Energy and gradient calculations at the Hartree–Fock (HF) and DFT level can be carried out in two ways: dscf and grad perform conventional calculations based on four–center two–electron repulsion integrals (ERI’s); ridft and rdgrad employ the RI–J approximation, as detailed below.
dscf and grad are modules for energy and gradient calculations at the HF and DFT level, which use an efficient semi–direct SCF algorithm. Calculation of the Coulomb and HF exchange terms is based on the conventional method employing four–center two–electron repulsion integrals (ERI’s). These modules should be used for HF and DFT calculations with exchange-correlation functionals including HF exchange contribution, e.g. B3–LYP, if further approximations (RI–J) are to be avoided. All functionalities are implemented for closed–shell RHF and open–shell UHF reference wavefunctions. Restricted open shell treatments (ROHF) are supported on the HF level only, i.e. not for DFT.
The most important special features of the dscf and grad modules are:
ridft and rdgrad are modules for very efficient calculation of energy and gradient at the Hartree–Fock (HF) and DFT level . Both programs employ the Resolution of the Identity approach for computing the electronic Coulomb interaction (RI–J). This approach expands the molecular electron density in a set of atom–centered auxiliary functions, leading to expressions involving three–center ERI’s only. This usually leads to a more than tenfold speedup for non–hybrid DFT compared to the conventional method based on four–center ERI’s (for example the dscf or grad module).
The combination of RI–J for Coulomb–interactions with a case–adapted conventional exchange treatment reduces the scaling behaviour of the (conventional) exchange evaluation required in HF–SCF and hybrid DFT treatments. Usage of ridft and rdgrad for HF and hybrid DFT is of advantage (as compared to dscf and grad) for larger systems, where it reduces computational costs significantly.
The most important special features of the ridft and rdgrad modules are:
All algorithms implemented in dscf, grad, ridft, and rdgrad modules can exploit molecular symmetry for all finite point groups. Typically, the CPU time is proportional to 1∕NG, where NG is the order of the nuclear exchange group. Another important feature is a parallel implementation using the MPI interface.
Additionally dscf and ridft modules include the following common features:
RI-techniques can also be used for the Hartree–Fock exchange part of the Fock matrix (RI-HF). This is done by the ridft-module, if the keyword $rik is found in the control file. In this case ridft performs a Hartree–Fock-SCF calculation using the RI- approximation for both J and K, if suitable auxiliary basis sets (which differ from that used for fitting of the Coulomb part only) are specified. This is efficient only for comparably large basis sets like TZVPP, cc-pVTZ and larger.
HF-exchange can also be calculated semi-numerically . The calculation of 4c-2e-Integrals is split into an analytical and a numerical part. The latter is evaluated on a dft-type integration grid. The semi-numerical calculation scales better with system size than RIK and is suitable for large molecules and large basis sets.