7.1 Basis sets and their selection

Basis sets are the mathematical building blocks used to describe electron orbitals in quantum chemistry calculations. They come in various types, from simple minimal sets to more complex extended sets, each offering different levels of accuracy and computational cost.

Choosing the right basis set is crucial for balancing accuracy and efficiency in electronic structure calculations. Extended sets with polarization and diffuse functions provide more precise results, especially for properties like molecular geometries and intermolecular interactions.

Orbital Types and Minimal Basis Sets

Slater-Type and Gaussian-Type Orbitals

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• Slater-type orbitals (STOs) resemble atomic orbitals and have a cusp at the nucleus
  • Provide a more accurate description of the electron density near the nucleus
  • Computationally expensive due to the presence of exponential functions
• Gaussian-type orbitals (GTOs) are approximations to STOs and have a smooth behavior at the nucleus
  • Computationally more efficient as they involve simple Gaussian functions
  • Require a larger number of basis functions to achieve the same accuracy as STOs (carbon atom, water molecule)

Minimal Basis Sets

• Minimal basis sets contain the minimum number of basis functions required to describe the electrons in an atom
  • Use one basis function for each atomic orbital in the ground state configuration
  • Examples include STO-3G and STO-6G, which approximate each STO with 3 or 6 GTOs, respectively
• Minimal basis sets are computationally inexpensive but have limited accuracy
  • Often insufficient for describing chemical bonding and electronic properties accurately
  • Suitable for initial geometry optimizations or qualitative studies (small organic molecules, transition metal complexes)

Extended Basis Sets

Split-Valence Basis Sets

• Split-valence basis sets use multiple basis functions for each valence atomic orbital
  • Provide flexibility in describing the electron density distribution in different bonding environments
  • Denoted as X-YZG, where X represents the number of GTOs for core orbitals, and Y and Z represent the number of GTOs for valence orbitals (6-31G, cc-pVDZ)
• Split-valence basis sets improve the description of chemical bonding and molecular geometry
  • Allow for a more accurate representation of electron density in the valence region
  • Commonly used in routine quantum chemical calculations (organic molecules, transition metal complexes)

Polarization and Diffuse Functions

• Polarization functions add higher angular momentum basis functions to the basis set
  • Improve the description of electron density distortion in molecules
  • Denoted by adding asterisks () or letters (d, p, f) to the basis set name (6-31G, cc-pVDZ)
• Diffuse functions are large-sized Gaussian functions with small exponents
  • Improve the description of long-range interactions and electron density far from the nuclei
  • Denoted by adding a plus sign (+) to the basis set name (6-31+G, aug-cc-pVDZ)
• Polarization and diffuse functions are essential for accurate calculations of properties such as
  • Molecular polarizabilities, electron affinities, and intermolecular interactions (hydrogen bonding, van der Waals complexes)

Correlation-Consistent Basis Sets

• Correlation-consistent basis sets are designed to systematically converge to the complete basis set limit
  • Provide a hierarchy of basis sets with increasing size and accuracy
  • Denoted as cc-pVXZ, where X represents the cardinal number (D, T, Q, 5, 6) indicating the size of the basis set
• Correlation-consistent basis sets are optimized for post-Hartree-Fock methods that include electron correlation
  • Suitable for high-accuracy calculations of molecular properties and energetics
  • Widely used in benchmark studies and for calibrating lower-level methods (small molecules, transition states)

Basis Set Limitations

Basis Set Superposition Error (BSSE)

• Basis set superposition error arises from the inconsistent treatment of basis functions in molecular complexes
  • Occurs when the basis functions of one molecule artificially improve the description of the electron density of another molecule
  • Leads to an overestimation of the interaction energy and an underestimation of the intermolecular distance
• BSSE can be mitigated by using larger basis sets or applying counterpoise correction
  • Counterpoise correction involves calculating the energy of each molecule with the full basis set of the complex
  • The difference between the corrected and uncorrected energies gives an estimate of the BSSE (hydrogen-bonded complexes, weakly interacting systems)
• BSSE is more pronounced in smaller basis sets and can significantly affect the accuracy of intermolecular interaction energies
  • Important to consider when studying molecular clusters, host-guest systems, or adsorption phenomena