# Basis Sets¶

• Basis sets are a linear combination of atomic orbitals
• Linear combinations of these will combine to form molecular orbitals (LCAO)
• The basis set itself is a set of mathematical functions from which the overall wavefunction is constructed

## SCF¶

• The SCF allows for the initial guess of $$\Psi$$ (the initial combination of AOs) to be formed into a best guess
• This can then be used to determine the properties of the molecule

## The Wavefunction¶

• $$\Psi$$ is a mathematical expression that determines the probability of electron distribution in 3D space
• From it we can determine some important properties about the electrons, such as:
• The energy
• The angular momentum
• The orbital orientation within the shape
• $$\Psi$$ being a wavefunction has a polarity and can be combined to form in and out of phase combinations

## Slater Type Orbitals (STOs)¶

• Are based on the mathematical function $$e^{−αr}$$
• They are a complete set and represent the electron distribution well
• The cannot be cleanly integrated and cannot have nodes (kind of important)

## Gaussian Type Orbitals (GTOs)¶

• Are based on the mathematical function $$e^{−αr^2}$$
• Also have a complete set but they don’t represent the electron distribution as well at STOs
• The can have nodes and are much more easily integrated

## The Solution¶

• Use multiple GTOs to approximate an STO
• Thanks to the LCAO principle, it’s possible to linearly combine GTOs

## Minimal Basis Sets - STO-nG¶

• Single $$\zeta$$ - the valence electrons are represented by only a single function
• Are only good for spherical orbitals
• In STO-3G, three specific GTOs are used to describe the orbital

## Pople Basis Sets¶

6-311G+(d)

• Uses 6 primitive basis sets to describe the core electrons
• All 6 gaussians use the same α
• Uses three separate functions to describe the valence orbitals
• Each ζ split uses a different α in the gaussians
• Adds an extra (larger) function on each atom of Z>2 to account for loosely held electrons, long range interactions, excited states and transition states
• ++ adds them form H as well
• For bonding atoms adds extra functions of angular momentum as specified. First letter to Z>2 the second to Z≤2 atoms
• (d,p) adds p functions for Z≤2 atoms as well
• (2df,pd) adds 2d and 1f functions to Z>2 and a p and d function to Z≤2

## Correlation Consistent Basis Sets - Dunning¶

• Designed mathematically so that they will systematically converge to the Complete Basis Set (CBS) limit
• This limit is the maximum accuracy that a basis set can provide
• The target of this convergence is different for different property calculations and is typically “trained” with empirical data to achieve the results desired

aug-cc-pVTZ

• Is correlation consistent
• Is polarised (has higher angular momentums)
• Has a split valence number - Valence Triple Zeta
• aug- are augmented with a diffuse function

## Dunning Vs Pople¶

• A parallel can be drawn with a few basis sets
• cc-pVDZ ≈ 6-31G(d,p)
• cc-pVTZ ≈ 6-311G(2df,2pd)

## Cost¶

• Number of GTOs used is roughly:
• Minimal < Split Valence < Polarised < Diffuse
• The increase in the number of integrals is approximately $$N^4$$ where $$N$$ is the number of basis functions
• Each iteration of the energy minimisation process iterates on the exponents (parameters) of the orbitals to define a new basis function for the next iteration