Chemistry · Lesson

Approximate electronic-structure methods and DFT

Separate electron/spin inputs, a fixed variational model, balanced reaction accounting and numerical convergence.

By the end, you can

Count electrons and selected spin occupations separately.
Compare nested trial spaces for the same Hermitian Hamiltonian.
Balance an assigned reaction before combining electronic and thermal contributions.
Distinguish numerical refinement from a change of physical model.

An electronic-structure calculation starts with a physical model

An electronic calculation needs nuclear identities and positions, electron number, spin specification and other relevant conditions. A commonly used fixed-nuclei approximation treats an electronic problem for chosen nuclear positions. Nuclear motion or other effects require additional treatment.

A charge state and spin state are different specifications. A reported electronic energy also needs the method, reference convention and numerical settings before it can be compared with another result.

Input states and variational comparisons need a fixed problem

The CO occupation example specifies charge and multiplicity independently. Removing an electron changes the total electron count; selecting a different multiplicity changes the spin account. Neither operation supplies an energy ordering from bookkeeping alone. The fixed-nuclei positions, interactions and electronic method would also be needed for a calculation.

The separate two-state matrix illustrates the variational principle in its restricted setting. A larger nested trial space offers every previously allowed vector, so its minimum cannot be higher for the same Hamiltonian. A strict lowering requires useful additional freedom; zero coupling in the practice case gives no lowering.

Approximate exchange-correlation treatments, pseudopotentials and numerical representations can change the problem or its implementation. A variational statement about one fixed Hermitian operator cannot certify every comparison carrying the same software or method label.

Approximation and numerical representation have different roles

A finite basis or grid represents functions numerically. The chosen physical approximation additionally decides how interactions and correlation are treated. Improving the numerical representation does not automatically remove every error of the physical model.

A convergence check changes appropriate numerical settings while holding a defined target problem. Agreement within a tolerance supports that check, rather than exactness for every property. Compare energies on a consistent basis and chemical account.

A result can be numerically consistent and still have limited meaning

The formal nitrile-to-amine account retains the full atom, charge and electron inventory, including two H₂ molecules. Assigned electronic energies must be combined on that stoichiometric basis. Standard-state and thermal corrections then answer a different question from the bare electronic-energy difference.

In the original refinement case, the exact limiting expression is given, allowing both successive changes and known toy errors to be calculated. Actual calculations usually lack that privileged knowledge. A small change between two settings supports a specific convergence check; it does not generally prove an exact error bound.

Numerical convergence within one physical approximation leaves that approximation’s limitations. A separate model can converge to a different limit. Suitable comparisons should identify the target property, chemical state, reference conventions and uncertainty of any reference data.

Successive change and known toy error use different reference values

Refinement g	Assigned E / E_h	Known error / kJ mol⁻¹	Change from prior row / kJ mol⁻¹
2	-76.412	21.003997	No prior row
4	-76.418	5.250999	15.752998
8	-76.4195	1.312750	3.938249
16	-76.419875	0.328187	0.984562
32	-76.41996875	0.082047	0.246141

Original assigned E(g)=−76.420+0.032/g² E_h. The limiting error is available because the function is stipulated; no actual software convergence is claimed. Values use the CODATA 2022 central Hartree conversion.

Density functional methods use a density-based framework

Density functional theory relates ground-state properties to an electron density within its formal setting. Practical calculations use approximations and numerical procedures, including a chosen exchange-correlation treatment. Different approximations can perform differently across systems and properties.

Orbital-based implementations provide useful computational constructions. An orbital value, ground-state energy difference, free energy and measured spectral transition are different quantities. Additional assumptions may be needed to connect them.

An electronic result needs a tested interpretation

A calculation can compare defined structures, states or reactions under its method. A reaction energy requires a balanced chemical account and compatible reference states. Geometry, electronic energy, thermal corrections, solvent treatment and other contributions must be kept identifiable.

Use suitable reference data or higher-level comparisons where relevant, while recognizing their own uncertainty and scope. The original comparisons below are assigned energies and convergence sequences; they do not constitute a validated calculation of a real reaction or spectrum.

Worked examples

Charge and spin are separate inputs

For fixed CO nuclei, compare neutral CO with selected singlet multiplicity and CO⁺ with selected doublet multiplicity. Use the collinear highest-M_S occupation convention. What electron and spin counts follow?

Nuclear charges sum to 6 + 8 = 14. N_e=ΣZ−Q gives 14 electrons for neutral CO and 13 for its cation. Formal CO→CO⁺+e⁻ retains C/O nuclei and balances charge: 0=+1−1.
Multiplicity is 2S+1. The selected singlet has S=0; n_α+n_β=14 and n_α−n_β=0 give 7/7. The selected doublet has S=1/2; sum 13 and difference one give 7/6.
These selected inputs do not calculate a ground state, relative stability or ionization energy. This occupation account alone also does not establish an eigenstate of S² or include spin-orbit coupling.

Neutral: 14 electrons, selected S=0, occupations 7/7. Cation: 13 electrons, selected S=1/2, occupations 7/6.

Enlarging one fixed trial space

Assign the real Hermitian matrix H=[[-1.85,−0.18],[−0.18,−1.42]] E_h in two orthonormal symbolic basis states. Compare the first basis vector alone with the full two-state space.

The one-dimensional Rayleigh quotient is −1.85 E_h. For a=−1.85, d=−1.42 and b=−0.18, eigenvalues are (a+d)/2 ± √[((a−d)/2)²+b²].
The full-space values are −1.915401498 and −1.354598502 E_h. Their sum is −3.27 E_h and product 2.5946 E_h², matching trace and determinant.
The minimum lowers by 0.065401498 E_h. This comparison uses the same Hamiltonian and nested trial spaces. It does not state that arbitrary changes of functional, model or numerical settings in approximate density-functional calculations must lower every energy.

The two-state minimum is −1.915401498 E_h; the fixed nested-space comparison lowers it by 0.065401498 E_h.

A reaction energy needs a balanced account

Use the formal neutral account CH₃−C≡N + 2 H−H → CH₃−CH₂−NH₂. Assign compatible gas-phase toy energies −132.314, −1.113 and −134.551 E_h for ethanenitrile, H₂ and ethanamine. At 320.0 K assign ΔZPE=+6.80 kJ mol⁻¹, further thermal enthalpy +4.60 kJ mol⁻¹ and ΔS=−0.0520 kJ mol⁻¹ K⁻¹. Find ΔE_el, ΔH and ΔG per written reaction.

Both sides contain C₂H₇N, charge zero and 26 electrons. Reactants have 22 + 2×2 = 26 electrons; product has 2×6 + 7 + 7 = 26. This full formal balance does not give a reaction procedure or actual yield.
ΔE_el=−134.551−(−132.314)−2(−1.113)=−0.011000 E_h. With the CODATA 2022 central Hartree value and exact N_A, E_hN_A/1000≈2625.49964 kJ mol⁻¹, giving −28.8804960 kJ mol⁻¹. The Hartree value is a measured-constant estimate, not an exact SI constant.
ΔH=−28.8804960 + 6.80 + 4.60 = −17.4804960 kJ mol⁻¹. ZPE and the additional thermal term are kept separate to avoid counting either twice.
ΔG=ΔH−TΔS=−17.4804960−320.0(−0.0520)=−0.8404960 kJ mol⁻¹. The assigned compatible state conventions matter. No rate, mechanism or kinetic feasibility follows from this ΔG.

Assigned ΔE_el=−28.8804960, ΔH=−17.4804960 and ΔG=−0.8404960 kJ mol⁻¹ per written reaction.

Numerical agreement leaves model error separate

Assign E(g)=−76.420 + 0.032/g² E_h for positive dimensionless refinement index g. Compare g=2,4,8,16,32 using a successive-change tolerance 0.500 kJ mol⁻¹. A separate model has limiting energy −76.414 E_h. What is established?

The energies are −76.412, −76.418, −76.419500, −76.419875 and −76.41996875 E_h. The g=4→8 change is 0.001500 E_h, or 3.93824946 kJ mol⁻¹.
The g=8→16 change is 0.000375 E_h, or 0.984562365 kJ mol⁻¹. The g=16→32 change is 0.00009375 E_h, or 0.246140591 kJ mol⁻¹, the first listed pair below 0.500.
The actual error at g=32 in this invented function is 0.00003125 E_h, or 0.082046864 kJ mol⁻¹. We know this only because its exact limiting function was assigned; successive agreement alone is not a general rigorous error bound.
The two assigned model limits differ by 0.006000 E_h, or 15.7529978 kJ mol⁻¹. Refining g within one model does not remove that separate model difference. g is not an actual software cutoff or basis size.

The first listed passing successive pair is 16→32; a separate 15.7529978 kJ mol⁻¹ model-limit difference remains.

Try it yourself

For neutral CO, select a triplet rather than a singlet under the same highest-M_S convention. Find N_e, S, n_alpha, n_beta.
Show the answer and reasoning
N_e=14, S=1, n_α=8 and n_β=6.
Changing the selected spin specification does not change the net charge or nuclear inventory.
Keep both diagonal values but set the off-diagonal coupling to zero. What are the two-state minimum and lowering relative to the first trial state?
Show the answer and reasoning
Minimum −1.85 E_h, lowering zero, upper eigenvalue −1.42 E_h.
Enlarging the nested trial space cannot raise the minimum for the same Hermitian Hamiltonian, but strict lowering is not guaranteed.
Keep the assigned energies and corrections but use Delta S = -0.0400 kJ mol-1 K-1 at 320.0 K. Find Delta G per written reaction.
Show the answer and reasoning
ΔG=−4.6804960 kJ mol⁻¹ per written reaction.
Only the -T Delta S term changes. The formal atom/charge balance and electronic reaction energy are retained.
Tighten only the successive-change tolerance to 0.100 kJ/mol and append g = 64. Which newly listed pair passes?
Show the answer and reasoning
32→64 passes: change 0.061535148 kJ mol⁻¹; known toy error at 64 is 0.020511716 kJ mol⁻¹.
A tighter numerical check may require a larger refinement index, while the separate assigned model-limit difference remains.

What to carry forward

Charge determines electron number; selected multiplicity specifies a separate spin condition.
A variational comparison needs the same Hamiltonian and nested spaces.
Electronic, enthalpy and Gibbs-energy accounts need compatible stoichiometry and reference states.
Converging a numerical representation does not remove every physical-model error.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.