Wavefunction Separability and Size Consistency of the Energy

It is perhaps useful to compare the exponential ansatz of Eq. [2.31] with the analogous expansions of other wavefunctions. In the configuration interaction (CI) approach,[85,86] for example, a linear excitation operator is used instead of an exponential,

$$\Psi_{\rm CI} = \left(1 + \hat{C} \right) \Phi_0,$$ (2.32)

where $\hat{C}$ is a linear combination of cluster-like operators defined similarly to $\hat{T}$, viz.,

$$\hat{C} \hat{C}_1 + \hat{C}_2 + \ldots$$ =

$$\sum_{ia} c_i^a a_{a}^{\dagger} a_{i}^{\ } + \frac{1}{4} \sum_{ijab} c_{ij}^{ab} a_{a}^{\dagger} a_{b}^{\dagger} a_{j}^{\ } a_{i}^{\ } + \ldots .$$ = (2.33)

Truncation of $\hat{C}$ at the single- and double-excitation level (CISD) leads to a wavefunction with exactly the same number of amplitudes (c_i^a and c_ij^ab) as that needed for the CCSD wavefunction (t_i^a and t_ij^ab). However, the latter implicitly includes higher excitation levels (triples and quadruples) by the inclusion of $\hat{T}$ products in the power series expansion of $e^{\hat{T}}$. Such products are commonly referred to in the literature as disconnected wavefunction contributions.^2.5 Both the CI and CC methods will produce exact wavefunctions if one does not truncate $\hat{C}$ (full CI) or $\hat{T}$ (full CC). In fact, in the limit of exact linear and exponential wavefunction expansions, a relationship between the CI and CC amplitudes may be developed[5] that reveals the factorization of each level of CI excitation into connected and disconnected components, e.g.,

$$\hat{C}_2 = \hat{T}_2 + \frac{1}{2} \hat{T}_1^2.$$ (2.34)

The two different forms of the excitation operator in CI and CC theory have significant consequences for both the energy and wavefunction as the number of electrons is increased or as the (molecular) system is separated into fragments.

Consider the structure of the coupled cluster and configuration interaction wavefunctions for a generic system involving two infinitely separated (and therefore non-interacting) components Xand Y. If the molecular orbitals used to define the cluster functions $\hat{T}$ and $\hat{C}$ are localized on each of the two fragments -- a choice which will not affect the energy associated with either the reference determinant, $\Phi_0$, or the correlated wavefunction, $\Psi_{CI}$ or $\Psi_{CC}$ -- then the cluster operators may be separated into components involving intrafragment excitations only, i.e.,

$$\hat{T} = \hat{T}_X + \hat{T}_Y \ \ \ {\rm and} \ \ \ \hat{C} = \hat{C}_X + \hat{C}_Y.$$ (2.35)

For example, the amplitudes t_ij^ab or c_ij^ab, in which orbitals $\phi_i$ and $\phi_a$ are localized on fragment X and orbitals $\phi_j$ and $\phi_b$ are localized on fragment Y, will be zero. Thus, the total coupled cluster exponential operator may be written as a product of independent coupled cluster operators for each fragment, viz.[87]

$$\Psi_{CC} = e^{\hat{T}} \Phi_0 = e^{\hat{T}_X + \hat{T}_Y} \Phi_0 = e^{\hat{T}_X} e^{\hat{T}_Y} \Phi_0.$$ (2.36)

Since the reference determinant, $\Phi_0$, is factorizable into determinants isolated on each fragment (in the localized orbital description), the total coupled cluster wavefunction may be written as a product of coupled cluster wavefunctions for each of the separated fragments.^2.6 As a result, the sum of the coupled cluster energies computed for each fragment separately is the same as that computed for the ``supermolecule'' in which the fragments are included together in the calculation,

E_CC = E_CC^X + E_CC^Y. (2.37)

This property of the coupled cluster energy is commonly known as ``size consistency''[89].

For the configuration interaction wavefunction, however, multiplicative separability is not possible:

$$\Psi_{CI} = \left( 1 + \hat{C} \right) \Phi_0 = \left( 1 + \hat{C}_X + \hat{C}_Y \right) \Phi_0.$$ (2.38)

As a result, the CI energy is not size consistent, and the sum of the energies of the separated fragments differs from the CI energy of the supermolecule,

$$E_{CI} \neq E_{CI}^{X} + E_{CI}^{Y}.$$ (2.39)

In the event that the CI cluster operator, $\hat{C}$, is not truncated, however, it is possible to write the resulting full CI wavefunction as a product of wavefunctions for each separated fragment, since the linear operator may be transformed into an exponential using a generalized form of Eq. [2.34].

Consider the classic example of an ensemble of hydrogen molecules. Both the CCSD and CISD wavefunctions are exact (within the given one-electron basis set) for a single H₂ molecule since there are only two electrons to be correlated. However, errors are introduced in the CI energy in the case of two (or more) non-interacting H₂units due to the lack of multiplicative separability of the wavefunction. The size consistent CCSD method, on the other hand, produces the correct total energy, regardless of the number of non-interacting H₂ monomers in the system, since the total coupled cluster wavefunction may be written as a product of separated wavefunctions, each of which is exact for the given hydrogen molecule.

Some caution should be exercised in the application of the size consistency concept when applied to open-shell fragments, however. As Taylor has recently pointed out[81], a given method may be size consistent for some systems but not for others. For example, the spin-restricted Hartree-Fock (RHF) approach is size consistent for the dissociation of the hydrogen fluoride in its $^3\Pi$ excited state into atoms,

$${\rm HF} (^3\Pi) \rightarrow {\rm H} (^2S) + {\rm F} (^2P),$$ (2.40)

since the single determinant wavefunction can correctly describe the high-spin electronic states in both the supermolecule and the separated fragments. The RHF method is not size consistent, however, when describing the dissociation of the ground state of HF, into these same atomic states,

$${\rm HF } (^1\Sigma^+) \rightarrow {\rm H} (^2S) + {\rm F} (^2P).$$ (2.41)

This size-inconsistency occurs because the two open-shell electrons on the atoms must be singlet-coupled to produce the correct dissociation limit, and a supermolecule, two-determinant approach is therefore required. This difficulty also applies to coupled cluster or perturbation-based wavefunctions that use the RHF determinant as a reference; these methods cannot be size consistent for a given molecular system unless the reference wavefunction is size consistent.

A more general property of the coupled cluster energy which is related to size consistency is ``size extensivity.'' This is a strictly mathematical characteristic of the wavefunction which relates to scaling of the computed energy with respect to the number of correlated electrons and the resulting energy dependence of the wavefunction amplitude equations. Size extensivity is not dependent on the system under study, and it applies to all regions of the potential energy surface -- not just to the fragmentation limit. We will return to this topic later in the chapter after we have discussed the algebraic and diagrammatic techniques needed to derive working coupled cluster equations.