4. Spin densities and magnetic states
Position in the course: Lesson 4 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
Magnetism changes the variables of the density-functional problem. In collinear spin DFT, two spin densities describe electrons with respect to a chosen quantization axis. Total density and spin density are their sum and difference. Their integrals give electron number and spin imbalance. This treatment describes spin polarization but does not automatically include orbital magnetism, spin–orbit coupling, or a continuously varying spin direction.
Vary the functional independently with respect to the two spin densities. The Hartree potential depends only on their sum, whereas exchange–correlation produces separate spin potentials. The orbitals therefore solve two coupled equations through the common electrostatic density and spin-dependent exchange–correlation energy. A restricted nonmagnetic calculation imposes equal spin channels; an unrestricted calculation lets them differ. Symmetry constraints should represent a physical question, not merely a convenient initial guess.
A ferromagnetic state has aligned local spin polarization; an antiferromagnetic state can have zero net magnetization and large opposing local moments. A zero total moment therefore does not imply a nonmagnetic density. Local moments depend on how space or orbitals are partitioned. Compare both the integrated moment and the spin-density pattern; use the same projection convention when comparing codes or structures.
Start from several plausible arrangements, such as nonmagnetic, parallel, and alternating spin initializations. Compare their fully converged energies with identical numerical settings, examine whether moments survived, and identify possible metastability. Small cells may exclude long-period order. A converged broken-symmetry determinant can approximate a magnetic phase while failing to represent an exact spin eigenstate of a finite molecule. Spin contamination and symmetry restoration belong to the many-electron discussion; they are not fixed by reducing an SCF tolerance.
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. Energy comparisons must use matched conditions
Comparing a ferromagnetic primitive cell with an antiferromagnetic supercell requires expressing energies per the same number of formula units and matching sampling density. The same nominal mesh dimensions do not mean the same reciprocal-space resolution when cell sizes differ. Distinguish constrained spin calculations from unconstrained relaxation: a fixed spin imbalance asks for the minimum under a restriction, while a relaxed spin state asks a different question. Total moment can change discontinuously as occupations switch. Examine several starting states and their final densities, because a label on an input file does not guarantee the converged state retains that label.
3. Worked example
A cell with N↑=6 and N↓=4 has N=10 and a spin-only moment of 2 μB in this convention. Two sublattices with moments +2 and −2 μB sum to zero yet form an antiferromagnetic pattern. These are electron-counting examples, not predicted material moments.
4. Exercises with explained solutions
Exercise. Show that a physical collinear spin density must satisfy |m|≤n.
Explained solution. Since n↑=(n+m)/2 and n↓=(n−m)/2 must both be nonnegative, n+m≥0 and n−m≥0. Combining them gives −n≤m≤n. A trial density violating this bound cannot represent nonnegative spin populations.
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
The lowest energy among a few initial guesses is only the lowest state found. Ordering temperature cannot be read directly from a zero-temperature energy difference without a magnetic model and statistical treatment.
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