Efficient Explicitly Correlated Coupled-Cluster Approximations

Abstract

Explicitly correlated MP2-F12 and CCSD(T)-F12 methods are reviewed. We focus on the CCSD(T)-F12x (x = a,b) approximations, which are only slightly more expensive than their non-F12 counterparts. Furthermore, local approximations in the LMP2-F12 and LCCSD-F12 methods are described, which make it possible to treat larger molecules than with standard coupled-cluster methods. We demonstrate the practicability of F12 methods by large benchmark calculations for various properties, including reaction energies, vibrational frequencies, and intermolecular interactions. In these calculations, the newly developed VnZ-F12 orbital and OPTRI auxiliary basis sets by Peterson et al. are compared to other previously used basis sets. The accuracy and efficiency of local approximations is demonstrated for reactions of large molecules.

Appendix: The CCSD Equations in Expanded Form

The CCSD residuals take the explicit form

$${R}^{ij}_{ab} = K^{ij}_{ab} + [{\bf K}({\bf D}^{ij})]_{ab} + \sum_{k,l} \alpha_{ij,kl} {D}^{kl}_{ab} + {G}^{ij}_{ab}+{G}^{ji}_{ba},$$

(21-143)

$$\begin{aligned} {\bf G}^{ij} &= {\bf T}^{ij} {\bf X} - \sum_{k} {\bf T}^{ik} \beta_{kj} - \sum_k [{\bf K}({\bf D}^{ij})^{(k)} + {\bf k}^{ijk}] {{\bf t}^k}^\dagger\\&+ \sum_k \left[ {\tilde {\bf T}}^{ik}{\bf Y}^{kj} - \frac{1}{2} {\bf T}^{ki} {\bf Z}^{kj} - ({\bf T}^{ki} {\bf Z}^{kj} )^\dagger \right], \end{aligned}$$

(21-144)

$$\begin{aligned} {\bf r}^i &= {\bf f}^i + [{\bf R}^\dagger- \sum_k (2{\bf J}^{kk}-{\bf K}^{kk})]{\bf t}^i - \sum_{k,l} {\bf T}^{lk} {\bf l}^{kli} \\ &+ \sum_k\left[ {\bf K}({\tilde {\bf D}}^{ik})^{(k)} + {\tilde {\bf T}}^{ik}{\bar {\bf f}}^k- {\bf t}^k\beta_{ki} \right], \end{aligned}$$

(21-145)

with the intermediates:

$$\alpha_{ij,kl}= K^{ij}_{kl} + \mathrm{tr} ({\bf C}^{ij} {\bf K}^{lk}) + ({{\bf t}^i}^\dagger {\bf k}^{klj} + {{\bf t}\,^j}^\dagger {\bf k}^{lki}),$$

(21-146)

$$\beta_{ki} = f_{ki}+{{\bf f}\,^k}^\dagger {\bf t}^i+ \sum_l \left[\mathrm{tr}({\bf L}^{kl} {\bf T}^{li})+{{\bf t}^l}^\dagger ({\bf l}^{lki}+{\bf L}^{lk} {\bf t}^{i})\right],$$

(21-147)

$${\bar {\bf f}}\,^k = {\bf f}\,^k+ \sum_l {\bf L}^{kl} {\bf t}^l,$$

(21-148)

$${\bf R} = {\bf f} - \sum_{k,l} {\bf L}^{kl} {\bf T}^{lk},$$

(21-149)

$${\bf X} =\, {\bf R} - \sum_k {\bar {\bf f}}\,^{k} {{\bf t}^k}^\dagger + \sum_k [ 2 {\bf J}({\bf E}^{kk}) - {\bf K}({\bf E}^{kk})],$$

(21-150)

$$\begin{aligned} {\bf Y}^{kj} &= {\bf K}^{kj} + {\bf K}({\bf E}^{kj}) - \frac{1}{2}\left[ {\bf J}^{kj} + {\bf J}({\bf E}^{kj}) \right]\\ &+ \frac{1}{4} \sum_l {\bf L}^{kl} {\tilde {\bf T}}^{lj} - \frac{1}{2} \sum_l ({\bf l}^{klj} + {\bf L}^{kl}{\bf t}\,^j ){{\bf t}^l}^\dagger, \end{aligned}$$

(21-151)

$${\bf Z}^{kj} = {\bf J}^{kj} + {\bf J}({\bf E}^{kj}) - \sum_l \left[ \frac{1}{2} {\bf K}^{lk} {\bf T}^{jl} + ({\bf k}^{lkj} + {\bf K}^{lk}{\bf t}\,^j){{\bf t}^l}^\dagger \right].$$

(21-152)

Glossary

$A_{kl,mn} = \langle kl|F_{12} \hat {g}_{12} \hat Q_{12}F_{12}|mn\rangle$

Part of MP2-F12 B-matrix

$B_{kl,mn} = \langle kl|F_{12}\hat{Q}_{12}\hat{g}_{12}\hat{Q}_{12}F_{12}|mn\rangle$

MP2-F12 B-matrix

$C_{\mu i}$

MO coefficients

$C^{kl}_{ab} = \langle kl|F_{12}\hat{Q}_{12} \hat{f}_{12}|ab\rangle$

MP2-F12 coupling terms

$D^{ij}_{rs} = \delta_{rc}\delta_{sd} \left(T^{ij}_{cd}+ t^i_c t^j_d\right) + \delta_{ri} t^j_s + \delta_{sj} t^{i}_r,$

Composite amplitudes

$E^{ij}_{rs} = \delta_{ri} t^j_s$

Used in contractions of integrals with singles

$F^{ij}_{\alpha\beta} = \langle\alpha\beta| F_{12} | ij\rangle$

Integrals overF_12

$\bar F^{ij}_{\alpha\beta} = \langle\alpha\beta| F_{12} | kl\rangle T^{ij}_{kl}$

Contracted integrals overF_12

$G^{ij}_{\alpha\beta}$

Intermediates in CCSD-F12 residuals

$H_{mn,op} = \langle mn |F_{12} \hat{Q}_{12} r_{12}^{-1} \hat{Q} _{12}F_{12}|op \rangle$

Matrix elements in CCSD-F12 energy expression

$J^{kl}_{\alpha\beta} = \langle \alpha k | r_{12}^{-1} | \beta l\rangle$

Two-external Coulomb integrals

$K^{kl}_{\alpha\beta} = \langle \alpha\beta | r_{12}^{-1} | k l\rangle$

Two-external exchange integrals

$\bar K^{rs}_{\alpha\beta} = \langle \bar \alpha \bar \beta | r_{12}^{-1} |\tilde r \tilde s\rangle$

Dressed integrals

$K^F_{ij,kl}= \langle ij|r_{12}^{-1}F_{12}|kl\rangle$

Integrals over$r_{12}^{-1} F_{12}$

$R_{\alpha\beta}$

Intermediates in CCSD-F12 residual

R ^ij _kl

MP2-F12 residual for explicitly correlated amplitudes

R ^ij _ab

MP2-F12 or CCSD-F12 doubles residual

$S_{mn,op} = \langle mn|F_{12}\hat{Q}_{12}F_{12}|op\rangle$

Overlap of explicitly correlated configurations

$\bar S_{ij,kl} = \tilde T^{ij}_{mn} S_{mn,op} T^{kl}_{op}$

Contracted overlap matrix

$\tilde S^{(ij)}_{rs}= \langle \tilde r | \tilde s\rangle, \;\;r,s \in [ij]$

Overlap matrix of PAOs in domain [ij]

T ^ij _kl

Amplitudes of explicitly correlated configurations

T ^ij _ab

Doubles amplitudes, virtual space

${\cal T}^{ij}_{\alpha\beta}= \langle \alpha\beta | \hat Q_{12} F_{12} | kl \rangle T^{ij}_{kl}$

Amplitudes of explicitly correlated terms in the complete virtual space

${\cal U}^{ij}_{\alpha\beta} = \delta_{\alpha c} \delta_{\beta d} T^{ij}_{cd} + {\cal T}^{ij}_{\alpha \beta}$

Composite doubles amplitudes in complete space

$U^{kl}_{\alpha\beta} = \langle kl|[F_{12},\hat t_1 + \hat t_2]|\alpha\beta\rangle$

Commutator integrals used in MP2-F12

$U^F_{kl,mn} = \langle kl|[F_{12},\hat t_1 + \hat t_2]F_{12}|mn\rangle$

Commutator integrals used in MP2-F12

$V^{ij}_{kl} = \langle kl|F_{12}\hat{Q}_{12}r_{12}^{-1}|ij\rangle$

Integrals used in MP2-F12

$\bar V^{ij}_{\alpha\beta} = \langle\alpha\beta|r_{12}^{-1} \hat{Q} _{12}F _{12}|kl \rangle T^{ij}_{kl}$

Contracted integrals used in CCSD-F12

$\bar W^{ij}_{\alpha\beta} = \langle\alpha\beta| r_{12}^{-1}F_{12} | kl\rangle T^{ij}_{kl}$

Contracted integrals used in CCSD-F12

$W^{(ij)}_{\tilde \mu \tilde a }, \;\;\tilde \mu,\tilde a \in [ij]$

Transformation from PAO basis to orthonormal basis in domain [ij]

$X_{\alpha\beta}$,$Y^{kj}_{\alpha\beta}$,$Z^{kj}_{\alpha\beta}$

Intermediates in CCSD-F12 residuals

$Y_{kl,mn} = \langle kl | F_{12}\hat n_{12} \hat{Q}_{12} F_{12} | mn\rangle$

Intermediates in MP2-F12 residuals

$f_{\alpha\beta} = h_{\alpha\beta} + 2 J^{kk}_{\alpha\beta} - K^{kk}_{\alpha\beta}$

Closed-shell Fock matrix

$f^i_{\alpha} = f_{\alpha i}$

One-external Fock matrix elements

$\bar f_{\alpha\beta}$

Dressed Fock matrix

$\bar f^i_{\alpha} = \bar f_{i\alpha}$

One-external dressed Fock matrix elements

$\tilde f_{rs}^{(ij)} = f_{\tilde r \tilde s},\;\;\tilde r,\tilde s \in [ij]$

Fock matrix in PAO basis in domain [ij]

$k^{kli}_\alpha = K^{kl}_{\alpha i}$

One-external integrals

t ⁱ _a

CCSD singles amplitudes

r ⁱ _a

CCSD singles residuals