11 3: Extended Basis Sets Chemistry LibreTexts

In the Supporting Information the reader can find full Crystal17(9) inputs for all the calculations presented

in the following, including the explicit definition of the atomic

basis sets. Basis functions can be considered as representing the atomic orbitals

of the atoms and are introduced in quantum chemical calculations because the

basis test set

equations defining the molecular orbitals are otherwise very difficult to solve. The molecular spin-orbitals that are used in the Slater determinant usually are expressed as a linear combination of some chosen functions, which are called basis functions. The fact that one function can be represented by a linear combination of other functions is a general property. All that is necessary is that the basis functions span-the-space, which means that the functions must form a complete set and must be describing the same thing. For example, spherical harmonics cannot be used to describe a hydrogen atom radial function because they do not involve the distance r, but they can be used to describe the angular properties of anything in three-dimensional space.

In some cases coefficients are premultiplied by a normalization constant for a Gaussian primitive, but in most cases it is assumed that is already normalized (and this is the correct way!). You have to be prepared for surprises when entering explicit basis sets from the literature. When specifying structure of the basis sets for the entire molecule, slashes are used to separate information for different atoms (or rows, if basis sets for a given row have the same structure for all atoms). For example, the basis set for water would be given as (10s,5p,1d/5s,1p) [4s,2p,1d/2s,1p] in which case the contractions for oxygen atoms are (10,5p,1d) [4s,2p,1d] and for the hydrogen (5s,1p) [2s,1p].

In practice, however, these exponents are estimated “using well established rules of thumb or by explicit optimization” (Dunning, 1989). The segmented contractions are disjointed, i.e., given primitive appears only in one contraction. Occasionally, one or two primitives may appear in more than one contraction, but this is an exception to the rule. The general contractions, on the contrary, allow each of the primitives to appear in each basis function (contraction).

1 Introduction to Basis Sets

In these cases, the wavefunctions of the system in question are represented as vectors, the components of which correspond to coefficients in a linear combination of the basis functions in the basis set used. In practice, plane-wave basis sets are often used in combination with an ‘effective core potential’ or pseudopotential, so that the plane waves are only used to describe the valence charge density. This combined method of a plane-wave basis set with a core pseudopotential is often abbreviated as a PSPW calculation.

This means that one primitive is shared (i.e. doubled) between two contractions, 6- and 3-contraction in this case. It would make little sense to share a primitive between 6- and 1- or 3- and 1-contrac- tion since such contraction whould yield the basis set of the same quality as “undoubled” one. In some cases the smallest exponent from the first contraction is repeated in the next contraction as the largest one. In the above case, the basis set formaly represents a general contraction, but since only one function is doubled, it is used frequently in programs that do not support general contractions.

The reason for their popularity is not that they are better, but simply, that the most popular ab initio packages do not implement efficient integral calculations with general contractions. The computer code to perform integral calculations with general contractions is much more complex than that for the segmented case. BSSE effects are reduced much more considerably by the increasing

of the basis set quality, rather than by the optimization of the exponents,

so that BSSE is quite similar for pob- or dcm- basis sets.

Clearly, the notation giving the number of primitives in each contraction as (abcd…) is not really useful here. It is especially true with newer sets implementing general contractions, where each primitive has all nonzero coefficients in practically every column. Examination of f-type functions shows that there are 10 possible Cartesian Gaussians, which introduce , and type contamination. This is a major headache since some programs remove these spurious functions and some do not. Of course, the results obtained with all possible Cartesian Gaussians will be different from those obtained with a reduced set. Both are covalently

bound systems but differ by hybridization (sp3 and sp2), as well as crystalline

(3D Vs 2D) and electronic (insulator and conductor) structures.

These are extended Gaussian basis functions with a small exponent, which give flexibility to the “tail” portion of the atomic orbitals, far away from the nucleus. Diffuse basis functions are important for describing anions or dipole moments, but they can also be important for accurate modeling of intra- and inter-molecular bonding. The segmented basis sets are usually structured in such a way that the most diffuse primitives (primitives with the smallest exponent) are left uncontracted (i.e. one primitive per basis function). More compact primitives (i.e. those with larger exponents) are taken with their coefficients from atomic Hartree-Fock calculations and one or more contractions are formed. Sometimes different contractions share one or two functions (the most diffuse function(s) from the first contraction enter the next one). In modern computational chemistry, quantum chemical calculations are typically performed using a finite set of basis functions.

A minimal basis set is one in which, on each atom in the molecule, a single basis function is used for each orbital in a Hartree–Fock calculation on the free atom. For atoms such as lithium, basis functions of p type are also added to the basis functions that correspond to the 1s and 2s orbitals of the free atom, because lithium also has a 1s2p bound state. For example, each atom in the second period of the periodic system (Li – Ne) would have a basis set of five functions (two s functions and three p functions). To add to the confusion, the coefficients are sometimes listed either as original coefficients in atomic orbitals or are renormalized for the given contraction.

However, the convergence of the energy does not imply convergence of other properties, such as nuclear magnetic shieldings, the dipole moment, or the electron momentum density, which probe different aspects of the electronic wave function. A minimal basis set is when one basis function for each atomic orbital in the atom, while a double-\(\zeta\), has two two basis functions for each atomic orbital. Correspondingly, a triple and quadruple-\(\zeta\) set had three and four basis functions for each atomic orbital, respectively. The selection of a basis set for quantum chemical calculations is very

What is Basis Test Set?

important. It is sometimes possible to use small basis sets to obtain good

chemical accuracy, but calculations can often be significantly improved by the

addition of diffuse and polarization functions.

basis test set

The use of double zeta functions in basis sets is especially important because without them orbitals of the same type are constrained to be identical even though in the molecule they may be chemically inequivalent. For example, in acetylene the \(p_z\) orbital along the internuclear axis is in a quite different chemical environment and is being used to account for quite different bonding than the \(p_x\) and \(p_y\) orbitals. With a double zeta basis set the \(p_z\) orbital is not constrained to be the same size as the \(p_x\) and \(p_y\) orbitals. This is particularly

important in solid-state calculations where the use of atom-centered

diffuse functions is more delicate and sometime useless. Gaussian-type orbital basis sets are typically optimized to reproduce the lowest possible energy for the systems used to train the basis set.

If this is systematically pursued, it

leads to a “theoretical model chemistry”,363 that is, a

well-defined energy procedure (e.g., Hartree-Fock) in combination with a well-defined basis set. Similarly to plane-wave basis sets an LAPW basis set is mainly determined by a cutoff parameter for the plane-wave representation in the interstitial region. In the spheres the variational degrees of freedom can be extended by adding local orbitals to the basis set.

Band structure comparison

between plane waves (Quantum Espresso)

and CRYSTAL in the graphene case with dcm[Cgraph] and POB basis sets with PBE as the DFT functional. The dcm[NaCl] basis for Cl, optimized in the rocksalt structure,

features significantly more contracted exponents, as far as s- and p-functions are concerned, while

the d exponent becomes more diffuse. As reported

in Table 4, a stronger

basis test set

contraction is observed in exponents of the s-type

  • Also, the possibility of employing basis sets specifically calibrated

    on a given system allowed us to easily reach the HF complete basis

    set limit for LiH which has been a long debated issue and for diamond.

  • The smallest of these are called minimal basis sets, and they are typically composed of the minimum number of basis functions required to represent all of the electrons on each atom.
  • A minimal basis set is one in which, on each atom in the molecule, a single basis function is used for each orbital in a Hartree–Fock calculation on the free atom.
  • It can be shown that the molecular orbitals of Hartree–Fock and density-functional theory also exhibit exponential decay.
  • Obviously, the best results could be obtained if all coefficients in Gaussian expansion were allowed to vary during molecular calculations.

orbitals https://www.globalcloudteam.com/ in going from the molecular def2 to the bulk metal and then

the ionic NaCl. In this case we had to remove the most diffuse p-function (0.03 au) in order to ensure convergence, but

at difference with the pob-TZVP case, we were able to keep all the d-functions in. Radial part of some Gaussian functions of carbon

from def2-TZVP

(def2), pob-TZVP (pob), and two different dcm-TZVP (dcm[Cdiam] for diamond, dcm[Cgraph] for graphene) basis sets.

Gaussian 16 can generate an appropriate fitting basis automatically from the AO basis, or you may select one of the built-in fitting sets. To compensate for this problem, each STO is replaced with a number of Gaussian functions with different values for the exponential parameter. Linear combinations of the primitive Gaussians are formed to approximate the radial part of an STO.

Adding a single polarization function to 6-311G (i.e. 6-311G(d)) will result in one d function for first and second row atoms and one f function for first transition row atoms, since d functions are already present for the valence electrons in the latter. Similarly, adding a diffuse function to the 6-311G basis set will produce one s, one p, and one d diffuse functions for third-row atoms. Many standard basis sets have been carefully optimized and tested over the

years. In principle, a user would employ the largest basis set available in

order to model molecular orbitals as accurately as possible. In practice, the

computational cost grows rapidly with the size of the basis set so a compromise

must be sought between accuracy and cost.

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