Hartree-Fock and DFT Calculations

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 [45]. 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:

- An UHF implementation [46] with automatic
generation of optimal start vectors by solving the HF instability
equations [47] in the AO basis (see the keyword
`$scfinstab`for detailed information). - Occupation number optimization using (pseudo-Fermi) thermal smearing.

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 seminumerically [48]. 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 seminumerical calculation scales better with system size than RIK and is suitable for large molecules and large basis sets.

- Prerequisites
- How to Perform a Calculation
- Background Theory
- Exchange-Correlation Functionals Available
- Restricted Open-Shell Hartree-Fock

- Two-component Hartree-Fock and DFT Calculations

- Using the Douglas-Kroll-Hess (DKH) Hamiltonian
- Periodic Electrostatic Embedded Cluster Method

- Empirical Dispersion Correction for DFT Calculations