# 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
- D=Double, T=Triple, Q=Quadruple, 5=5…

- 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