Density Functional Theory (DFT)

OpenQuantum solves the Kohn–Sham equations on an atom-centered numerical grid. The Kohn–Sham Fock operator replaces the wavefunction-theory exchange operator with a density-dependent exchange–correlation (XC) potential, and the total electronic energy is

$$E_{\text{KS}}[\rho ]=T_s[\rho ]+J[\rho ]+V_{\text{ne}}[\rho ]+E_{\text{xc}}[\rho ]+V_{\text{nn}},$$

where $T_s$ is the non-interacting kinetic energy, $J$ the classical Coulomb energy, $V_{\text{ne}}$ the electron–nuclear attraction, $V_{\text{nn}}$ the nuclear repulsion, and $E_{\text{xc}}$ the exchange–correlation contribution evaluated by numerical quadrature.

Functional rungs

OpenQuantum supports four rungs of Jacob's ladder. The semilocal kernels read the spin densities ${\rho }_{\sigma }$, gradient invariants ${\sigma }_{\sigma {\sigma }^{\prime }}=\nabla {\rho }_{\sigma }\cdot \nabla {\rho }_{{\sigma }^{\prime }}$, and (for meta-GGAs) kinetic-energy densities ${\tau }_{\sigma }=\frac{1}{2}{\sum }_i|\nabla {\varphi }_{i\sigma }{|}^2$ at every quadrature point.

The exact-exchange fraction is wired through the SCF Fock builder. Range-separated exchange splits the Coulomb operator with the error function and contracts short- and long-range fractions independently against the same density.

M06-family correlation

M06-L and M06-2X correlation use the Voorhis–Scuseria (VS98) reduced-gradient working factor combined with a polynomial baseline:

$$E_c^{\sigma \sigma }[\rho ]=\int {\rho }_{\sigma }{\epsilon }_c({\rho }_{\sigma },1)(G_{ss}({\chi }_{\sigma })+h({\chi }_{\sigma },z_{\sigma }))D_{\sigma }d\mathbf{r},$$

$$E_c^{\alpha \beta }[\rho ]=\int (\rho {\epsilon }_c(\rho ,\zeta )-{\rho }_{\alpha }{\epsilon }_c({\rho }_{\alpha },1)-{\rho }_{\beta }{\epsilon }_c({\rho }_{\beta },1))(G_{ab}({\chi }^2)+h(\chi ,z))d\mathbf{r},$$

with reduced gradients ${\chi }_{\sigma }^2={\sigma }_{\sigma \sigma }/{\rho }_{\sigma }^{8/3}$, $z_{\sigma }={\tau }_{\sigma }/{\rho }_{\sigma }^{5/3}-C_F$, $C_F=\frac{3}{5}(6{\pi }^2{)}^{2/3}$, self-interaction screen $D_{\sigma }=1-{\chi }_{\sigma }^2/(4(z_{\sigma }+C_F))$, and PW92 spin-resolved correlation ${\epsilon }_c$. The polynomial baseline keeps the working factor positive at high reduced gradient, so M06 correlation integrates physically without any pointwise guard.

The working function $h(x,z)$ is

$$h(x,z)=\frac{u_0}{w}+\frac{u_1{x}^2+u_{2}z}{w^2}+\frac{u_3{x}^4+u_4{x}^{2}z+u_5{z}^2}{w^3},w=1+a_x{x}^2+a_{z}z.$$

The polynomial baseline has the form $G(s^2)={\sum }_{i=0}^4{c}_i(\gamma s^2/(1+\gamma s^2){)}^i$ with ${\gamma }_{ss}=0.06$ (same-spin) and ${\gamma }_{ab}=0.0031$ (opposite-spin).

Atom-centered grid

The molecular grid is a Becke-style sum of atom-local quadratures, each a direct product of a Mura–Knowles radial scheme and a Lebedev angular grid.

• Radial. Mura–Knowles maps the unit interval to $[0,\infty )$ via $r=-R\ln (1-x^3)$, with $R=7\text{Bohr}$ for the alkali/hydrogen column and $R=5\text{Bohr}$ elsewhere.
• Angular. Genuine Lebedev–Laikov grids of degree 3, 5, 7, 11, 17, 23, and 29 (point counts 6, 14, 26, 50, 110, 194, 302). The dispatcher routes any requested order to the largest exact Lebedev set that does not exceed the request; orders above 302 fall back to a Fibonacci-sphere distribution. Each Lebedev order is gated by an explicit spherical-moment exactness test.
• Pruning. Below the SuperFine level, near-nucleus and tail radial shells drop to lower angular orders, with denser tails kept for heavy atoms.
• Partition. A compact Stratmann–Scuseria-style cell function $f(\mu )$, built from a sixth-degree polynomial scaled by $X_{\text{scal}}=0.86$ with a $3003/2048$ normalization, replaces Becke's iterated step. The cell weight is identically zero outside $|\mu |>0.852$ and goes smoothly to zero at the partition tail, which removes spurious tail points without further screening.

The integration loop screens AO shells whose product bound falls below $10^{-16}$ and accumulates the XC potential matrix only over active AO pairs whose pointwise product (or gradient bound) exceeds $10^{-24}$.

Empirical dispersion

Dispersion-bearing functionals add an empirical pair correction to the total energy. OpenQuantum exposes three models with the same element coverage (Z = 1–54, including transition metals).

• D2. $E_{\text{disp}}=-s_6{\sum }_{i<j}{f}_6(R_{ij})C_6^{ij}/R_{ij}^6$ with a Fermi-type damping function. Used by ωB97X-D.
• D3 zero-damping. Adds an estimated $C_8$ term and uses the standard zero-damping factor for both $R^{-6}$ and $R^{-8}$ contributions.
• D4 charge-/coordination-scaled. Estimates per-atom partial charges from Pauling electronegativities, scales pair coefficients by local coordination numbers, and applies a rational damping form to $R^{-6}$ and $R^{-8}$ terms.

The model is selected via the functional metadata tag (D2, D3, D4) or called explicitly through the dispersion API.

Restricted, unrestricted, and open-shell DFT

Closed-shell systems use a restricted Kohn–Sham (RKS) Fock build that contracts the spin-summed XC potential $(v_{{\rho }_{\alpha }}+v_{{\rho }_{\beta }})/2$, $(v_{{\sigma }_{aa}}+v_{{\sigma }_{ab}}+v_{{\sigma }_{bb}})/4$, and $(v_{{\tau }_{\alpha }}+v_{{\tau }_{\beta }})/2$ against the AO basis. Open-shell systems use an unrestricted Kohn–Sham (UKS) Fock build with spin-resolved XC matrices for α and β. Both share the same grid construction and screening pipeline.

Solvent-coupled DFT

Kohn–Sham calculations with implicit solvation combine the XC potential and the reaction-field contribution in the same effective Fock matrix. OpenQuantum provides dedicated SCF entry points:

• rks_scf_with_reaction_field — RKS with PCM/ddCOSMO/SMD solvent.
• uks_scf_with_reaction_field — UKS with PCM/ddCOSMO/SMD solvent.

At each SCF iteration the XC potential matrix (from grid quadrature) and the reaction-field contribution (from the cavity surface charges) are added to the core Hamiltonian alongside the Coulomb and exchange terms. The solvent-coupled SCF supports all functional rungs including range-separated hybrids.

Analytical gradients and Hessians with DFT

The analytic nuclear gradient for Kohn–Sham DFT includes the XC contribution from grid differentiation alongside the HF-like terms (nuclear, Coulomb, exchange). When solvent is active, the analytical gradient additionally includes the PCM contributions (grad_nuc + grad_solver + grad_qv).

The analytical Hessian for DFT extends the CPHF formalism to include XC response terms from grid differentiation. The XC contribution to the Hessian contains both the first-order XC potential response (through the CPHF equations) and the second-order XC kernel evaluated on the grid. With solvent, the full PCM Hessian (hess_nuc + hess_solver + hess_qv) is added, including the CPHF solvent response (AXPCM kernel).