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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

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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