Counting whole electrons exposes impossible molecular calculations

A fractional average is not the problem

When a molecule pulls apart, approximate calculations can assign its fragments electron statistics that no physical state could produce. The difficulty is not simply a fractional average: a region that contains two electrons half the time and three electrons half the time legitimately averages two and a half.

The problem arises when a calculation combines that average with too little fluctuation. Any integer-valued count averaging two and a half must have a variance of at least one quarter. A prediction below that threshold cannot describe actual electron counts.

In an arXiv preprint, Maho Nakata develops this observation into a hierarchy of mathematical checks for molecular calculations. The work combines proofs, constructed counterexamples and numerical calculations—not laboratory experiments.

Checking pairs without tracking everything

The target is variational second-order reduced-density-matrix theory. Rather than work directly with the full many-electron wavefunction, this approach optimizes a smaller mathematical object containing information about electron pairs. Necessary consistency conditions restrict that optimization, but incomplete conditions can admit impossible candidates and energies that are too low.

Nakata examines the real-space pair density: a description of where pairs of distinct electrons occur. Its one-electron counterpart describes the average spatial distribution of electrons. The central question is whether a proposed pair density can arise from any probability distribution of a specified whole number of electrons.

The method divides space into cells and lists every possible allocation of electrons among them. A linear program—a constrained optimization problem—then asks whether probabilities assigned to those allocations can reproduce both the proposed average populations and pair counts.

If not, the program produces an inequality that all valid allocations obey but the candidate violates. Adding this inequality as a constraint, called a cut, excludes the offending candidate from the molecular calculation.

Finer cells give stronger tests

A region and its complement recover the basic integer-count checks used in the preceding paper. More cells can reveal inconsistencies that every individual-region test misses. Jointly tracking up-spin and down-spin electron counts can also detect failures hidden when those counts are checked separately.

The theoretical hierarchy becomes tighter as cells are subdivided. Each level retains every representable pair density, while progressively excluding impossible candidates. Under sufficiently fine refinement, the hierarchy characterizes pair densities that admit a symmetric many-particle probability-measure extension.

The optimization also measures a discrepancy, rather than merely returning a pass or fail. At a fixed one-electron density, its optimum is the smallest total absolute change in the cell-pair counts needed to make them consistent. It supplies a lower bound on the corresponding change needed in the full pair density. A constructive approximation supplies an upper bound, separated from the lower one by a coarse-graining error that vanishes under refinement.

An impossibility that standard checks miss

To show that these constraints add genuinely different information, Nakata constructs candidate data for five same-spin electrons in ten Gaussian-type orbitals arranged around two centers. The data pass all positivity conditions through degree three, including established conditions called T1 and T2.

Nevertheless, a half-space contains an average of two and a half electrons with an inferred variance of about 0.01125, far below the required 0.25. No five-electron probability distribution can reproduce that pair density.

This is a deliberately constructed mathematical counterexample, not evidence that a particular molecule selects exactly those data.

Better energies, with important limits

Numerical molecular tests show practical improvements. For the positively charged nitrogen molecule at a separation of four ångströms, the basic relaxation gives each atomic region the correct average of six and a half electrons, but too little count fluctuation. Cuts restore the minimum allowed variance. The energy error relative to full configuration interaction in the same minimal orbital basis falls from about 138 to 22 millihartree; a millihartree is one thousandth of a hartree, an energy unit used in atomic calculations.

These numerical results are not certified error bounds. Nor does passing every tested partition prove representability: a finer partition can still expose a violation. Enumerating allocations also grows rapidly more expensive as cells are added.

Most importantly, even the complete spatial hierarchy does not guarantee the exact quantum energy. It checks pair-density probability consistency, not all compatibility requirements involving the one-particle reduced density matrix, off-diagonal information or kinetic energy. The preprint therefore offers a systematic way to diagnose and exclude one class of impossible electron statistics—not a complete solution to molecular electronic structure.