7. Forces, stress, and structural optimization
Position in the course: Lesson 7 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
A geometry optimization asks for a stationary point on an electronic energy surface. It is a nested problem: electrons must be sufficiently converged at each nuclear geometry before their forces guide nuclear motion. Small changes in electronic energy are not a substitute for force convergence. An optimizer can faithfully minimize a noisy or biased energy surface and still produce an unreliable geometry.
Differentiate an expectation value with respect to a nuclear coordinate. One term differentiates the Hamiltonian; two terms differentiate the state. For an exact normalized eigenstate, the state-response terms cancel by the eigenvalue equation and normalization, giving the Hellmann–Feynman result. In a moving finite basis, the accessible subspace changes with geometry, so basis derivatives yield Pulay contributions. Incomplete electronic convergence introduces additional residual-force errors.
Stress describes energy response to cell strain, with sign and volume conventions varying among codes. Changing a plane-wave cell changes which reciprocal vectors lie below cutoff, producing basis incompleteness effects commonly called Pulay stress. Check cell optimization with a sufficiently converged cutoff; a stable total energy at a fixed cell may not establish a stable equilibrium volume. For surfaces, artificial cell volume also complicates conversion of stress into meaningful surface quantities.
Distinguish finding a minimum from merely finding a stationary point. A vanishing gradient can occur at a saddle. Vibrational Hessian eigenvalues help diagnose stability, while constrained coordinates require analysis in the allowed subspace. Multiple starting structures may be necessary. Bond dissociation, spin changes, and phase transitions can create competing electronic solutions or non-smooth approximate surfaces. Record force thresholds, energy thresholds, cell constraints, symmetry constraints, and electronic tolerances together.
2. Derivation step by step
Read each equality with its assumptions. Atomic units are used for DFT equations unless another unit is stated; TB parameters retain explicit energy and length units. The conjugate transpose is denoted by a dagger, and a prime on a coordinate denotes a separate integration variable.
2.1. A force check is also a unit check
An energy in eV differentiated with respect to a coordinate in ångström gives eV/Å. A misplaced length conversion can produce a consistent-looking geometry step with incorrect forces. Central finite differences should keep every other coordinate and cell parameter fixed, and each displaced electronic problem must be converged. Compare several step sizes: a quadratic truncation regime at larger steps and a noisy regime at smaller steps are expected. Constraints complicate the comparison because the reported force may have been projected onto allowed motion. Stress comparisons need a documented strain definition and sign convention, with electron and nuclear contributions included consistently.
3. Worked example
For E=E0+K(R−R0)²/2 with K=4 eV/Ų, a displacement of 0.05 Å gives F=−0.20 eV/Å and energy increase 0.005 eV. The force is linear in displacement while the energy change is quadratic; this explains why small energy changes can coexist with appreciable forces.
4. Exercises with explained solutions
Exercise. Derive a central-difference force and its leading truncation order.
Explained solution. Expand E(R±h)=E±hE′+h²E″/2±h³E‴/6+…. Subtract and divide by 2h: E′+[h²/6]E‴+…. Thus F≈−[E(R+h)−E(R−h)]/(2h), with O(h²) truncation. Making h too small amplifies electronic numerical noise.
Further check. State the units and the allowed regime for every parameter in the worked example. Change one assumption and identify which derivation step must be revisited. A correct explanation names the affected constraint, operator, or boundary condition rather than merely saying that the answer changes.
5. Misconceptions and limitations
Harmonic stability is local. It does not determine a transition barrier, finite-temperature free energy, or the global lowest-energy phase.
The illustration is an original teaching schematic. It is not output from a numerical materials simulation.
6. Connections and sources
Related: localized-basis theory · Molecular electronic structure
ABACUS · VASP · Quantum ESPRESSO · CP2K
- Hohenberg–Kohn, ground-state density theorem (1964)
- Kohn–Sham, self-consistent orbital equations (1965)
- PBE, constrained GGA construction (1996)
The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.
7. Related theory and practice
Quantum mechanics · Molecular methods · Tight binding · Molecular dynamics · VASP · Quantum-Espresso · CP2K · ABACUS