Basis set (chemistry)
The basis set can either be composed of atomic orbitals (yielding the linear combination of atomic orbitals approach), which is the usual choice within the quantum chemistry community; plane waves which are typically used within the solid state community, or real-space approaches.
In modern computational chemistry, quantum chemical calculations are performed using a finite set of basis functions.
The physically best motivated basis set are Slater-type orbitals (STOs), which are solutions to the Schrödinger equation of hydrogen-like atoms, and decay exponentially far away from the nucleus.
However, hydrogen-like atoms lack many-electron interactions, thus the orbitals do not accurately describe electron state correlations.
[2] Basis sets typically come in hierarchies of increasing size, giving a controlled way to obtain more accurate solutions, however at a higher cost.
A minimal basis set may already be exact for the gas-phase atom at the self-consistent field level of theory.
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.
Minimal basis sets typically give rough results that are insufficient for research-quality publication, but are much cheaper than their larger counterparts.
In contrast, minimal basis sets lack the flexibility to adjust to different molecular environments.
The notation for the split-valence basis sets arising from the group of John Pople is typically X-YZg.
[5] In this case, X represents the number of primitive Gaussians comprising each core atomic orbital basis function.
Since then, correlation-consistent or polarization-consistent basis sets (see below) have been developed which are usually more appropriate for correlated wave function calculations.
[6] For Hartree–Fock or density functional theory, however, Pople basis sets are more efficient (per unit basis function) as compared to other alternatives, provided that the electronic structure program can take advantage of combined sp shells, and are still widely used for molecular structure determination of large molecules and as components of quantum chemistry composite methods.
More recently these 'correlation-consistent polarized' basis sets have become widely used and are the current state of the art for correlated or post-Hartree–Fock calculations.
These core-valence sets (cc-pCVXZ) can be used to approach the exact solution to the all-electron problem, and they are necessary for accurate geometric and nuclear property calculations.
Manninen and Vaara have proposed completeness-optimized basis sets,[10] where the exponents are obtained by maximization of the one-electron completeness profile[11] instead of minimization of the energy.
In 1974 Bardo and Ruedenberg [13] proposed a simple scheme to generate the exponents of a basis set that spans the Hilbert space evenly [14] by following a geometric progression of the form:
The main advantage of a plane-wave basis is that it is guaranteed to converge in a smooth, monotonic manner to the target wavefunction.
In addition, certain integrals and operations are much easier to program and carry out with plane-wave basis functions than with their localized counterparts.
When this property is combined with separable pseudopotentials of the Kleinman-Bylander type and pre-conditioned conjugate gradient solution techniques, the dynamic simulation of periodic problems containing hundreds of atoms becomes possible.
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.
In the spheres the variational degrees of freedom can be extended by adding local orbitals to the basis set.
The atomic-like representation in the spheres allows to treat each atom with its potential singularity at the nucleus and to not rely on a pseudopotential approximation.
The disadvantage of LAPW basis sets is its complex definition, which comes with many parameters that have to be controlled either by the user or an automatic recipe.
Another consequence of the form of the basis set are complex mathematical expressions, e.g., for the calculation of a Hamiltonian matrix or atomic forces.
Real-space approaches offer powerful methods to solve electronic structure problems thanks to their controllable accuracy.
Various methods have been proposed for constructing the solution in real space, including finite elements, basis splines, Lagrange sinc-functions, and wavelets.
Moreover, in the case of wavelets and finite elements, it is easy to use different levels of accuracy in different parts of the system, so that more points are used close to the nuclei where the wave function undergoes rapid changes and where most of the total energies lie, whereas a coarser representation is sufficient far away from nuclei; this feature is extremely important as it can be used to make all-electron calculations tractable.
For example, in finite element methods (FEMs), the wave function is represented as a linear combination of a set of piecewise polynomials.
