Self-Interaction Error

The self-interaction problem of KS-DFT is a result of, as the name suggests, the methodology including contributions of an electron interacting with itself. In HF theory this is not an issue as, although electrons interact with themselves in the Coloumb term, the effect of self-interaction is directly cancelled out by the exchange term.

In ‘pure’ DFT, although there is an exchange term, it does not cancel out the self-interaction error exactly and leads to DFT predicting impossible quantities such as a two electron energy for a Hydrogen atom [46, 47]. The effects of the self-interaction error can be reduced through the use of hybrid functionals which contain a percentage of HF exchange, see Eqns 2.39 and 2.40. However, despite the inclusion of HF exchange in hybrid functionals the self interaction error is still present in the methodology and can still cause issues when modelling complexes containing radicals or involving significant charge-transfer [46].

There have been attempts to remedy the problems of the self-interaction error within the KS-DFT framework, most notably by Perdew and Zunger [48]. However xc-functionals which account for the self-interaction energy whilst also providing accurate results for more general chemistry applications comparable to popular functionals such as B3LYP are few and far be-tween.

2.5 CASSCF/RASSCF

The Complete Active Space Self Consistent Field (CASSCF) and its derivative Restricted Ac-tive Space Self Consistent Field (RASSCF) are methods that have been developed over the last 30 years to better describe the static correlation of a molecular system whilst attempting to keep the computational cost as low as possible [49].

The CASSCF method involves selecting chemically important ‘active’ orbitals with which to do a full Configuration Interaction (CI) calculation [50] upon; this involves building the wave-function of the molecule as a linear combination of a series of configurations as shown in Eqn 2.49, where ci are the CI coefficients.

|Ψi = c₀|Ψ₀i +X

c^a_i|Ψ^a_ii +X

c^ab_ij|Ψ^ab_iji +X

c^abc_ijk|Ψ^abc_ijki + ... (2.49) In this equation the first term is the reference configuration, the second is a sum of the single excitations, the third is a sum of the double excitations and so on. The reference configuration is generally the Hartree-Fock wavefunction. The CASSCF (and RASSCF) approach is therefore an example of a multi-configurational wavefunction based method.

In the CASSCF methodology two parameters are optimised; firstly the CI coefficients, de-scribing the mixing of each electron configuration to the overall wavefunction. Secondly, the CASSCF molecular orbitals are optimised using an SCF method similar to HF.

To illustrate how a multi-configurational wavefunction can be constructed consider the simple example of Benzene. Benzene has three π bonding & three π anti-bonding molecular orbitals, as illustrated in Figure 2.5.

Figure 2.5: Energy ordering of Benzene π orbitals. Orbitals generated with CASSCF(6/6) active space with isosurface of 0.35 a.u.

A ground state reference wavefunction can be constructed by occupying the three lowest molec-ular orbitals with six electrons (Eqn 2.50), this is the same as a Hartree-Fock calculation.

Ψ₀ = |...π₀²π²₁π₁²π₂⁰π⁰₂π₃⁰i (2.50) As per Eqn 2.49 we can further expand the molecular wavefunction beyond a simple Hartree-Fock calculation by the addition of single excitations, of which there are nine in this benzene example (Eqn 2.51), double excitations (Eqn 2.52) and so on.

Ψ^S₁ = |...π₀²π₁²π₁¹π₂¹π₂⁰π₃⁰i+|...π₀²π₁²π₁¹π₂⁰π₂¹π₃⁰i+|...π₀²π₁²π₁¹π₂⁰π₂⁰π₃¹i+|...π₀²π₁¹π₁²π₂¹π₂⁰π₃⁰i+... (2.51) Ψ^D₂ = |...π₀²π₁²π₁⁰π₂²π₂⁰π₃⁰i + |...π²₀π⁰₁π²₁π⁰₂π²₂π⁰₃i + |...π₀⁰π₁²π₁²π₂⁰π₂⁰π₃²i + |...π²₀π¹₁π¹₁π¹₂π¹₂π₃⁰i + ...

(2.52)

The Benzene molecular wavefunction is then built as a linear combination of the configurations and the CI coefficients determined variationally, this is done by solving a series of CI secular equations, shown in Eqn 2.53.

Some of the diagonal elements in the matrix presented in Eqn 2.53 can be set to zero in accor-dance with Brillouin’s Theorem [51] and gives the CI matrix a block diagonal structure.

It can be seen even in the simple example of Benzene that the computational expense of a CASSCF calculation is dictated by not only the size of the system but also the number of electrons and orbitals that are included in the active space. As the number of active electrons and orbitals are increased the total number of configurations increases and can be calculated by Eqn 2.54 [3];

N = m!(m + 1)!

(ⁿ₂)!(ⁿ₂ + 1)!(m − ⁿ₂)!(m − ⁿ₂ + 1)! (2.54)

In Eqn 2.54; N is the total number of configurations, m is the number of orbitals and n is the number of electrons [3]. It can be seen that the total number of configurations (and hence the computational cost) increases factorially meaning that realistically only a limited number of electrons and orbitals may be considered. Unfortunately computer processing power has only progressed so far to the point that a rough maximum of 18 orbitals and 18 electrons can be practically considered within the active space, which is still a significant increase on the number that would have been able to be considered a decade ago. Fortunately the presence of symmetry within a molecule can help as this reduces the total number of configurations which are present

in the description of the wavefunction.

18 orbitals and electrons can often be an acceptable limit for examining smaller molecules, however for larger molecules, especially those with a large conjugated π-system, it can begin to affect the accuracy of simulations. This is where the Restricted Active Space Self Consistent Field (RASSCF) method can be effectively utilised.

In the RASSCF approach instead of the solitary CAS there are now three spaces; RAS1, RAS2 and RAS3 [49]. The RAS2 is similar to the CAS in that all excitations are permitted within this space (although a RAS2 typically, but not always, contains fewer orbitals and electrons than a CAS would). The RAS1 space is then chosen where all selected orbitals are doubly occupied and l excitations are allowed from the RAS1. The RAS3 is composed of formally unoccupied orbitals with m excitations permitted into the RAS3, thus in the RAS1 and RAS3 spaces a truncated CI calculation is performed. Generally up to triple excitations (l,m=3) are used as this recovers a lot of the static correlation that you would obtain from a Full CI calculation [52] but crucially cuts the computational cost. A schematic of the CAS/RASSCF partitioning is shown in Figure 2.6.

Figure 2.6: Schematic of orbital partitioning in the Hartree-Fock (left), CASSCF (middle) and RASSCF (right) methodologies.