6.12 Coupled-Cluster Methods 6.12.4 Second-Order Approximate Coupled Cluster Singles and Doubles (CC2)6.12.6 Size-Consistent Brillouin-Wigner Second-Order Approximate Coupled Cluster Singles and Doubles (BWs-CC2)

6.12.5 Stochastic CC2 and CCSD

(July 4, 2026)

An implementation of stochastic RI-CC2 (sRI-CC2) is available in libgmbpt. Currently, codes support RI-CC2 and sRI-CC2 property calculations, including ground- and excited-state (singlet and triplet) energies, analytical gradients for ground and excited states, oscillator strengths, and derivative couplings. The main advantage of sRI-CC2 is its reduced scaling ( ${\cal{O}}({N^{3}})$ , to be compared with ${\cal{O}}({N^{5}})$ of RI-CC2), which facilitates applications to much larger molecules. These codes are under development. Users are advised to refer to the sample jobs named libgmbpt_sricc $\ast$ and libgmbpt_thc_sricc $\ast$ for guidance.

Note: sRI-CC2 currently does not support frozen core: N_FROZEN_CORE must be set to zero.

Note: sRI-CC2 currently does not support geometry optimization and can only be used with JOBTYPE=SP.

RI-CC2 and sRI-CC2 are deployed by setting METHOD = CC2 and specifying AIX_BASIS appropriately. SRI-CC2 is only available in libgmbpt, so one should set CCMAN2 = -1 in the $rem section, and add do_sri = 1 in the $development section. Then, in the $development section, add the keyword sri to control the use of sRI: 0 for deterministic RI calculations, 1 for sRI calculations with an ${\cal{O}}({N^{3}})$ scaling, and 2 for partial sRI calculations. In some cases, such as gradient calculations, the sRI approach introduces significant noise. To mitigate such problems, one can use the partial sRI scheme with a suboptimal ${\cal{O}}({N^{4}})$ scaling but improved accuracy.

The keyword sri_ntheta indicates the number of sRI orbitals. It acts as a prefactor in the scaling and larger values give better accuracy.

Example 6.31 sRI-CC2 excitation energy of the water molecule.

$molecule
    0 1
    O 0.0 0.0 0.0
    H 0.0  0.75410300 -0.56492300
    H 0.0 -0.75410300 -0.56492300
$end

$rem
    CCMAN2        -1                ! use code in libgmbpt
    method        CC2
    BASIS         cc-pvdz
    AUX_BASIS     RIMP2-cc-pVDZ
    EE_SINGLETS   [1]
    N_FROZEN_CORE 0                 ! frozen core (FC) electrons
$end

! below for sRI development
$development
    do_sri        1
    sri           1                 ! if use sRI, 0-RI, 1-complete sRI, 2-partial sRI
    nsinglets     1                 ! number of singlet states
!    ntriplets    0                 ! number of triplet states
    sri_ntheta    5000              ! number of stochastic orbitals
    sri_nseed     0                 ! index to generate stochastic seed : 0~23
$end

Further improvements of sRI-CC2 are afforded by a tensor hypercontraction implementation (THC-sRI-CC2) with an ${\cal{O}}({N^{3}})$ scaling and reduced stochastic noise for the same set of properties. THC-sRI-CC2 is invoked by adding thc = $1$ in the $rem section and setting sri = $2$ in the $development section. The rest of the keywords are the same as in sRI-CC2.

Example 6.32 THC-sRI-CC2 ground-state analytical gradient of the water molecule.

$molecule
    0 1
    O 0.0 0.0 0.0
    H 0.0  0.75410300 -0.56492300
    H 0.0 -0.75410300 -0.56492300
$end

$rem
    CCMAN2        -1                ! use code in libgmbpt
    method        CC2
    BASIS         cc-pvdz
    AUX_BASIS     RIMP2-cc-pVDZ
    N_FROZEN_CORE 0                 ! frozen core (FC) electrons, n
    CC_EOM_PROP   true              ! properties calculation
    thc           1                 ! if use THC
$end

! below for sRI development
$development
    do_sri 1
    sri 2                           ! 2-THC-sRI
    sri_ntheta 5000                 ! number of stochastic orbitals for energy
    sri_grad_ntheta 5000            ! number of stochastic orbitals for gradient solution
    sri_nseed 0                     ! index to generate stochastic seed : 0~23
    gradient_gs 1                   ! ground_state gradient calculation
$end

Finally, THC-sRI-CCSD is available for calculations of ground- and excited-state energies with an ${\cal{O}}({N^{3}})$ scaling. One can invoke THC-sRI-CCSD by adding sriccsd = 1 in the $development section.

Example 6.33 THC-sRI-CCSD ground-state energy of the water molecule.

$molecule
    0 1
    O 0.0 0.0 0.0
    H 0.0  0.75410300 -0.56492300
    H 0.0 -0.75410300 -0.56492300
$end

$rem
    CCMAN2         -1                ! use code in libgmbpt
    method         CC2
    BASIS          cc-pvdz
    AUX_BASIS      RIMP2-cc-pVDZ
    N_FROZEN_CORE  0                 ! frozen core (FC) electrons, n
    CC_CONVERGENCE 6
    thc            1                 ! if use THC
$end

! below for sRI development
$development
    do_sri         1
    sri            2                 ! if use sri, 0-THC, 2-THC-sRI
    sri_ntheta     5000              ! number of stochastic orbitals
    sri_nseed      0                 ! index to generate stochastic seed : 0~23
    sriccsd        1                 ! CCSD
$end

For further technical details, see Refs. 1554 Zhao C., Lee J., Dou W.
J. Phys. Chem. A
(2024), 128, pp. 9302. Link , 1556 Zhao C. et al.
J. Chem. Theory Comput.
(2024), 20, pp. 5188. Link , 1557 Zhao C. et al.
J. Chem. Phys.
(2025), 163, pp. 024102. Link , 1555 Zhao C., Li C., Dou W.
J. Chem. Phys.
(2025), 163, pp. 234101. Link .

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6.12.5 Stochastic CC2 and CCSD