6 Hartree–Fock and DFT Calculations for Molecular Systems

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:

• Selective storage of the most time consuming and frequently used integrals. The integral storage is controlled by two threshold parameters, $thize and $thime, related to integral size and computational cost.

• Efficient convergence acceleration techniques for energy calculations. They include standard methods for convergence acceleration (DIIS), which reduce the number of SCF iterations needed as well as methods to reduce the effort within each iteration when the calculation is almost converged (integral prescreening and differential density scheme).

ridft and rdgrad are modules for very efficient calculation of energy and gradient at the Hartree–Fock (HF) and DFT level [80]. 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:

• A very efficient semi-core algorithm for energy calculation. The most expensive three–center integrals are kept in memory which significantly reduces the computational time for small and middle sized molecules. The amount of stored integrals is controlled by simply specifying the amount of free memory using the keyword $ricore.

• Multipole accelerated RI for Coulomb (MARI–J) linear scaling (\(O(N)\)) method for large molecules. It significantly reduces calculation times for molecules with more than 1000 basis functions.

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 \(1/{N_G}\), where \(N_G\) 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:

• Several approaches to solving SCF equations to reach convergence.

  • The default is Pulay’s DIIS algorithm (see $scfdiis for details) with various options for damping (see $scfdamp).

  • Fractional occupation by using (pseudo-) Fermi thermal smearing to automatically optimize occupations and spin state (see $fermi). Fermi can be used in combination with DIIS.

  • Augmented Roothaan-Hall (ARH) using second-order information for a variety of cases where DIIS leads to non-convergence or converges to a saddle point (see $arh).

• An UHF implementation [81] with automatic generation of optimal start vectors by solving the HF instability equations [82] in the AO basis (see the keyword $scfinstab for detailed information).

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 [83]. 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.