Comparable Energies and Thermodynamic Cycles

An energy difference is meaningful only after the quantity, reference states, coefficients, and method protocol are defined. A reaction energy requires a balanced stoichiometric cycle. A binding or adsorption energy requires explicit separated references and a deformation policy. A voltage requires an electron-transfer convention. A Gibbs energy requires thermal, entropic, and standard-state terms that are absent from a bare electronic calculation.

Tako returns calculator/model energies for fixed geometries and optimized endpoints. Molecular vibration analysis separately returns harmonic zero-point energy (ZPE). Tako does not currently return translational, rotational, vibrational thermal, or electronic entropy contributions; enthalpy; Gibbs free energy; symmetry-number corrections; quasi-RRHO corrections; standard-state conversion; or a combined $E+\mathrm{ZPE}$ field. The energy_ev label carries none of those quantities.

Energy is a model-defined state function

For nuclear geometry $\mathbf R$, charge $q$, spin state $S$, method $M$, and environment/model settings $\Theta$, write the returned energy as

The subscript and arguments matter. GFN2 and an MLIP use different energy zeros and approximations. GFN2 with and without dispersion or implicit solvent defines different effective surfaces. Two MLIP assets may be trained to different reference energies. Even within one method, changing charge, spin, cell, composition, or periodicity changes the physical state.

Absolute total energies from unrelated methods must not be subtracted. Method comparison should construct the same balanced difference independently within each method:

then compare $\Delta E_{M_1}$ and $\Delta E_{M_2}$. Mixing $E_{M_1}$ and $E_{M_2}$ inside one cycle destroys cancellation of method-specific energy zeros.

Fixed-geometry and optimized energies

A single point evaluates one supplied geometry. A relative conformer energy at fixed geometry protocol is

The geometries may have been generated by another method, but that fact must be recorded. Evaluating several methods on identical geometries isolates energy-model differences; optimizing separately under each method compares complete method protocols, where both surface shape and energy expression change.

An optimization result’s final_energy_ev is the energy of its final accepted geometry whether or not the optimizer converged. Scientific use requires converged: true and the requested force gate, plus structural inspection. The public tako.analysis.energyEv helper rejects explicit top-level converged: false; direct field access does not.

An MD final_energy_ev is an instantaneous endpoint potential energy and is accepted by the generic helper. It is not an ensemble average, enthalpy, or free energy. Comparing MD endpoint energies as though they were optimized minima is generally invalid.

Reaction-energy bookkeeping

For a reaction

use signed stoichiometric coefficients ($\nu_i>0$ products, $\nu_i<0$ reactants):

Before evaluating, construct a balance table:

Every element must balance. Net charge and electron bookkeeping must match the modeled process. When electrons, protons, bulk atoms, gas reservoirs, or solvent species are implicit, define their chemical potentials rather than silently omitting them.

Tako’s tako.analysis.reactionEnergy performs only

It does not validate finite values, units, coefficients, species, composition, charge, spin, method, convergence, or balance. Multiplicity must be represented by repeated/scaled values before the call. Empty arrays return zero; nonfinite values propagate. The helper is arithmetic, not a thermochemistry engine.

Binding and association energies

For association $A+B\rightarrow AB$,

Under this convention, negative means electronically/model-energetically favorable association. The sign convention must be stated; some fields report positive dissociation energy instead.

Reference geometries define the question. If $E_A$ and $E_B$ are separately relaxed, the result includes deformation of fragments upon binding. A frozen-fragment interaction energy uses monomer geometries extracted from the complex:

The deformation contribution is

and $\Delta E_{\mathrm{bind}}=\Delta E_{\mathrm{int}}+\Delta E_{\mathrm{def}}$ under consistent definitions.

For atom-centered basis methods, basis-set superposition error and counterpoise conventions can be important. Tako’s current GFN2/MLIP workflows expose no counterpoise correction. Periodic adsorption adds cell, coverage, slab thickness, vacuum, dipole, and lateral-interaction policies.

Adsorption energies

A common adsorption cycle is

All terms must use compatible calculator/model settings. The clean slab should use the same cell, atom constraints, and surface stoichiometry as the combined system. The adsorbate reference might be an isolated molecule, half a gas molecule, or a chemical-potential reservoir; each defines a different number.

Coverage is part of the state because periodic images interact. Comparing sites is most robust within an identical supercell and adsorbate count. A site ranking can change after relaxation; unconverged or collapsed candidates must be rejected rather than sorted.

Tako has no public adsorption-energy helper. Example workflows compute this expression in ordinary JavaScript. The workflow exporter currently degrades some adsorption-selection nodes (Known Limitations), so exported application scripts are code requiring inspection.

Voltage and intercalation cycles

For insertion of $x_2-x_1$ alkali atoms between host compositions, a zero-temperature model-energy voltage against bulk metal is commonly

If energies are in eV and $z$ is transferred electrons per inserted atom, division by the electron count gives volts numerically. This remains a model/electronic-energy voltage, not necessarily the finite-temperature equilibrium voltage $-\Delta G/(nF)$.

The metal reference must be calculated consistently and normalized per atom. Host phases require consistent composition, cell/relaxation policy, and magnetic/electronic states. A lower convex hull is constructed from formation energies versus composition; off-hull structures are metastable under that model. Tako’s lowerHull helper operates on supplied composition/energy data, not raw structures, and does not validate phase provenance.

Harmonic zero-point energy

For real harmonic vibrational frequencies $\omega_k$,

Tako molecular vibration output returns zero_point_energy_hartree and zero_point_energy_ev inside result.vibrations. It does not return an electronic energy in the same vibration result and does not combine the stages. Construct

only when both calculations use the same final geometry, method surface, charge, spin, constraints, and frequency interpretation.

Imaginary modes are not stable harmonic oscillators. A minimum used for thermochemistry should have no meaningful imaginary internal frequency. Near-zero translations/rotations and constrained modes require a declared policy. Tako reports harmonic ZPE but no quasi-harmonic or hindered-rotor replacement.

For a reaction or binding correction,

ZPE can be significant for bond breaking, proton transfer, isotope effects, and hydrogen-rich association. It is not a universal constant shift.

Thermal internal energy and enthalpy

Ideal-gas molecular thermochemistry commonly decomposes thermal internal energy as

Enthalpy is

For one ideal-gas molecule, the molar $pV$ term corresponds to $RT$. Condensed phases, periodic solids, surfaces, and solvated species require different treatment. Adding ideal-gas translational terms to a periodic slab is physically wrong.

Tako does not calculate these thermal contributions or $H(T)$. The finite electronic-temperature/free-energy component inside a calculator is not molecular nuclear thermochemistry and is not a Gibbs energy.

Entropy and Gibbs free energy

At temperature $T$ and pressure/standard state $p^\circ$,

An ideal-gas rigid-rotor/harmonic-oscillator entropy includes translational, rotational, vibrational, and possibly electronic terms:

This requires molecular mass, moments of inertia, rotational symmetry number, electronic degeneracy, temperature, pressure, and a policy for low frequencies. Very low harmonic modes make RRHO vibrational entropy grow unphysically; quasi-RRHO, quasi-harmonic, or hindered-rotor models regularize them with additional assumptions.

Tako returns frequencies but no entropy or Gibbs correction. It does not determine rotational symmetry numbers, distinguish linear/nonlinear rotor partition functions for thermochemistry, or expose temperature/pressure in vibration settings. Any reported $\Delta G$ must therefore come from an external documented procedure.

Standard states and solution corrections

Gas-phase standard pressure and solution standard concentration are not interchangeable. At ideal conditions, converting a species from gas standard state $p^\circ$ to solution standard concentration $c^\circ$ introduces a term involving $RT\ln(c^\circ RT/p^\circ)$, with sign and stoichiometric multiplication depending on the convention. Association reactions are especially sensitive because the number of independent molecules changes.

Implicit-solvation energy from a single-point method is not by itself a complete solution free energy. A cycle can require gas-phase thermal terms, solvation free energies for each species, standard-state conversion, conformational populations, and concentration/protonation equilibria. Use one coherent convention for every term.

Conformer and phase ensembles

When several conformers $i$ are thermally accessible, the ensemble free energy is

where $g_i$ accounts for justified degeneracy. This ensemble free energy weights every accessible conformer, not only the lowest model energy. Missing conformers bias the result, and duplicate conformers should not be counted as independent degeneracy.

For solids, phase stability uses free energy per consistent formula unit and may require phonon free energies, configurational entropy, magnetic/electronic contributions, pressure, and finite-size convergence. Tako’s current phonon result provides harmonic mode information but not a temperature-dependent phonon free-energy table.

Unit discipline

Tako stores core energies in Hartree and eV; the exact Script conversion used by energyEv is 27.211386245988 eV per Hartree. Preserve full precision in machine-readable artifacts and round only reports.

Useful conversions include

An energy difference in eV per transferred elementary charge is numerically volts. This identity does not decide electron count or make an electronic difference a Gibbs energy.

Uncertainty and significant figures

Energy differences carry model error, numerical error, structure/reference error, sampling error, and thermodynamic-model error. Tight SCF or optimizer convergence reduces only numerical components. Displayed precision can exceed the method’s chemical accuracy.

Report sensitivity to relevant conformers, cells, constraints, state choices, grid/model versions, and correction schemes. When comparing a series, correlated errors may cancel; when combining unrelated species, cancellation can be weak. Validate representative cycle terms against a more defensible method or experiment when possible.

A defensible reporting table

If only Tako model energies are used, say “model-energy difference at 0 K without ZPE, thermal, entropic, or standard-state corrections.”

The interface procedure is Read and Compare Energies. Programmatic helper contracts and safe cycle construction are documented separately in Energy Analysis with Tako Script.

References and credit

• S. Grimme, “Supramolecular Binding Thermodynamics by Dispersion-Corrected Density Functional Theory,” Chem. Eur. J. 18, 9955–9964 (2012), DOI: 10.1002/chem.201200497, for low-frequency/association thermochemistry context.
• C. J. Cramer, Essentials of Computational Chemistry, 2nd ed., Wiley (2004), for statistical-thermodynamic cycles, standard states, and interpretation.