A 4D Electrostatic Model for Fast Computational Chemistry: Using Explicit Electron Internal Structure.

A 4D Electrostatic Model for Fast Computational Chemistry: Using Explicit Electron Internal Structure. Dr David A. Sinclair Cambridge, UK [email protected] 6th December 2025 Abstract Electrons have internal structure, you cannot have spin without internal structure, you cannot have chemistry without spin, computing with out an explicit model for spin has made chemists’ lives hard. We present a deterministic framework for computational chemistry that replaces the probabilistic wave function with a classical electromagnetic model of the electron possessing internal temporal structure. By recognizing the electron as a half-photon electromagnetic field configuration oscillating between two temporal poles separated by δ0=λC/2≈1.21 pm, we resolve the classical radiation catastrophe and recover atomic stability without quantum axioms. The electron’s energy resides entirely in the half-photon field oscillation at frequency ω0=mec2/ℏ, not in electrostatic self-energy of the temporal poles. Angular momentum conservation (L=ℏ) makes the electron stiff against stretching but compressible under strong nuclear fields, with compression energy E(δ) = ℏc/δ increasing hyperbolically as δ < δ0. This temporal structure naturally regularizes the nuclear Coulomb potential, predicting the hydrogen ground state to 0.18% accuracy. The exchange interaction emerges as magnetic phase-locking between antiparallel electron pairs, calibrated from Cooper pair coherence lengths in superconductors. This mechanism reproduces the lithium ionization energy (0.8% error) and derives nitrogen’s sp3geometry from magnetic dipole packing constraints. The Pauli exclusion principle arises from two physical mechanisms: (1) compressed electrons cannot phase-lock with uncompressed electrons due to frequency mismatch, and (2) half-photon analytic paths resist spatial overlap. The model offers O(N2) computational scaling for molecular systems, presenting a classical alternative to density functional theory. Python code for all of the calculations in the paper is available at https://s-hull.org/4D Chemistry Suite.py 1 Introduction For over a century, computational chemistry has relied on the Schr¨odinger equation [1]. While highly successful, this approach entails heavy computational costs (O(N4) or worse for high accuracy methods like coupled-cluster) and conceptual abstractions—probability clouds, exchangecorrelation functionals [2], and mysterious “exchange forces”—that obscure the underlying physical mechanics of bonding. It is not credible to assert that the electron contains no internal structure and is a circularly symmetric distribution of something when it has measurable intrinsic angular momentum. This paper proposes a return to a deterministic, classical electromagnetic foundation. We argue that the failure of classical physics in 1911—the “Radiation Catastrophe” predicted by Larmor’s formula—was not due to the failure of Maxwell’s equations [3], but due to an incorrect assumption about the electron’s motion through space. The model for the electron chosen in this paper is the simplest plausible object you could make from half a photon that has spin. 1 4D Electrostatics in Computational Chemistry D. A. Sinclair 1.1 The Radiation Catastrophe Resolution The Larmor formula for radiated power from an accelerating charge is: PLarmor =e2a2 6πε0c3(1) where ais the spatial acceleration. Classical orbital motion requires a= 0, leading to continuous radiation and atomic collapse on timescales of ∼10−11 seconds. However, if the electron is not a point particle orbiting in space, but an electromagnetic field configuration oscillating in time [5], then its spatial center of mass can remain stationary: rcm(t) = const ⇒a= 0 ⇒Prad ∝a2= 0 (2) This allows us to develop 4D Electrostatics, where electrons maintain interesting electromagnetic interactions without requiring spatial motion. The electron’s internal energy manifests as temporal oscillation, not kinetic energy of spatial motion. 2 The Half-Photon Electron Model 2.1 Physical Picture We model the electron as a trapped electromagnetic field configuration derived from splitting a photon at one of its potential zeros [5]. The photon follows a helical “analytic path”: P(u) = [u, |sin(u)|cos(u),|sin(u)|sin(u)] (3) where u=x−ct is the traveling wave coordinate. This path exhibits three potential zeros per wavelength: at u=−π, 0, π. When the photon splits at the central zero (u= 0), creating an electron and a positron [?], each half-photon reflects at its unbound end. The speed of light constraint forces this reflection into the time dimension, creating a temporal loop structure. The electron thus consists of a “half-photon” propagating alternately forward and backward in time, between two temporal poles separated by interval 2δt. 2.2 Temporal Pole Structure The electron’s charge −eis distributed between two temporal poles: •Forward-time pole: Charge −e/2 at time t0+δt •Backward-time pole: Charge −e/2 at time t0−δt Both poles share the same spatial location rcm, but are separated in time by 2δt. This temporal separation is not a classical motion through space—the electron’s center of mass remains fixed. Instead, it represents the internal oscillation frequency of the trapped electromagnetic field. 2.3 Energy Storage in the Half-Photon •A charge ecirculating at frequency f •In a loop of radius r=ℏ/(mec) •Creates magnetic moment µ= 2µB 2 4D Electrostatics in Computational Chemistry D. A. Sinclair No quantum axioms needed for the base value just geometry. Earlier work [5] incorrectly suggested that the electron’s rest mass mec2arose from electrostatic potential energy between the temporal poles. Calculating the required separation for V=e2/(4πε0r) = mec2yields the classical electron radius re≈2.8 fm, which contradicts the structural requirement for spin ℏ/2 and magnetic moment µB, which requires spatial extent ∼λC≈2400 fm. The electron’s rest mass energy resides entirely in the half-photon electromagnetic field oscillation, not in electrostatic self-energy between the poles. The energy is: Erest =ℏω0=mec2(4) where ω0is the internal oscillation frequency of the half-photon loop. The electrostatic interaction between poles could contribute at most a small fraction (∼α≈1/137) of the electron’s mass. This is analogous to how a photon’s energy E=ℏωresides in its oscillating electromagnetic field, not in the potential energy between its positive and negative potential regions. 3 The 4D Coulomb Law 3.1 Minkowski Spacetime Distance The effective interaction distance between two events in spacetime uses the Minkowski metric. For events at the same spatial location rbut separated in time by ∆t, the invariant interval is: d2 4D= (∆r)2−(c∆t)2(5) For static spatial separation ∆r=rand temporal separation ∆t, taking the spatial signature convention: d4D=pr2+ (cδt)2(6) 3.2 Nuclear Attraction Potential An electron’s two temporal poles at (r, t0±δt) interact with a nuclear charge +Ze at (0, t0). Each half-charge −e/2 sees effective distances: d+=pr2+ (cδt)2(7) d−=pr2+ (cδt)2(8) The total nuclear attraction potential is: Vnuc(r) = −Ze 4πε01/2 d+ +1/2 d−=−Ze2 4πε0pr2+ (cδt)2(9) Using the compact notation δ=cδt: Vnuc(r) = −Ze2 4πε0√r2+δ2(10) This potential provides natural regularization: as r→0, the potential approaches the finite value Vnuc(0) = −Ze2/(4πε0δ), preventing the classical collapse catastrophe without quantum postulates. 3 4D Electrostatics in Computational Chemistry D. A. Sinclair 3.3 Electron-Electron Repulsion For two electrons at spatial separation r12, we must account for four cross-terms between their temporal poles. Electron 1 has poles at (r1, t ±δt) and electron 2 at (r2, t ±δt). The four interaction terms are: V++ =(e/2)(e/2) 4πε0pr2 12 + 02=e2 16πε0r12 (11) V+−=(e/2)(e/2) 4πε0pr2 12 + (2cδt)2(12) V−+=(e/2)(e/2) 4πε0pr2 12 + (2cδt)2(13) V−− =(e/2)(e/2) 4πε0pr2 12 + 02=e2 16πε0r12 (14) Summing these: Vee(r12) = e2 8πε0 1 r12 +1 pr2 12 + 4δ2!(15) The electron-electron potential has a slightly modified 1/r character at small separations, preventing complete spatial overlap while maintaining long-range Coulombic behavior. 3.4 Calibration: The Temporal Radius δ0 The equilibrium temporal radius is determined by the electron’s intrinsic angular momentum requirement. As derived in detail in Section 4, the half-photon structure carries angular momentum: L=p⊥·δ=ℏ(16) From the half-photon model [6], the effective radius of the helical circulation is approximately δ. For a photon-like structure moving at speed cin its internal frame, the transverse momentum is p⊥=ℏ/δ. The energy-momentum relation for the circulating structure is: E=p⊥c=ℏc δ(17) At equilibrium (free electron), this energy equals the rest mass: mec2=ℏc δ0 (18) Solving for δ0: δ0=ℏ mec=λC 2π·π=λC 2(19) where λC=h/(mec)=2.426 pm is the Compton wavelength. Thus: δ0=λC 2= 1.213 pm (20) In atomic units (ℏ=me=e= 1, c= 137.036): δ0= 0.02293 a0(21) This value is not a free parameter—it is determined by the fundamental constants and the requirement that the electron carry angular momentum ℏ. 4 4D Electrostatics in Computational Chemistry D. A. Sinclair 4 Angular Momentum Conservation and Electron Stiffness 4.1 Physical Mechanism The electron’s temporal structure is stiff against changes in δdue to conservation of its intrinsic angular momentum L=ℏ. This stiffness is not postulated—it emerges directly from the electromagnetic field dynamics of the half-photon circulation. For a helical electromagnetic structure of effective radius δcarrying a circulating current-like flow with transverse momentum p⊥, the angular momentum is: L=p⊥·δ(22) Since angular momentum is conserved, L=ℏ= constant, which requires: p⊥=ℏ δ(23) The energy associated with this transverse momentum, for a photon-like structure propagating at speed c, is: E(δ) = p⊥c=ℏc δ(24) 4.2 Compression Energy Derivation If an external field (such as a strong nuclear potential) compresses the electron from its equilibrium radius δ0to a smaller radius δ < δ0, the energy increases hyperbolically. The compression energy is: ∆Eelastic(δ) = E(δ)−E(δ0) = ℏc1 δ−1 δ0(25) Substituting ℏc=mec2δ0(from Eq. 4): ∆Eelastic(δ) = mec2δ0 δ−1(26) This energy represents the cost of increasing the internal circulation frequency to maintain L=ℏat a smaller radius. In practical units: ∆Eelastic(δ) = 511 keV ×δ0 δ−1(27) = 511 keV ×1.213 pm δ−1(28) For small compressions (δ≈δ0−ϵwhere ϵ≪δ0), the Taylor expansion yields: δ0 δ0−ϵ= 1 + ϵ δ0 +ϵ2 δ2 0 +O(ϵ3) (29) The quadratic term gives the harmonic approximation: ∆Eelastic ≈mec2 δ2 0 ϵ2=1 2keff(δ0−δ)2(30) where the effective stiffness constant is: keff =2mec2 δ2 0≈2300 eV/˚ A2(31) 5 4D Electrostatics in Computational Chemistry D. A. Sinclair 4.3 Asymmetric Behavior: Stiff vs. Compressible Critical point: The electron is: •Stiff against stretching: For δ > δ0, the energy would decrease as E∝1/δ. However, the electron cannot stably extend beyond δ0because this would require it to emit energy (as radiation) to reach a lower energy state. In practice, δ≤δ0is a strong constraint. •Compressible: For δ < δ0, compression is possible if the potential energy gain from approaching the nucleus exceeds the elastic energy cost. This occurs naturally in high-Z atoms where the core electrons compress significantly. This asymmetry explains why: 1. The Compton wavelength sets a maximum natural size for the electron structure 2. Core electrons in heavy atoms can be significantly compressed (δ∼0.3δ0) 3. Valence electrons typically remain near δ≈δ0 5 Hydrogen: Validation of the Model 5.1 Variational Calculation For hydrogen (Z= 1), we use the standard variational method with a 1s trial wavefunction: ψ1s(r) = sZ3 eff πa3 0 e−Zeffr/a0(32) where Zeff is the variational parameter and a0= 0.529 ˚ A is the Bohr radius. The kinetic energy expectation value is: ⟨T⟩=ℏ2 2meZψ∗∇2ψ d3r=Z2 eff 2a2 0 ℏ2 me =Z2 eff ×13.606 eV (33) The potential energy with the 4D Coulomb law (Eq. 10) is: ⟨V⟩=−e2 4πε0Z|ψ(r)|21 pr2+δ2 0 d3r(34) This integral must be evaluated numerically. For the optimal Zeff: Zopt eff ≈0.9982 (35) Eground =⟨T⟩+⟨V⟩ ≈ −13.58 eV (36) Compared to the experimental value Eexp =−13.606 eV: Error = |Ecalc −Eexp| |Eexp|=0.026 13.606 = 0.18% (37) This remarkable agreement validates both: 1. The magnitude of δ0= 1.213 pm 2. The 4D Coulomb potential form 6 4D Electrostatics in Computational Chemistry D. A. Sinclair 6 Magnetic Phase-Locking: The Physical Nature of Exchange 6.1 Conceptual Foundation In standard quantum mechanics, the “exchange interaction” is a mathematical artifact arising from wavefunction antisymmetry. It has no classical analog and is often presented as a purely quantum phenomenon. However, in the half-photon model, exchange is a physical magnetic force. The temporal loop structure creates a magnetic dipole moment. For a current loop of area Acarrying current I: µ=IAˆ n(38) For the electron’s helical circulation [6], detailed field theory calculations show: |µ|=µB=eℏ 2me = 9.274 ×10−24 J/T (39) The direction of µdepends on the spin orientation: •Spin-up (↑): North pole at forward-time pole, µ= +µBˆ z •Spin-down (↓): North pole at backward-time pole, µ=−µBˆ z 6.2 Near-Field Magnetic Coupling For two electrons with antiparallel spins (↑↓), the magnetic dipoles are oriented antiparallel. In the standard dipole-dipole interaction: Vdipole(r) = µ0 4πr3[µ1·µ2−3(µ1·ˆ r)(µ2·ˆ r)] (40) For antiparallel alignment along the z-axis at separation ralong z: Vdipole(r) = µ0 4πr3(−µ2 B−3µ2 B) = −µ0µ2 B πr3(41) However, this 1/r3scaling is too weak to explain atomic bonding or Cooper pairing at distances r∼1˚ A to 100 nm. 6.3 The 1/r Scaling: Near-Field Enhancement The key insight from superconductivity [7] is that Cooper pairs bind at coherence lengths ξ∼100 nm despite Coulomb repulsion VC∝+e2/r. For a stable bound state, the magnetic attraction must scale as: Vmag(r)∝ −1 r(42) This 1/r scaling likely arises from near-field interference effects when the interacting structures are not point dipoles but extended helical currents of length ∼λC. The magnetic fields of two phase-locked helices can couple through their entire length, creating an effective interaction much stronger than the far-field dipole approximation. From the Cooper pair force balance condition: VCoulomb +Vmag = 0 at r=ξ(43) we deduce: e2 4πε0ξ=αmag ξ(44) implying αmag =e2/(4πε0) = ke(the Coulomb constant). 7 4D Electrostatics in Computational Chemistry D. A. Sinclair 6.4 Calibrated Magnetic Potential Including finite-size softening and geometric enhancement, the magnetic phase-locking potential is: Vmag(r) = −λkee2 qr2+δ2 mag ·S(s1, s2) (45) where: •λ≈0.42 is the geometric enhancement factor (from wavefunction overlap) •δmag ≈0.05a0is the magnetic softening radius •S(s1, s2) is the spin coupling function: S(s1, s2) = (1 if s1·s2<0 (antiparallel) 0 if s1·s2>0 (parallel) (46) The parameters λand δmag are calibrated from helium (see Section 7) and then applied without modification to all subsequent atoms, demonstrating the universality of the magnetic phase-locking mechanism. 7 Helium: Calibrating Magnetic Exchange 7.1 The Challenge Helium (Z= 2) presents the first critical test for exchange forces. With two electrons in the 1s orbital, we must account for: 1. Nuclear attraction: Vnuc = 2 ×(−2e2/(4πε0√r2+δ2)) 2. Electron-electron repulsion: Vee(r12) from Eq. 15 3. Magnetic phase-locking: Vmag(r12) for the antiparallel pair 7.2 Variational Setup Using a simplified Hylleraas-type wavefunction: ψ(r1, r2, r12) = Ne−α(r1+r2)(1 + βr12) (47) The total energy functional is: E[α, β, δ1, δ2] = ⟨T⟩+⟨Vnuc⟩+⟨Vee⟩+⟨Vmag⟩+ ∆Eelastic(δ1)+∆Eelastic(δ2) (48) 7.3 Without Magnetic Binding Setting Vmag = 0, optimization yields: Eno mag calc ≈ −61.2 eV (49) Eexp =−79.01 eV (50) Missing energy: ∆E≈17.8 eV (51) This 22% deficit is the signature of missing exchange energy. 8 4D Electrostatics in Computational Chemistry D. A. Sinclair 7.4 With Calibrated Magnetic Binding Enabling Vmag from Eq. 45 with: λ= 0.42 (52) δmag = 0.05a0≈0.026 ˚ A (53) the optimization yields: Ewith mag calc ≈ −79.34 eV (54) Error: |Ecalc −Eexp| |Eexp|=0.33 79.01 = 0.42% (55) The magnetic term contributes exactly the missing ∼18 eV, validating the physical picture of exchange as magnetic phase-locking. 7.5 Physical Interpretation The calibration reveals: 1. Coupling strength: Magnetic attraction is comparable to Coulomb forces at atomic scales (αmag =ke) 2. Enhancement factor: λ≈0.42 represents quantum wavefunction overlap effects in a classical framework 3. Length scale: δmag ≈0.026 ˚ A is consistent with the electron’s magnetic core structure Once calibrated from helium, these parameters are fixed for all subsequent applications. 8 Lithium: Exchange as Magnetic Attraction 8.1 The Lithium Test Lithium (Z= 3) provides a definitive test of the exchange mechanism. Standard electrostatic shielding predicts that the 2s valence electron sees an effective nuclear charge: Zshielding eff =Z−(Zcore) = 3 −2 = 1 (56) This would give an ionization energy: IEshielding =Z2 eff ×13.6 eV = 12×13.6 eV = 13.6 eV (57) Accounting for the larger orbital radius (n= 2): IEshielding ≈13.6 n2=13.6 4= 3.4 eV (58) However, the experimental ionization energy is: IEexp = 5.39 eV (59) Quantum mechanics attributes this ∼2 eV difference to “wavefunction penetration” and exchange effects. 9 4D Electrostatics in Computational Chemistry D. A. Sinclair 4. “Exchange interaction” is magnetic phase-locking between antiparallel pairs, calibrated from superconductor data 5. Pauli exclusion emerges from frequency mismatch (compressed vs. uncompressed) and half-photon path interference 6. VSEPR geometry arises from magnetic dipole packing on spherical shells The model offers O(N2) computational scaling, potentially enabling materials simulations at length scales currently inaccessible to DFT. More fundamentally, it suggests that the ”quantum” phenomena of atomic structure may be emergent properties of classical electromagnetic field configurations with proper temporal dynamics. Future work will extend this framework to molecular bonding, chemical reactions, and condensed matter systems, establishing whether the 4D electrostatic approach can serve as a practical alternative to quantum chemistry for computational materials design. Acknowledgments The author acknowledges tolerance of friends and family. References [1] E. Schr¨odinger, “Quantisierung als Eigenwertproblem,” Ann. Phys. 384, 361 (1926). [2] W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Phys. Rev. 140, A1133 (1965). [3] J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999). [4] D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press (2018). [5] D. A. Sinclair, “Maxwell’s Electron: A Massless Dynamic Field Model,” DOI 10.5281/zenodo.17019696 (2025). [6] D. A. Sinclair, “Geometric Torque and Polarization Rotation: A Structured-Photon Model for Birefringence,” (2025). [7] D. A. Sinclair, “Magnetic Coupling of Half-Photon Electrons: A Phase-Locking Model for Superconductivity,” DOI 10.5281/zenodo.17176279 (2025). [8] N. Bohr, “On the Constitution of Atoms and Molecules,” Philos. Mag. 26, 1 (1913). [9] W. Heitler and F. London, “Wechselwirkung neutraler Atome und hom¨oopolare Bindung nach der Quantenmechanik,” Z. Phys. 44, 455 (1927). [10] W. Pauli, “¨ Uber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren,” Z. Phys. 31, 765 (1925). [11] G. W. F. Drake, “High Precision Calculations for Helium,” in Springer Handbook of Atomic, Molecular, and Optical Physics, pp. 199-219 (Springer, 2006). [12] A. Kramida et al., NIST Atomic Spectra Database (ver. 5.10), National Institute of Standards and Technology, Gaithersburg, MD (2022). Available: https://physics.nist.gov/ asd [13] A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications (1996). 16 4D Electrostatics in Computational Chemistry D. A. Sinclair A Appendix A: Detailed Energy Functional for Multi-Electron Systems For a system of Nelectrons with positions {ri}, spins {si}, and temporal radii {δi}, the complete 4D Hamiltonian is: H4D= N X i=1 [Vnuc(ri, δi) + Eelastic(δi)] + X i<j [Vee(rij) + Vmag(rij, si, sj)] (84) where: Vnuc(ri, δi) = −Ze2 4πε0qr2 i+δ2 i (85) Eelastic(δi) = mec2δ0 δi−1(86) Vee(rij) = e2 8πε0  1 rij +1 qr2 ij + 4δ2 0 (87) Vmag(rij, si, sj) = −λkee2 qr2 ij +δ2 mag ×(1si·sj<0 0si·sj>0(88) Optimization proceeds by minimizing H4Dwith respect to all degrees of freedom {ri, δi} subject to spin assignments. B Appendix B: Numerical Implementation Details B.1 Units and Constants We use atomic units throughout: ℏ=me=e= 1 (89) a0= 0.529177 ˚ A (90) Eh= 27.2114 eV (Hartree) (91) c= 137.036 (92) ke= 1 (in a.u.) (93) In these units: δ0= 1/(mec)=1/137.036 = 0.02293 a0(94) keff = 2mec2/δ2 0= 2 ×137.0362/0.022932≈7.13 ×108Eh/a2 0(95) B.2 Optimization Strategy 1. Initial guess: Place electrons on spherical shells at radii rn=n2a0 2. Spin assignment: Aufbau filling with Hund’s rules 3. Variational loop: 17 4D Electrostatics in Computational Chemistry D. A. Sinclair (a) Fix {δi}, optimize {ri}(geometric relaxation) (b) Fix {ri}, optimize {δi}(compression relaxation) (c) Iterate until convergence (∆E < 10−6eV) 4. Constraints: •δi≤δ0(no stretching) •δi≥0.3δ0(prevent numerical collapse) Typical convergence requires 10 −50 iterations for atoms Z≤10. C Appendix C: Comparison with Quantum Mechanics Table 2: Conceptual Mapping: QM ↔4D Electrostatics Quantum Mechanics 4D Electrostatics Wavefunction ψ(r) EM field on analytic path Probability density |ψ|2Energy density uEM Schr¨odinger equation Maxwell equations + field structure Kinetic operator −∇2Confinement energy 1/r2 Exchange integral Magnetic phase-locking Pauli exclusion Frequency mismatch + path interference Spin ±ℏ/2 Temporal loop orientation Uncertainty principle Finite temporal extent δ0 Quantized energy levels Stable field configurations The 4D model reproduces QM results not by invoking quantum axioms, but by recognizing that electrons are structured electromagnetic fields with specific geometric constraints. The “mystery” of quantum behavior dissolves when the temporal dimension is properly included in classical electromagnetism. D Appendix D: Deriving the Magnetic Dipole moment of the Electron For the avoidance of doubt we derive the magnetic dipole moment of the proposed electron. This should end any pointless speculation on the existence of magnetic monopoles. D.1 Circulation Current Model The Maxwell Electron has a helical structure with the charge circulating in a transverse loop. This circulation creates a magnetic dipole moment. 18 4D Electrostatics in Computational Chemistry D. A. Sinclair D.1.1 Half-Photon Energy and Frequency The half-photon has energy equal to the electron rest energy: E=mec2= 0.511 MeV (96) The angular frequency is: ω=E ℏ=mec2 ℏ= 7.763 ×1020 rad/s (97) The linear frequency is: f=ω 2π=mec2 2πℏ= 1.236 ×1020 Hz (98) The period is: T=1 f= 8.093 ×10−21 s=8.093 zs (zeptoseconds) (99) D.1.2 Circulation Geometry The Compton wavelength determines the scale: λC=h mec=2πℏ mec= 2.426 pm (100) The circulation radius is: rcirc =λC 2π=ℏ mec= 0.386 pm (101) The circulation area is: A=πr2 circ =πℏ mec2 = 4.685 ×10−25 m2(102) D.1.3 Effective Current The charge ecirculates with period T/2 (half-photon completes one loop): I=e T/2= 2ef = 2e·mec2 2πℏ=emec2 πℏ(103) Numerically: I= 39.59 A (104) D.2 Magnetic Moment Calculation The magnetic moment of a current loop is: µ=I×A(105) 19 4D Electrostatics in Computational Chemistry D. A. Sinclair Substituting our expressions: µ=emec2 πℏ×πℏ mec2 (106) =emec2 πℏ×πℏ2 m2 ec2(107) =eℏ me (108) = 2 ×eℏ 2me (109) = 2µB(110) where µB=eℏ/(2me) is the Bohr magneton. D.3 Result µ= 2µB(111) This exactly reproduces the Dirac prediction of g= 2 for the electron magnetic moment! Table 3: Magnetic Moment Comparison Source Moment Difference Maxwell Electron (circulation) 2.0000 µB— Dirac equation prediction 2.0000 µB0.0000 µB Experimental (with QED) 2.0023 µB0.0023 µB The small anomalous magnetic moment (g−2) ≈0.0023 may result from the helical path being slightly deformed as it resists the compressive force exerted by stretching spacetime. Or there could be some rest mass associated with the stress in space time associated with the discontinuity. You are free to invent your own explanation as the discrepancy is not thought large enough to influence 4D electrostatics calculations needed for chemistry. The magnetic moment is not a mysterious “intrinsic spin” - it emerges naturally and inevitably from the geometric circulation of the electromagnetic field in the helical structure. This is pure classical electromagnetism: 20