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Theory

Electron Number Distribution Function

Given a molecule partitioned into \(m\) non-overlapping, space-filling fragments \(\Omega_1, \ldots, \Omega_m\), the Electron Number Distribution Function (EDF) is the joint probability of finding exactly \(n_1\) electrons in \(\Omega_1\), \(n_2\) electrons in \(\Omega_2\), …, and \(n_m\) electrons in \(\Omega_m\):

\[p(n_1, n_2, \ldots, n_m) = \binom{N}{n_1, n_2, \ldots, n_m} \int_{\Omega_1^{n_1} \otimes \cdots \otimes \Omega_m^{n_m}} |\Psi|^2 \, d\mathbf{r}_1 \cdots d\mathbf{r}_N\]

where the multinomial coefficient accounts for the indistinguishability of electrons and the integration runs over all assignments of \(n_i\) electrons to each region \(\Omega_i\). The EDF satisfies:

\[\sum_{n_1 + \cdots + n_m = N} p(n_1, \ldots, n_m) = 1\]

Atomic Overlap Matrix

The central computational object is the Atomic Overlap Matrix (AOM):

\[S^A_{ij} = \int_{\Omega_A} \phi_i(\mathbf{r}) \, \phi_j(\mathbf{r}) \, d\mathbf{r}\]

where \(\phi_i\) are molecular orbitals (natural orbitals for CASSCF) and \(\Omega_A\) is the basin of atom \(A\). The AOM satisfies the sum rule:

\[\sum_A S^A_{ij} = \delta_{ij}\]

(which edf verifies via TOLAOM).

Single-determinant case

For a single Slater determinant of \(N\) spin-orbitals \(\{\phi_i\}\), the EDF probability of configuration \((n_1, \ldots, n_m)\) is a permanent of the AOM:

\[p(n_1, \ldots, n_m) = \frac{1}{N!} \sum_{\sigma \in S_N} \prod_{i=1}^N S^{A_{\sigma(i)}}_{ii}\]

where the sum is over all permutations \(\sigma\) that assign \(n_k\) orbitals to fragment \(k\). In practice this is evaluated as a sum of products of matrix permanents.

For the two-group case (\(m = 2\)), a binomial recurrence relation (Cançes, Keriven, Lodier, Savin, Theor. Chem. Acc. 111, 373, 2004) allows the full EDF to be computed in \(O(N^2)\) time instead of \(O(N!)\).

Multi-determinant (CASSCF) case

For a CASSCF wavefunction

\[|\Psi\rangle = \sum_I C_I |\Phi_I\rangle\]

the EDF is a double sum over determinant pairs:

\[p(n_1, \ldots, n_m) = \sum_{I,J} C_I C_J \, p_{IJ}(n_1, \ldots, n_m)\]

where \(p_{IJ}\) is the cross-determinant contribution. The total computation scales as \(O(N_\text{det}^2)\) in the number of determinants. The EPSDET keyword truncates pairs with \(|C_I C_J| < \epsilon\), and NDETS/EPSWFN trim the determinant expansion.

Average populations and delocalization indices

Average population of fragment \(i\):

\[\langle n_i \rangle = \sum_{\mathbf{n}} n_i \, p(\mathbf{n}) = \sum_j S^i_{jj}\]

(the trace of the fragment AOM over occupied orbitals).

Two-fragment delocalization index \(\delta(i,j)\):

\[\delta(i,j) = 2\left[\langle n_i n_j \rangle - \langle n_i \rangle \langle n_j \rangle\right] = -2 \sum_{k,l} S^i_{kl} S^j_{lk}\]

where the second expression holds for single-determinant wavefunctions (Eq. 28 of J. Chem. Phys. 126, 094102, 2007). The delocalization index measures electron sharing: \(\delta \approx 1\) for a single covalent bond, \(\delta \approx 2\) for a double bond.

Localization index: \(\delta(i,i) = 2[\langle n_i^2 \rangle - \langle n_i \rangle^2]\) is the variance of the electron population in fragment \(i\). The percentage localization is \(\delta(i,i)/\langle n_i \rangle \times 100\).

Fuzzy atomic partitions

When QTAIM or ELF basins are unavailable, edf computes the AOM internally using a fuzzy partition in which the sharp basin indicator function is replaced by a smooth weight \(w_A(\mathbf{r}) \geq 0\) with \(\sum_A w_A(\mathbf{r}) = 1\):

\[S^A_{ij} = \int w_A(\mathbf{r}) \, \phi_i(\mathbf{r}) \, \phi_j(\mathbf{r}) \, d\mathbf{r}\]

Available weight functions:

Keyword Weight \(w_A\)
mulliken Mulliken (step at AO center)
lowdin Löwdin (symmetric orthogonalization)
mindef Minimally Deformed Atoms (Fernández Rico et al.)
mindefrho MinDef with \(w_A = \rho_A/\rho\)
netrho Net density (eliminates cross-center terms)
promrho Promolecular: \(w_A = \rho_A^{\rm atom}/\rho\)
heselmann Heselmann chemical localization weights
becke Becke fuzzy cells

The numerical integrals use Lebedev angular quadrature (controlled by LEBEDEV) and a radial Becke-type grid (NRAD, IRMESH).

Key references

  • Francisco, E.; Martín Pendás, A.; Blanco, M. A. "EDF: Computing electron number probability distribution functions in real space from molecular wave functions." Comput. Phys. Commun. 178, 621 (2008).
  • Cançes, E.; Keriven, R.; Lodier, F.; Savin, A. "How electrons guard the space." Theor. Chem. Acc. 111, 373 (2004).
  • Bader, R. F. W. Atoms in Molecules: A Quantum Theory. Oxford University Press, 1990.
  • Becke, A. D. "A multicenter numerical integration scheme for polyatomic molecules." J. Chem. Phys. 88, 2547 (1988).
  • Heselmann, A. "Improved description of chemical bonding from a localized molecular orbital perspective." J. Chem. Theory Comput. 12, 2720 (2016).
  • Matito, E.; Solà, M.; Salvador, P.; Duran, M. "Electron sharing indexes at the correlated level." Faraday Discuss. 135, 325 (2007). [Eq. 28 for DI]