Classes of DFT functional

1.4.3.1 Local Density Approximation

The simplest approximation to represent the exchange term is by the Local Density Approximation (LDA), which treats the local density as a uniform electron gas. This allows the Dirac formula[69] shown in Equation 1.13 to be used to obtain the exchange energy:

_{[ ] ∫} ⁄

( )

⁄

Equation 1.13

This approximation is for a uniform electron gas and ignores the spin properties of electrons and so the Local Spin Density Approximation (LSDA) was formulated by Slater in 1951[70] and resulted in the exchange functional, shown in Equation 1.14:

_{[ ]} ⁄ _{∫ (} ⁄ ⁄ ₎ Equation 1.14

where and represent the spin up and spin down densities. The LDA and LSDA terms are typically used interchangeably, as for closed shell systems the two methods are essentially equal. The correlation energy of these models proved difficult to obtain separately from the exchange energy, but has been achieved by the use of Monte Carlo methods with several different electron densities of a homogeneous electron gas.[71, 72] A popular formulation for this functional was developed by S. Vosko, L. Wilk and M. Nusair,[73] which is known as the VWN functional. Other popular functionals include that formulated by J. Perdew and A. Zunger in 1981 (PL functional )[74] and that of Perdew and Wang in 1992 (PW functional).[75] It has been found that these approximations (LDA and LSDA) allow the calculation of values similar to those obtained via the use of HF, which is based on wave mechanics. However, it has also been found that these types of functionals do not correlate well with experimental data.

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1.4.3.2 Generalised Gradient Approximation

The main limitation of the LDA method is that the electron density is a non-uniform electron gas. One approach developed to improve this inconsistency was to consider not just the electron density at any point but to include the gradient of the density i.e. . It has been found that this type of approximation can yield accurate results using various calculations.[76] Two routes for its development were taken by separate research groups run by Becke and Perdew. Becke took a route which was partially empirical, with large molecular training set used to allow numerical fitting procedures.[77-82] Examples of popular functionals which were developed in this way are the Perdew-Wang (PW)[83] _{and Becke88}

(B)[84] functionals. The second route was undertaken by Perdew and involved a more rational-based approach.[75, 85-90] Perdew’s route was linked more closely to quantum mechanics than that of Becke, with common functionals in this class as Becke86 (B86),[78] Perdew 86 (P)[87] and Perdew-Burke-Ernzerhof (PBE).[89] The first class of functionals was found to perform well at predicting atomisation energies and reaction barriers for molecular reactions, whereas the second class of functionals was better at predicting solid- state properties.[80, 90] Several different correlation functionals have been developed, with popular GGA correlation functionals being Perdew 86 (P86),[87] and Perdew-Wang (PW91).[91] The Lee-Yang-Parr functional (LYP)[92] which is based on the Colle-Salvetti correlation energy formula[93] is also in this class of functionals and has been extremely popular with researchers, with over 45000 citations. Combinations of the exchange and correlation functionals have been proposed, which form standard functionals. Such combinations are the PBE functional, which uses both the PBE exchange and PBE correlation functionals,[75, 89] BLYP which combines the B88 exchange with the LYP correlation functional and BP86 which combines the B88 exchange with the VWN and P86 correlation functionals. This last functional combination is also recommended by the authors of the Turbomole package[94] for the whole of the periodic table.[76] Whilst the inclusion of gradients to form GGA functionals improves the match between experimental values with calculated values compared to those obtained from LSDA functionals, GGA functionals are still not accurate enough for several chemical systems. For this reason hybrid density functionals have been developed.

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1.4.3.3 Hybrid functionals

Hybrid functionals utilise the same methodology as that used in GGA functionals, but include a percentage of HF exchange. The percentage of HF energy which is needed cannot be obtained from first principles and so is generally obtained empirically. The most widely used scheme to represent the form of hybrid functionals is that proposed by Becke in 1993,[95] as shown in Equation 1.15:

Equation 1.15

where the local and gradient functionals can be varied, along with the constants to . It is these three constants which are determined by fitting to experimental data (generally). The term uses GGA functionals previously described. The popular B3LYP functional

from Becke[95] adopts the form shown in Equation 1.16:

Equation 1.16

The B3LYP name originates from the use of the B exchange functional with the LYP correlation functional, combined with 3 empirical parameters. These parameters were determined by optimising the results obtained with the G1 database of molecules.[96, 97] A contribution from the VWN correlation function is also included (with a coefficient of 0.19). Another hybrid functional of note is the PBE0 functional from Adamo in 1999 which does not include adjustable parameters; this functional takes the form shown in Equation 1.17:

_{Equation 1.17}

where the correlation description has contributions from both the PBE and PW functionals described in Section 1.4.3.1. The value of the single parameter (0.25) was derived from work by Perdew in 1996 which showed that the parameter value of 0.25 taken from fourth- order perturbation theory (MP) leads to sufficiently accurate results.[98]

The B3LYP functional is popular and is often used in theoretical investigations, although investigations into its performance have revealed that it does not perform as well as other functionals for models involving transition metals.[99-101] However, good results are obtained when used for main group elements as recommended by the authors of Turbomole.[94]

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1.4.3.4 Dispersion corrected functionals

The classes of DFT functionals described so far all suffer from the same failing; modelling long-range electron correlations such as dispersive effects or van der Waals forces.[102] This arises from the form of these functionals; long-range interactions of the London form − C6/R6 are needed (where C6 represents a coefficient for a given atom and R is the distance),

whereas the functionals result in the long-range interaction decreasing exponentially.[66] A significant piece of work by Grimme reported in 2006,[102] added an empirical dispersion correction to several functionals, with improved results obtained from the B-97D functional. Further corrections were reported in 2010 under the term GD3 and this work detailed corrections for many more functionals.[103] At the time of writing, the latest version of the Gaussian software (Gaussian 09 Rev D.01) has 19 functionals with the correction applied available with simple keywords.

1.4.3.5 Double hybrid functionals

An alternative method was also presented by Grimme in 2006, where the long-range correlation energy from second order Möller-Plesset perturbation theory (MP2) was also included.[104] The first functional of this type reported in the work was named B2PLYP; this family is referred to as double-hybrid functionals. A significant efficiency was determined with the calculation of the MP2 term; the Kohn-Sham orbitals and eigenvalues calculated in the GGA part of the functional are used for the MP2 calculation. This saving decreases the computational cost of the overall functional as new orbitals are not needed for the MP2 term. The double hybrid functionals adopt the general form for the exchange-correlation energy shown in Equation 1.18 (note, the correlation term is referred to by

Grimme as ):

_{Equation 1.18}

where the term is calculated via:

_{∑ ∑}[ ]

Equation 1.19

with Equation 1.19 expressed in spin-orbital form and is the standard second-order Möller- Plesset correlation term. The difference to the standard term is that the Kohn-Sham orbitals

41 and corresponding eigenvalues are used, the indices and represent the single occupied-virtual replacements and the regular two-electron integrals over the KS orbitals are represented by the term in brackets. The work of Grimme assessed the performance of the B2PLYP functional with the G2 standard benchmark set and very good results were obtained.[104] _{The development of this type of functional is currently an active research area}

and several functionals exist, such as mPW2PLYP[105] and PBE0-DH from Adamo.[106]

1.4.3.6 Jacob’s Ladder and DFT

The performance of the different classes of functionals was summarised by Perdew in 2001,[107] with the concept of a DFT functional “Jacob’s Ladder”. In this, the lowest accuracy is defined as the HF method, with the highest being a level of theory that describes chemical properties very accurately. A series of five rungs then represent the different classes of functional described previously. The ladder proposed by Adamo in 2012 is illustrated in Figure 1.10, with examples of each functional type included alongside.[108]

Figure 1.10: Jacob’s Ladder for DFT functionals proposed by Adamo in 2012[108]

In its original concept, the final rung of the ladder involved the modelling of the nonlocal correlation to give accurate results. Recent work by Adamo has proposed that this rung has been achieved.[108] However, other researchers are not convinced, as stated by Becke in 2014[66] who pointed out that double hybrid functionals have formal scaling as an order higher than HF or traditional DFT functionals (when no approximations are used).

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