2 4.3 The local density approximation
32 Chapter 2 Method review and the dimer potential which means that
Becke corrected this by adding a new term of the form
x 2 _(3pl/3 X 1 + 6(3x sinh- 1 x Vp p4/3 (2.4 7) (2.48) (2.49)
which has one adjustable parameter which was originally chosen by fitting to the exchange energies of 6 noble gas atoms.
2.4.6 Density gradient corrections
The use of the density gradient in Becke's correction of the exchange energy can be easily motivated. The shift from a continuous system of constant density to the finite regime of molecules with their sharply varying density means that simple knowledge of the density is no longer sufficient. Naturally enough, the electronic density in molecules varies due to the inherent shell structure; this may only be reproduced to a degree by DFT by the inclusion of a IV pi dependence. Other functionals have since been introduced with the general form
(2.50)
which are commonly known as generalised gradient approximation (GGA) function also These are the most commonly used functionals in computational chemistry, in particular the hybrid functional3 B3LYP is widely popular and very successful for certain types of molecules (generally light and organic). This is not so surprising as these functionals are often fitted to experimental data for the systems of interest. LDA and the Becke GGA corrected version (often labeled BP86, after Becke-Perdew and the year of publication) are however not parametrised in such a way.
2.4.7 The Perdew-Wang 91 functional
In a recent paper, Langlet
et al.
[Langlet et al. , 2004] describe "an interesting prop erty of the PW9 1 functional" . They have compared terms arising from symmetry- 3The term hybrid is used to indicate that Hartree-Fock exchange is included to correct for defi ciencies in the underlying model.2.5.
M011er-Plesset perturbation theory
33 adapted perturbation theory (SAPT) for the exchange repulsion and dispersion, which they define aswith a term calculated from DFT which they call the Pauli repulsion. This they define as the repulsion due to the antisymmetrisation of the wavefunction, through a density deformation of the monomer orbitals, a change in kinetic energy, and consequent changes in the effective exchange and correlation potentials. This term is designed to exclude the purely electrostatic interaction of the monomers' charge densities, and the orbital interaction that is obtained by solving the Kohn-Sham equations for the antisymmetrised wavefunction.
Both the SAPT and DFT derived terms are considered to approximate the dis persion interaction of a van der Waals molecule.
In calculations on a number of weakly interacting systems, they find the correla tion between these two terms to be remarkably better with PW91 than for any other functional. They do not propose any theoretical basis for this coincidence, but do suggest that it may be more than j ust a "fortuitous cancellation" that makes this functional particularly appropriate for vdW-bonded systems.
An examination of the original papers by Perdew and Wang shows that the parametrisation of this functional was originally aimed at improving the represen tation of the correlation energy for a uniform electron gas [Per dew and Wang, 1991a, Perdew and Wang, 1991b, Perdew and Wang, 1992] . Although no fundamental ex planation is given, the numerical evidence of reference [Langlet et al. , 2004] leads us to expect that this functional may describe the vdW bonding in small mercury systems better than others.
2 . 5 M011er-Plesset perturbat ion theory
M!ZIller Plesset perturbation theory is the most common application of many-body perturbation theory to the calculation of electron correlation effects. It uses the Hartree-Fock wavefunction as the reference wavefunction and adds correlation as a perturbation to the Fock operator:
34 Chapter 2. Method review and the dimer potential The perturbation operator V is defined for correlation as
• -1 •
V
= - VHF i<j L r;/ - L VHF(i) i<j (2.53) (2.54)i.e. the difference between the exact Coulomb interaction and the Hartree-Fock in teraction. Applying this to the reference wavefuction W H F gives
E
(F + V) Iw) = E Iw) ,
EHF IWHF)
E�O) + E�l) + E�2) + . . .
(2.55) (2.56) (2.57) where EHF is the ground state HF energy, and W HF =
10
> is the ground state wavefunction.The first order correction is
E(l) 0 =
(011710)
(2 .58)(0
L rij - 10)
-(0
L VHF (i)0)
(2.59)i<j
1
2
I.=
(ij l lij) - L (ilvHF li) (2.60)�J
1
- 2
I.=
(ij l lij) (2.6 1 )�J
(2.62) Comparison with equation (2. 18) shows that the HF energy is the sum of the zeroth and first order energies.
E(O) o + Eel) 0 1 L Ei - 2 L (ijl lij) i ij (2.63) (2.64) (2.65)
where the second term comes from the antisymmetry of the wavefunction. Therefore the next useful correction comes at second order.
2.6.
Coupled cluster theory
The second order energy is
E(2) 0 10) = In)
I(OIVln)r.
L
E(O) _ E(O) ,n>O
0 n IWHF) , Iw:b) 35 (2.66) (2.67) (2.68) wheren
labels the sum over all doubly excited states. Here the doubly excited wavefuntion Iw:b) uses indices a, b for unoccupied and r, s for occupied orbitals. This then gives us the second order perturbative correlation to the Hartree-Fock energy, and is commonly known as MP2 (for M011er P lesset in second order) . If the expansion is carried out to higher orders then MP3, MP4, and so on are obtained. As the expense of the calculation scales asNn+3
for an MPn calculation4 without a great improvement in accuracy, calculations are not usually carried out to higher than fourth order. If higher accuracy is required, a coupled cluster calculation is usually preferred.2 . 6 Coupled cluster theory
The best way of including electron correlation in a calculation is to include all pos sible configurations of the electrons in the wavefunction (consisting of Slater deter minants) . Therefore the wavefunction is written as a sum over the ground state plus all the possible singly excited and doubly excited determinants and so on to the lim iting possibility of a full configuration interaction. This sum would be modulated by coefficients which would control the contribution of each determinant to the wave function. However in practice this is only possible for the smallest systems, and some truncation of the series of excited wavefunctions is necessary. The problem with this is that any such truncation results in a wavefunction that is no longer size-extensive, that is, that the energy of two non-interacting atoms is no longer the same as the sum of their energies calculated separately, for example. Obviously for molecules this is a considerable challenge, and has led to the development of coupled cluster theory, where the excited states are included in the wavefunction in such a way as to keep the calculations size-extensive. The excitation from the ground state into the various