2. Kohn–Sham orbitals, self consistency, and total energy
Position in the course: Lesson 2 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
Kohn–Sham theory introduces a fictitious noninteracting system with the same ground-state density as the interacting target. Its orbitals provide an efficient kinetic-energy representation. They are not trajectories of individual electrons. Define Ts by minimizing the noninteracting kinetic energy consistent with n. Put the difference between the true internal energy and Ts plus the classical Hartree term into Exc. Thus exchange–correlation includes a kinetic correlation contribution as well as nonclassical interaction effects.
For orthonormal occupied orbitals, vary the energy with respect to their complex conjugates. Differentiating Ts gives −∇²φ/2. The two identical contributions from the symmetric Hartree double integral cancel its factor one half and give vHφ. Differentiating Exc gives vxcφ for a local density-functional representation. The orthonormality multipliers initially form a Hermitian matrix; a unitary rotation among the orbitals diagonalizes it and gives canonical orbital energies.
The potential depends on the density and the density depends on its eigenfunctions. An SCF iteration therefore starts from an input density, builds the potential, solves the orbital equations, occupies levels, and constructs an output density. Mixing updates the guess. Convergence requires the input/output residual to shrink, not merely the displayed energy to stop changing. Near stationarity, density errors may give only second-order energy changes, hiding a significant residual.
To obtain the total energy from the eigenvalues, multiply each orbital equation by its conjugate, integrate, and sum occupations. The resulting sum contains Ts+∫nv+2EH+∫nvxc. Subtract one Hartree energy and the exchange–correlation potential integral, then add Exc and nuclear repulsion. The double-counting correction is why the band-energy sum cannot be used as a molecular binding energy. Occupations must sum to N, with spin convention stated.
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 linearized SCF stability test
Let the density map be G and consider a scalar error mode with slope λ near a fixed point. Linear mixing nnew=(1−α)n+αG(n) multiplies that error by 1−α+αλ. Convergence of this mode requires its absolute value below one. For λ=−3, α=0.2 gives a factor 0.2, while α=0.8 gives −2.2 and grows with alternating sign. This explains why aggressive mixing can oscillate. In real systems different reciprocal-space modes have different slopes; preconditioning and history-based schemes treat those modes more intelligently than one scalar parameter. Changing the mixing affects the route to a fixed point, not the intended physical energy functional.
3. Worked example
Take Ts=5, ∫nv=−12, EH=4, Exc=−3, ∫nvxc=−4, ENN=2 in arbitrary consistent energy units. The eigenvalue sum is 5−12+8−4=−3. Correcting it gives −3−4−3+4+2=−4, matching direct evaluation 5−12+4−3+2. These numbers are a bookkeeping example, not a calculation for a chemical system.
4. Exercises with explained solutions
Exercise. Why does the Hartree potential contribute twice EH to the eigenvalue sum?
Explained solution. Insert vH(r)=∫n(r′)/|r−r′|dr′ into ∫nvH. The result is the full symmetric double integral, whereas EH is half that integral. The factor follows from pair-energy counting, not from spin degeneracy.
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
Different self-consistent states can be local stationary solutions. Try physically distinct spin or charge initial guesses when metastability is relevant. SCF convergence does not establish the accuracy of the functional.
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