2.4 Post Hartree-Fock techniques

The limitations of the Hartree-Fock method may be reduced by going beyond the ansatz of a single-determinant wavefunction. There are two broad categories of such approaches: those based on perturbation theory and those based on the variational principle. Among the latter approaches is the configuration interaction method which will be covered here because the focus of this thesis is on obtaining accurate many-body wavefunctions.^2.2

2.4.1 Configuration interaction

Configuration interaction (CI) methods are one of the conceptually simplest methods for solving the many-body Hamiltonian. Although theoretically elegant, in principle exact, and relatively simple to implement, in practice full CI can be applied to only the smallest of systems.

The basis for CI methods is the simple observation that an exact many-body wavefunction, $\Psi$, may be written as a linear combination of Slater determinants, $D_{k}$,

$$\Psi=\sum_{k=0}^{\infty}c_{k}D_{k} \;\;\;,$$ (2.10)

where the $D_{k}$ fully span the Hilbert space of the wavefunction. The determinants can be any complete set of $N$-electron antisymmetric functions but are typically constructed from Hartree-Fock orbitals such that $D_0$ is the ground-state Hartree-Fock determinant.

The Hartree-Fock ``reference determinant'' $D_0$ is by definition the best single-determinant approximation to the exact wavefunction $\Psi$. In most electronic systems, the Hartree-Fock energy accounts for the majority of the exact total energy, and the missing correlation energy is small. If the coefficients $c_{k}$ are normalised then typically $c_0 \approx 1$ and all remaining $c_{k}$ are very small. A very large number of configurations is required to yield energies and wavefunctions approaching the exact many-body wavefunction. In practice the expansion must be limited on physical grounds, as the total number of determinants is

$$k_{max} = \frac{M!}{N!(M-N)!} \;\;\;,$$ (2.11)

where the length of the expansion $k_{max}$ is given in terms of the number of electrons, $N$, and the number of basis states, $M$, in the expansion ($M >> N$).

The scientific problem in adapting the CI method into a practical one is to obtain the best wavefunction, and hence lowest CI energy, with the shortest expansion length. A typical approach would be to truncate the expansion after only double or quadruple excitations from the reference determinant, where an excitation consists of replacing a ground state occupied orbital by an unoccupied one. These levels of truncation are the CI singles-doubles (CISD) and CI singles-doubles-triples-quadruples (CISDTQ) methods. A formidable number of terms are still left in the expansion. Accurate applications of the methods are consequently limited due to their computational cost.

When performed within a finite reference space, an additional problem with the method becomes apparent: the methods lack ``size-extensivity'' and do not perform equally well in systems of differing size. As the size of system increases, the proportion of the electronic correlation energy contained within a fixed reference space (such as all single and double excitations) decreases. The lack of size-extensivity results in a non-cancellation of errors when systems of different sizes are compared, resulting in difficulties when interaction or bonding energies are required.

Despite these limitations, CI represents a controlled (and variational) improvement to the ground-state wavefunction, and may therefore be used in the determinantal parts of trial wavefunctions in QMC. Details of this approach are given in chapter 3.

2.4.2 Other methods

The lack of size extensivity and substantial cost of the CI method has led to the development of several related methods.

The coupled-cluster (CC) method[7,3] is one of the most important practical advances over the CI method. Although non-variational, it resolves the problem of size extensivity, and is often very accurate, but more expensive than (limited) CI. The CC method assumes an exponential ansatz for the wavefunction

$$\Psi_{CC}=\exp(\hat{T}) \Psi_{HF} \;\;\;,$$ (2.12)

where the coupled-cluster wavefunction is given by an excitation operator acting on a reference wavefunction, usually the Hartree-Fock determinant $\Psi_{HF}=D_{0}$. The operator $\hat{T}$ is generates k-fold excitations from a reference state

$$\hat{T}=\sum_{k} \hat{T}_{k} =\hat{T}_{1}+\hat{T}_{2} +\ldots\;\;\;,$$ (2.13)

so that, for example,

$$\hat{T}_{2}\Psi_{HF}=\sum t_{ij}^{ab}D_{ij}^{ab} \;\;\;.$$ (2.14)

The operator $\hat{T}_{2}$ generates excitations from pairs of occupied states, $ij$, to pairs of virtual states, $ab$, from a reference Hartree-Fock determinant $D$. The expansion coefficients $t_{ij}^{ab}$ are determined self-consistently.

A ``coupled-cluster doubles'' wavefunction is written as

$$\Psi_{CCD}=\exp(T_2)\Psi_{HF}= (1+\hat{T}_{2}+\hat{T}^{2}_{2}/2+\hat{T}^{3}_{2}/{3!}+\ldots)\Psi_{HF}\;\;\;,$$ (2.15)

$$\Psi_{CCD}=D_{0}+\sum t_{ij}^{ab}D_{ij}^{ab}+\frac{1}{2}\sum\sum t_{ij}^{ab}t_{kl}^{cd}D_{ijkl}^{abcd}+\ldots \;\;\;.$$ (2.16)

The CC expansion is usually terminated after all double excitations or all quadruple excitations have been included. It may be shown[3] that this expansion is size consistent. By including many excitation terms in the expansion, CC methods are computationally very expensive relative to HF calculations. Formally, CC singles-doubles scales as the sixth power of the number of basis states included in the expansion, and calculations including up to quadruple excitations scale as the tenth power of the number of states.

The key limitation of the CC methods are their rapid increase in computational cost with system size. This presently limits the methods to small molecular systems.

2.4.3 Limitations

Hartree-Fock is a simple theory which satisfies the commonly known features of fermionic wavefunctions. The theory generates wavefunctions that are antisymmetric with respect to the exchange of two electron positions and includes exchange between like-spin electrons. The cost of a Hartree-Fock calculation formally scales with the cube of the number of basis functions, but depending on implementation the scaling can be between linear and quartic with system size. It is insufficiently accurate for quantitative predictions of the properties of many compounds. By neglecting electron correlation, interaction energies are typically very poor. A Hartree-Fock wavefunction is a well-controlled approximation to the many-body wavefunction, and for this reason Hartree-Fock continues to be widely used: it is often predictably accurate or inaccurate, and therefore useful for determining qualitative information such as trends in a structural parameter with system size.

Almost all post Hartree-Fock methods share the combined limitations of a poor scaling with system size and a strong basis set dependence. In practice, post Hartree-Fock methods typically scale with the fourth or higher power of the number of basis states included in the calculation, limiting their application to small systems. The basis states depend on the underlying basis set used in their numeric expansion, equation 2.9, and it is commonly found that use of improved methods requires an improved basis set, further increasing the cost of calculation. CI and CC-based methods effectively transform the electron-correlation problem into a basis set problem, where the basis set is the set of molecular orbitals derived from a Hartree-Fock (or similar) calculation.