For solving the electronic structure problem, either within the density functional theory (through the Kohn-Sham equations) or within wavefunction-based methods such as the Hartree-Fock and post-Hartree-Fock approaches, one needs to choose a mathematical representation for the one-electron orbital. The Kohn-Sham equa-tions are solved self-consistently with the electron density generated by the sum of orbital densities. The basis functions{η}can be either composed of atomic orbitals or plane waves.
Atomic orbitals
Atomic orbitals (AO) are the localized basis functions centered on different atoms.
In the early days of quantum chemistry the so-called Slater type orbitals (STOs) were used as basis function due to their similarity with the eigenfunctions of the hydrogen atom. The STOs have the general form,
ηST O =N Ylm(Θ, ϕ)rn−1e−ζr (2.34) whereN is the normalization factor,nis the principal quantum number, andYlm are the spherical harmonics that describe the angular part of the function. ζ controls the width of the orbitals; the large ζ gives a tight function, while a small ζ gives a diffuse function. Such functions that resemble hydrogen atomic orbitals are very suitable for expanding molecular orbitals as they have a correct shape near and far from the nucleus (decay likee−ξr). However, from a computational point of view the exponential part is inconvenient in the course of integrals in the self-consistent field procedure, which drastically decreases the speed of computation. For this reason, typically Gaussian type functions, also called as Gaussian type orbitals (GTOs) are used,
ηGT O =N Ylm(Θ, ϕ)r2n−2−le−ζr2 (2.35) Accordingly, the integrals are much easier to compute because the product of Gaus-sians are Gaussian; for example, four center integrals can be reduced to one single center. But a single Gaussian does not accurately reproduce the sharp cusp at the center of an exponential, neither does it have the correct asymptotic decay.
The GTO inaccuracies can be solved to some extent using linear combinations of Gaussian orbitals, referred as contracted GTOs (CGTOs) which are defined as,
ηCGT O =
∑J j=1
NjYlm(Θ, ϕ)r2n−2−le−ζjr2dj (2.36) where dj is the the contraction coefficient for the primitive with exponentζj, andJ is the total number of gaussian primitives used to represent the basis function. A combination of n Gaussians to mimic an STO is often called as “STO-nG” basis.
The STO-3G basis is the minimal basis set, which contracts three Gaussian primitive functions to represent one Slater atomic orbital. Increasing the number of primitive basis functions can enhance the quality of contracted GTO in description of atomic orbitals. For instance, if two sets of basis functions for the description of each atomic orbital are employed, it is called “double-ζ” or “double-zeta”. This can be further
expanded to “triple-ζ”, and “quadruple-ζ”, if three, and four basis functions for each atomic orbital are used, respectively.
In principle, the core electrons attend less in chemical reaction, and it is mainly the valence electrons which contribute to the properties such as chemical bonding or reactivity of chemical systems. Thereby, the so-called “split-valence-type” basis set can be employed, which treats each core AO with a minimal basis set, and the valence AOs with a larger basis set. For example, the split-valence double zeta basis set, 4-31G, employs four gaussian primitives for each core atomic orbital, and two sets of three and one CGTO functions for each valence orbital. This approach is also used for larger molecules, which allows to reduce the computational cost.
Including higher angular momentum orbitals, denoted as polarization functions, can augment the basis sets. For example, the only occupied orbital for the hydrogen atom is s-type, and if the p-type or d-type basis functions are added to the hydrogen atom, it is then considered as polarization basis functions. The basis functions with polarization allow for the atomic electron densities to be polarized, and particularly give a better representation of the electron density in bonding regions where a dis-tortion of the atomic orbitals from their original symmetry occurs. Furthermore, the flexibility of the basis sets can be enhanced by adding extra basis functions (usu-ally of s-type or p-type) with a small exponentζ to account for the diffusion effect.
This can be important for an accurate description of electron density far away from nucleus particularly in anions, and strongly electronegative atoms.72
Plane waves
The other basis function, especially suited for periodic systems in solids are plane waves (PW). They are by nature periodic, and form a complete orthonormal set of functions that can be used to expand the wavefunctions. The Bloch’s theorem facilitates the use of plane waves (PW) as basis functions. According to the Bloch theorem, it is not necessary to determine the electronic wave function everywhere in an infinite periodic system, rather it is sufficient to know it in the unit cell. In neighboring cells, the electronic wave function is exactly the same but for a phase factor due to the translational symmetry, thus the electronic wave function in a periodic system can be written as,
ψn,k(r) = un,k(r)eik.r (2.37) wherek is a wave vector, andn is the band index. un,k(r) is arbitrary function but
always with the periodicity of the primitive cell, i.e.,
un,k(r+R) = un,k(r) (2.38)
for any lattice vector R. Since un,k(r)is a periodic function, it can be expanded in terms of Fourier series,
un,k(r) = 1
√Ω
∑
G
cn,k,G eiG.r (2.39)
with Ωas the volume of the direct lattice, andGis the reciprocal lattice vector that satisfies the requirement,
1
2π|G.R| ∈N (2.40)
Thereby, the electronic wavefunction is written as a linear combination of plane waves,
ψn,k(r) = 1
√Ω
∑
G
cn,k,G ei(k+G).r (2.41)
By the use of Bloch’s theorem, the problem of determining the wavefunction over an infinitely large periodic system has been mapped onto the one of solving for the wavefunction in terms of an infinite number of possible value for k within the first Brillouin zone of the periodic cell. In principle, the electronic wavefunction at each k-point is an infinite discrete plane wave basis set. In practice, the Fourier expansion of the wave function in Eq. (2.41) is truncated by keeping the maximum value of plane wave vector for the contribution to the kinetic energy less than a given cut-off energy,
1
2|k+G|2 ≤Ecut (2.42)
For metallic systems, where rapid changes in electronic structure may occur along the energy band that crosses the Fermi level, a large number of k-points are needed to sample the first Brillouin zone. The nonmetallic systems are less sensitive to the quality of k-point sampling, and it is often sufficient to consider only one particular high symmetry wave vector k = 0, the so called Γ point. The maximum value for reciprocal lattice vector at Γ point is obtained to be |Gmax| = √
2Ecut, which corresponds to a sphere with radius Gmax centered at the origin of the reciprocal lattice. All reciprocal lattice vectors inside this sphere are included into the basis set. The truncation of the basis set at a finite cutoff energy will lead to an error in the computed total energy and its derivatives, which can be reduced by increasing the value of the cutoff energy in systematic way. In principle, the cutoff energy
should be increased until the calculated total energy converges within the required tolerance.
In contrast to the Gaussian basis functions, the plane waves are orthonormal and site independent and hence are free of basis set superposition errors. Total energies and forces on the atoms can be calculated using computationally efficient fast Fourier transform techniques, which transform a physical quantity from real-space (R-space) to reciprocal-space (G-space). A major disadvantage of PWs over atom-center basis function is that tens or hundreds of thousands of basis functions must be used to get the same level of accuracy as can be obtained with a few hundred Gaussian basis functions. Specifically, a huge number of plane waves are needed to describe regions of high electronic density near atomic cores that leads to the necessity for employing pseudopotentials as will be described in the next section.