Coupled-cluster theory in quantum chemistry

Today, coupled-cluster theory offers the most accurate results among the practical ab initio electronic-structure theories applicable to moderate-sized molecules. Though it was originally proposed for problems in physics, it has seen its greatest development in chemistry, enabling an extensive range of applications to molecular structure, excited states, properties, and all kinds of spectroscopy. In this review, the essential aspects of the theory are explained and illustrated with informative numerical results.

Figure 1

The nature of quantum chemical problems.

Figure 2

Graphical examples of the selected single $\left({\Phi }_i^a\right)$, double $\left({\Phi }_{ij}^{ab}\right)$, and triple $\left({\Phi }_{ijk}^{abc}\right)$ excitations due to the $T_1$ $\left({\mathrm{C}}_1\right)$, $T_2$ $\left({\mathrm{C}}_2\right)$, and $T_3$ $\left({\mathrm{C}}_3\right)$ operators, respectively. Electrons from all possible occupied orbitals can be excited to all possible unoccupied orbitals, so $T_3{\Phi }_0={\sum }_{{}_{a<b<c}^{i<j<k}}{t}_{ijk}^{abc}{\Phi }_{ijk}^{abc}$.

Figure 3

(Color online) Performance of theories for the correlation energy in small molecules. Graphed is the percentage of the full correlation energy achieved by the CI, CC, and MBPT theories, as a function of the level of approximation. To facilitate comparisons, the ordinate gives the size-scaling parameter of the approximation $\alpha ={\alpha }_n+{\alpha }_N+{\alpha }_{it}$ in the computational cost function $n^{{\alpha }_n}{N}^{{\alpha }_N}{N}_{it}^{{\alpha }_{it}}$. Shown are MBPT (solid circles), approximations (2, 3, 4, 5, 6); CI (solid squares), approximations SD-SDTQ; and CC (stars), approximations SD-SDTQ. The correlation energy is defined with respect to the Hartree-Fock energy for the given basis set, and the full correlation energies are obtained from the FCI calculations quoted in Table .

Figure 4

Diagrammatic form of the $f_N$, $W_N$, and $T_n$ operators. The labels at the bottom of the $f_N$ and $W_N$ operators refer to the changes in the excitation level caused by that form of the operator.

Figure 5

Skeleton diagrams for the second- and third-order MBPT energies.

Figure 6

Generation of the $W{T}_2$ and $W{T}_2^2∕2$ contributions to the $T_2$ amplitude.

Figure 7

The choices made for the spin-orbital form: (a) RHF (closed-shell molecule with doubly occupied spatial orbitals), (b) UHF (closed-shell spin polarized situation), (c) UHF open-shell triplet, and (d) ROHF (triplet maximum double occupancy).

Figure 8

Diagrammatic representation of the CCD method in the closed-shell spatial orbital form together with the corresponding algebraic expression. Summation over repeated upper and lower indices assumed. $P(ia∕jb)$ implies a sum of two components differing by permutaion of $ia$ and $jb$.

Figure 9

Diagrammatic representation of the CCD method in the spin-expanded open-shell form together with corresponding algebraic expressions. Summation over repeated upper and lower indices assumed. The antisymmetrizer is defined as $P(pq∕rs)=1+\left(qp\right)\left(sr\right)-\left(qp\right)\left(rs\right)-\left(pq\right)\left(sr\right)$. The capital letters $A$, $B$, $I$, $J$, etc. refer to $\alpha$ spin-orbitals, and lower case letters $a$, $b$, $i$, $j$, etc. denote $\beta$ spin-orbitals.

Figure 10

Diagrammatic representation of the CCD method in the generic spin-orbital form together with the corresponding algebraic expression. Summation over repeated upper and lower indices is assumed. The antisymmetric permutation operator $P(pq∕rs)$ is defined as $P(pq∕rs)=P\left(pq\right)\left(rs\right)=1+\left(qp\right)\left(sr\right)-\left(qp\right)\left(rs\right)-\left(pq\right)\left(sr\right)$.

Figure 11

Diagrams representing the single excitation and single contributions to the double-excitation CCSD equations. Diagrams occurring in the $T_2$ equation plus those present in the CCD model (Fig. 10) form the full $T_2$ equation in the CCSD method.

Figure 12

Diagrammatic CCSDT equations with total factorization of nonlinear terms: (a), (b), and (c) represent the $T_1$, $T_2$, and $T_3$ equations, respectively.

Figure 13

Diagrams representing connected quadruple contributions to the double-, triple-, and quadruple-excitation equations in the CCSDTQ method.

Figure 14

Diagrammatic form of intermediates introduced in CCSDT and CCSDTQ models with total factorization of nonlinear terms. The ${}^{*}$ indicates that the intermediate is preceded by a $1∕2$ to avoid overcounting.

Figure 15

Diagrammatic CCSD equations with total factorization of nonlinear terms.

Figure 16

Additional diagrams to the CCSD model that define the CCSDT-1 method.

Figure 17

Additional diagrams to the CCSD model that define the CCSDT-3 method.

Figure 18

Normal distribution functions of the deviations from experiment of the calculated bond distances $\left(pm\right)$ for a set of $31\text{molecules}$ in the (a) cc-pVTZ basis set and (b) cc-pVQZ basis set (68).

Figure 19

Normal distribution functions of the deviations from experiment of the calculated bond distances $\left(pm\right)$ for a set of $31\text{molecules}$ for the CCSD(T) method in the cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pCVQZ basis sets (68).

Figure 20

Normal distribution of errors (kcal/mol) for calculated equilibrium atomization energies compared to experiment for a set of $16\text{molecules}$ containing first-row atoms in the cc-pCVQZ basis set (11).

Figure 21

Normal distribution of errors (kcal/mol) for calculated reaction enthalpies compared to experiment for a set of 13 reactions containing first-row atoms in the cc-pCVQZ basis set (11).

Figure 22

(Color online) Normal distribution of dipole moment errors $\left(D\right)$ related to the CCSD(T)/aug-cc-pVQZ reference numbers for a set of 11 polar closed-shell molecules for the CCSD(T) method in the cc-pVXZ $(X=D,T,Q)$ and cc-pCVQZ basis sets (10).

Figure 23

(Color online) Vibrational frequencies of ozone (DZP basis set). Experimental values are 1135, 1089, and $716{\mathrm{cm}}^{-1}$ for symmetric stretch, asymmetric stretch, and bending modes, respectively (280; 17).

Figure 24

(Color online) Vibrational frequencies of ozone (cc-pVTZ basis set). See Fig. 23 for the experimental values.

Figure 25

(Color online) Potential-energy curves for the ${\mathrm{F}}_2$ molecule obtained with various CC methods in a cc-pVDZ basis set.

Figure 26

The SCF potential curves for the ${\mathrm{F}}_2$ molecule.

Figure 27

(Color online) Potential-energy curves for the ${\mathrm{N}}_2$ molecule. (a) CC results compared to full CI in a cc-pVDZ basis set (frozen core). (b) Errors compared to CI for UHF-CC methods as a function of $R$.

Figure 28

(Color online) Potential-energy curves for the HF molecule in the cc-pVDZ basis set. The horizontal axis represents the internuclear distance normalized by the equilibrium bond length $(R_e=1.733\mathrm{a.u.})$ (133).

Figure 29

Potential-energy curves for the ${\mathrm{N}}_2$ molecule in the cc-pVDZ basis set. The horizontal axis represents the internuclear distance normalized by the equilibrium bond length $(R_e=2.074\mathrm{a.u.})$ (133).

Figure 30

(Color online) Potential-energy curves for the ${\mathrm{N}}_2$ molecule in the cc-pVDZ basis set obtained with the FCI and $\mathrm{R}\text{−}\mathrm{CCSD}\left({\mathrm{TQ}}_f\right)$ methods.

Figure 31

(Color online) Potential-energy curves for the ${\mathrm{N}}_2$ molecule in a STO-3G basis set obtained with the FCI, ECCSD, ECCSD(TQ), CCSD, and CR-CCSD(TQ) methods.

Figure 32

Skeleton diagrams for the ${\Lambda }_1$ and ${\Lambda }_2$ equations for CCSD.

Figure 33

Diagrams representing the one-particle reduced density matrix for CCSD: (a) particle-particle elements, (b) hole-hole elements, (c) hole-particle elements, and (d) particle-hole elements.

Figure 34

Diagrammatic form of the EE-EOM-CCSD equations.

Figure 35

Diagrammatic form of the three-body elements of $\overline{H}$ used in the EE-EOM-CCSD equations.

Figure 36

The diagrammatic representation of the one-, two-, and three-body elements of $\overline{H}$.

Figure 37

Classification of the one-particle states.

Figure 38

(Color online) $\mathrm{C}{\mathrm{H}}^{+}$ potential-energy curves obtained with the FCI and 5R-SU CCSD methods.

Figure 39

(Color online) Vibrational frequencies of ozone (multireference methods, DZP basis set). See Fig. 23 for the experimental values.

Figure 40

A graphical representation and algebraic expressions for the $S^{(0,0)}$, $S^{(1,0)}$, $S^{(0,1)}$, and $S^{(1,1)}$ operators at the CCSD level. In the definition of the $S_2^{(1,1)}$ operator, the case in which both creation lines are active is excluded (this is denoted by ${}^{\prime }$).

Figure 41

The diagrammatic representation of the FS-CCSD equations for the (1,0) and (0,1) sectors in the skeleton form: (a) an equation for the $S_1$, (b) an equation for the $S_2$, and (c) an expression for the one-particle part of the $H_{\mathrm{eff}}$. Active lines are designated by a circle.

Figure 42

The diagrammatic representation of the $S_2$ equation for the (1,1) sector in skeleton form.

Figure 43

The diagrammatic expression for the two-particle part of $H_{\mathrm{eff}}$.

Figure 44

Matrix representation of the intermediate Hamiltonian in diagrammatic form.