Forces between molecules

The force field, term by term

A molecular-dynamics program moves atoms by a force field: springs for bonds, springs for the angles between them, a twist term for each bond, charges, and a pair force between atoms that aren't bonded. In our world that pair force is usually Lennard-Jones, 4ε[(σ/R)¹² − (σ/R)⁶]. Each term comes from somewhere in 3D physics. Here is each one worked out again with the force pinned at 1/r. All of it follows from one fact: the pinned potential is the Fourier transform of 4π²/|k|³ in 4D.

So the pair force for the molecular-dynamics view is V(R) = A e^(−βR) − B/R⁵ − C₆/R⁶, plus charges where there are any.

Neutral atoms attract, permanently

A neutral atom's charge, seen in Fourier space, starts at S k²/8, where S = Σ⟨r²⟩ over its electrons. Two such atoms meet through S_A S_B k⁴/64 times 4π²/|k|³, which is proportional to |k|, and |k| transforms in 4D to −3/(4π² R⁵). So any two neutral atoms attract as −3 S_A S_B / 64 R⁵. Nothing fluctuates and nothing is polarized: it's the plain electrostatics of two spherical clouds, the shell theorem's failure again. In 3D the same step gives k⁴·4π/k², which has no reach at all. That's why our neutral spherical atoms don't feel each other until they touch.

Checked against an exact calculation: two model atoms (a nucleus in a Gaussian cloud) whose 4D interaction is integrated directly come within 1.3% of the formula at twenty-four cloud-widths apart, closing in as they separate. Closer in, the exact pull is stronger than the formula (by 14% at eight cloud-widths), so the formula undersells it at contact.

At the distances where our noble-gas atoms sit together, this permanent pull alone is many times deeper than their whole van der Waals well. The 4D noble gases would hardly be gases.

That assumes 4D atoms sit about where ours do. Computed, they sit a little closer, 4.5 to 5.25 bohr apart.

Dispersion: still R⁻⁶, a little stronger, computed

Two atoms' electrons jiggle, and each jiggle polarizes the other: London's force. Two dipoles meet through ∂ᵢ∂ⱼ(1/R) = (3R̂ᵢR̂ⱼ − δᵢⱼ)/R³, which is the same in any dimension because the potential is 1/r. So dispersion stays −C₆/R⁶. Its strength comes from the sum of the squares of that tensor's parts: (−2)² along the line between the atoms plus 1 for each direction across it. That's 6 in our world and 7 in 4D, so London's 3/2 becomes 7/4. Checked by solving two coupled 4D oscillators exactly: their zero-point energy drops by 7/4's prediction to one part in ten million.

C₆ itself comes from how easily each atom polarizes at every frequency, by Casimir and Polder: C₆ = (tr T²/2π) ∫ α_A(iω) α_B(iω) dω, which is 3/π in our world and 7/2π here. We compute α(iω) by the coupled response. The field distorts each orbital, the distorted cloud pushes back on every electron, and the two are solved together, with each electron's push on itself taken out, as in the ground state. The checks: hydrogen comes out exact in both worlds (α = 4.5, and 34.805 in 4D, which the same closed form gives). At high frequency α → (number of electrons)/ω², as the Thomas–Reiche–Kuhn sum rule says. And helium gives 1.322, exactly time-dependent Hartree–Fock's value. Against our measured noble gases:

α within 9% and C₆ within 11%, on both sides. So, in 4D:

Two atoms, computed

To know where two atoms sit and how hard they hold on, the repulsion has to be computed, and that means two clouds overlapping. We use Gordon and Kim's electron-gas model on the solver's own atoms. Each atom keeps its cloud. Their electrostatics is computed exactly, nucleus and cloud, with the 4D pull at every point. The overlap costs the extra kinetic energy and gains the extra exchange energy of an electron gas at the summed density. Those need the free electron gas's formulas in four dimensions, which we derived: kinetic (4π/3)ρ^(3/2), exchange −16/(15√π) ρ^(5/4). They agree with the general-D formulas collected by Loos and Gill (2016). The same derivation in 3D gives Dirac's 0.7386. Dispersion is added from the computed C₆ (damped close in, as Tang and Toennies do). Correlation is left out.

First, in our world, end to end, every part computed, against the measured wells:

It puts the wells within 5% of where they are, but makes them too deep, by 1.4 to 1.8 times. (Helium, a known weak spot of the method, is left out.) So below, each 4D well's depth is read as a range: the computed depth divided by 1.8, up to the computed depth.

The wells are deep, and what digs them is electrostatics. At the bottom of each, electrostatics alone pulls harder than the whole well is deep, and the overlap's repulsion holds the atoms apart. From element 12 on, dispersion adds a quarter to two-fifths more. By the rule our noble gases follow, boiling at 0.7 of the well depth, 4D's second noble gas, element 12, would boil somewhere between 400 and 730 K, so it's a liquid or a solid at room temperature. That rule undercounts, though. In 4D most of a noble gas's hold comes from beyond its nearest neighbours, which the well depth leaves out. Simulated (chapter 12), element 12 condenses below 4,700 to 8,900 K, and by the same reckoning element 22 below about 10,000 to 40,000 K. These are noble gases only in having full shells. Element 2, 4D's helium, is the one that might stay a gas at room temperature.

The repulsion's steepness agrees with the rule β ≈ 2√(2I) to within 20%. The fitted A, β and B, with the computed C₆, give the molecular-dynamics view its formula.

Twisting

Across a bond, each end's other bonds point to the corners of a regular simplex of the space across it: a triangle in 3D, a tetrahedron in 4D. Over 200,000 random twists of the far end, the least energy is the far tetrahedron inverted through the bond (staggered) and the most is the two lined up (eclipsed).

Salt water screens by a power law

In salt water the ions crowd round any charge and cancel its field. In our world the cancelled field fades exponentially, e^(−κr)/r: two ions a few nanometres apart in seawater barely feel each other. That exponential needs the transform to be smooth in k², and 4π²/|k|³ isn't. Screening gives 1/(|k|³/4π² + λ) = 1/λ − |k|³/(4π²λ²) + …, and the |k|³ term leaves a tail of −r⁻⁷, of the opposite sign to the charge. In this universe a dissolved ion is overscreened: past a few screening lengths it looks faintly like its opposite, and the effect fades only as a power.

The tail comes from the Fourier asymptotics of a non-smooth transform, the same machinery that gives the R⁻⁵ above, which the exact two-atom calculation confirms. The salt case itself isn't integrated numerically: its tail is far too small to resolve beside the 1/r it cancels.

What the molecular-dynamics view gets, and what's open

Ready: the bonded terms with 4D's angles and three-way twists, charges at 1/r, the permanent R⁻⁵ with S from the solver, and, for hydrogen and the noble gases, dispersion's C₆ and the repulsion's A and β, all computed. Open: open-shell atoms' polarizabilities (the response assumes closed shells; that's next, for Pxygen), pairs of unlike atoms and molecules (the same methods, more runs), correlation, and retardation (in our world dispersion turns to R⁻⁷ beyond a few hundred bohr; through the lens it hasn't been worked out). Hydrogen bonds are computed, for Xater, in the next chapter. Also open: the three-body forces that make our liquids non-additive.