Mastering Basis Sets in Theoretical Chemistry: Physical Meaning, Types, Applications, and BSSE Correction

 

Basis Set in Theoretical Chemistry: An Introduction

In theoretical chemistry, the concept of a basis set plays a fundamental role in the calculation of molecular properties. A basis set is a collection of functions used to approximate the wavefunction of a molecule. The wavefunction represents the quantum mechanical state of a molecule, and its calculation is the foundation for the prediction of molecular properties such as bond lengths, bond angles, and energies. The choice of basis set significantly affects the accuracy and computational cost of the calculation. Therefore, selecting the most suitable basis set is critical for obtaining reliable and accurate results.

We will be looking at...

• ·         why we use basis sets.
• ·         the physical meaning of basis sets.
• ·         why to use STOs and GTOs.
• ·         how we use basis sets.
• ·         basis set notation.
• ·         choosing a basis set
• ·         the quality of basis sets.
•       BSSE
• ·         methods vs basis set’ll look at…

Basis Set?

·         Set of mathematical functions (basis functions) which represents the electronic wavefunction in HF/DFT.

• ·         Help in solving the Schrödinger equation.
• ·         Linear combination of basis function approximate total electronic wavefunction (Ψ) and molecular orbitals.
• ·         Types of used atomic orbitals : Gaussian-type orbitals (GTOs), Slater-type orbitals (STOs) and Numerical atomic orbitals
• ·         Most often people uses GTOs

Why Basis Set?

We want one or both of the following :-

Ø  The electronic energy of molecule.

Ø  The wavefunction for molecule so that we can calculate other properties.

We satisfy our necessity by solving the stationary state Schrödinger equation.

ĤΨ = EΨ

·         Solving the Schrödinger equation to obtain Ψ of hydrogenic atom is easy.

·         But for more than two particles we are forced to make guesses for Ψ.

·         One guess is to use functions that are similar to the formulae obtained already (s, p, d, f etc. atomic orbitals (AO’s)).

·         Basis sets, very loosely, as sets of functions like s, p, d, f, etc. that will be used to describe the behavior of electrons in all systems whether they be hydrogenic or not.

How to Understand Basis Set

Physical Meaning and Use of basis set

Ø  Several (usually) basis functions describe the electron distribution around an atom.

Ø  Combining atomic basis functions yields electron distribution in whole molecule.

Ø  Basis sets are used to approximate Ψ.

Ø  The bigger and better the basis set, the closer we get to Ψ, and hence E.

Ø  Nowadays, almost everyone utilizes gaussian functions in basis sets.

Ø  One or more gaussian-type functions are used for each AO in each atom in the molecule of interest.

 

Slater functions are used in semiempirical calculations, like EHM and other semiempirical methods. Modern molecular ab initio program employ Gaussian functions.

Types of Basis Sets

There are various types of basis sets, and each has its own advantages and disadvantages. Below are the most commonly used types of basis sets in theoretical chemistry.

Minimal Basis Set:

A minimal basis set consists of a small number of functions, usually one per atom. These functions are generally atomic orbitals such as 1s, 2s, 2p, and so on. The advantage of a minimal basis set is that it is computationally efficient, making it suitable for calculations on large molecules. However, the accuracy of the results obtained from a minimal basis set is often limited.

Split Valence Basis Set:

A split valence basis set consists of two sets of functions, a core set, and a valence set. The core set contains the functions that are shared by all atoms in the molecule, while the valence set contains functions that are unique to each atom. The advantage of a split valence basis set is that it provides a more accurate representation of the electron density of the molecule than a minimal basis set.

Polarized Basis Set:

A polarized basis set includes additional functions to account for the polarization of electron density around an atom caused by the presence of nearby atoms. These functions are generally more diffuse than those in a minimal or split valence basis set. The inclusion of polarization functions improves the accuracy of the results and is particularly important for calculations involving anions or cations.

Diffuse Basis Set:

A diffuse basis set includes functions that are more diffuse than those in a polarized basis set. These functions are used to describe the electron density in the outer regions of the molecule, such as in the case of anions or cations. A diffuse basis set is particularly useful for calculations that involve the prediction of properties such as dipole moments and electron affinities.

Pseudopotential Basis Set:

A pseudopotential basis set is used to represent the electron density of an atom by using pseudopotentials instead of the actual electron wavefunctions. The advantage of a pseudopotential basis set is that it significantly reduces the computational cost of the calculation, particularly for heavy elements. However, the accuracy of the results obtained from a pseudopotential basis set is generally lower than that of a full-electron basis set.

Conclusion

In summary, the choice of basis set is an important consideration in theoretical chemistry calculations. The selection of an appropriate basis set depends on the level of accuracy required, the size of the molecule being studied, and the computational resources available. Each type of basis set has its own advantages and disadvantages, and selecting the most suitable basis set for a given calculation requires careful consideration.

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