The previous section detailed the use of DFT in order to add correlation to the description of a molecular system using semi-empirical functionals. The next two sections will focus on introducing correlation using anAb Initioapproach.
The first thing to look at is what makes up correlation. Correlation can be divided into two separate effects. The first of these effects is dynamical correlation. This describes what happens to a wave function when two electrons approach each other. The other type of correlation is non-dynamical correlation, also known as static correlation. Static corre- lation is due to near degeneracy of different electron configurations in the wave function. In order to fully describe a system, both forms of correlation must be properly accounted for.
Non-dynamical correlation can be included in a system quite easily by allowing a mixing of more than one configuration in the wave function. Methods for doing this will be discussed in the next section. Dynamical correlation is very difficult to include in a system. It takes many different configurations in order to properly describe these interactions.
There are broadly three classes of method for including dynamical correlation in a wave function, all closely related in theory. These methods are perturbation theory, con- figuration interaction (CI) and coupled cluster.
In perturbation theory it is assumed that the new problem only differs slightly from a problem that has already been solved, either exactly or approximately. In this case cor- related wave functions differ only in the inclusion of correlation compared to an already solved wave function such as Hartree-Fock.
Perturbation theory solves the new problem by introducing a small change or pertur- bation into the problem. In the case of correlation this is done by defining a two part Hamiltonian consisting of a reference Hamiltonian,H0, and a perturbation,H′, with the
magnitude ofλ. The relationship is shown in (2.41).
H=H0+λH′ (2.41)
The perturbed wave function is given by (2.42).
HΨ=WΨ (2.42)
As the perturbation increases to a finite value the new energy (W) and wave function also changes continuously. The new wave function and energy can be expanded in a power series in the perturbation parameterλas shown in (2.43).
W =λ0W0+λ1W1+λ2W2+λ3W3+. . .
Ψ=λ0Ψ0+λ1Ψ1+λ2Ψ2+λ3Ψ3+. . . (2.43)
Inserting the results from (2.42) and (2.43) into the Schr¨odinger equation and collect- ing terms of the same order inλgives (2.44).
H0|Ψ0⟩=W0|Ψ0⟩
(H0−W0)|Ψ1⟩= (W1−H1)|Ψ0⟩
(H0−W0)|Ψ2⟩= (W1−H1)|Ψ1⟩+W2|Ψ0⟩ (2.44)
It can be assumed that the perturbed wave functions are orthogonal to the zeroth order wave function. This orthogonality allows a normalisation of the total wave function such that⟨Ψ|Ψ0⟩=1. This normalisation is used to produce energy expressions up to second
W0=⟨Ψ0|H0|Ψ0⟩
W1=⟨Ψ0|H1|Ψ0⟩
W2=⟨Ψ0|H1|Ψ1⟩ (2.45)
The most popular form of perturbation theory is Møller-Plesset Perturbation Theory (MP) which chooses the sum over Fock operators as the zeroth-order Hamiltonian and uses the exactVee operator minus twice the⟨Vee⟩operator. There are different orders of perturbation theory based on how far along the power series the perturbation extends. The most popular form of (MP) is second order Møller-Plesset (MP) Perturbation theory (MP2) as it is computationally cheap, although not as cheap as DFT, and seldom gives a more accurate result as it often overshoots the correlation energy.
The second method is Configuration Interaction (CI) which is similar to mixing mil- lions of different configurations. In CI the wave function is built from a linear combination of Configuration State functions (CSF). These CSF’s are constructed using a ground state Hartree-Fock determinant expanded with a number of excited state determinants.
A CI method is generally limited by the number of additional determinants used in the construction of the CSF’s. For example if any occupied orbital is replaced by a virtual orbital this is a single excitation, two is a double excitation and so on. CI methods are usually limited by the number of excitations allowed. Just single excitations are no use as all matrix elements between the single excitations and the Hartree-Fock determinants are all zero. but CID (just double excitations) and CISD (singles and doubles) are popular forms.
CI is generally less widely used than both MP2 and coupled cluster methods as it is extremely expensive computationally (although no worse than coupled cluster), especially at higher excitation levels or when used on larger systems. It only produces better results than MP2 in small systems and does not scale correctly with the number of electrons.
ence treatment of correlation. It is size-extensive (it scales correctly with the number of electrons) giving it the advantage over CI methods and gives a much better result than MP2. Unfortunately these advantages come at the expense of computational speed.
In the perturbation theory method outlined above all types of correction to the refer- ence wave function are added to a given order (2,3,4, etc.). The idea with coupled cluster methods is to include all corrections of a given type to infinite order. To do this an ex- citation operator Tas shown in (2.46) is used to act on a Hartree-Fock reference wave functionΦ0whereTigenerates allith excited Slater determinants. As shown in (2.47).
T=T1+T2+T3+···+TNelec (2.46) T1Φ0= occ
∑
i vir∑
a tiaΦai T2Φ0= occ∑
i<j vir∑
a<b ti jabΦabi j (2.47)A CI wave function can be generated by allowing Tto work on a Hartree-Fock wave function. The corresponding CC wave function is defined as the exponential of the exci- tation operator acting on the same Hartree-Fock wave function. Both are shown in (2.48).
ΨCI= (1+T)Φ0 = (1+T1+T2+T3+T4+···)Φ0 ΨCC=eTΦo Where eT=1+T+1 2T 2+1 6T 3+. . . = ∞
∑
k=0 1 K!T k (2.48)The exponential operatoreTcan be also be written as in (2.49) where each term gener- ates all excitations of a given order. For example the second term in parentheses generates
all triple excitations (triples). eT=1+T1+ (T2+ 1 2T 2 1) + (T3+T2T1+ 1 6T 3 1) + (T4+T3T1+ 1 2T 2 2+ 1 2T2T 2 1+ 1 24T 4 1). . . (2.49)
So far, this wave function generates an exact solution equivalent to a full CI. However, as with a full CI, this is just not possible for anything but the smallest systems and in the same way that a CI is usually truncated to only include certain excitations, the same is applied to CC methods. For example CCSD, one of the most popular CC methods, includes only contributions fromT=T1+T2.
The addition of a full treatment of triples (CCSDT) causes a lot of added computa- tional complexity to coupled cluster calculations; however for only a small amount of extra computational work over CCSD, a perturbative treatment of triples can be used. The perturbative treatment uses equations derived from MP4 and MP5, but using CCSD amplitudes (t) rather than perturbation coefficients, this gives CCSD(T).