Single-Point Energies and Properties

A single-point calculation evaluates a model at one supplied geometry without optimizing atomic positions or the periodic cell. “Single point” describes the nuclear-coordinate operation, not a cheap or incomplete electronic calculation: a self-consistent calculator may still perform many electronic iterations, and requested analyses can cost more than the underlying energy evaluation.

For a structure $(\mathbf R,\mathbf h)$ and calculator state $\mathcal M$, the central value is

When implemented, the same evaluation supplies coordinate and strain derivatives,

The coordinates remain fixed even when the forces are large. A single point therefore answers “what does this model predict here?” It does not answer “where is the nearby minimum?”

Fixed geometry as a scientific condition

The geometry may be fixed because it is experimentally measured, generated by another method, selected from a trajectory, placed along a scan, or intentionally held away from equilibrium. That provenance determines how the result may be interpreted.

A large force on an unrelaxed imported structure is not necessarily a calculation failure. It is evidence that the structure is not stationary on the chosen surface. Conversely, a small force does not prove global stability; it only describes the local gradient and can be influenced by constraints, numerical noise, or a flat surface.

When comparing fixed geometries, keep the geometry policy explicit. Evaluating several conformers at their own optimized geometries compares relaxed states. Evaluating them all at geometries optimized by another method isolates a different question: the response of the single-point model to a common externally prepared geometry set.

Property routing

Energy and forces form the structural core retained by Tako. Other channels depend on the calculator and system:

Machine-learned interatomic potentials in Tako provide structural energy/derivative channels, not orbitals, charge populations, electron densities, or excited states. GFN2 supplies additional electronic analyses where the operation implements them. A property is available only when the calculator provides it.

Energy components and energy zeros

The returned total depends on the calculator definition. A semi-empirical electronic result can include repulsion, electronic band terms, electrostatics, dispersion, solvation, and finite-electronic-temperature contributions according to its selected variant. An MLIP reproduces the label convention learned from its training data. The numerical zero is therefore method-specific.

Absolute totals from unlike calculators must not be subtracted. Within one calculator, a difference is interpretable only when atom counts, composition, charge/spin state, periodic normalization, model asset, and relevant settings form a balanced comparison. For a reaction written as $\sum_i\nu_i A_i=0$,

is meaningful only if every term belongs to the same declared electronic-energy convention and the stoichiometric coefficients balance the modeled particles. Thermal, entropic, standard-state, electrode, or reservoir terms are not created by the subtraction.

Forces as diagnostics

The maximum atomic-force norm,

is a compact diagnostic, but it hides which atoms and directions dominate. Inspect the force vectors or per-atom values when a result matters. A single close contact can dominate $F_{\max}$; equal maximum values can describe completely different spatial patterns.

For constrained structures, Tako stores constraint-applied forces in forces_ev_per_angstrom. When the structure has selective-dynamics constraints, the runtime obtains forces through the constrained atoms path rather than publishing the calculator’s unmodified raw force array. Frozen components are therefore absent from this stored diagnostic; the single-point result does not provide a second raw-force field. This convention must be distinguished from geometry optimization traces whose convergence implementation can count unmasked anchor forces.

Energy and force consistency is especially important for dynamics, optimization, and path calculations. A conservative model derives forces from the energy gradient. A direct-force MLIP may predict an energy and forces through separate heads; successful inference does not guarantee exact integrability or zero work around a closed loop.

Periodic stress

Stress is a tensor, not a single pressure. Hydrostatic pressure is derived from the trace under a chosen sign convention; shear components describe shape-driving response. Before comparing stress values, verify tensor order, units, tension/compression sign, cell convention, and whether the calculator returns Cauchy stress or a related virial quantity.

A molecule placed in an arbitrary large periodic box does not acquire a useful bulk stress merely because a tensor is returned. Stress interpretation requires a physically meaningful periodic volume. For slab models, vacuum makes three-dimensional volume normalization dependent on the chosen cell height; surface stress requires an appropriate two-dimensional convention.

Electronic populations and densities

Atomic charges are assignments produced by a partitioning rule. They are useful for trends within one consistent protocol—such as charge redistribution across a homologous series—but different population schemes need not agree atom by atom. Excess digits do not remove this model dependence.

DOS and PDOS plots combine discrete or sampled states with a chosen energy alignment and broadening. The plotted curve is derived data. Preserve the underlying state energies, weights, occupations, projection labels, and broadening parameters when making a quantitative claim.

Real-space grids add spatial discretization. A cube value is meaningful only together with origin, axes, dimensions, spacing, units, and source structure. Comparing two density grids requires compatible registration or an explicitly defined interpolation. Visual isosurfaces are thresholded views and can change dramatically when the isovalue changes.

Self-consistent convergence

For an electronic calculator, a single point can fail before any requested analysis is produced. Self-consistent iterations seek a density or charge state compatible with the Hamiltonian it generates. Difficult geometries, small gaps, open-shell states, inappropriate occupations, strong charge separation, or inconsistent charge/spin choices can destabilize the iteration.

An SCF threshold is a numerical condition on successive electronic states or energies; it is not a bound on chemical accuracy. Loosening it may produce a result while degrading gradients and derived properties. Before modifying convergence controls, inspect geometry, electron count, charge, spin, occupations, and whether the selected method is suitable.

Downstream analyses must be associated with the converged electronic state actually used. If the energy converges but a requested post-processing stage fails, distinguish partial electronic completion from a complete property contract.

Sequential analyses and partial results

After the core energy/force evaluation, the GFN2 runtime processes requested analyses sequentially in this order: DOS, PDOS, charges, electron-density cube, charge-density cube, NCI cubes, frontier-orbital cubes, spin density, ESP, density derivatives, Fukui functions, ELF, and WFN. It emits a cumulative partial result after the core evaluation and after each successfully completed analysis. The MLIP branch emits its structural result as a partial result before normal completion.

A later analysis can fail after earlier stages succeeded. Such a failure rejects the overall worker operation; it does not convert the cumulative partial result into a successful final result. Artifacts already promoted from partial data may remain useful for diagnosis, but their presence does not establish that the complete requested property contract finished. Preserve the log, identify the failed stage, and label any retained earlier output as partial.

Cancellation terminates the active request and provides no restartable electronic or post-processing checkpoint. A new single point begins the calculator evaluation again, even if an earlier partial energy or DOS artifact remains in the workspace. Reset and abnormal worker completion likewise do not provide numerical resume state.

A comparable fixed-geometry series

Consider an adsorption scan in which one fragment is translated relative to a frozen surface. A defensible series keeps the surface cell, atom identities and ordering, calculator asset, charge/spin state, property request, and numerical thresholds fixed. The independent variable—the displacement or separation—must be stored with each geometry.

The resulting curve is a constrained fixed-geometry scan. Its minimum is the lowest sampled point under that construction, not a relaxed adsorption geometry. Forces perpendicular to the scan coordinate reveal whether hidden degrees of freedom strongly favor relaxation. A subsequent constrained or unconstrained optimization answers a different question and should be labelled as a separate stage.

Failure and acceptance

Acceptance requires a correct fixed structure, a suitable and fully identified calculator, completed numerical evaluation, available authoritative artifacts, and an interpretation consistent with the property’s definition. The interface procedure is Run a Single-Point Calculation; the operation contract is Single-Point Calculations with Tako Script; calculator construction is in Calculator Construction and Validation; exact generated settings are in Single-point settings.

References

1. F. Jensen, Introduction to Computational Chemistry, 3rd ed., Wiley (2017).
2. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, Dover (1996).
3. A. Hjorth Larsen et al., “The atomic simulation environment—a Python library for working with atoms,” Journal of Physics: Condensed Matter 29, 273002 (2017).
4. E. R. Johnson et al., “Revealing noncovalent interactions,” Journal of the American Chemical Society 132, 6498–6506 (2010).