Density Matrices in Electronic Structure

Electronic reduced-density matrices compress the information needed for one- and two-body observables without storing a many-electron wavefunction in every subsequent calculation. They are not the full density operator on the Fock space. Their index conventions, normalization, and representability constraints must be explicit before arrays from different programs can be compared.

Helpful background. Configuration interaction supplies a determinant representation and the second-quantized Hamiltonian.

A fixed spin-orbital convention

Use orthonormal spin orbitals and a normalized state with exactly $N$ electrons. Define

$$\gamma_{pq}=\langle a_q^\dagger a_p\rangle,\qquad \Gamma_{pq,rs}=\langle a_r^\dagger a_s^\dagger a_q a_p\rangle.$$

Then $\gamma=\gamma^\dagger$, $\operatorname{Tr}\gamma=N$, and the eigenvalues of $\gamma$ lie in $[0,1]$. The two-body tensor is antisymmetric under $p\leftrightarrow q$ and $r\leftrightarrow s$, and $\Gamma_{pq,rs}^*=\Gamma_{rs,pq}$.

For

$$H=\sum_{pq}h_{pq}a_p^\dagger a_q +\frac14\sum_{pqrs}\langle pq\Vert rs\rangle a_p^\dagger a_q^\dagger a_s a_r,$$

the energy is

$$E=\sum_{pq}h_{pq}\gamma_{qp} +\frac14\sum_{pqrs}\langle pq\Vert rs\rangle\Gamma_{rs,pq}.$$

This formula fixes the array contractions completely. Transposing a density without transposing the corresponding integral convention can silently change complex-valued observables.

Contraction identities and natural orbitals

The number operator gives the contraction

$$\sum_q\Gamma_{pq,rq}=(N-1)\gamma_{pr}, \qquad \sum_{pq}\Gamma_{pq,pq}=N(N-1).$$

For example, insert $\sum_q a_q^\dagger a_q$ between $a_r^\dagger$ and $a_p$. After $a_p$ acts, the state contains $N-1$ particles, yielding the first identity. These are exact tests of normalization and index order for a fixed-particle-number state; fluctuating-particle ensembles require modified expectations.

Diagonalizing $\gamma$ gives natural orbitals and their occupations. A determinant has occupations zero or one and satisfies $\gamma^2=\gamma$. Correlated pure states generally have fractional occupations. A spin-summed spatial density instead permits occupations up to two, so a value of $1.8$ is not intrinsically unphysical. Do not mix these two conventions.

Hermiticity, trace, and occupation bounds are necessary checks, but they do not establish that an arbitrary pair $(\gamma,\Gamma)$ comes from an $N$-electron state. Two-particle representability imposes further positivity and consistency conditions. The Coleman review explains why optimizing a two-particle density directly is a constrained many-body problem.

Determinant and correlated examples

For a determinant with occupied spin orbitals $1$ and $2$,

$$\gamma=\operatorname{diag}(1,1,0,\ldots),\qquad \Gamma_{pq,rs} =\gamma_{pr}\gamma_{qs}-\gamma_{ps}\gamma_{qr}.$$

Thus $\Gamma_{12,12}=1$, $\Gamma_{12,21}=-1$, and the ordered-pair trace is $2$, agreeing with $N(N-1)$.

Now use $|\Psi\rangle=(|12\rangle+|34\rangle)/\sqrt2$ with two electrons in four spin orbitals. A one-body operator cannot connect determinants differing by two orbitals, so $\gamma=\tfrac12 I_4$. Its trace remains two but it is not idempotent. The two-body tensor retains coherence between the two determinants, which the one-body density alone cannot reconstruct.

For CI coefficients, assemble densities by applying annihilation and creation operators with the same sign convention used for the Hamiltonian action. Independently compare the contracted energy with $c^\dagger Hc$; agreement tests substantially more than the trace.

Numerical limits

Small negative occupations at roundoff scale can reflect assembly noise; larger violations require investigation. Symmetrizing a density or clipping eigenvalues can make a plot look physical while breaking contraction identities and energies. Report the original defects before applying any controlled repair.

Natural-orbital truncation should be checked against target energies and properties. Small occupations do not guarantee negligible correlation contributions when integrals or response denominators are large.

Exercises

What are $\operatorname{Tr}\gamma^2$ for the determinant and the equal two-determinant superposition above?

Solution

The determinant gives $1^2+1^2=2$. The superposition gives $4(1/2)^2=1$. Both have $\operatorname{Tr}\gamma=2$, showing that electron count alone does not diagnose correlation.

References

• Coleman, A. J., “Structure of fermion density matrices,” Reviews of Modern Physics 35, 668–686 (1963), doi:10.1103/RevModPhys.35.668 — occupation constraints and representability.
• Szabo, A., and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover, 1996 — density matrices and electronic energy contractions.