The coupled-cluster family: from MP2 to CCSDT

Between second-order perturbation theory and full configuration interaction lies a ladder of approximations that all descend from the same coupled-cluster (CC) equations: keep the doubles but not the singles (CCD), drop the nonlinear terms (LCCD, CEPA(0)), add pair-specific shifts (CEPA(1)), keep only the quadratic terms that restore size extensivity (QCISD), approximate the doubles to first order (CC2), add triples perturbatively (CCSD(T), QCISD(T)), iteratively but approximately (CC3), or exactly (CCSDT), or rotate the orbitals until the singles vanish (BCCD). Every one of them is a statement about which terms of the CCSD amplitude equations matter, and the cleanest way to learn the family is to run all of them on one small molecule, in one basis, with one set of integrals, and look at the numbers side by side.

vibe-qc exposes the whole ladder through run_job(method=...) on a closed-shell RHF reference; the coupled-pair and QCI variants share the canonical CCSD kernel (CCSDOptions.cc_variant), CC3 and CCSDT run in an exact determinant-space representation, and all of them are production within the envelopes stated in the CCSD user guide (see Methods: theory and status). This page runs twelve of them on water in 6-31G, adds CC3 and CCSDT in a minimal basis where their combinatorial cost stays under a minute, and uses a frozen-core CISD to show what size extensivity buys.

Setup

The 6-31G basis has no registered RI auxiliary basis, so the run passes density_fit=False and every method works on the same exact four-index integrals; that also makes the CCSD(T) number directly comparable with the determinant-space CC3 and CCSDT below. run_job freezes the oxygen 1s core by default for all of these methods.

import time
import vibeqc as vq

ANG = 1.8897259886
h2o = vq.Molecule([
    vq.Atom(8, [0.0, 0.0, 0.0]),
    vq.Atom(1, [0.0,  0.7572 * ANG, 0.5865 * ANG]),
    vq.Atom(1, [0.0, -0.7572 * ANG, 0.5865 * ANG]),
], 0, 1)

ladder = ["mp2", "cc2", "cepa(1)", "ccd", "lccd", "ccsd", "qcisd", "cepa(0)",
          "ccsd(t)", "qcisd(t)", "bccd", "bccd(t)"]
rows = []
for method in ladder:
    t0 = time.perf_counter()
    res = vq.run_job(h2o, basis="6-31g", method=method, density_fit=False,
                     output="h2o_" + method.replace("(", "").replace(")", ""), verbose=0)
    wall = time.perf_counter() - t0
    post = res.mp2 if method == "mp2" else res.ccsd      # MP2Result or CCSDResult
    rows.append((method, post.e_hf, post.e_total - post.e_hf, post.e_total, wall))

print(f"{'method':9s} {'E_ref / Ha':>12s} {'E_corr / Ha':>12s} {'E_total / Ha':>14s} {'wall / s':>9s}")
for method, e_ref, e_corr, e_tot, wall in rows:
    print(f"{method:9s} {e_ref:12.6f} {e_corr:12.6f} {e_tot:14.6f} {wall:9.1f}")

On a vibe-qc 0.17.10 development build (Apple M5 Max) each method takes two to five seconds, RHF included:

method      E_ref / Ha  E_corr / Ha   E_total / Ha  wall / s
mp2         -75.983974    -0.127814     -76.111788       2.7
cc2         -75.983974    -0.128585     -76.112559       2.0
cepa(1)     -75.983974    -0.133233     -76.117208       2.0
ccd         -75.983974    -0.133791     -76.117765       2.0
lccd        -75.983974    -0.133959     -76.117934       1.7
ccsd        -75.983974    -0.134471     -76.118446       1.7
qcisd       -75.983974    -0.134612     -76.118586       3.0
cepa(0)     -75.983974    -0.134814     -76.118789       2.2
ccsd(t)     -75.983974    -0.135456     -76.119431       1.7
qcisd(t)    -75.983974    -0.135502     -76.119476       2.7
bccd        -75.983157    -0.135170     -76.118328       4.8
bccd(t)     -75.983157    -0.136265     -76.119423       4.3

The wall times were measured with other jobs running and vary by a factor of two between runs; the energies do not.

What just happened

Every row is the same RHF reference (\(E_\text{ref} = -75.983974\) Ha) followed by a different treatment of the correlation problem in the same 12 active orbitals (13 minus the frozen core). The post-SCF result is attached as res.mp2 (an MP2Result: e_hf, e_correlation, e_total) or res.ccsd (a CCSDResult: e_hf, e_ccsd_correlation, e_t, e_total, n_iter, cc_trace), and res.energy_total is the correlated total energy in both cases while res.energy stays the SCF energy. Read the table from the top.

MP2 to CC2 to CCSD. MP2 recovers \(-127.8\) mHa, CC2 \(-128.6\) and CCSD \(-134.5\). CC2 is MP2’s iterative cousin: its doubles are correct only to first order in the fluctuation potential, so at a single geometry it sits within a millihartree of MP2; its value is that the singles are treated exactly, which makes it a proper model for response properties and excitation energies where MP2 has none. The 6 mHa between them and CCSD is the effect of the infinite-order doubles.

CCD and LCCD versus CCSD. Freezing the singles (CCD, \(-133.8\)) costs 0.7 mHa relative to CCSD: singles are small for a molecule at its equilibrium geometry with a good RHF reference, and their role is to relax the orbitals in the presence of correlation. Linearising the doubles (LCCD, \(-134.0\)) overshoots CCD slightly. That is the general rule: the quadratic \(T_2^2\) terms that LCCD drops are net repulsive (they contain the exclusion-principle-violating corrections), so every linearised method recovers too much correlation energy.

The coupled-pair family. CEPA(0), which is LCCSD, overshoots further (\(-134.8\), beyond CCSD) for the same reason, while CEPA(1) (\(-133.2\)) applies Meyer’s pair-specific energy shifts that mimic the missing quadratic terms and lands above CCSD. CCSD is bracketed between the two, which is the historical reason coupled-pair methods were considered good approximations to it before CCSD was affordable. The CEPA numbers depend on the choice of occupied orbitals (vibe-qc uses canonical orbitals; see the user guide); the CEPA(0) value does not.

QCISD and CCSD. QCISD (\(-134.6\)) sits 0.14 mHa below CCSD. QCISD is CISD plus the two quadratic terms needed for size extensivity; it lacks the remaining nonlinear terms of CCSD, and it is known to overestimate correlation slightly relative to CCSD for exactly the reason LCCD does.

Triples. The (T) increment is \(-0.985\) mHa on CCSD and \(-0.890\) mHa on QCISD (res.ccsd.e_t); note that QCISD(T) uses the original Pople-Head-Gordon-Raghavachari convention \(E[T] + 2E_{ST}\) on QCISD amplitudes, whereas CCSD(T) uses \(E[T] + E_{ST}\), so the two triples increments are defined differently and the totals should not be compared term by term. Both land within 0.05 mHa of each other here, which is typical for a well-behaved closed-shell molecule.

Brueckner orbitals. BCCD changes the reference: its \(E_\text{ref}\) (\(-75.983157\) Ha) is the energy of the Brueckner determinant, 0.8 mHa above RHF, because the orbitals have been rotated until the singles amplitudes vanish rather than to minimise the determinant’s energy. Its correlation energy (\(-135.2\) mHa) is correspondingly larger, and the total BCCD energy is within 0.12 mHa of CCSD; add the triples and BCCD(T) is within 0.01 mHa of CCSD(T). This is the expected behaviour when singles are small; Brueckner orbitals earn their cost when the RHF reference is poor (spin contamination, symmetry breaking, large \(T_1\)), where BCCD(T) is markedly more stable than CCSD(T).

Variation: CC3 and CCSDT in a minimal basis

CC3 and CCSDT live in an exact determinant-space representation whose size grows combinatorially with the number of triples, so vibe-qc labels them energy-only and benchmark-scale. In 6-31G the CC3 job on water runs for minutes; in STO-3G, with two virtual orbitals, the whole hierarchy takes seconds and shows the ordering exactly.

for method in ("ccsd", "ccsd(t)", "cc3", "ccsdt"):
    t0 = time.perf_counter()
    res = vq.run_job(h2o, basis="sto-3g", method=method, density_fit=False,
                     output="h2o_sto3g_" + method.replace("(", "").replace(")", ""), verbose=0)
    if hasattr(res, "ccsd"):
        e_corr, e_tot, extra = res.ccsd.e_total - res.ccsd.e_hf, res.ccsd.e_total, ""
    else:                                        # CC3Result / CCSDTResult
        e_corr, e_tot = res.e_correlation, res.energy
        extra = f"  ({res.n_doubles} doubles, {res.n_triples} triples, {res.n_frozen_core} frozen)"
    print(f"{method:8s} E_corr = {e_corr:.6f}  E_total = {e_tot:.6f} Ha  {time.perf_counter() - t0:5.1f} s{extra}")

ccsd     E_corr = -0.049360  E_total = -75.012383 Ha    2.4 s
ccsd(t)  E_corr = -0.049428  E_total = -75.012451 Ha    1.8 s
cc3      E_corr = -0.049428  E_total = -75.012451 Ha   16.5 s  (76 doubles, 96 triples, 1 frozen)
ccsdt    E_corr = -0.049454  E_total = -75.012477 Ha   17.3 s  (76 doubles, 96 triples, 1 frozen)

In this tiny space the perturbative (T) and the iterative CC3 triples agree to \(10^{-7}\) Ha, and full CCSDT adds another 0.026 mHa. The ordering \(|E_\text{CCSD}| < |E_\text{CCSD(T)}| \approx |E_\text{CC3}| < |E_\text{CCSDT}|\) is the usual one near equilibrium; CC3 separates from CCSD(T) when the triples become large (stretched bonds, multiple bonds, open shells), where the iterative treatment keeps its accuracy and the perturbative one degrades. CC3 saves nothing over CCSDT at this size because the dense representation regenerates its approximate triples at every iteration; the asymptotic scalings in the theory section are what matter for anything larger. Both return a SolverResult-like object (CC3Result, CCSDTResult) with e_reference, e_correlation, energy and the residual norms, and res.energy is the correlated total energy for these two, unlike the wrapped results above.

Variation: CISD and size extensivity

CISD is the one method on the page that is not size-extensive. Run it in the same frozen-core space (active_space=(12, 8): 12 orbitals, 8 electrons, which freezes the lowest orbital):

cisd = vq.run_job(h2o, basis="6-31g", method="cisd", active_space=(12, 8),
                  output="h2o_cisd", verbose=0)
rhf = vq.run_job(h2o, basis="6-31g", method="rhf", output="h2o_rhf", verbose=0)
print(f"{cisd.method}: E_corr = {cisd.energy - rhf.energy:.6f} Ha")

cisd(ndet=1425): E_corr = -0.129228 Ha

CISD recovers \(-129.2\) mHa, less than LCCD (\(-134.0\)) although it contains every singly and doubly excited determinant that CCSD does. The difference is the unlinked terms: in CISD the doubles equation carries a term \(-E_\text{corr}\,c_{ij}^{ab}\) that grows with the size of the system and damps the amplitudes, so the fraction of the correlation energy CISD recovers decreases as molecules get bigger and the energy of two non-interacting molecules is not twice that of one. Every coupled-cluster and coupled-pair variant above removes that term (CEPA methods approximately, QCISD and the CC methods exactly), which is why their energies are additively separable and why CISD has left the production toolbox. The determinant CISD here is a Python solver and takes about a minute; it is on the page for the comparison, not for routine use.

Theory

Coupled-cluster theory (Čížek, Jiří (1966). *On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods.* Journal of Chemical Physics, **45**(11), 4256–4266. doi:[10.1063/1.1727484](https://doi.org/10.1063/1.1727484)) writes the correlated wavefunction as an exponential of excitation operators acting on the reference determinant,

\[ |\Psi\rangle = e^{\hat T}|\Phi_0\rangle, \qquad \hat T = \hat T_1 + \hat T_2 + \hat T_3 + \cdots, \qquad \hat T_2 = \tfrac14\sum_{ijab} t_{ij}^{ab}\,a_a^\dagger a_b^\dagger a_j a_i , \]

with \(i,j\) occupied and \(a,b\) virtual spin orbitals. Projecting the similarity-transformed Schrodinger equation \(\bar H = e^{-\hat T}\hat H e^{\hat T}\) onto the reference and onto the excited determinants gives the energy and the amplitude equations,

\[ E = \langle\Phi_0|\bar H|\Phi_0\rangle, \qquad 0 = \langle\Phi_{ij}^{ab}|\bar H|\Phi_0\rangle, \quad 0 = \langle\Phi_i^{a}|\bar H|\Phi_0\rangle, \;\ldots \]

Because \(\bar H\) terminates at fourth order in \(\hat T\) (the Baker-Campbell-Hausdorff expansion of a two-body Hamiltonian), the CCSD doubles residual is a finite polynomial: linear terms (\(\langle ab\|ij\rangle\), the Fock-operator terms, the particle-particle and hole-hole ladders and the ring terms in \(t_2\)), quadratic terms in \(t_2\) (the “disconnected quadruples” \(\tfrac12 T_2^2\) contribution and the exclusion-principle-violating terms), and the terms in \(t_1\) up to \(t_1^4\). The exponential guarantees that every term is connected, and connectedness is what makes the energy size-extensive: the energy of two non-interacting fragments is the sum of their energies at every truncation level of \(\hat T\).

Each member of the family is a rule for which of those terms to keep:

Method

Approximation to the CCSD (or CCSDT) equations

CCD

\(\hat T = \hat T_2\); the singles equation is dropped and \(t_1 = 0\) (Čížek, Jiří (1966). *On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods.* Journal of Chemical Physics, **45**(11), 4256–4266. doi:[10.1063/1.1727484](https://doi.org/10.1063/1.1727484)).

LCCD

CCD with the doubles residual truncated to the terms linear in \(t_2\); the equations become a linear system, equivalent to summing all double-excitation diagrams of many-body perturbation theory to infinite order.

CEPA(0) = LCCSD

Singles and doubles, all nonlinear terms dropped.

CEPA(1), (2), (3)

The linear residual plus a pair-specific shift \(-\Delta_{ij}\,t_{ij}^{ab}\) that models the missing exclusion-principle-violating quadratic terms; \(\Delta_{ij}\) is built from the pair correlation energies \(e_{kl}\) (CEPA(1): \(\tfrac12\sum_k(e_{ik}+e_{jk})\); CEPA(2): \(e_{ij}\); CEPA(3): \(-e_{ij}+\sum_k(e_{ik}+e_{jk})\)), following Meyer (Meyer, Wilfried (1973). *PNO-CI Studies of electron correlation effects. I. Configuration expansion by means of nonorthogonal orbitals, and application to the ground state and ionized states of methane.* Journal of Chemical Physics, **58**(3), 1017–1035. doi:[10.1063/1.1679283](https://doi.org/10.1063/1.1679283)) in the closed-shell convention tabulated by Wennmohs and Neese (Wennmohs, Frank and Neese, Frank (2008). *A comparative study of single reference correlation methods of the coupled-pair type.* Chemical Physics, **343**(2–3), 217–230. doi:[10.1016/j.chemphys.2007.07.001](https://doi.org/10.1016/j.chemphys.2007.07.001)).

QCISD

CISD made size-extensive by adding the quadratic \(\tfrac12\hat T_2^2\) term to the doubles equation and \(\hat T_1\hat T_2\) to the singles equation, with the CI-like energy expression (no \(\hat T_1^2\) contribution); the other nonlinear terms of CCSD are absent (Pople, John A., Head-Gordon, Martin, and Raghavachari, Krishnan (1987). *Quadratic configuration interaction. A general technique for determining electron correlation energies.* Journal of Chemical Physics, **87**(10), 5968–5975. doi:[10.1063/1.453520](https://doi.org/10.1063/1.453520)).

CC2

The singles equation is the full CCSD one; the doubles equation keeps only the terms of first order in the fluctuation potential, evaluated with the \(\hat T_1\)-dressed Hamiltonian, plus \([\hat F, \hat T_2]\) (Christiansen, Ove, Koch, Henrik, and Jørgensen, Poul (1995). *The second-order approximate coupled cluster singles and doubles model CC2.* Chemical Physics Letters, **243**(5–6), 409–418. doi:[10.1016/0009-2614(95)00841-Q](https://doi.org/10.1016/0009-2614(95)00841-Q)). Its energy is correct to second order, like MP2, but the singles are treated to all orders.

CCSD

All singles and doubles terms (Purvis, George D. III and Bartlett, Rodney J. (1982). *A full coupled-cluster singles and doubles model: The inclusion of disconnected triples.* Journal of Chemical Physics, **76**(4), 1910–1918. doi:[10.1063/1.443164](https://doi.org/10.1063/1.443164)).

CCSD(T)

CCSD plus the non-iterative triples correction \(E_{[T]} + E_{ST}\): the fourth-order triples energy from converged \(t_2\) and the fifth-order singles-triples coupling (Raghavachari, Krishnan, Trucks, Gary W., et al. (1989). *A fifth-order perturbation comparison of electron correlation theories.* Chemical Physics Letters, **157**(6), 479–483. doi:[10.1016/S0009-2614(89)87395-6](https://doi.org/10.1016/S0009-2614(89)87395-6)). QCISD(T) uses \(E_{[T]} + 2E_{ST}\) on QCISD amplitudes.

CC3

Triples included iteratively, with the triples equation approximated to leading order (the \(\hat T_1\)-dressed Hamiltonian acting on \(\hat T_2\)) and no triples-triples coupling beyond the Fock term (Koch, Henrik, Christiansen, Ove, et al. (1997). *The CC3 model: An iterative coupled cluster approach including connected triples.* Journal of Chemical Physics, **106**(5), 1808–1818. doi:[10.1063/1.473322](https://doi.org/10.1063/1.473322)).

CCSDT

The full singles, doubles and triples equations (Noga, Jozef and Bartlett, Rodney J. (1987). *The full CCSDT model for molecular electronic structure.* Journal of Chemical Physics, **86**(12), 7041–7050. doi:[10.1063/1.452353](https://doi.org/10.1063/1.452353)).

BCCD, BCCD(T)

CCD in the basis of Brueckner orbitals, rotated iteratively until the CCSD singles amplitudes vanish, so that the reference determinant has maximum overlap with the correlated wavefunction; (T) is then evaluated on that reference (Handy, Nicholas C., Pople, John A., et al. (1989). *Size-consistent Brueckner theory limited to double substitutions.* Chemical Physics Letters, **164**(2–3), 185–192. doi:[10.1016/0009-2614(89)85013-4](https://doi.org/10.1016/0009-2614(89)85013-4)).

The cost follows the highest excitation level and the iteration structure. With \(o\) occupied and \(v\) virtual orbitals, MP2 is a single \(O(o^2 v^3)\) transformation; CC2 iterates an \(O(o^2 v^3)\) step; CCD, LCCD, CEPA(\(n\)), QCISD and CCSD iterate the \(O(o^2 v^4)\) particle-particle ladder; the (T) correction is one non-iterative \(O(o^3 v^4)\) pass; CC3 iterates an \(O(o^3 v^4)\) step, so it costs a CCSD(T) at every iteration; and CCSDT iterates \(O(o^3 v^5)\) with \(o^3 v^3\) amplitudes to store. That is why (T) is the production triples method and CC3 and CCSDT are benchmarks: on water in 6-31G the ladder up to BCCD(T) runs in seconds, CC3 in minutes, and a triple-zeta CCSDT on anything larger than a diatomic is out of reach without a dedicated tensor engine.

Where the family helps is the accuracy hierarchy it gives you at a known price: near-equilibrium closed-shell molecules with a good RHF reference, where CCSD(T) is within a kilocalorie of the exact answer in a large basis and the cheaper rungs tell you how far from convergence you are. Where it fails is where the single reference fails: stretched or multiple bonds, biradicals, transition-metal multiplets, where the \(T_1\) diagnostic grows, (T) can overshoot badly, and the iterative triples of CC3 or, better, a multireference treatment (CASSCF and CASPT2) takes over. The coupled-pair and QCI variants are not more accurate than CCSD; they are on the page because they are cheaper rungs with a clear diagrammatic meaning, because they remain the workhorses of local-correlation methods (LPNO-CEPA), and because the sign of their deviation from CCSD tells you which terms of the equations are doing the work.

Resources

References

  • Coupled-cluster theory. J. Cizek, “On the correlation problem in atomic and molecular systems,” J. Chem. Phys. 45, 4256 (1966), Čížek, Jiří (1966). *On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods.* Journal of Chemical Physics, **45**(11), 4256–4266. doi:[10.1063/1.1727484](https://doi.org/10.1063/1.1727484). The exponential ansatz and CCD.

  • CCSD. G. D. Purvis and R. J. Bartlett, “A full coupled-cluster singles and doubles model,” J. Chem. Phys. 76, 1910 (1982), Purvis, George D. III and Bartlett, Rodney J. (1982). *A full coupled-cluster singles and doubles model: The inclusion of disconnected triples.* Journal of Chemical Physics, **76**(4), 1910–1918. doi:[10.1063/1.443164](https://doi.org/10.1063/1.443164).

  • CCSD(T). K. Raghavachari, G. W. Trucks, J. A. Pople and M. Head-Gordon, “A fifth-order perturbation comparison of electron correlation theories,” Chem. Phys. Lett. 157, 479 (1989), Raghavachari, Krishnan, Trucks, Gary W., et al. (1989). *A fifth-order perturbation comparison of electron correlation theories.* Chemical Physics Letters, **157**(6), 479–483. doi:[10.1016/S0009-2614(89)87395-6](https://doi.org/10.1016/S0009-2614(89)87395-6).

  • CCSDT. J. Noga and R. J. Bartlett, “The full CCSDT model for molecular electronic structure,” J. Chem. Phys. 86, 7041 (1987), Noga, Jozef and Bartlett, Rodney J. (1987). *The full CCSDT model for molecular electronic structure.* Journal of Chemical Physics, **86**(12), 7041–7050. doi:[10.1063/1.452353](https://doi.org/10.1063/1.452353).

  • CC2. O. Christiansen, H. Koch and P. Jorgensen, “The second-order approximate coupled cluster singles and doubles model CC2,” Chem. Phys. Lett. 243, 409 (1995), Christiansen, Ove, Koch, Henrik, and Jørgensen, Poul (1995). *The second-order approximate coupled cluster singles and doubles model CC2.* Chemical Physics Letters, **243**(5–6), 409–418. doi:[10.1016/0009-2614(95)00841-Q](https://doi.org/10.1016/0009-2614(95)00841-Q).

  • CC3. H. Koch, O. Christiansen, P. Jorgensen, A. M. Sanchez de Meras and T. Helgaker, “The CC3 model: An iterative coupled cluster approach including connected triples,” J. Chem. Phys. 106, 1808 (1997), Koch, Henrik, Christiansen, Ove, et al. (1997). *The CC3 model: An iterative coupled cluster approach including connected triples.* Journal of Chemical Physics, **106**(5), 1808–1818. doi:[10.1063/1.473322](https://doi.org/10.1063/1.473322).

  • QCISD and QCISD(T). J. A. Pople, M. Head-Gordon and K. Raghavachari, “Quadratic configuration interaction. A general technique for determining electron correlation energies,” J. Chem. Phys. 87, 5968 (1987), Pople, John A., Head-Gordon, Martin, and Raghavachari, Krishnan (1987). *Quadratic configuration interaction. A general technique for determining electron correlation energies.* Journal of Chemical Physics, **87**(10), 5968–5975. doi:[10.1063/1.453520](https://doi.org/10.1063/1.453520).

  • CEPA. W. Meyer, “PNO-CI studies of electron correlation effects. I.,” J. Chem. Phys. 58, 1017 (1973), Meyer, Wilfried (1973). *PNO-CI Studies of electron correlation effects. I. Configuration expansion by means of nonorthogonal orbitals, and application to the ground state and ionized states of methane.* Journal of Chemical Physics, **58**(3), 1017–1035. doi:[10.1063/1.1679283](https://doi.org/10.1063/1.1679283); F. Wennmohs and F. Neese, “A comparative study of single reference correlation methods of the coupled-pair type,” Chem. Phys. 343, 217 (2008), Wennmohs, Frank and Neese, Frank (2008). *A comparative study of single reference correlation methods of the coupled-pair type.* Chemical Physics, **343**(2–3), 217–230. doi:[10.1016/j.chemphys.2007.07.001](https://doi.org/10.1016/j.chemphys.2007.07.001), the shift conventions and the benchmark that places CEPA(1) beside CCSD.

  • Brueckner CCD. N. C. Handy, J. A. Pople, M. Head-Gordon, K. Raghavachari and G. W. Trucks, “Size-consistent Brueckner theory limited to double substitutions,” Chem. Phys. Lett. 164, 185 (1989), Handy, Nicholas C., Pople, John A., et al. (1989). *Size-consistent Brueckner theory limited to double substitutions.* Chemical Physics Letters, **164**(2–3), 185–192. doi:[10.1016/0009-2614(89)85013-4](https://doi.org/10.1016/0009-2614(89)85013-4).

  • CISD and size consistency. J. A. Pople, R. Seeger and R. Krishnan, “Variational configuration interaction methods and comparison with perturbation theory,” Int. J. Quantum Chem. 12, 149 (1977), Pople, J. A., Seeger, R., and Krishnan, R. (1977). *Variational configuration interaction methods and comparison with perturbation theory.* International Journal of Quantum Chemistry, **12**(S11), 149–163. doi:[10.1002/qua.560120820](https://doi.org/10.1002/qua.560120820).

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