5. CC models and comments

5.1. Standard CC models

With the equations that we have presented in the previous section, the standard models of CC theory can be easily defined. Basically a given CC model is defined by the truncation level of the cluster operator (a bit like in the CI models). If only single excitations are included, \(\hat{T} = \hat{T}_1\), the CC energy is equal to the HF energy and this model, denoted CCS, is not relevant here (it can however be used to calculate excited state properties). The next model is the most common in the standard CC hierarchy. It includes single and double excitations and it is denoted CCSD.

As you can imagine, the next steps consist in including triples, CCSDT, and quadruples, CCSDTQ, etc. This allows for a systematic convergence to the FCI solution which is the main selling point of CC theory (together with the size-extensivity property described before).

5.2. Computational cost

However, as we get closer to the FCI solution, the computational cost of solving the CC amplitude equations increases drastically. In particular the scaling of the method with the size of the system, N, goes as follows ,

  • CCSD already scales as \(N^6\),
  • CCSDT scales as \(N^8\),
  • and CCSDTQ as \(N^{10}\)…
_images/water.png

To be more concrete, if it takes 1 minute to perform a CCSD calculation on 1 water molecule, it would take 120 years to perform the same calculation on a cluster of 20 water molecules.

A lot of work has been done (and is still going on) to lower the computational cost of CC models and to develop intermediates in the CC hierarchy.

One particularly popular and successful method is the CCSD(T) model in which, a standard CCSD calculation is performed and corrected by the effect of triples which are included in a perturbative manner (i.e., by relying on arguments from many-body perturbation theory). This method scales as \(N^7\) with the system size but recover most of the effects of a full CCSDT calculation. Due to the success of the CCSD(T) model, it is often described as the gold-standard of quantum chemistry.

5.3. What is CC theory used for?

As we have seen CC theory has the following properties:

  • Fast and systematic convergence to the FCI solutions.
  • Size-extensivity of the energy, (wave-function and energy strictly separable).
  • Fast increase of the computational scaling with the truncation level, i.e., high computational cost.
  • Single-reference method. CC cannot be applied when more than one electronic configuration is important to describe a system, e.g. when breaking bonds. In other words, the (HF) reference needs to provide a qualitatively good description of the system under consideration.

So what can it be used for?

The ground-state CC theory that has been introduced here can be used to calculate very accurate ground-state energies of small molecules near their equilibrium geometry.

In addition molecular properties (dipole moments, polarizabilities…) and excitation energies can be calculated by combining CC theory with response theory (see also the equation-of-motion EOM-CC methods).

Being able to calculate energies and properties accurately and with a systematic convergence to the FCI solutions makes it possible to benchmark more affordable models (like DFT) that can be applied to larger molecules.

It should also be mentioned that since the late 90s, a lot of effort has been put in the development of more affordable CC models and a lot of progress has been made.