LESSON NOTES · 02

Self-consistency and the total-energy ledger

Prerequisites: Density-to-Kohn–Sham lesson; elementary linear stability

1. A nonlinear fixed-point problem

Use the fixed-nuclei, atomic-unit conventions of the first lesson. The potential depends on the density obtained from its own eigenfunctions. Solving one eigenproblem is therefore not a self-consistent DFT solution. Occupations \(0\leq f_i\leq1\) refer here to spin orbitals; spin-restricted spatial orbitals can instead carry \(0\leq f_i\leq2\). Never mix conventions.

\[ n_{in}^{(m)}\to v_{KS}[n_{in}^{(m)}]\to\{\phi_i,\varepsilon_i\}\to n_{out}^{(m)}=\sum_i f_i\sum_\sigma|\phi_i|^2,\qquad\sum_i f_i=N. \]

2. Residual and density mixing

Monitor density residual, electron count and energy change together. A numerically converged solution need not be the lowest-energy electronic state; compare relevant spin or symmetry solutions. Smearing and finite-temperature work must distinguish internal energy and the appropriate free energy. Mixing blends input and output densities rather than blindly replacing one with the other.

\[ r^{(m)}=n_{out}^{(m)}-n_{in}^{(m)},\qquad n_{in}^{(m+1)}=n_{in}^{(m)}+\alpha r^{(m)},\quad0<\alpha\leq1. \]

3. Why mixing can stabilize a mode

Near a fixed point \(n^*\), linearize the SCF map \(F\). An error mode with Jacobian eigenvalue \(\lambda\) is multiplied by \(\mu=1-\alpha+\alpha\lambda\). Contraction requires \(|\mu|<1\). For \(\lambda=-2\), choosing \(\alpha=0.3\) gives \(\mu=0.1\), whereas an unmixed iteration multiplies error by \(-2\). For \(\lambda=1.2\), every positive mixing parameter gives \(\mu>1\). Thus simple mixing is a stability tool, not a guarantee; large-cell charge sloshing may need preconditioning.

\[ \delta n^{(m+1)}\simeq[(1-\alpha)I+\alpha J_F]\delta n^{(m)},\qquad\mu=1-\alpha+\alpha\lambda. \]

4. Derive the energy correction

Assume a local multiplicative KS potential and consistent orbitals and density. The occupied eigenvalue sum contains the full Hartree potential expectation, which equals twice \(E_H\). It also contains \(\int nv_{xc}\), generally unequal to \(E_{xc}\). Subtract the unwanted pieces and restore the intended energy terms. For a nonlocal hybrid generalized-KS operator, change the corresponding subtraction; do not directly transplant this formula. During SCF, distinguish input and output densities in the evaluated energy.

\[ \begin{aligned}\sum_i f_i\varepsilon_i&=T_s+\int v_{ext}n+\int v_Hn+\int v_{xc}n,\\\int v_Hn&=2E_H,\\E_{tot}&=\sum_i f_i\varepsilon_i-E_H+E_{xc}-\int nv_{xc}+E_{NN}.\end{aligned} \]

Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.

Worked energy ledger

Toy values in hartree: \(\sum f_i\varepsilon_i=-10\), \(E_H=4\), \(E_{xc}=-3\), \(\int nv_{xc}=-4\), \(E_{NN}=2\). Carefully retain each sign: total energy is \(-11\) hartree, not \(-10\). These numbers teach bookkeeping and are not computed molecular data.

\[ E_{tot}=-10-4-3+4+2=-11\ \mathrm{hartree}. \]

Check yourself

Does converged total energy equal the eigenvalue sum?

Answer and reasoning

Generally no; interaction corrections are required.

Does a small energy change guarantee a small density error?

Answer and reasoning

No. Near stationarity energy errors can be second order in density error. Check residuals explicitly.

References and further reading

• Kohn & Sham (1965), original self-consistent equations
• Burke et al., The ABC of DFT, university teaching text

Derivations and numerical examples here are original teaching constructions, not copied passages or reported research data.

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