4. The Coupled Cluster equations¶

4.1. The exponential ansatz¶

The size-extensivity problem of truncated CI methods can be linked to the linear parametrization of the CI wave-function. The CC wave-function, on the other hand relies on a exponential parametrization, which introduces the non-linearity required to have size-extensive energies (see appendix CI and CC for the hydrogen dimer). The CC wave-function is usually written as

\[| CC \rangle = \exp (\hat{T}) | \Phi_0 \rangle\]

where the cluster operator \(\hat{T}\) is given by

\[\hat{T} = 1 + \hat{T}_1 + \hat{T}_2 \dots \hat{T}_N\]

\(N\) is the total number of electrons and each cluster operator \(\hat{T}_{\mu}\) produces all the determinants that differ in orbital occupation by \(\mu\) electrons with respect to a reference determinant (e.g. HF). We refer to this process as (virtual) excitations since it excites or promotes \(\mu\) electrons from spin-orbitals that are occupied (\(i, j...\)) in the reference determinant to virtual (unoccupied) spin-orbitals (\(a, b...\)). For example, the singles and doubles operators acting on a reference determinant lead to

\[\hat{T}_1 | \Phi_0 \rangle = \sum_i^{occ} \sum_a^{vir} t_i^a | \Phi_i^a \rangle\]

\[\hat{T}_2 | \Phi_0 \rangle = \sum_{i<j}^{occ} \sum_{a<b}^{vir} t_{ij}^{ab} | \Phi_{ij}^{ab} \rangle\]

where the \(t\) coefficients are usually called the CC amplitudes and are the equivalent of the expansion coefficients of the CI wave-function, i.e., they are the CC wave-function parameters.

Note

The mathematical formalism behind the (virtual) excitation processes is called second-quantization. It is at the basis of quantum field theory and extensively used in the development of CC theory. For more details on the usage of second-quantization in quantum chemistry, see [Helgaker2000]

The CC exponential ansatz can then be expanded explicitly

\[\exp(\hat{T}) | \Phi_0 \rangle = (1 + \hat{T} + \frac{1}{2!} \hat{T}^2 + \frac{1}{3!} \hat{T}^3 + \cdots ) | \Phi_0 \rangle.\]

This expansion reveals the non-linearity of the parametrization. We see for examples that even if the cluster operator is truncated to include only double excitations, higher excitation will remain in the expansion through the non-linear terms like \(\hat{T}_2 \hat{T}_2\) which correspond to a quadruple excitation.

In CC theory, those terms are the key to size-extensivity. They are also responsible for the fast convergence of CC models to the FCI limit, since for an equivalent number of parameters (e.g. between truncated CI and CC) the CC expansion will include higher excitations indirectly through those non-linear terms.

4.2. The CC Schrödinger equation¶

Now that we have seen the expression of the CC wave-function we can insert it into the electronic Schrödinger equation to determine the CC ground-state energy.

\[H | CC \rangle = E_{CC} | CC \rangle\]

For reasons that we will not get into here, it is not possible to have an efficient formulation of CC theory by relying on the variational principle to minimize the energy. Instead, the CC Schrödinger equation is projected against the reference state \(\Phi_0\). It is also common to work with a similarity transformed Hamiltonian,

\[H^{\hat{T}} = \exp(-\hat{T}) H \exp(\hat{T})\]

which can be shown to preserve the eigenvalue spectrum of the original Hamiltonian but it is not a Hermitian operator anymore. We now have

\[\begin{split}\exp(-\hat{T}) H | CC \rangle &= \exp(-\hat{T}) E_{CC} | CC \rangle \\ \exp(-\hat{T}) H \exp(\hat{T}) | \Phi_0 \rangle &= E_{CC} \exp(-\hat{T}) \exp(\hat{T}) | \Phi_0 \rangle \\ H^{\hat{T}} | \Phi_0 \rangle &= E_{CC} | \Phi_0 \rangle.\end{split}\]

Projection against the reference determinant leads to the CC energy,

\[E_{CC} = \langle \Phi_0 | H^{\hat{T}} | \Phi_0 \rangle\]

or more explicitly

\[E_{CC} = \langle \Phi_0 | \exp(-\hat{T}) H \exp(\hat{T}) | \Phi_0 \rangle.\]

By expanding the exponentials and using the algebra of the second-quantization operators (hidden in the cluster operators), one arrives at the following explicit expression

\[E_{CC} = E_{\Phi_0} + \sum_i^{occ} \sum_a^{vir} t_i^a \langle \Phi_0| H | \Phi_i^a \rangle + \sum_{i<j}^{occ} \sum_{a<b}^{vir} (t_{ij}^{ab} + t_i^a t_j^b - t_i^b t_j^a ) \langle \Phi_0| H | \Phi_{ij}^{ab} \rangle\]

where \(E_0\) is the energy corresponding to the reference wave-function \(\Phi_0\). When the reference wave-function is the HF determinant the expression for the CC energy simplifies to

\[E_{CC} = E_{HF} + \sum_{i<j}^{occ} \sum_{a<b}^{vir} (t_{ij}^{ab} + t_i^a t_j^b - t_i^b t_j^a ) (ia||jb)\]

due to Brillouin’s theorem.

4.3. The CC amplitudes¶

From the last equation we see that the CC energy if (of course) depending on the CC wave-function parameters (the amplitudes). As stated previously these are not obtained variationally but by projection techniques. To determine the amplitudes we project the CC Schrödinger equation against the set of excited determinants (up to the truncation level), i.e., for a coupled cluster with single and double excitations (CCSD) we have to solve the following non-linear equations

\[\begin{split}\langle \Phi_i^a | \exp(-\hat{T}) H \exp(\hat{T}) | \Phi_0 \rangle &= 0, \\ \langle \Phi_{ij}^{ab} | \exp(-\hat{T}) H \exp(\hat{T}) | \Phi_0 \rangle &= 0.\end{split}\]

For higher truncation levels we need to project over excited determinants of higher excitation ranks. Note that the right-hand-side of the equation is zero due to orthogonality between the reference determinant and any excited determinant.

The left-hand-side of the equation can be derived into more explicit expressions, i.e, in terms of amplitudes and integrals (like the final expression for the energy) which can be implemented into computer programs.

The CC amplitude equations form a set of non-linear equations which have to be solved iteratively before the CC energy can be calculated.

Note

The equations presented here correspond to standard CC theory for which the reference wave-function must be a single slater determinant. In general this determinant is chosen to be the HF wave-function.