# Ab Initio¶

- Ab initio methods utilise the HF approximations
- BO approximation - PES
- HF approximation (mean field approximation) - SCF
- LCAO

## HF Theory¶

- In HF Theory, the energy of a system is the sum of five different energy terms

- \(E_T\) is the electron kinetic energy
- \(E_V\) is the coulombic nuclear-electron attraction energy
- \(E_J\) is the coulombic electron-electron repulsion energy
- \(E_K\) is the electron-electron exchange energy
**(Energy released when electrons in degenerate orbitals exchange position)** - \(E_C\) is the correlation energy
**(Results from the instantaneous interaction of electrons and is neglected by the HF approximation)**

Note

From the BO approximation, the Hamiltonian term \(\widehat{T_n}\) doesn’t exist (nuclear kinetic energy) and \(\widehat{V_{nn}}\) can be reduced to a constant (\(E_{NN}\))

## The Wavefunction¶

- LCAO is in effect

- \(\Phi_i\)= the molecular orbital
- \(c_{\mu i}\)= the molecular orbital coefficient - a weighted contribution of how much each basis function effects the final molecular orbital
- \(\chi_\mu\)= the basis function
- The basis function here is \(Ne^{−\zeta r^2}\)
- \(N\) in the basis set is a normalisation constant \(C_i\) in this unit
- \(\zeta\) is the exponent, providing the width of the Gaussian function
- \(r\) is the spacial location of the orbital
- The amount of zeta determines the amount of functions that can be optimised
- E.g. Triple \(\zeta\) would be \(\chi=(Ne^{−\zeta_1r^2})+(Ne^{−\zeta_2r^2})+(Ne^{−\zeta_3r^2})\)

E.g.

## Electron Spin¶

- We also need to consider the spin of electrons (\(\alpha\) and \(\beta\))
- In closed shell systems (systems where all the electrons are paired in orbitals, \(\alpha\) and \(\beta\) electrons have the same spacial coordinates (\(r\)) for simplicity
- In open shell systems we need to be a bit more careful with this

## RHF/ROHF/UHF¶

- Restricted Hartree-Fock (RHF)
- For closed shell systems all the spin up and spin down orbitals have the same spatial coordinates

- For closed shell systems all the spin up and spin down orbitals have the same spatial coordinates
- Restricted Open-Shell Hartree-Fock (ROHF)
- All paired electrons are treated are RHF and any unpaired electrons have different functionals for the occupied component and the unoccupied virtual component

- All paired electrons are treated are RHF and any unpaired electrons have different functionals for the occupied component and the unoccupied virtual component
- Unrestricted Hartree-Fock (UHF)
- All electrons are considered to have their own orbital, without considering paired and unpaired electrons in the same orbital

## Advantages and Disadvantages of UHF¶

- Advantages
- Accounts for the influence that unpaired electrons have on the paired electrons
- They have a habit of changing spin densities

- Provides qualitatively description of bond breaking/forming
- Models open shell systems more accurately than RHF

- Accounts for the influence that unpaired electrons have on the paired electrons

- Disadvantages
- Computationally more expensive
- Can lead to spin contamination - where the wavefunction isn’t always made of one spin up and one spin down electron. This leads to an improper wavefunction being formed

## Electron Correlation¶

- An inherent error within HF neglects a portion of the energy of the system. This is known as correlation error
- Of the three remaining Hamiltonian terms after the BO approximation, they are broken down as such

- Where:
- \(E_T\)= electron kinetic energy
- \(E_V\)= electron potential energy
- \(E_J\)= coulombic repulsion energy
- \(E_K\)= exchange energy
- \(E_C\)= correlation energy

- Correlation energy is always negative as it’s the process by which electrons minimise their energy
- Is sensitive to the amount of electron pairs and comes in two types
- Dynamic Correlation - which is the dance that electrons make to try and avoid each other
- Static Correlation - is the energy associated with electrons being able to change their configuration as needed to minimise their energy (accounted for in CI)

- Calculating this is the single most important thing in quantum chemistry

## Møller-Plesset¶

- Perturbation methods like MP theory assume that the problem we’d like to solve differs only slightly from a problem that we’ve already solved
- The energy is calculated to various orders of approximation
- MP2 - second order
- MP3 - third order
- Etc..

- The computational cost greatly increases with each successive order
- The series is not even guaranteed to converge. * The job may never finish and unlike SCF will not necessarily give you an error
- In general only MP2 is recommend
- MP2 approximately includes single and double excitations

## Configuration Interaction¶

- To account for this, the next generation of ab initio methods accounts for this and are called “post HF methods”
- One of the ways this works is to look at Configuration Interaction (CI)
- This theory considers that the actual composition of a quantum system accounts for all the possible “configurations” (determinants). In this case, that means that a linear combination of all the possible configurations needs to be calculated to account for this process.

```
Single Double Triple Quadruple
```

This number is the amount of electrons that are excited in the process

## Coupled Cluster¶

Rather than using a linear combination of configurations like in CI, CC uses exponential expansion

In CI, we start with the HF determination of the wavefunction \(\Phi\) and correct it with a linear combination of terms calculated from the different determinants (in CI, this operator is denoted C)

In CC, a similar correction is applied but uses the Taylor series to get there (in CC, this operator is denoted \(T\))

- The subscript on each operator is the amount of excited electrons
- In “Full CI” all electrons are distributed among all the orbitals as massive cost, so it is common to “truncate” the process and limit it to a certain number of excited electrons

## Frozen Core Approximation (FC)¶

- Since correlating electrons is such a costly and time consuming process, freezing the core electrons which have a mostly negligible impact on the bonding of a molecule can save a huge amount of time
- This is mostly applicable for first row elements atoms as the higher up, the more electrons there are and the more they can interact.
- Some processes rely on core electrons though, so it is really important to understand what you’re doing

By default MP# calculations freeze the core, and to overcome this you need to add
`mp2(full)`

## Cost¶

As a comparison, here is a cost table of how computational time where \(N=\) the number of basis functions

- DFT is similar to HF (~\(N^3\))

Method | Scaling Cost |
---|---|

HF | N^2 |

MP2 | N^5 |

CCSD | N^6 |

CCSD(T) | N^7 |

- While CISD is an approach that can be used, it has a few shortfalls that make it inefficient
- QCISD fixes these issues but is more costly than CCSD which is more accurate
- “CISD is not sufficient, CISDTQ is too expensive “

## Semiempirical Methods (AM1, PM3, RM1, etc.)¶

- Make a fair few simplifications
- Only look at the valence electrons, so as to minimise the amount of functions needed
- Don’t look at long range interactions
- Really important in protein modelling, as there is just far too much going on to account for everything

- Parameterise lots of properties using experimental results
- Be aware that these are often trained, and thus may not account for systems outside of the training dataset
- Also only limited to ground state applications for the same reason

- Use a minimal basis set, such as 3-21G
- Employ a non iterative solution process, that is it won’t use guess and check methods such as with SCF