Basis Set Input
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Basis Set Input
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CRYSTAL performs ab initio calculations on periodic systems within the linear combination of atomic orbitals (LCAO) approximation. That is, the crystalline orbitals (CO) are treated as linear combinations of Bloch functions (BF), \[ \begin{equation} \label{eq:COdef} \psi_{i}({\bf r};{\bf k}) = \sum_{\mu} a_{\mu,i} ({\bf k}) \phi_{\mu}({\bf r};{\bf k}) \end{equation} \] \[ \begin{equation} \label{BFdef} \phi_{\mu}({\bf r};{\bf k}) = \sum_{\bf g} \varphi_{\mu} ({\bf r}-{\bf A}_\mu-{\bf g})\; e^{i {\bf k} \cdot {\bf g}} \end{equation} \]
defined in terms of local functions, hereafter indicated as atomic orbitals (AO). Those local functions are expressed as linear combination of a certain number of Gaussian type functions (GTF). \[ \begin{equation} \label{AOdef} \varphi_\mu ({\bf r}-{\bf A}_\mu - {\bf g}) = \sum_j^{n_G} d_j\; G( \alpha_j; {\bf r}-{\bf A}_\mu - {\bf g}) \end{equation} \]
They are characterized by the same centre A, with fixed coefficients, d, and exponents,
alpha, defined in the input.
| r | coordinate of an electron |
| g | direct lattice vector the sum over g is extended to the all lattice vectors (infinite) of direct lattice. |
| k | lattice vector defining a point in the reciprocal lattice |
| A | coordinate of an atom in the reference cell |
| a | variational coefficients. They multiply the BF; the sum over μ is limited to the number of basis functions |
| d | coefficients of the primitive gaussians in the contraction, fixed for a given basis set; the sum over j is limited to the number of functions in the contraction |
The AOs belonging to a given atom are grouped into shells.
The shell can contain either all AOs with the same quantum numbers, n and l
(for instance 3s, 2p, 3d shells), or all the AOs with the same principal quantum number n and different l (sp shells; exponent of s and p gaussians are the same).
A single, normalized, s-type GTF, the adjoined gaussian,
is associated with each shell. The exponent of the adjoined gaussian is
the smallest exponent of the gaussians in the contraction.
The adjoined gaussian is used to estimate the AO
overlap and select the level of approximation to be adopted for the
evaluation of the integrals.
This chapter discusses briefly the basis set
input section. The basis set definition is the first step to uniquely
define the level of calculation. The molecular/crystalline basis set
must be balanced, that means each centre must have the same variational freedom in describing the electrons formally attributed to the centre.
Basis sets of different quality on different atoms (minimal basis sets
on some atoms and split valence + polarization on others) may give
spurious effects, exploited during the SCF iterations, and driving to
solution not converging.
Few simple examples will be shown to explain how the basis set has to be specified in the CRYSTAL input.
The definition of the basis set is in the second input block. Basis set and initial electronic configuration must be given for each atom with a different conventional atomic number in the crystal structure input. CRYSTAL can use either general basis sets, including s, p, d, f functions or standard Pople basis sets (internally stored). All electrons and effective core pseudopotentials (ECP) calculations can be performed. In the latter case, the ECP must be inserted in this section as well.
When the basis set input has been specified, several optional keywords can be used, related to modification of the electronic configuration, use of ghost functions, and printing options.
The basis set input format is strictly related to the mathematical definition of basis set given above.
For each atom (as many blocks as different types of atoms in the crystal structure) it must be specified:
the conventional atomic number and the number of shells ns of the atomic basis set
for each shell (ns blocks of records), type of basis set (0-1-2), type of shell (0-1-2-3-4),
number of primitives GTF ng, shell electronic charge, scale factor
for each primitive (ng records - optional - basis set type 0 only)
exponent, contraction coefficient, [contraction coefficient]
The definition of atomic basis sets ends with the record:
99 0
that is the conventional atomic number 99 with a 0 shell. Optional basis set keywords may follow.
The "conventional atomic number" links the basis set to the atoms entered in geometry input.
Basis set input is closed by the keyword END
In CRYSTAL three basis set types are available:
| 0 | general basis set: exponent and contraction coefficients defined in input; |
| 1 | Pople STO-nG type basis set; |
| 2 | Pople 3(6)-21G type basis set; |
The shell types available correspond to:
| shell type code | shell type | AO | AO order | max shell charge |
|---|---|---|---|---|
| 0 | s | 1 | s | 2 |
| 1 | sp | 4 | s, x, y, z | 8 |
| 2 | p | 3 | x, y, z | 6 |
| 3 | d | 5 | 2z2-x2-y2, xz, yz, x2-y2, xy | 10 |
| 4 | f | 7 | (2z2-3x2-3y2)z, (4z
2-x2-y2)x, (4z2-x2-y2)y,
(x 2-y2)z, xyz, (x2-3y2)x, (3 x2-y2)y |
0 - polarization only |
d shells include 5 d orbitals, f shells include 7 orbitals.
For sp shells two contraction coefficients must be specified, for
s and p AO, respectively.
Standard polarization functions can be added to 3(6)-21G basis sets of atoms up to Z=18, by inserting a record describing the polarization shell.
The formal shell electronic charge is the number of electrons attributed to each shell as initial electronic configuration. The electronic configuration of the atoms is used in the calculation of the atomic wave function only (when the guess for SCF is a superposition of atomic densities). It may correspond to a neutral atom or to an ion (for MgO, Mg and O, or Mg++ and O--). The net charge in the cell must be zero, the cell must be neutral.
Example 1: General basis set input for MgO (Hartree-Fock) with a STO-3G Pople basis set (H - Ne: W.J. Hehre, R.F. Stewart and J.A. Pople, J. Chem. Phys. 2657 (1969). # Na - Ar: W.J. Hehre, R. Ditchfield, R.F. Stewart, J.A. Pople, J. Chem. Phys. 2769 (1970)).
MgO bulk |
MgO bulk |
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Since CRYSTAL14, a set of internally stored pre-defined basis sets are available by using the keyword BASISSET.
Note that when specifying this keyword, the END to close the GEOMETRY input section and the keywords 99 0 and END that close the "standard" basis set input section (see above), are no more necessary.
The dataset of available basis sets includes (available atomic numbers in parentheses):
| Keyword | Description | |
|---|---|---|
| STO-3G | Pople's STO-3G minimal basis set (1--53) | |
| STO-6G | Pople's STO-6G minimal basis set (1--36) | |
| POB-DZVP | POB double-\(\zeta\) valence + polarization set for solid state systems (1--35, 49, 74) | |
| POB-DZVPP | POB double-\(\zeta\) valence basis set + a double set of polarization functions for solid state systems (1--35, 49, 83) | |
| POB-TZVP | POB triple-\(\zeta\) valence + polarization basis set for solid state systems (1--35, 49, 83) |
The same example used for MgO examples 1 and 2, can be obtained by specifying two keywords: BASISSET and STO-3G.
MgO bulk |
The conventional atomic number, NAT, links the basis set with the atoms defined in the crystal structure.
| NAT<200: all-electron BS | Given Z, NAT=Z, NAT'=Z+100 |
| NAT>200: valence-electron BS | Given Z, NAT=Z+200, NAT'=Z+300. A core pseudopotential (ECP) must be defined |
A maximum of two different basis sets may be given for the same
chemical species in positions not symmetry related, using the
conventional atomic number NAT and NAT'.
Atoms with equal conventional atomic number are associated with the same basis set.
The atomic number Z is given by the remainder of the division of the conventional atomic number by 100 (Example: NAT=108, Z=8, Oxygen, all electron; NAT=208, Z=8, Oxygen, ECP).
A conventional atomic number 0 defines ghost atoms, that is points in space with an associated basis set, but lacking a nuclear charge.
Here an example is reported concerning the use of different conventional atomic numbers for the same atom, but in non-equivalent positions.
In the following example (test 35 of CRYSTAL test cases), a three-layer slab model of the MgO(001) surface is created (SLABCUT) and a CO molecule is added (ATOMINSE) upon the surface to simulate an adsorption process. Two different atomic basis sets are used for the oxygen atom: in MgO the oxygen (NAT=8, Z=8) basis set is optimized for O--, in CO molecule the oxygen (NAT=108, Z=8) basis set is a standard molecular one.
TEST35 - MGO SLAB (001), 3 LAYER + CO ADSORPTION |
Exercise: Use the MgO bulk input to create a Mg defect in 32 atoms super cell of MgO. Then specify in the input deck that the first-neighbors of the defect have a different basis set with respect to the other oxygen atoms.
CRYSTAL can perform valence-electron only calculations with the aid of
effective core pseudopotentials (ECP). The idea behind pseudopotentials
is to treat the core electrons as effective averaged potentials rather
than actual particles. Thus, pseudopotentials are modifications to the
Hamiltonian.
Atoms with conventional atomic number > 200 have
a valence only basis set, being the core electrons described by an
effective core pseudopotential.
The definition of the ECP is inserted in the atomic
basis set input block, after the record giving the conventional atomic
number and before the records giving the variational basis set.
The variational basis set must be the basis set supplied with the ECP
The following ECPs are available as internal data in the CRYSTAL code:
| ECP | Keyword |
|---|---|
| Hay and Wadt large core | HAYWLC |
| Hay and Wadt small core | HAYWSC |
| Durand and Barthelat | BARTHE or DURAND |
Warning: The data defining the pseudopotentials were included in CRYSTAL92 and never modified.
The keyword INPUT allows to enter pseudopotentials from external data, when available. This is the case for Stuttgart-Dresden pseudopotentials for which updated data can be found at the web site: http://www.uni-stuttgart.de/theochem/
See CRYSTAL User Manual (subject index ECP) for complete explanation of user defined ECP.
A conventional atomic number 0 defines ghost atoms, that is points in
space with an associated basis set, but lacking a nuclear charge. Two
are the main applications of ghost atoms.
The first application is linked to the transformation of an atom in a "ghost", by removing the nuclear
charge and the shell electron charge attributed to that centre, but
leaving the basis set centred at the atomic position. This can be
accomplished after the basis set definition by the use of the optional
keyword GHOSTS. The number of atoms to be transformed and the corresponding atomic
labels must be specified. If the system is forced to maintain the original symmetry (KEEPSYMM),
all the atoms symmetry related to the given one are turned into ghosts.
Attention must be paid to the neutrality of the cell.
Ghost atoms and ghost functions can be very useful when treating defects like
vacancies in materials (e.g. F center in oxides, C vacancy in diamond),
because they allow a build up of charge density in the defective region
by giving variational freedom in that site.
The first step when using a periodic model for a
local defect in a crystalline materials the creation of a super cell,
to guarantee no interaction between defects in neighboring cells.
See tutorial on defects in
materials.
Note: the keyword ATOMREMO (see the geometry input section) allows to create a vacancy, like GHOSTS, but the basis functions are also removed, as well as nuclear charge and electronic charge.
A second application is the use of ghost functions to evaluate the basis set superposition error (BSSE) in periodic systems by means of the counterpoise method. For instance, to evaluate the adsorption energy of a monolayer of CO on the (001) surface of MgO, the difference between the energy of CO+MgO and the sum of the energy of CO monolayer and clean MgO (001) surface, computed with the basis set of the atoms of the two separate layers only, gives an incorrectly high value.
\(E_{adsorption} = E_{CO+MgO} - (E_{CO} +E_{MgO})\)
As the basis set used in the calculation are generally far from complete,
both the adsorbate layer and the surface layer may use additional
variational freedom offered by each others basis functions to lower
their energy of the complex system. This gives a non-physical
stabilizing contribution to the energy of the surface-adsorbate complex
(and may also lead to artificial charge transfer if the basis set
description of the two systems is unbalanced). Hence there may be an
error (BSSE) in the interaction energy which is connected with the
superposition of the basis functions of the two subsystems.
Within the counterpoise method, the
correction to the BSSE can be computed by means of the GHOSTS keyword.
In the example above, the two contributions of CO monolayer and clean
MgO (001) surface are computed starting from the geometry of the
complex system, and transforming into ghosts the atoms of the surface
and the CO monolayer, respectively.
See test35 (MgO+CO) and test36 (CO monolayer with Mg and O ghosts)
A quite peculiar application of GHOSTS is to add bond basis functions. An example is supplied by test 28 of CRYSTAL test cases. The variational basis set includes s functions only: To simulate p functions on Lithium, s functions are added around the Li atom. Warning: this is a numerical test, created to compare the results of CRYSTAL with other programs handling s functions only (Mulliken population analysis is based on the assumption that the basis functions are centred on an atomic position). The ghost atom is inserted in the geometry input, as an atom with conventional atomic number 0.
TEST28 - LIH - GHOST ATOMS TO SIMULATE P FUNCTIONS ON LI |
NEIGHBORS OF THE NON-EQUIVALENT ATOMS |
CHEMOD
It may be useful to allow atoms with the same basis set to have different
electronic configurations. The formal shell charges attributed in the
basis set input may be modified for selected atoms, identified by their
label, by inserting the keyword CHEMOD.
The user is advised to run a TESTGEOM, to be secure of what
atoms/labels must be altered.
An example is an oxygen vacancy (GHOST) in a MgO(001) 3-layers slab (SLABCUT) with a Li defect
(ATOMSUBS). The CRYSTAL input (TEST37) file follows:
TEST37 - MGO-LI WITH OXYGEN VACANCY |
**** ATOMS BELONGING TO THE SUPERCELL |
ATOMS IN THE ASYMMETRIC UNIT 12 - ATOMS IN THE UNIT CELL: 24 |
The electronic configuration of the atoms is Mg++, O
--, Li neutral. When Li substitute Mg
(the cell is not neutral).
An oxygen "ghost" is created,
(all shell charges set to 0) and the charge of one electron is attributed to the ghost.
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In summary, a 3-layers slab is cut out from the MgO bulk, the 2D cell is
then enlarged by defining a 2x2 super cell, and a Li defect is finally
created by substituting a Mg atom. After the definition of the
defective structure, the basis set is given in input for each element
type. An ionic configuration has been used for Mg and O whereas an
atomic electronic configuration has been used for Li. To create the
oxygen vacancy the GHOSTS keyword is then used by leaving the
ghost functions into the vacancy. However, the basis set of the vacancy
is defined for an oxygen atom, so the corresponding electronic
configuration must be changed to define an unpaired electron in the
vacancy. Further, the initial atomic configuration for Li is modified
to be ionic.
The choice of the basis set is a fundamental step in defining the level of calculation and its accuracy. This is of particular importance when treating periodic systems where a large variety of chemical bonding can be found.
The choice of the basis set is a compromise between accuracy and costs. Nevertheless, we think that the accuracy must be the main goal of ab initio calculations. So, good quality basis sets should be always used in spite of their computational cost to avoid producing meaningless numbers.
More detailed discussions about the choice of the basis set in Chapter "Basis set" of CRYSTAL User's Manual.
A library of basis sets for periodic systems is available at the
CRYSTAL web site.
A click-able periodic table is shown. A text file with a list of basis
sets is displayed by clicking on each element type. References to
publications and hints about optimization, where relevant, are also
supplied.
An introductory guide to Gaussian basis set in solid-state electronic structure calculations has been written by Mike Towler (TCM Group, University of Cambridge, UK). The library contains basis sets for almost all the atoms beyond zinc in the periodic table. However, it contains basis sets which have never been used in published calculations, and also sets which have simply been optimized in atomic SCF calculations but not tested in a crystalline compound.
The EMSL
library supplies a wide selection of atomic basis sets optimized for
molecules, good starting point to generate an atomic basis set to be
used in a crystalline compound.:
The utility program
gaustocry converts a basis set input in GAUSSIAN
format in CRYSTAL format one.
Exercise: Take a molecular 6-31G(d,p) basis set from the EMSL site for H, C, N, and O, in the Gaussian code format, and translate it in the CRYSTAL code format.