What basis set should I use in ADF?
***********************************

This question is hard to answer in general, but a few broad
suggestions can be made.

Note:

  This page discusses the basis set requirements for DFT calculations.
  For a discussion of this question for GW calculations, see also the
  discussion in the GW page input block.


ZORA or non-relativistic calculation?
=====================================

By default, scalar relativistic effects with ZORA are included.

If you are doing a ZORA calculation, you will need the ZORA basis
sets, which can be found in $AMSHOME/atomicdata/ADF/ZORA. You may also
use the all-electron basis sets from the ET or AUG directory, but be
aware that these were optimized for non-relativistic calculations.
Currently the ZORA basis sets cover the entire periodic table and,
besides all-electron basis sets, offer a choice of frozen cores. At
present the ZORA directory does not contain basis sets with very
diffuse functions, which may be required in calculations of
hyperpolarizabilities or high-lying excitation energies, but for the
lighter elements (H-Kr) you can certainly use the all-electron basis
sets from the ET or AUG directory. Warning: in a ZORA calculation, use
only the frozen core basis sets coming from the
$AMSHOME/atomicdata/ADF/ZORA directory, or use all-electron basis
sets.

If you do not use ZORA, your basis sets should come from the
directories SZ, DZ, DZP, TZP, TZ2P, or one of the ET or AUG basis
sets. For many of the heavy elements only ZORA basis sets are
available, but for such elements it would be inadvisable to do non-
relativistic calculations anyway. For light elements the ZORA and
normal basis sets should be identical except for the description of
the frozen core. Usually the ZORA basis sets contain much steeper
basis and fit functions to accurately describe the core region.


Large or small molecule?
========================

For standard calculations (energies, geometries, etc.) we recommend
the following hierarchy of basis sets: SZ < DZ < DZP < TZP < TZ2P <
TZ2P+ < ET/ET-pVQZ < ZORA/QZ4P, where the largest and most accurate
basis is on the right. Not all basis sets are available for all
elements. For small negatively charged atoms or molecules, like
\text{F}^- or \text{OH}^- , basis sets with extra diffuse functions
are needed, as available in the AUG or ET/QZ3P-nDIFFUSE directories.
For example, the standard basis sets, or even the large ZORA/QZ4P
basis set, will often not be large enough for the accurate calculation
of such anions.

In general it is advisable to use the best basis set that you can
afford in terms of CPU time and memory. If you want to optimize the
geometry or calculate the atomization energy of a diatomic molecule
there is little reason not to use the very large ZORA/QZ4P basis, or
(for light elements) a similarly large ET basis (we recommend the ET-
pVQZ basis). If you are studying a molecule with 100 atoms or more,
the use of such large basis sets not only becomes prohibitive because
of the required CPU time and memory, but it also is much less needed
than for smaller systems. In medium-sized or large molecules even the
moderately large basis sets will prove to be quite adequate because of
the effect of basis set sharing. Each atom benefits from the basis
functions on its many neighbors. Additionally, if a large basis
contains diffuse functions, linear dependency problems may occur. See
also the input key DEPENDENCY. In many cases basis DZ or DZP will give
acceptable accuracy for calculations on large systems. If you are
inexperienced it may be prudent to test a few different basis sets to
get a feel for the size of basis set effects. To get a rough idea for
the size of various basis sets, we mention here the number of
functions for all-electron basis sets from the directories ZORA/SZ up
to ZORA/QZ4P. For carbon, the number of functions is 5 (basis
ZORA/SZ), 10 (DZ), 15 (DZP), 19 (TZP), 26 (TZ2P), 43 (QZ4P). The same
numbers for hydrogen are: 1 (SZ), 2 (DZ), 5 (DZP), 6 (TZP), 11 (TZ2P),
21 (QZ4P). These numbers arise because ADF uses "pure" d and f
functions. In other words, 5 instead of 6 d functions are used and 7
instead of 10 f functions. Note that especially the jump from TZ2P to
QZ4P is quite steep.

Broadly speaking we can state that the single zeta basis SZ is hardly
ever sufficient to get more than a qualitative picture and should be
used only when larger basis sets are not affordable. The double zeta
basis DZ performs already much better and may give quite reasonable
results, for example in geometry optimizations on large molecules.
However, in more subtle situations, for example if hydrogen bonds are
important, it is advisable to use at least one set of polarization
functions. This is the double zeta polarized DZP basis set. Basis set
TZP extends the valence space (but not the core space which remains
double zeta) to triple zeta. In basis TZ2P an additional polarization
function is added. For example, hydrogen gets a d polarization
function in addition to its p polarization function and carbon gets an
f polarization function on top of a d polarization function. Several
tests have shown that often the second polarization function is of
more use when it has an l value one higher than the first polarization
function. This is reflected in the choice just described. If another
polarization function is needed it is usually best to add another one
of the lowest l value (2p+1d for hydrogen, 2d+1f for carbon). This
choice has been made in the ET basis ET-QZ3P. There, sometimes even
three d polarization functions were added, for example 3 p functions
for Be, and 3 d functions for S. The reason for this is that S can
occur in hypervalent species such as SF_6 , which put special demands
on the basis set. In the case of Be, the unoccupied p level is so
close in energy to the occupied ones that it is sometimes called a
valence level. Semantics aside, it is clear that a proper description
of the p level of Be is very important and it is therefore not strange
to add a third p function. In general, the unoccupied levels for the
atoms on the left side of the periodic table are close to the occupied
ones. This makes it necessary to add a few extra functions for the
lowest unoccupied levels in order to get a description that
corresponds to the general level of accuracy one expects from the
hierarchical basis set names SZ-TZ2P. The basis set quality for a
particular subdirectory is now rather uniform throughout the periodic
system. At the same time we have attempted to increase the number of
functions in a systematic fashion so that each element is described by
at least as many functions of a particular l value as its predecessor.

The ZORA/QZ4P basis sets might be roughly described as core triple
zeta, valence quadruple zeta, with 4 polarization functions (2 d and 2
f functions for C, 2 p and 2 d for H). The fit sets corresponding to
these basis sets are also much larger than the fit sets found in basis
sets SZ-TZ2P. If one has doubts about the adequacy of a fit set for a
certain element, this can be tested by replacing its fit set by the
large one from the QZ4P directory, see the subkey FitType of the key
BASIS. In the ZORA/QZ4P basis sets, the choice for the exponents of
the polarization functions was done in a systematic, but somewhat
hand-waving manner. For this reason the exponents were always rounded
to half integers. Also the geometric mean of the exponents usually
does not coincide with the choices made in directories SZ-TZ2P and the
ET basis sets. However, the fact that two polarization functions (with
reasonable exponents) are present instead of a single one is far more
important. A reasonable intermediate basis set, in size between TZ2P
and QZ4P might be envisaged in which a single polarization function is
added, as described above. This is roughly the choice for the
polarization functions made in the ET directory ET-QZ3P.


Frozen core or all-electron?
============================

In general we recommend the use of frozen core basis sets for LDA and
GGA functionals if available. Especially for the heavier atoms the
number of functions is much smaller than for their all-electron
counterparts. Our tests indicate that the error made by invoking the
frozen core approximation is usually clearly smaller than the
difference with respect to slightly higher quality basis sets. For the
ZORA/QZ4P basis sets, only all-electron basis sets are available, as
these are intended for near basis set limit calculations only in which
the CPU time is not a major concern.

All-electron basis sets are required in case of SAOP, meta-GGA and
meta-hybrid functionals, functionals that use LibXC, post-KS
calculations like GW, RPA, MP2 or double hybrids. We recommend using
all-electron basis sets in case of Hartree-Fock or (range-separated)
hybrids. The X2C and RA-X2C relativistic method requires the use of
all-electron basis sets.

Geometry optimizations involving atoms with a too large frozen core
may give rise to numerical problems. In such cases it is recommendable
to use a smaller frozen core. In previous occurrences we have removed
such basis sets.

For accurate results on properties like nuclear magnetic dipole
hyperfine interactions (ESR), nuclear quadrupole coupling constants,
and chemical shifts (NMR), all-electron basis sets are needed on the
atoms of interest. For such properties tight functions might be
necessary for high accuracy, especially in a ZORA calculation.


Diffuse functions needed?
=========================

For small negatively charged atoms or molecules, like \text{F}^- or
\text{OH}^- , basis sets with extra diffuse functions are needed, as
available in the AUG or ET/QZ3P-nDIFFUSE directories. For example, the
standard basis sets, or even the large ZORA/QZ4P basis set, will often
not be large enough for the accurate calculation of such anions.

For accurate results on properties like polarizabilities,
hyperpolarizabilities, and high-lying excitation energies, also
diffuse functions are needed. This is especially true for calculations
on small molecules. In larger molecules the nature of the relevant
virtuals is much more 'molecular', much less Rydberg-like, so that the
normal basis sets may be sufficient. Basically all properties
calculated through the RESPONSE keyword may require diffuse functions.
If you use the EXCITATIONS keyword, the necessity of diffuse functions
depends on the type of excitation you are interested in. The lowest
excitations do not require diffuse functions, but Rydberg excitations
do.

In case of diffuse basis functions the risk of linear dependency in
the basis increases. This can be checked, and corrected for, using the
DEPENDENCY keyword. It is recommended to use this keyword for all
calculations involving diffuse functions. A good default setting is

DEPENDENCY bas=1d-4

However, it may be advisable to experiment a bit with the parameter,
especially if many linear dependent combinations of AOs are removed.
Using too many diffuse functions on a large molecule will lead to
insurmountable numerical problems. In such a case it is not only
useless, but even harmful, to add many diffuse functions.

In the previous release only some basis sets were provided that
contained diffuse functions. These were gathered in the directory
Vdiff. Now several ET basis sets have been developed for the elements
up to Ar containing some or many diffuse functions. We recommend using
these instead of the Vdiff directory. Most of these basis sets are
quite large and not very suitable for large molecules.

In ADF2005.01 augmented basis sets were added in the AUG directory,
especially devised for use in TDDFT calculations, such that one can do
a reasonably accurate calculation of excitation energies with a
relatively small basis set, see D.P. Chong [1].


Normal or even-tempered basis?
==============================

For normal calculations (these form the vast majority) we recommend
the use of the optimized basis sets in the directories SZ-TZ2P and,
for ZORA calculations, ZORA/SZ-QZ4P. These should be sufficient in
accuracy for even very demanding users and are available for the
entire periodic system (in the case of the ZORA basis sets). They are
also available with a frozen core variety, which saves much CPU time.

The ET basis sets, on the other hand, are available only in all-
electron form at the moment. Furthermore, most are pretty large
(larger, but also better than TZ2P). Additionally, relatively large
basis set superposition errors were detected for molecules containing
atoms in the row K-Kr. For this reason we only recommend ET basis sets
for the elements H-Ar at the moment. There they have yielded quite
nice, near basis set limit, results for the G2 test set. For these
light elements the ET basis sets can be comparable in quality to the
ZORA/QZ4P basis sets, even though they are smaller. The ET basis sets
are considered to be especially useful when diffuse functions are
required. In that case it is very easy to adapt the original ET basis
and fit set. The utilities provided for this in ADF will be described
below in an Appendix. The ET basis set utilities will also prove
useful for users who want to experiment with making their own basis
sets, or who have very special demands on the basis and fit. The
provided utilities automate much of the work needed to make new
atomicdata files.


Recommendations for Double-hybrids and MP2
==========================================

Double-hybrid and MP2 calculations are correlated-electron
calculations, and consequently they converge slower to the complete
basis set limit than independent-electron methods. Also for large
molecules, calculations with basis sets smaller than TZ2P are usually
not very precise. We thus recommend TZ2P as a minimum requirement. The
special, pseudo-correlation-consistent basis sets Corr/TZ3P and
Corr/QZ6P usually give better results. They are similar to TZ2P and
QZ4P, respectively, but they contain more polarization functions. They
can also be used to extrapolate the **correlation energy** to the
complete basis set limit according to

   E_{c}^{CBS} = E_{c}^{QZ} -  \frac{1}{N^{QZ}_{bas} }\frac{E_{c}^{QZ}
   - E_{c}^{TZ}}{\frac{1}{N^{QZ}_{bas}} - \frac{1}{N^{TZ}_{bas}}}\;.

where E_{c}^{QZ} and E_{c}^{TZ} are the values of the correlation
energies calculated with Corr/TZ3P and Corr/QZ6P, respectively, and
N^{QZ}_{bas} and N^{TZ}_{bas} are the corresponding number of basis
functions. In a double-hybrid calculation, one should only extrapolate
the correlation contribution to the total energy and use the KS
contribution from the calculation with the Corr/QZ6P basis set.

The dependency key should always be used, also when non-augmented
basis sets are used (augmented basis sets should be avoided):

   DEPENDENCY
      Bas 5e-4
   END

is usually sufficient.


What accuracy do the basis sets give?
=====================================

Tests on many diatomics were performed to assess the various basis
sets. We now document the results of some of these tests, in order to
give a feeling for the quality that can be obtained from the various
basis sets. See also van Lenthe and Baerends [2].

Tests for non-relativistic calculations on 36 diatomics containing
oxygen, namely the oxides of the first 36 elements (H-Kr). All-
electron basis sets were used. The ZORA/QZ4P basis set was used to
define the basis set limit result. The numbers in the table refer to
bonding energies in eV. Differences were taken between the QZ4P
results and the results in smaller basis sets. By construction, the
errors in the QZ4P column are zero.

+------------------+------------------+------------------+------------------+------------------+------------------+
|                  | QZ4P             | DZ               | DZP              | TZP              | TZ2P             |
+------------------+------------------+------------------+------------------+------------------+------------------+
| Average error    | 0.0              | 1.33             | 0.39             | 0.18             | 0.06             |
+------------------+------------------+------------------+------------------+------------------+------------------+
| Average absolute | 0.0              | 1.33             | 0.39             | 0.18             | 0.06             |
| error            |                  |                  |                  |                  |                  |
+------------------+------------------+------------------+------------------+------------------+------------------+
| Maximum error    | 0.0              | 2.84             | 1.07             | 0.41             | 0.13             |
+------------------+------------------+------------------+------------------+------------------+------------------+
| Worst case       | all              | SO               | BeO              | FO               | O_2              |
+------------------+------------------+------------------+------------------+------------------+------------------+

A few comments may help explain this table.

The oxides were used as a small test set because their equilibrium
bond lengths are known in many cases. Also, they have a large
influence on the electronic structure of the molecule, so that they
also test the adequacy of the polarization functions.

The errors in the small basis sets are systematic, because the
isolated atoms are described reasonably well, but the molecular energy
is not deep enough. For this reason the average errors and average
absolute errors are (nearly) always equal.

Test calculations on 100 diatomics containing oxygen, using all-
electron ZORA basis sets. Many basis sets for (very) heavy elements
are included here, which could not be included in the table above. The
numbers have the same interpretation as above and are again in eV.

+--------------+--------------+--------------+--------------+--------------+--------------+--------------+--------------+
|              | QZ4P         | DZ           | DZ           | TZP          | TZP          | TZ2P         | TZ2P         |
+--------------+--------------+--------------+--------------+--------------+--------------+--------------+--------------+
|              | ae           | fc           | ae           | fc           | ae           | fc           | ae           |
+--------------+--------------+--------------+--------------+--------------+--------------+--------------+--------------+
| Average      | 0.00         | 0.95         | 1.07         | 0.20         | 0.20         | 0.05         | 0.05         |
| error        |              |              |              |              |              |              |              |
+--------------+--------------+--------------+--------------+--------------+--------------+--------------+--------------+
| Average      | 0.00         | 0.98         | 1.07         | 0.20         | 0.21         | 0.05         | 0.05         |
| absolute     |              |              |              |              |              |              |              |
| error        |              |              |              |              |              |              |              |
+--------------+--------------+--------------+--------------+--------------+--------------+--------------+--------------+
| Maximum      | 0.00         | 2.86         | 2.83         | 0.74         | 0.74         | 0.19         | 0.17         |
| error        |              |              |              |              |              |              |              |
+--------------+--------------+--------------+--------------+--------------+--------------+--------------+--------------+
| Worst case   | all          | SO           | SO           | OgO          | OgO          | ThO          | OgO          |
+--------------+--------------+--------------+--------------+--------------+--------------+--------------+--------------+

Again we place a few comments on these frozen core and all-electron
results.

The trends are very similar to those in the previous table for the
lighter elements.

The frozen core results are very satisfactory, as they are very close
to the results with the corresponding all-electron basis sets. The
error introduced by the frozen core approximation is typically much
smaller than the one introduced by basis set incompleteness.

The average errors are quite comparable to those from the previous
table. The heavier elements do not seem to be much more difficult than
the lighter ones.

For heavy elements no reliable ET basis set is yet available for
comparison.

More results, all-electron, non-relativistic on roughly 140 different
diatomics at experimental or 'reasonable' equilibrium geometries.

+---------------------------+---------------------------+---------------------------+---------------------------+
|                           | QZ4P                      | DZ                        | TZP                       |
+---------------------------+---------------------------+---------------------------+---------------------------+
| Average error             | 0.00                      | 0.89                      | 0.11                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| Average absolute error    | 0.00                      | 0.89                      | 0.11                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| Maximum error             | 0.00                      | 2.84                      | 0.32                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| Worst case                | all                       | SO                        | O_2                       |
+---------------------------+---------------------------+---------------------------+---------------------------+

Only the non-relativistic basis sets DZ and TZP are fairly complete
for heavier elements.

Also for these general diatomics (not just oxides) the average and
maximum errors have decreased substantially, especially for basis TZP.

Same table, but now for frozen core basis sets. In all these tests the
smallest frozen core files were employed (i.e. the largest basis).

+---------------------------+---------------------------+---------------------------+---------------------------+
|                           | QZ4P                      | DZ                        | TZP                       |
+---------------------------+---------------------------+---------------------------+---------------------------+
| Average error             | 0.00                      | 0.73                      | 0.13                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| Average absolute error    | 0.00                      | 0.75                      | 0.16                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| Maximum error             | 0.00                      | 2.87                      | 1.80                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| Worst case                | all                       | SO                        | ThO                       |
+---------------------------+---------------------------+---------------------------+---------------------------+

The frozen core approximation has little influence on the accuracy for
the basis DZ, but a somewhat larger effect on the basis TZP. This is
especially due to certain worst cases, such as ThO.

ZORA, all-electron, over 240 diatomics

+----------------------+----------------------+----------------------+----------------------+----------------------+
|                      | QZ4P                 | DZ                   | TZP                  | TZ2P                 |
+----------------------+----------------------+----------------------+----------------------+----------------------+
| Average error        | 0.00                 | 0.70                 | 0.11                 | 0.02                 |
+----------------------+----------------------+----------------------+----------------------+----------------------+
| Average absolute     | 0.00                 | 0.70                 | 0.11                 | 0.03                 |
| error                |                      |                      |                      |                      |
+----------------------+----------------------+----------------------+----------------------+----------------------+
| Maximum error        | 0.00                 | 2.83                 | 0.44                 | -0.16                |
+----------------------+----------------------+----------------------+----------------------+----------------------+
| Worst case           | all                  | SO                   | I_2                  | Cr_2                 |
+----------------------+----------------------+----------------------+----------------------+----------------------+

The average error goes down very nicely from 0.70 to 0.11 to 0.03 eV
when going from DZ to TZP to TZ2P. The average error in basis TZ2P is
clearly below 1 kcal/mol (the famous chemical accuracy). Errors due to
deficiencies in current XC functionals are still much larger than
this. As a consequence, the ZORA/TZ2P basis will be more than adequate
for all standard calculations.

It is to be expected that these conclusions will not dramatically
change if larger test molecules are used. For geometry optimizations
as well the improved basis sets SZ-TZ2P and ZORA/SZ-TZ2P should be
more than sufficient for all standard cases. The ZORA/QZ4P can be
considered a very safe (though expensive) option for basis set limit
calculations.

[1] D.P. Chong, *Augmenting basis set for time-dependent density
    functional theory calculation of excitation energies: Slater-type
    orbitals for hydrogen to krypton*, Molecular Physics 103, 749
    (2005)

[2] E. van Lenthe and E.J. Baerends, *Optimized Slater-type basis sets
    for the elements 1-118*, Journal of Computational Chemistry 24,
    1142 (2003)
