# Basis set¶

Band represents the single-determinant electronic wave functionus 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 Description: Definition of the basis set `Type`

Type: Multiple Choice Default value: DZ Options: [SZ, DZ, DZP, TZP, TZ2P, QZ4P] Description: The basis sets to be used. `Core`

Type: Multiple Choice Default value: Large Options: [None, Small, Medium, Large] Description: Size of the frozen core.

## 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]

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 Single-Zeta-SCF-Restart link xxx.**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

## Available Basis Sets¶

The basis set files, containing the definition of the basis set, are located in `$ADFHOME/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.

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-Uuo(Z=104-118) | ae .5d .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
ByAtomType # Non-standard block. See details.
...
End
End
```

`Basis`

Type: Block Description: Definition of the basis set `Folder`

Type: String Description: 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’ `ByAtomType`

Type: Non-standard block Description: Definition of the basis set for specific atom types (one definition per line). Format: ‘AtomType Type=Type Core=Core’. Example: ‘C.large_basis Type=TZ2P Core=None’

See also: Example: Multiresolution.

## 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]
Radius float
Delta float
End
```

`SoftConfinement`

Type: Block Description: 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 Default value: Auto Options: [Auto, Basic, Normal, Good, VeryGood, Excellent] Description: 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 Unit: Bohr Description: Explicitely specify the radius parameter of the Fermi-Dirac function. `Delta`

Type: Float Unit: Bohr Description: 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 n^{th}type, which in turn has atoms at positions defined by the n^{th}`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:- the number of atomic shells (1s,2s,2p,etc.) and the nr. of core-shells (two integers on one line).
- 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

- \(E(A+B)\)
- \(E(A)\)
- \(E(B)\)

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

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

See also

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. |