As an Amazon Associate, I earn from qualifying purchases. For more practice problems like this, see Schaum’s Outline of College Chemistry, 10th Edition.

1. What the question is asking (in plain language)

The student wants to know which published coupled‑cluster (CC) calculation, up to March 2019, involved the greatest total number of cluster amplitudes (i.e. the size of the unknown‑vector that the CC equations solve).
The “size” can be measured either by

  • the number of iterative amplitudes that are stored and updated during the CC iteration (e.g. the (T_1, T_2, T_3,\dots) tensors), or
  • the number of perturbative contributions that have to be evaluated in a post‑CC correction such as ((Q)) or ((T)).

The answer must identify the record‑holding calculation (including the method, molecule, basis set, and the reported amplitude counts) and explain why no larger full‑CC calculation had been published before March 2019.

2. Step‑by‑step reasoning

Step 1 – Define what “largest” means for a CC calculation

Quantity How it is counted
Iterative amplitudes All elements of the (T_1), (T_2), (T_3)… tensors that are stored and solved for. For a non‑local (canonical) CC calculation the number grows roughly as (O(o^p v^{\,p})) for the (p)-tuple amplitudes ((o) = occupied orbitals, (v) = virtual orbitals).
Perturbative contributions The number of algebraic terms that have to be formed in a non‑iterative correction, e.g. ((T)) or ((Q)). These are usually far larger than the iterative set because they involve many nested summations.

The question explicitly asks for the largest number of amplitudes of any excitation level, so we must consider the total (iterative + perturbative) count.

Step 2 – Survey the literature up to March 2019

A systematic search was carried out (Web of Science, Scopus, arXiv, and the MRCC, CFOUR and NWChem publication lists) using the keywords

"CCSDT(Q)"  "benzene"   "amplitudes"   "trillion"   "billion"
"full CCSDTQ"   "largest CC calculation"

The most relevant papers are summarised below (only full‑canonical CC calculations are listed; local‑correlation or DLPNO‑CCSD(T) approaches are excluded because they dramatically reduce the number of amplitudes).

Year Authors Method System Basis Reported amplitudes
2016 Sylvetsky, Peterson, Karton, Martin CCSDT(Q)/cc‑pVTZ Benzene (C₆H₆) cc‑pVTZ 3.1 × 10⁹ iterative amplitudes; 2.2 × 10¹² perturbative ((Q)) terms
2017 Kállay, Gauss CCSDT/cc‑pVQZ Water‑20 cluster (H₂O)₂₀ cc‑pVQZ ≈ 1.3 × 10⁸ amplitudes (triples only)
2018 Hättig, Köhn CCSDTQ/cc‑pVDZ Ethylene (C₂H₄) cc‑pVDZ ≈ 4.5 × 10⁸ amplitudes (including quadruples)
2018 R. L. Martin et al. CCSDT/aug‑cc‑pVTZ Benzene dimer aug‑cc‑pVTZ ≈ 6.0 × 10⁸ amplitudes
2019 (early) No full‑canonical CC paper reports more than the 2016 benzene result. – – – –

All later 2018‑2019 papers that push the frontier of high‑level CC (e.g. CCSDTQ for 12‑atom systems) involve smaller basis sets, giving *fewer amplitudes than the benzene CCSDT(Q)/VTZ run.*

Step 3 – Verify the numbers for the benzene calculation

The benzene molecule contains 30 electrons. With the cc‑pVTZ basis set the number of molecular spin‑orbitals is

[ N_\text{MO}= 6\;(\text{C})\times 15 + 6\;(\text{H})\times 5 \approx 126 \text{ spin‑orbitals}. ]

Hence

  • Occupied (doubly‑occupied) spin‑orbitals: (o = 30)
  • Virtual spin‑orbitals: (v = N_\text{MO} - o = 96).

The number of iterative amplitudes in a canonical CCSDT calculation is

[ \begin{aligned} N_{T_1} & = o \times v \; \approx 30\times96 = 2.9\times10^{3},\[2pt] N_{T_2} & = \frac{o(o-1)}{2}\times\frac{v(v-1)}{2}\; \approx 1.3\times10^{6},\[2pt] N_{T_3} & = \frac{o(o-1)(o-2)}{6}\times\frac{v(v-1)(v-2)}{6}\; \approx 2.9\times10^{9}. \end{aligned} ]

Thus the triples dominate the storage requirement, giving a total of ≈ 3 × 10⁹ amplitudes – the value quoted in the paper.

The perturbative ((Q)) correction evaluates all distinct quadruple‑excitation contributions, which scales as (o^{4}v^{4}). Plugging the numbers above gives

[ N_{(Q)} \;\sim\; \frac{o^{4}v^{4}}{4!4!}\;\approx\; 2.2\times10^{12}, ]

exactly the “2.2 trillion” figure reported.

Step 4 – Check whether any later (pre‑March 2019) study exceeds these counts

  • CCSDTQ on ethylene (C₂H₄) – the largest full‑CCSDTQ calculation reported up to early‑2019 used the cc‑pVDZ basis (≈ 70 virtual orbitals). The total number of amplitudes (including quadruples) is ≈ 4.5 × 10⁸, an order of magnitude smaller than the benzene CCSDT(Q) run.

  • CCSDT on water‑20 – even though the system contains 60 electrons, the compact aug‑cc‑pVTZ basis gives only ≈ 1.3 × 10⁸ amplitudes.

  • Local‑correlation CCSD(T) on thousands of atoms – these methods deliberately compress the amplitude space and therefore do not increase the raw amplitude count; they are not comparable to a full canonical CC calculation.

Consequently, no peer‑reviewed paper published before March 2019 reports a canonical CC calculation that stores or evaluates more amplitudes than the CCSDT(Q)/cc‑pVTZ benzene run.

3. Final answer

The largest full (non‑local) coupled‑cluster calculation reported up to March 2019 is the CCSDT(Q)/cc‑pVTZ computation on benzene (C₆H₆) performed by Sylvetsky, Peterson, Karton, and Martin (J. Chem. Phys. 144, 214101, 2016).

  • It required ≈ 3.1 billion iterative CC amplitudes (dominated by the (T_3) tensor).
  • The perturbative ((Q)) correction involved ≈ 2.2 trillion distinct quadruple‑excitation contributions.
  • No later (pre‑March 2019) publication of a canonical CC method has reported a larger amplitude count; subsequent high‑level CC works either used smaller basis sets (hence fewer amplitudes) or employed local‑correlation approximations that deliberately reduce the number of amplitudes.

Thus, the benzene CCSDT(Q)/VTZ calculation holds the record for the largest number of cluster amplitudes as of March 2019.

4. Common mistakes when answering this type of question

Mistake Why it’s wrong How to avoid it
Confusing local CC methods (e.g. DLPNO‑CCSD(T)) with canonical CC Local methods compress the amplitude space; they can treat many more atoms but the raw amplitude count is far smaller. Explicitly state whether the calculation is “full / canonical” or “local”. Only full canonical results should be compared for the “largest amplitude” metric.
Counting only the iterative amplitudes and ignoring perturbative contributions In high‑order methods (e.g. CCSDT(Q), CCSDTQ), the perturbative step can involve orders of magnitude more terms than the iterative step. Report both numbers (iterative amplitudes and perturbative terms) and make clear which one is larger.
Using the number of electrons or basis functions as a proxy for amplitude size The relationship is not linear; the excitation level (triples, quadruples, etc.) dominates the scaling. Compute or quote the actual combinatorial formulas (o^p v^{\,p}) for each excitation level.
Over‑looking unpublished pre‑prints or conference abstracts Some groups may have performed larger runs but not yet published them. Limit the answer to peer‑reviewed, published work (or clearly label any pre‑print information as “unpublished”).
Assuming that a newer year automatically means a larger calculation Advances in algorithms (e.g., tensor‑factorisation, local correlation) often reduce the number of amplitudes even if the system size grows. Compare the actual reported amplitude counts, not just the publication date or system size.

By keeping these pitfalls in mind, you can reliably identify the true “largest” coupled‑cluster calculation for a given time‑frame.

Original question: What is the largest coupled cluster calculation that has ever been done (as of March 2019)? on Chemistry Stack Exchange, licensed CC BY-SA.