This is a double-\(\zeta\) basis sets and includes split-valence set (inner and valence) and linear combination of two orbitals of same type, but with different effective charges (i.e., \(\zeta\)). The most important basis sets are contracted sets of atom-centered Gaussian

functions. The number of basis functions used depends on the number of core and

valence atomic orbitals, and whether the atom is light (H or He) or heavy

## GENERAL CONTRACTIONS. TERMS AND NOTATION

(everything else). Contracted basis sets have been shown to be computationally

efficient and to have the ability to yield chemical accuracy (see Appendix B).

Basis sets have been constructed from Slater, Gaussian, plane wave and delta

functions. Slater functions were initially employed because they are considered

“natural” and have the correct behavior at the origin and in the asymptotic

regions. However, the two-electron repulsion integrals (ERIs) encountered when

using Slater basis functions are expensive and difficult to evaluate. However, while codes

incorporating delta functions are simple, thousands of functions are required

## SEGMENTED CONTRACTIONS. TERMS AND NOTATION

to achieve accurate results, even for small molecules. Plane waves are widely

used and highly efficient for calculations on periodic systems, but are not so

convenient or natural for molecular calculations.

Since the different orbitals of the split have different spatial extents, the combination allows the electron density to adjust its spatial extent appropriate to the particular molecular environment. In contrast, minimal basis sets lack the flexibility to adjust to different molecular environments. The original contractions derived from atomic Hartree-Fock calculations are frequently augmented with other functions. The polarization functions are simply functions having higher values of L than those present in occupied atomic orbitals for the corresponding atom. At least for me, there is some ambiguity here, since for lithium, the p-type functions are not considered polarization functions, while for sulphur, the d-functions are considered polarization functions. In both cases these orbitals are not populated in the ground electronic state of the atom.

The only problem with general contractions is that only a few programs support them. The code for integral package is much more complicated in this case, since it has to work on a block of integrals at each time, to compute the contribution from the given primitive set only once. Of course, you can always enter general contractions as “user defined segmented basis sets,” by repeating the same primitives over and over again in different contractions. This will cost you, however, immensely in computer time at the integral computation stage. Remember, the time required for calculating integrals is proportional to the 4th power in the number of Gaussian primitives, and most programs assume that primitives entering different contractions are different.

A common feature of all real-space methods is that the accuracy of the numerical basis set is improvable, so that the complete basis set limit can be reached in a systematical manner. Unfortunately, calculating integrals with STOs is computationally difficult and it was later realized by Frank Boys that STOs could be approximated as linear combinations of Gaussian-type orbitals (GTOs) instead. Because the product of two GTOs can be written as a linear combination of GTOs, integrals with https://www.globalcloudteam.com/ Gaussian basis functions can be written in closed form, which leads to huge computational savings (see John Pople). In the example above, corresponding exponents for s- and p-type contractions are equal but coefficients in s- and p-type contractions are different. Gaussian primitives are normalized here since coefficients for basis functions consisting of one primitive (last row) are exactly 1.0. The basis set above represents the following contraction (16s,10p) [4s,3p] or (6631,631).

Some of the most widely used basis sets are those developed by Dunning and coworkers,[7] since they are designed for converging post-Hartree–Fock calculations systematically to the complete basis set limit using empirical extrapolation techniques. In a minimum basis set, a single basis function is used for each atomic orbital on each constituent atom in the system. Density fitting can be made the default for jobs using pure DFT functionals by adding the DensityFit keyword to the route section (-#-) line in the Default.Route file. Fitting is faster than doing the Coulomb term exactly for systems up to several hundred atoms (depending on basis set), but is slower than exact Coulomb using linear scaling techniques (which are turned on automatically with exact Coulomb) for very large systems. The exponents for polarization functions cannot be derived from Hartree-Fock calculations for the atom, since they are not populated.

Consult the literature,

including pertinent

review articles,363, 411, 206, 254, 427

in order to aid your selection. In Figure Figure55 we compare,

for graphene, the electronic band structure computed with Gaussian

basis sets with the bands for the same systems as obtained from a

plane wave code55 using a considerably

high cutoff. It is evident that the bands at the triple-ζ level

are different from the reference ones, specially in the Γ and M points of the Brillouin zone. A considerably better agreement is attained by

using the dcm[Cgraph]-QZVP

basis (right panel of Figure Figure55). We believe this is strong evidence of the possibility of

reaching converged results with Gaussian basis sets and the effectiveness

of a system-specific optimization scheme.

Similarly, the inclusion of exact HF exchange in hybrid HF/DFT

calculations leads to a steep increase in computational time. Gaussian-type

basis sets are less commonly adopted for the quantum chemical treatment

of solids, with respect to plane waves. The price to pay is the mandatory definition

of a basis set for each atomic species, that is ultimately left in

the hands of the end user. When molecular calculations are performed, it is common to use a basis composed of atomic orbitals, centered at each nucleus within the molecule (linear combination of atomic orbitals ansatz).

Post validation of the specific functionality, developers perform the white box testing for loop holes such as memory leaks, statement testing and path testing to ensure each line of the code that is being written is tested. The basis test set is a series of tests done on the internal structures of a component in software. The weighted sets aim to capture core-valence correlation, while neglecting most of core-core correlation, in order to yield accurate geometries with smaller cost than the cc-pCVXZ sets. Basis Test Set is set of tests derived from the internal structure of a component in order to achieve 100% coverage of a specific criterion.

For example, an active area of research in industry involves calculating changes in chemical properties of pharmaceutical drugs as a result of changes in chemical structure. As shown in Table 5it can be seen that in all cases dcm- energies

are significantly lower than pob- ones, and quite surprisingly the

dcm[Cl2]-TZVP and dcm[Na]-TZVP basis sets seem to perform

well also in the ionic case. Our goal is to devise

a suitable algorithm for a system-specific optimization of the exponents

αj and contraction coefficients dj as in eq 3. Taking inspiration from the well-known Direct

Inversion of Iterative Subspace (DIIS) algorithm of Pulay,33,34 we describe in the following our Basis-set DIIS (BDIIS) method. The two equations are the same except for the value of \(\zeta\) which accounts for how large the orbital is.

Please do not confuse general contrac- tions with a term “general basis set” used in some program manuals to denote “user defined segmented basis sets”. Obviously, the best results could be obtained if all coefficients in Gaussian expansion were allowed to vary during molecular calculations. Moreover, the computational effort (i.e. “CPU time”) for calculating integrals in the Hartree-Fock procedure depends upon the 4th power in the number of Gaussian primitives. However, all subsequent steps depend upon the number of basis functions (i.e. contractions).

Gaussian primitives are usually obtained from quantum calculations on atoms (i.e. Hartree-Fock or Hartree-Fock plus some correlated calculations, e.g. CI). Typically, the exponents are varied until the lowest total energy of the atom is achieved (Clementi et al., 1990). In others, the exponents are related to each other by some equation, and parameters in this equation are optimized (e.g. even-tempered or “geometrical” and well-tempered basis sets). The primitives so derived describe isolated atoms and cannot accurately describe deformations of atomic orbitals brought by the presence of other atoms in the molecule. Basis sets for molecular calculations are therefore frequently augmented with other functions which will be discussed later.

This way, the flexibility of the basis set can be focused on the computational demands of the chosen property, typically yielding much faster convergence to the complete basis set limit than is achievable with energy-optimized basis sets. A minimum basis set is one in which a single basis function is used for each orbital in a Hartree-Fock calculation on the atom. However, for atoms such as lithium, basis functions of p type are added to the basis functions corresponding to the 1s and 2s orbitals of each atom. For example, each atom in the first row of the periodic system (Li – Ne) would have a basis set of five functions (two s functions and three p functions).

In Table 1 we compare the exponents of the original

def2-TZVP, the original pob-TZVP, the recently revised pob-rev2 basis

set, and our dcm-TZVP basis specifically optimized for diamond, graphene,

and carbyne with the PBE functional. For brevity, we will refer to

the latter two basis sets as dcm[Cdiam]-TZVP, dcm[Cgraph]-TZVP, and dcm[Ccby]-TZVP, respectively. Figure Figure33 shows a corresponding graphical representation of the radial

component of some of the involved gaussians. The first striking effect

observed is the overall contraction of exponents with respect to the

- Linear combinations of the primitive Gaussians are formed to approximate the radial part of an STO.
- These are either relativistic basis sets (all-electron, ECP)

or nonrelativistic basis sets, collected from other sources (Dalton, EMSL etc). - For example, an active area of research in industry involves calculating changes in chemical properties of pharmaceutical drugs as a result of changes in chemical structure.
- Also, the use of atomic orbitals allows us to interpret molecular properties and charge distributions in terms of atomic properties and charges, which is very appealing since we picture molecules as composed of atoms.
- This is a double-\(\zeta\) basis sets and includes split-valence set (inner and valence) and linear combination of two orbitals of same type, but with different effective charges (i.e., \(\zeta\)).
- The plane waves in the simulation cell that fit below the energy criterion are then included in the calculation.

molecular basis. This is not unexpected15,19 and is to

be ascribed to the higher density of atoms in the solid-state phase.

The reason for including p-type functions in the Li and Be atoms, even in the minimal basis sets, is prac- tical, however. The reason for not including d-type functions for sulphur should be the same as for other atoms, i.e., you can obtain reasonable results without them. On the other hand, for solid-state calculations,2 plane waves,6−8 atom-centered Gaussians9 (or their combinations10), and numerical

basis sets11,12 are all popular choices. The

plane wave basis, that is naturally suited for nonlocal wave functions

such as in the uniform electron gas or in a metal, has the undeniable

advantage of a one-knob tuning of accuracy and cost through the kinetic

energy cutoff parameter. However, the correct description of local

orbitals, core states, or the void can result in a rather high computational

cost.