7.1 Functionalities of Riper

The riper module implements Kohn–Sham DFT with Gaussian-type orbitals (GTOs) and treats molecular and periodic systems of any dimensionality on an equal footing [203]. Its core is a combination of the resolution-of-identity (RI) approximation and the continuous fast multipole method (CFMM) for the electronic Coulomb term [204, 205, 206]. The exchange–correlation (XC) term is evaluated using an octree-based hierarchical numerical integration scheme [207]. riper offers computational efficiency and favorable scaling, approaching \(O(N)\) for Kohn–Sham matrix formation [204] and for analytic gradients [205, 206]. For very large molecular systems, a low-memory modification of the RI approximation (LMIDF) combines CFMM with a preconditioned conjugate-gradient solver to reduce memory usage with only a small timing overhead [208].

Relativistic effects can be treated with scalar-relativistic effective core potentials (ECPs) in a one-component (1c) setting or with spin–orbit ECPs in a two-component (2c) framework [179, 177, 180]; the latter supports current-dependent functionals (see Sec. 6.5).

7.1.1 Key Functionalities

riper provides the following capabilities:

• Kohn–Sham DFT for molecular systems and systems with 1D, 2D, and 3D periodicity

• Closed- and open-shell energies and gradients; structure optimization including optimization of cell parameters

• Two-component Kramers-restricted and Kramers-unrestricted energies, gradients, and structure optimization including optimization of cell parameters

• Metals and semiconductors via fractional occupations with Gaussian smearing

• Efficient \(\mathbf{k}\)-point sampling for periodic systems allowing consistent results across different definitions of unit cells

• Sequential and parallel runs (OpenMP parallelization for shared-memory computers, see Sec. 3.4.2)

• All LDA, GGA, and meta-GGA XC functionals, including an interface to the XCFun and Libxc libraries (see Sec. 6.3)

• DFT-D3 dispersion correction for energies and gradients (see Sec. 6.8)

• Favorable scaling for Kohn–Sham matrix formation approaching \(O(N)\)

• Memory-efficient calculations for very large molecular systems using the LMIDF scheme

• Real-time time-dependent DFT (RT-TDDFT) for molecular systems

• DFT-based embedding via frozen-density embedding (FDE) and projection-based embedding (PbE), coupled with RT-TDDFT and wavefunction-theory methods (via dscf and ricc22)

• Hartree–Fock (HF) and hybrid DFT (global and range-separated) for molecular and periodic systems

• Analytic stress tensor for periodic systems

• Density of states (DOS) calculations

• Band structure calculations with automatic \(k\)-path generation

• Property calculations on grids (electron density, molecular orbitals)

7.1.2 Current Limitations

• No analytical derivatives with hybrid exchange-correlation functionals yet

• Only \(C_1\) symmetry point group for molecules and \(P_1\) space group for periodic systems