# Basis set¶

Band represents the single-determinant electronic wave functions as a linear combinations of atom-centered basis functions (the basis set). See also: Wikipedia page on Basis Sets.

The basis sets in Band consists of NAOs (Numerical Atomic Orbitals, obtained by solving numerically the Kohn-Sham equations for the isolated spherical atoms) augmented by a set of STOs (Slater Type Orbitals).

The choice of basis set is very important, as it influences heavily the accuracy, the CPU time and the memory usage of the calculation. Band comes with 6 predefined types of basis sets: SZ, DZ, DZP, TZP, TZ2P, QZ4P (SZ: Single Zeta, DZ: Double Zeta, DZP: Double Zeta + Polarization, TZP: Triple Zeta + Polarization, TZ2P: Triple Zeta + Double Polarization, QZ4P: Quadruple Zeta + Quadruple Polarization). See the sections Which basis set should I use? and Available Basis Sets for more details.

To speed up the calculation, Band can use the frozen core approximation in which core orbitals are kept frozen during the SCF procedure (and the valence orbitals are orthogonalized against the frozen orbitals). One can run an all electron calculation by specifying Core None in the Basis input block. Note: some features, such as Hybrid functionals, are incompatible with the frozen-core approximation, and require an all electron (i.e. Core None) basis set.

## Basis input block¶

You can specify which basis set to use in the Basis input block.

Basis
Type [SZ | DZ | DZP | TZP | TZ2P | QZ4P]
Core [None | Small | Medium | Large]
End

Basis
Type: Block Definition of the basis set
Type
Type: Multiple Choice DZ [SZ, DZ, DZP, TZP, TZ2P, QZ4P] Basis set Select the basis set to use. SZ : Single Z DZ : Double Z DZP : Double Z, 1 polarization function TZP : Triple Z, 1 polarization function TZ2P : Triple Z, 2 polarization functions QZ4P : Quadruple Z, 4 polarization function The basis set chosen will apply to all atoms in your structure. If a matching basis is not found a better type might be used.
Core
Type: Multiple Choice Large [None, Small, Medium, Large] Select the size of the frozen core you want to use. Small, Medium, and Large will be interpreted within the basis sets available (of the selected quality), and might refer to the same core in some cases.

## Which basis set should I use?¶

The hierarchy of basis sets, from the smallest and least accurate (SZ) to the largest and most accurate (QZ4P), is SZ < DZ < DZP < TZP < TZ2P < QZ4P.

The choice of basis set is in general a trade off between accuracy and computation time: the more accurate the basis set, the more computationally demanding the calculation will be (both in term of CPU time and the memory usage).

As an example, in the following table we compare accuracy v.s. CPU time for a (24,24) carbon nanotube using different basis sets. “Energy error” is defined as the absolute error in the formation energy per atom using the QZ4P results as reference. [1]

Table 1 Accuracy and CPU time ratio for a (24,24) carbon nanotube using different basis sets
Basis Set Energy Error [eV] CPU time ratio (relative to SZ)
SZ 1.8 1
DZ 0.46 1.5
DZP 0.16 2.5
TZP 0.048 3.8
TZ2P 0.016 6.1
QZ4P reference 14.3

It is worthwhile noting that the error in formation energies are to some extend systematic, and they partially cancel each other out when taking energy differences. For example, if one considers the difference in energy between two carbon nanotubes variants ((24,24) and (24-0)) with the same number of atoms, the basis set error is smaller than 1 milli-eV/atom already with a DZP basis set, which is much smaller than the absolute error in the individual energies. The same consideration holds for reactions barriers: the error in the energy difference between different conformations is much smaller than the error in the absolute energies themselves.

Band gaps:

The following figure shows the convergence WRT basis set of band gaps (XC:PBE). While DZ is often inaccurate (since DZ lacks any polarization function, the description of the virtual orbital space is very poor), TZP captures the trends very well.

In general, since the basis set might have different effects on different properties, it is advisable to run a few simple calculations to get an idea of the effect of the basis set with your property of interest.

A summary of the basis sets:

• SZ: Single Zeta, the minimal basis set (only NAOs), serves mostly a technical purpose. The results are rather inaccurate, but it’s computationally efficient. It can be useful for running a very quick test calculation. See also SZ-SCF-Restart.
• DZ: The Double Zeta basis set is computationally very efficient. It can be used for pre-optimization of structures (that should then be further optimized with a better basis set). Since it has no polarization functions, properties depending on the virtual orbital space will be rather inaccurate.
• DZP: Double zeta plus polarization function. Only available for main group elements up to Krypton. For other elements a TZP basis set will be used automatically. This is a reasonably good basis set for geometry optimizations of organic systems.
• TZP: The Triple Zeta plus Polarization basis set offers the best balance between performance and accuracy. This is the basis set we would generally recommend.
• TZ2P: The Triple Zeta plus Double Polarization basis set is an accurate basis set. It is qualitatively similar to TZP but quantitatively better. It should be used when a good description of the virtual orbital space is needed.
• QZ4P: Quadruple zeta plus Quadruple Polarization. This is the biggest basis set available. It can be used for benchmarking.

Frozen core:

In general, the frozen core approximation does not influence the results significantly (especially if one uses a small frozen core). For accurate results on certain properties (like Properties at Nuclei) all electron basis sets are needed on the atoms of interest.

• For Meta-GGA XC functionals, it is recommended to use small or none frozen core (the frozen orbitals are computed using LDA and not the selected Meta-GGA)
• For optimizations under pressure, use small or none frozen core

The basis set files, containing the definition of the basis set, are located in $AMSHOME/atomicdata/Band. The next table gives an indication frozen core (fc) standard basis sets are available for the different elements in BAND. Note that all electron (ae) basis set are available for all basis sets types. Table 2 Available standard basis sets for non-relativistic and ZORA calculations H-Ubn (Z=1-120) Element frozen core SZ, DZ DZP TZP, TZ2P, QZ4P H-He (Z=1-2) ae Yes Yes Yes Li-Ne (Z=3-10) ae .1s Yes Yes Yes Na-Mg (Z=11-12) ae .1s .2p Yes Yes Yes Al-Ar (Z=13-18) ae .2p Yes Yes Yes K-Ca (Z=19-20) ae .2p .3p Yes Yes Yes Sc-Zn (Z=21-30) ae .2p .3p Yes Yes Ga-Kr (Z=31-36) ae .3p .3d Yes Yes Yes Rb-Sr (Z=37-38) ae .3p .3d .4p Yes Yes Y-Cd (Z=39-48) ae .3d .4p Yes Yes In-Ba (Z=49-56) ae .4p .4d Yes Yes La-Lu (Z=57-71) ae .4d .5p Yes Yes Hf-Hg (Z=72-80) ae .4d .4f Yes Yes Tl (Z=81) ae .4d .4f .5p Yes Yes Pb-Rn (Z=82-86) ae .4d .4f .5p .5d Yes Yes Fr-Ra (Z=87-88) ae .5p .5d Yes Yes Ac-Lr (Z=89-103) ae .5d .6p Yes Yes Rf-Og (Z=104-118) ae .5d .5f Yes Yes Uue-Ubn (Z=119-120) ae .5f Yes Yes • element name (without suffix): all electron (ae) • .1s frozen: 1s • .2p frozen: 1s 2s 2p • .3p frozen: 1s 2s 2p 3s 3p • .3d frozen: 1s 2s 2p 3s 3p 3d • .4p frozen: 1s 2s 2p 3s 3p 3d 4s 4p • .4d frozen: 1s 2s 2p 3s 3p 3d 4s 4p 4d • .4f frozen: 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f • .5p frozen: 1s 2s 2p 3s 3p 3d 4s 4p 4d 5s 5p (La-Lu) • .5p frozen: 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p (other) • .5d frozen: 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d • .6p frozen: 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 6s 6p (Ac-Lr) • .5f frozen: 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p Note Not all combinations of basis set Type and Core are available for all elements. If a specific combination is not available, Band will pick the first better basis set. ## More Basis input options¶ Basis Folder string PerAtomType Core [None | Small | Medium | Large] File string Symbol string Type [SZ | DZ | DZP | TZP | TZ2P | QZ4P] End PerRegion Core [None | Small | Medium | Large] Region string Type [SZ | DZ | DZP | TZP | TZ2P | QZ4P] End End  Basis Type: Block Definition of the basis set Folder Type: String Path to a folder containing the basis set files. This can be used for special use-defined basis sets. Cannot be used in combination with ‘Type’ PerAtomType Type: Block True Defines the basis set for all atoms of a particular type. Core Type: Multiple Choice [None, Small, Medium, Large] Size of the frozen core. File Type: String The path to the basis set file. The path can be absolute or relative to$AMSRESOURCES/Band. Specifying the path to the basis file explicitly overrides the automatic basis file selection via the Type and Core subkeys.
Symbol
Type: String The symbol for which to define the basis set.
Type
Type: Multiple Choice [SZ, DZ, DZP, TZP, TZ2P, QZ4P] The basis sets to be used.
PerRegion
Type: Block True Defines the basis set for all atoms in a region. If specified, this overwrites the values set with the Basis%Type and Basis%PerAtomType keywords for atoms in that region. Note that if this keyword is used multiple times, the chosen regions may not overlap.
Core
Type: Multiple Choice Large [None, Small, Medium, Large] Size of the frozen core.
Region
Type: String The identifier of the region for which to define the basis set. Note that this may also be a region expression, e.g. ‘myregion+myotherregion’ (the union of two regions).
Type
Type: Multiple Choice DZ [SZ, DZ, DZP, TZP, TZ2P, QZ4P] The basis sets to be used.

## Confinement of basis functions¶

It is possible to alter the radial part of the basis functions in order to make them more compact, which will in turn speeds up the calculation.

SoftConfinement
Quality [Auto | Basic | Normal | Good | VeryGood | Excellent]
Delta float
End

SoftConfinement
Type: Block In order to make the basis functions more compact, the radial part of the basis functions is multiplied by a Fermi-Dirac (FD) function (this ‘confinement’ is done for efficiency and numerical stability reasons). A FD function goes from one to zero, controlled by two parameters. It has a value 0.5 at Radius, and the decay width is Delta.
Quality
Type: Multiple Choice Auto [Auto, Basic, Normal, Good, VeryGood, Excellent] Confinement In order to make the basis functions more compact, the radial part of the basis functions is multiplied by a Fermi-Dirac (FD) function (this ‘confinement’ is done for efficiency and numerical stability reasons). A FD function goes from one to zero, controlled by two parameters. It has a value 0.5 at Radius, and the decay width is Delta. This key sets the two parameters ‘Radius’ and ‘Delta’. Basic: Radius=7.0, Delta=0.7; Normal: Radius=10.0, Delta=1.0; Good: Radius=20.0, Delta=2.0; VeryGood and Excellent: no confinement at all. If ‘Auto’, the quality defined in the ‘NumericalQuality’ will be used.
Radius
Type: Float Bohr Explicitely specify the radius parameter of the Fermi-Dirac function.
Delta
Type: Float Bohr Explicitely specify the delta parameter of the Fermi-Dirac function (if not specified, it will be 0.1*Radius).
• For geometry optimizations under pressure, Basic soft confinement is recommended.

## Manually specifying AtomTypes (expert option)¶

AtomType (block-type)

(Expert Option) Description of the atom type. Contains the block keys Dirac, BasisFunctions and FitFunctions. The key corresponds to one atom type. The ordering of the AtomType keys (in case of more than one atom type) is NOT arbitrary. It is interpreted as corresponding to the ordering of the Atoms keys. The n-th AtomType key supplies information for the numerical atom of the nth type, which in turn has atoms at positions defined by the nth Atoms key.

AtomType ElementSymbol
Dirac ChemSym
{option}
...
shells cores
shell_specification {occupation_number}
...
End
{BasisFunctions
shell_specification STO_exponent
...
End}
FitFunctions
shell_specification STO_exponent
...
End
END


The argument ElementSymbol to AtomType is the symbol of the element that is referred to in the Atoms key block.

Dirac (block-type)

Specification of the numerical (‘Herman-Skillman’) free atom, which defines the initial guess for the SCF density, and which also (optionally) supplies Numerical Atomic Orbitals (NOs) as basis functions, and/or as STO fit functions for the crystal calculation. The argument ChemSym of this option is the symbol of the element of the atom type. The data records of the Dirac key are:

1. the number of atomic shells (1s,2s,2p,etc.) and the nr. of core-shells (two integers on one line).
2. specification of the shell and its electronic occupation.

This specification can be done via quantum numbers or using the standard designation (e.g. ‘1 0’ is equivalent to ‘1s’). Optionally one may insert anywhere in the Dirac block a record Valence, which signifies that all numerical valence orbitals will be used as basis functions (NOs) in the crystal calculation. You can also insert NumericalFit followed by a number (max. $$l$$-value) in the key block, which causes the program to use numerical STO fit functions. For example NumericalFit 2 means that the squares of all s,p, and d NOs will be used as STO fit functions with $$l=0$$, since the NOs are spherically symmetric. If you insert Spinor, a spin-orbit relativistic calculation for the single-atom will be carried out.

The Herman-Skillman program generates all its functions (atomic potential, charge density, one-electron states) as tables of values in a logarithmic radial grid. The number of points in the grid, and the min. and max. r-value are defaulted at 3000, 0.000001, and 100.0 (a.u.) respectively. These defaults can be overwritten by specifying anywhere in the Dirac block the (sub)keys radial, rmin and rmax.

The program will do a spin-unrestricted calculation for the atoms in addition to the restricted one. The occupation of the spin-orbitals will be of maximum spin-multiplicity and cannot be controlled in the Dirac key-block.

BasisFunctions (block-type)
Slater-type orbitals, specified by quantum numbers $$n$$,:math:l or by the letter designation (e.g. 2p) and one real (alpha) per STO. One STO per record. Use of this key is optional in the sense that Slater-type functions are not needed if other basis functions have been specified (i.e. the numerical atomic orbitals, see key Dirac).
FitFunctions (block-type)

Slater-type fit functions, described in the same way as in BasisFunctions. Each FitFunctions key corresponds to one atom type, the type being the one of the preceding Dirac key. The selection choice of a ‘good’ fit set is a matter of experience. Fair quality sets are included in the database of the molecular program ADF.

Example:

AtomType C :: Carbon atom
Dirac C
3 1
VALENCE
1s
2s
2p 2.0
End
BasisFunctions
1s 1.7
...
End
FitFunctions
1s 13.5
2s 11.0
...
End
End

TestFunctions (block-type)
An optional subkey of the AtomType key block is TestFunctions which has the same format as the BasisFunctions and FitFunctions blocks. The TestFunctions block specifies STOs to be used as test functions in the numerical integration package. For the time being the $$l$$ value is ignored. A possible application is to include a very tight function, to increase the accuracy near a nucleus.

## Basis Set Superposition Error (BSSE)¶

The Ghost Atom feature enables the calculation of Basis Set Superposition Errors (BSSE). Normally, if you want to know the bonding energy of system A with system B you calculate three energies

1. $$E(A+B)$$
2. $$E(A)$$
3. $$E(B)$$

The bond energy is then $$E(A+B) - E(A) - E(B)$$

The BSSE correction is about the idea that we can also calculate E(A) including basis functions from molecule B.

You can make a ghost atom by simply adding “Gh.” in front of the element name, for instance “Gh.H” for a ghost hydrogen , “Gh.C” for a ghost Carbon atom.

You will get a better bonding energy, closer to the basis set limit by calculating

$$E(A+B) - E(\text{A with B as ghost}) - E(\text{B with A as ghost})$$

The BSSE correction is

$$E(A) - E(\text{A with B as ghost}) + E(B) - E(\text{B with A as ghost})$$

Footnotes

 [1] Computational details: Single Point calculation, ‘Good’ NumericalQuality, no frozen core, 7 k-points, XC functional: GGA:PBE. Calculation performed on a 24 cores compute node. 96 atoms in the unit cell.