Atom-Centered Basis Sets

Atom-centered basis sets are, as the name implies, functions that are centered on the atomic nuclei. They can be constructed using any mathematical function, but are usually

expressed as either Slater-type orbitals (STOs) or Gaussian-type orbitals (GTOs) (Figure 3-4).99-

100 _{The atomic orbitals are then represented by a linear combination of primitive functions (γ)}

multiplied by a corresponding coefficient 𝑑𝑛,

φ(𝛾₁ , 𝛾₂, … , 𝛾_𝑖 ) = ∑ 𝑑𝑛 𝛾𝑛 (𝛼, 𝑟) 𝑛

(3.43)

where 𝛾_𝑛 is a function of radius and Gaussian exponent (𝑟 and 𝛼, respectively).96, 101-102

The primary choices of primitive functions are the Slater and Gaussian functions that have the forms

𝛾𝑆𝑇𝑂_{(𝛼, 𝑟) = 𝑒}−𝛼𝑟 _(3.44)

𝛾𝐺𝑇𝑂_{(𝛼, 𝑟) = 𝑒}−𝛼𝑟2

(3.45)

While STOs are better suited for reproducing the hydrogen atomic orbitals, primarily near and far from the nucleus (the cusp and exponential decay, respectively), they are computationally ineffective because the product of two STOs does not yield a STO.1 On the other hand, GTOs are multiplicative, because the product of two GTOs yields a new single GTO. This has major

implications for computationally efficiency, as any multi-centered integrals can be reduced to simpler two-center integrals.103-105 Moreover, the use of GTOs permits analytical first- and second-derivatives of the energy to be taken, which has consequences for optimization of

88 molecular structure.106-107 Therefore, the computational cost of using more GTOs is negated by the advantages, and this is the reason most atom-centered basis sets are based on GTOs rather than STOs. For the remainder of the discussion only GTOs will be discussed.

89

Figure 3-4. Comparison between the hydrogen 1s radial wavefunction (black) and the two types of primitive basis functions, the STO (green) and GTO (blue).

90 As mentioned, computational accuracy can be increased through the use of larger basis sets, and the simplest solution to this is the choice of the number or primitive functions used for constructing the GTOs. Three STO-nG basis functions (a basis set constructed from n primitive Gaussian functions) are shown in Figure 3-5, and as the number of primitive Gaussian functions is increased the deviations from the hydrogenic 1s orbital are decreased.103 However, as

previously mentioned, simply reproducing the occupied atomic orbitals mathematically does not (and usually will not) result in a better simulation. Therefore, the number of GTOs per atomic orbitals are usually increased to provide more flexibility by the computational algorithms.

91

Figure 3-5. Visualization of the role that the number of primitive Gaussian functions has on the reproduction of the hydrogen 1s wavefunction.

92 The simplest case of using a single GTO for each occupied atomic orbital is what is known as a minimal basis set. Examples of minimal basis sets are the STO-nG and MIDI!108 sets, and are usually not very effective for most systems, with the exception of single-atom

calculations.109-111 Increasing the number of GTOs per atomic orbital results in the X-ζ

(X=double, triple, …) basis sets; for example a triple- ζ basis set for hydrogen would have three

1s GTOs. But while the primitive functions of a GTO are typically input by the user during a calculation and fixed, the molecular orbitals are built by varying the linear combination of the individual GTOs by adjusting of the GTO coefficients iteratively. Thus, defining multiple GTOs for a particular orbital allows for a greater degree of flexibility, enabling a more accurate

representation of the molecular orbitals to be found. For example, imagining that a hydrogen atom begins to be perturbed in some way (e.g. during the formation of a covalent bond or upon application of an electric field), the hydrogen radial distribution function would change

accordingly. If this type of system was modelled using a minimal basis set, this perturbation would not be captured. However if a larger basis set, such as a triple-zeta set, were used, the perturbation is more likely to be well modelled because of the flexibility attained. This example is shown graphically in Figure 3-6, where an arbitrary p-type hybridization has been applied to a hydrogen 1s orbital while the resulting best fit basis function was calculated using the STO-3G and triple- ζ 6-311G basis sets.101

93

Figure 3-6. Example of the flexibility of a triple-ζ basis, containing three separate s-type orbital functions (top right), for both the hydrogen 1s radial wavefunction as well as an sp- type orbital (top left) compared to a minimal basis set. The minimal set accurately

reproduces the unperturbed wavefunction (bottom left), but fails when the wavefunction deviates too much from the ideal 1s wavefunction (bottom right).

94 There are countless methods for improving basis set quality while reducing

computational cost. One of the most effective is the split-valence basis set formulation, where the core atomic orbitals are represented by a minimal basis (one GTO per core orbital) but have multiple GTOs for the valence orbitals. These basis sets are by far the most common atom- centered basis sets, with much of the pioneering work done by John Pople, whose basis sets have the notation96-97, 101-103

𝑊 − 𝑋𝑌𝑍 … 𝐺 (3.46)

where W is the number of primitive Gaussian functions used to describe the core GTO and XYZ are the number of primitive Gaussian functions used to describe the first and second (in the case of a double-ζ) and third (in the case of a triple-ζ) valence GTOs. For example, the split-valence triple-ζ 6-311G basis set of lithium has a single 1s GTO that is described by 6 primitive Gaussian functions, and three 2s GTOs described by three, one, and one primitive Gaussian functions, respectively.

Additionally, basis sets can be augmented with additional functions to further facilitate computational flexibility. The two most common methods involve inclusion of polarization or diffuse basis functions.102, 112-114 Polarization functions are orbitals that have an angular momentum higher than what is typically occupied, providing a more efficient means for simulating hybridized atomic orbitals.102, 115 Diffuse functions are very broad GTOs (small 𝛼 exponent) that are added somewhat arbitrarily to allow for large-scale displacements of the electron from the nuclei. Diffuse functions are necessary for anions and highly polarizable systems,114 but often cannot be used in condensed phase simulations due to the propensity for artificial overlap with neighboring molecules, often promoting spurious conduction.116-117

95 Polarization and diffuse functions are typically use the notations * and +, respectively. However for polarization functions in particular it is becoming more conventional to use the polarization orbital lettering explicitly, for instance for the split-valence triple-ζ basis set with polarization functions added to hydrogen (p function) and carbon (d function), the following notations are equivalent

6-311G(d,p) = 6-311G**

(3.47)

However, if two polarization functions are added there is no method of conveying this using the asterisk notation,

6-311G(2d,2p) ≠ 6-311G****

(3.48)

and therefore the alphanumeric labelling is preferred. Furthermore, confusion can arise when the added polarization function is not a d function (such as when adding a polarization function to a zinc atom), as both 6-311G(d) and 6-311G(f) are often applied interchangeably.

It is important to note that while the atomic orbitals are represented here in one-

dimension, atomic wavefunctions are three-dimensional and must be constructed with both the radial and angular components. Therefore, the GTOs are transformed into ‘real’ atomic

wavefunctions using the spherical harmonics, and then the molecular orbital built by varying the GTO coefficients,

φ(𝑟, 𝜃, 𝜙) = ∑ 𝑌_{𝑖,𝑚,𝑙} (𝜃, 𝜙)𝑑_𝑛 𝛾_𝑛(𝛼_𝑖 , 𝑟)

𝑛

96 This is rather trivial to perform, but this transformation has implications for computational

efficiency. The transformation from a single GTO to atomic wavefunctions generates (with the exception of s orbitals), multiple wavefunctions, i.e. 3, 5, and 7 atomic orbitals per GTO for p, d, and f-type orbitals, respectively. Because of this, computational cost scales by N4, where N is the number of basis functions. This is yet another reason why basis set selection is a very important decision when performing ab initio calculations.