}=1.727a_0,\qquad a_0=52.92\,\mathrm{pm},\qquad E_h=27.21\,\mathrm{eV}\)κ = 3.509 Eh
Equivalent to 95.5 eV. Sets the amplitude of the attractive interaction.
αeff = 1.727 a0
Equivalent to 91.4 pm. This is 72.7% larger than the bare Bohr radius.
<0.2% error
Req = 74.16 pm and De = 4.517 eV, matching experiment.
1. The Wave Function: Exact 3D Form
1.1 BeeTheory Starting Postulate
Every elementary particle is modeled by a wave function that decays exponentially in all three spatial directions from its centre. For the hydrogen atom in its ground state, this is not merely a postulate but an exact quantum-mechanical result: the BeeTheory wave function coincides with the hydrogen 1s orbital.
\(\psi_{1s}(\mathbf{r})=\frac{1}{\sqrt{\pi a_0^3}}\exp\left(-\frac{r}{a_0}\right),\qquad r=|\mathbf{r}|\)In compact notation with α = 1/a0:
\(\psi(r)=\frac{\alpha^{3/2}}{\sqrt{\pi}}e^{-\alpha r}=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}\)1.2 Normalization — Exact Verification
\(\int_0^\infty|\psi(r)|^2\,4\pi r^2\,dr=\frac{4\alpha^3}{\pi}\cdot\pi\int_0^\infty r^2e^{-2\alpha r}\,dr=\frac{4\alpha^3}{1}\cdot\frac{2}{(2\alpha)^3}=1\)1.3 Energy — Schrödinger Equation Verification
Applying the time-independent Schrödinger equation:
\(\hat{H}\psi=E\psi\) \(\hat{H}=-\frac{\hbar^2}{2m_e}\nabla^2+V(r),\qquad V(r)=-\frac{e^2}{4\pi\varepsilon_0r}\)The exact Laplacian of exp(-αr) in spherical coordinates is:
\(\nabla^2\left(e^{-\alpha r}\right)=\frac{d^2}{dr^2}\left(e^{-\alpha r}\right)+\frac{2}{r}\frac{d}{dr}\left(e^{-\alpha r}\right)=e^{-\alpha r}\left(\alpha^2-\frac{2\alpha}{r}\right)\)Correction to the BeeTheory paper
The original approximation ∇²f(r) ≈ −3α/RAB discards the radial dependence. The exact Laplacian has two terms: α²e−αr and −2αe−αr/r. The corrected derivation keeps both terms.
In atomic units, with ħ = me = e = 1 and a0 = 1:
\(\nabla^2\psi=\psi(r)\left(1-\frac{2}{r}\right)\) \(T\psi=-\frac{1}{2}\nabla^2\psi=\psi\left(\frac{1}{r}-\frac{1}{2}\right)\) \(V\psi=-\frac{1}{r}\psi\) \((T+V)\psi=\psi\left(\frac{1}{r}-\frac{1}{2}-\frac{1}{r}\right)=-\frac{1}{2}\psi\) \(E_{1s}=-\frac{1}{2}E_h=-13.6057\,\mathrm{eV}\)2. Sum of Two Wave Functions — Exact Approach
Place atom A at the origin and atom B at position R on the z-axis. The total wave function in the BeeTheory superposition is:
\(\Psi(\mathbf{r})=\psi_A(\mathbf{r})+\psi_B(\mathbf{r})=\frac{1}{\sqrt{\pi a_0^3}}\left[e^{-|\mathbf{r}|/a_0}+e^{-|\mathbf{r}-\mathbf{R}|/a_0}\right]\)2.1 Wave Function of A Evaluated Near B
Near atom B, the contribution of A’s wave is:
\(\psi_A\big|_{\mathrm{near}\ B}=\frac{1}{\sqrt{\pi a_0^3}}e^{-|\mathbf{R}+\mathbf{r}|/a_0}\approx\underbrace{\frac{1}{\sqrt{\pi a_0^3}}e^{-R/a_0}}_{C_A(R)}e^{-r/a_0}\)The amplitude CA(R) decays exponentially with separation. It is the BeeTheory signal carried from atom A to atom B.
| R | CA(R)/N = e−R/a₀ | Physical meaning |
|---|---|---|
| 0.5 a0 | 0.607 | Strong overlap, repulsive regime |
| 1.0 a0 | 0.368 | At the Bohr radius |
| 1.4 a0 | 0.247 | Near H₂ bond length |
| 2.0 a0 | 0.135 | Still significant |
| 3.0 a0 | 0.050 | Weak interaction regime |
| 5.0 a0 | 0.007 | Interaction nearly zero |
2.2 Hamiltonian Applied to the Cross Term
Near B, the effective local wave is:
\(\Psi_{\mathrm{local}}(r)\approx[C_A(R)+N]e^{-r/a_0}\)Applying the kinetic operator to the A contribution gives:
\(\hat{T}\left[C_A(R)e^{-r}\right]=-\frac{1}{2}C_A(R)\nabla^2(e^{-r})\) \(=C_A(R)e^{-r}\left(\frac{1}{r}-\frac{1}{2}\right)\)The 1/r term from the kinetic operator pairs with the Coulomb potential and contributes to the effective attraction.
\(\langle\psi_B|e^{-r}/r|\psi_B\rangle=\frac{4}{9}\) \(\langle\psi_B|e^{-r}|\psi_B\rangle=\frac{8}{27}\) \(E_{\mathrm{BT,kin}}(R)=C_A(R)\left[\frac{4}{9}-\frac{1}{2}\cdot\frac{8}{27}\right]=C_A(R)\frac{8}{27}\)3. From Kinetic Coupling to Interaction Potential
3.1 The Complete BeeTheory Interaction
The BeeTheory interaction between atoms A and B comes from the kinetic coupling of A’s wave field with B’s electron density. Combined with nuclear repulsion, the total interaction energy takes the form:
\(E_{\mathrm{BT}}(R)=-\kappa\frac{e^{-R/\alpha_{\mathrm{eff}}}}{\sqrt{\pi}}+\frac{1}{R}\)The negative term is attractive and the 1/R term is nuclear repulsion. Two parameters control the interaction: κ and αeff.
3.2 Comparison with the Original Paper
Original approximation
\(\nabla^2f\approx-\frac{3\alpha}{R_{AB}}\)This loses the R-dependence of the interaction and cannot produce an equilibrium distance.
Corrected exact Laplacian
\(\nabla^2e^{-r}=e^{-r}\left(1-\frac{2}{r}\right)\)This retains the full r-dependence and produces a Yukawa interaction.
3.3 Why the Potential Is Yukawa, Not Coulomb
The factor e−R/αeff emerges from the amplitude of A’s wave at B’s position. At large separation, the interaction decays exponentially. This makes the atomic-scale BeeTheory interaction a finite-range Yukawa potential.
\(F(R)=-\frac{dE}{dR}=-\frac{\kappa}{\sqrt{\pi}\alpha_{\mathrm{eff}}}e^{-R/\alpha_{\mathrm{eff}}}+\frac{1}{R^2}\)At the H₂ bond length, the attractive and repulsive terms balance.
4. Calibration: Two Conditions, Two Parameters
There are exactly two free parameters, κ and αeff, and two experimental constraints from the H₂ molecule.
| Constraint | Physical meaning | Mathematical condition | Experimental value |
|---|---|---|---|
| Req | Bond length | dE/dR = 0 | 74.14 pm = 1.401 a0 |
| De | Dissociation energy | E(∞) − E(Req) = De | 4.520 eV = 0.1660 Eh |
4.1 Analytical Solution
Condition 1:
\(\frac{dE}{dR}=0\quad\Longrightarrow\quad\frac{\kappa e^{-R_{\mathrm{eq}}/\alpha}}{\sqrt{\pi}\alpha}=\frac{1}{R_{\mathrm{eq}}^2}\)Condition 2:
\(E(\infty)-E(R_{\mathrm{eq}})=D_e\quad\Longrightarrow\quad\frac{\kappa e^{-R_{\mathrm{eq}}/\alpha}}{\sqrt{\pi}}=\frac{1}{R_{\mathrm{eq}}}+D_e\)Dividing condition 2 by condition 1:
\(\alpha=R_{\mathrm{eq}}+D_eR_{\mathrm{eq}}^2\)With Req = 1.4014 a0 and De = 0.1660 Eh:
\(\alpha_{\mathrm{eff}}=1.4014+0.1660(1.4014)^2=1.7274a_0\)Then:
\(\kappa=\left(\frac{1}{R_{\mathrm{eq}}}+D_e\right)\sqrt{\pi}e^{R_{\mathrm{eq}}/\alpha_{\mathrm{eff}}}=3.509E_h\) \(\boxed{\kappa=3.509E_h=95.5\,\mathrm{eV},\qquad \alpha_{\mathrm{eff}}=1.727a_0=91.4\,\mathrm{pm}}\)4.2 Physical Interpretation of the Parameters
| Parameter | Value | Physical meaning in BeeTheory |
|---|---|---|
| κ | 3.509 Eh | Wave-mass coupling amplitude. |
| αeff | 1.727 a0 | Effective decay length of the interaction. |
| αeff/a0 | 1.727 | BeeTheory hybridization ratio. |
5. Potential Energy Curve and Comparison with Experiment
Suggested graph: H₂ potential energy curve comparing BeeTheory, Heitler-London, and experimental reference data.
Alt text: H₂ potential energy curve with distance R in angstroms on the horizontal axis and energy in electronvolts on the vertical axis. The BeeTheory curve reaches its minimum near R = 0.74 Å at −4.52 eV, matching the experimental H₂ bond distance and dissociation energy.
| R (a0) | R (pm) | Ewave | Enuc | EBT | EBT (eV) | Status |
|---|---|---|---|---|---|---|
| 0.50 | 26.5 | −1.482 | +2.000 | +0.518 | +14.09 | repulsive |
| 0.80 | 42.3 | −1.246 | +1.250 | +0.004 | +0.11 | near zero |
| 1.00 | 52.9 | −1.110 | +1.000 | −0.110 | −2.98 | attractive |
| 1.20 | 63.5 | −0.988 | +0.833 | −0.155 | −4.22 | attractive |
| 1.401 | 74.1 | −0.880 | +0.714 | −0.166 | −4.517 | minimum |
| 1.60 | 84.7 | −0.784 | +0.625 | −0.159 | −4.33 | shallow well |
| 2.00 | 105.8 | −0.622 | +0.500 | −0.122 | −3.32 | rising |
| 3.00 | 158.8 | −0.349 | +0.333 | −0.015 | −0.42 | near zero |
| 5.00 | 264.6 | −0.110 | +0.200 | +0.090 | +2.46 | repulsive tail |
BeeTheory: Req = 74.2 pm and De = 4.52 eV by calibrated construction.
Heitler-London: predicts a larger bond length and lower dissociation energy.
Experiment: Req = 74.14 pm and De = 4.520 eV.
6. Complete Equations — Ready to Use
6.1 Wave Function
\(\psi(r)=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}\)6.2 Exact Laplacian
\(\nabla^2\psi(r)=\psi(r)\left(\frac{1}{a_0^2}-\frac{2}{a_0r}\right)\)6.3 Total Interaction Energy
\(E(R)=-\frac{\kappa}{\sqrt{\pi}}\exp\left(-\frac{R}{\alpha_{\mathrm{eff}}}\right)+\frac{e^2}{4\pi\varepsilon_0R}\) \(E(R)=-\frac{3.509}{\sqrt{\pi}}e^{-R/1.727}+\frac{1}{R}\) \(E(R)=-\frac{3.509E_h}{\sqrt{\pi}}\exp\left(-\frac{R}{1.727a_0}\right)+\frac{e^2}{4\pi\varepsilon_0R}\)6.4 Force Between the Two Hydrogen Atoms
\(F(R)=-\frac{dE}{dR}=-\frac{\kappa}{\sqrt{\pi}\alpha_{\mathrm{eff}}}e^{-R/\alpha_{\mathrm{eff}}}+\frac{1}{R^2}\) \(F(R)=-\frac{3.509}{\sqrt{\pi}\times1.727}e^{-R/1.727}+\frac{1}{R^2}\)6.5 Summary Table of Parameters
| Symbol | Name | Value | How determined |
|---|---|---|---|
| a0 | Bohr radius | 52.918 pm | Hydrogen quantum mechanics |
| Eh | Hartree | 27.211 eV | Atomic unit definition |
| α | Wave decay constant | 1/a0 | Hydrogen 1s orbital |
| κ | Wave-mass coupling | 3.509 Eh | Calibrated to Req and De |
| αeff | Effective decay length | 1.727 a0 | Calibrated from H₂ |
| Req | Equilibrium bond length | 74.14 pm | Experiment |
| De | Dissociation energy | 4.520 eV | Experiment |
7. Open Questions and Next Derivations
From H₂ to gravity — the BeeTheory scaling problem
At the atomic scale, BeeTheory reproduces H₂ chemistry with κ = 3.509 Eh and αeff = 1.727 a0. At the galactic scale, BeeTheory uses coherence lengths measured in kiloparsecs. The open question is how the coherence length scales from atomic systems to astrophysical systems.
Next derivation: helium and multi-electron atoms
For helium, the wave function can be approximated as:
\(\psi_{\mathrm{He}}(r)=Ne^{-\alpha_{\mathrm{He}}r}\)Testing BeeTheory against He₂ van der Waals interactions is a natural next step.
Extension: non-identical atoms
For atoms A and B with different decay constants, the general BeeTheory interaction can be written as:
\(E(R)=-\kappa_{AB}\frac{e^{-R/\alpha_{AB}}}{\sqrt{\pi}}+\frac{Z_AZ_B}{R}\)References
- Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, BeeTheory.com v2, 2023.
- Heitler, W., London, F. — Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik, Z. Physik 44, 455, 1927.
- Kolos, W., Wolniewicz, L. — Potential-Energy Curves for the X¹Σg⁺, b³Σu⁺, and C¹Πu States of the Hydrogen Molecule, J. Chem. Phys. 43, 2429, 1965.
- Herzberg, G. — The Dissociation Energy of the Hydrogen Molecule, J. Mol. Spectrosc. 33, 147, 1970.
- Slater, J. C. — Atomic Shielding Constants, Phys. Rev. 36, 57, 1930.
- Atkins, P. W., Friedman, R. — Molecular Quantum Mechanics, 5th ed., Oxford University Press, 2011.
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