6 Features

1. 1.

   conventional and density-fitting (resolution-of-identity) Hartree–Fock SCF (Ref. 138): restricted HF (RHF), unrestricted HF (UHF), and restricted open-shell HF (ROHF)

2. 2.

   conventional and density-fitting (resolution-of-identity) multi-configurational SCF (MCSCF)

3. 3.

   conventional and density-fitting (resolution-of-identity) Kohn–Sham (KS) density functional theory (DFT) (Ref. 71): restricted KS (RKS), unrestricted KS (UKS), restricted open-shell KS (ROKS); local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA (depending on kinetic-energy density and/or the Laplacian of the density), hybrid, range-separated hybrid, double hybrid (DH), third-order double hybrid, and range-separated double hybrid (RSDH) functionals (for the available functionals see the description of keyword dft); dispersion corrections

4. 4.

   time-dependent HF (TD-HF), time-dependent DFT (TD-DFT), TD-DFT in the Tamm–Dancoff approximation (TDA); currently only for closed-shell molecules with RHF/RKS reference using density-fitting, for LDA and GGA functionals and their hybrids and double hybrids even with range separation (Refs. 91, 90, RSDH, SCS_RSDH, and ADC_RSDH)

5. 5.

   density-fitting (resolution-of-identity) MP2, spin-component scaled MP2 (SCS-MP2), and scaled opposite-spin MP2 (SOS-MP2); currently only for RHF and UHF references (Ref. 71)

6. 6.

   density-fitting (resolution-of-identity) random-phase approximation (RPA, also known as ring-CCD, rCCD), direct RPA (dRPA, also known as direct ring-CCD, drCCD), second-order screened exchange (SOSEX), renormalized second-order perturbation theory (rPT2), and approximate RPA with exchange (RPAX2); currently the dRPA, SOSEX, rPT2, and RPAX2 methods are available for RHF/RKS and UHF/UKS references, while the RPA method is only implemented for RHF/RKS (Refs. 71 and 72)

7. 7.

   scaled-equation and down-scaled dRPA and SOSEX methods (sedRPA, seSOSEX, dsdRPA, dsSOSEX (Ref. 100); range-separated dRPA (Ref. 99).

8. 8.

   density-fitting (resolution-of-identity) second-order coupled-cluster singles and doubles (CC2), configuration interaction singles with perturbative correction for double excitations [CIS(D)], iterative doubles correction to configuration interaction singles [CIS(D${}_{\mathrm{\infty }}$)], and second-order algebraic diagrammatic construction [ADC(2)] approaches; spin-component scaled CC2, CIS(D), CIS(D${}_{\mathrm{\infty }}$), and ADC(2) [SCS-CC2, SCS-CIS(D), SCS-CIS(D${}_{\mathrm{\infty }}$), and SCS-ADC(2)]; scaled opposite-spin CC2, CIS(D), CIS(D${}_{\mathrm{\infty }}$), and ADC(2) [SOS-CC2, SOS-CIS(D), SOS-CIS(D${}_{\mathrm{\infty }}$), and SOS-ADC(2)] (Refs. 93, 94, 90, and 92)

9. 9.

   arbitrary single-reference coupled-cluster methods (Ref. 69): CCSD, CCSDT, CCSDTQ, CCSDTQP, …, CC($n$)

10. 10.

    arbitrary single-reference configuration-interaction methods (Ref. 69): CIS, CISD, CISDT, CISDTQ, CISDTQP, …, CI($n$), …, full CI

11. 11.

    arbitrary perturbative coupled-cluster models (Refs. 63, 8, and 65):

    • •

      CCSD[T], CCSDT[Q], CCSDTQ[P], …, CC($n-1$)[$n$]

    • •

      CCSDT[Q]/A, CCSDTQ[P]/A, …, CC($n-1$)[$n$]/A

    • •

      CCSDT[Q]/B, CCSDTQ[P]/B, …, CC($n-1$)[$n$]/B

    • •

      CCSD(T), CCSDT(Q), CCSDTQ(P), …, CC($n-1$)($n$)

    • •

      CCSDT(Q)/A, CCSDTQ(P)/A, …, CC($n-1$)($n$)/A

    • •

      CCSDT(Q)/B, CCSDTQ(P)/B, …, CC($n-1$)($n$)/B

    • •

      CCSD(T)${}_{\mathrm{\Lambda }}$, CCSDT(Q)${}_{\mathrm{\Lambda }}$, CCSDTQ(P)${}_{\mathrm{\Lambda }}$, …, CC($n-1$)($n$)${}_{\mathrm{\Lambda }}$

    • •

      CCSDT-1a, CCSDTQ-1a, CCSDTQP-1a, …, CC($n$)-1a

    • •

      CCSDT-1b, CCSDTQ-1b, CCSDTQP-1b, …, CC($n$)-1b

    • •

      CC2, CC3, CC4, CC5, …, CC$n$

    • •

      CCSDT-3, CCSDTQ-3, CCSDTQP-3, …, CC($n$)-3

12. 12.

    explicitly correlated MP2 (MP2-F12) using the 2B ansatz, the F + K commutator approximation, fixed amplitudes, spin-flipped geminals in open-shell calculations (Ref. F12pT)

13. 13.

    explicitly correlated CCSD and CCSD(T) utilizing the CCSD(F12*)(T+) model and the above MP2-F12 approximation (Ref. F12pT)

14. 14.

    multi-reference CI approaches, Davidson correction for MRCI (Ref. 70)

15. 15.

    multi-reference CC approaches using a state-selective ansatz (Ref. 70)

16. 16.

    arbitrary single-reference linear-response (equation-of-motion, EOM) CC methods (Ref. 62): LR-CCSD (EOM-CCSD), LR-CCSDT (EOM-CCSDT), LR-CCSDTQ (EOM-CCSDTQ), LR-CCSDTQP (EOM-CCSDTQP), …, LR-CC($n$) [EOM-CC($n$)]

17. 17.

    linear-response (equation-of-motion) MRCC schemes (Ref. 62)

18. 18.

    DFT/WFT embedding (Ref. 48): DFT-in-DFT, WFT-in-DFT, WFT-in-WFT, and WFT-in-WFT-in-DFT embedding (where WFT stands for wave function theory); currently LDA, GGA, meta-GGA, or hybrid functionals can be used as the DFT method; any WFT method implemented in Mrcc can by used in WFT-in-DFT type embedding calculations; WFT-in-WFT multi-level methods (via keyword corembed) is only available for local correlation methods.

19. 19.

    ONIOM approach with arbitrary number of layers [86]. Currently, only the integrated MO+MO (IMOMO) [57] part of ONIOM is available in the standalone version of Mrcc with mechanical embedding, electrostatic embedding, and automated link atom handling. Note that all types of the aforementioned DFT/WFT embedding techniques are also supported as top-level ONIOM calculations.

The CI and CC approaches listed above are available with RHF, UHF, standard/semi-canonical ROHF, MCSCF orbitals. Density-fitting with CI or high-order CC is currently enabled only for RHF references. For the perturbative CC approaches with ROHF reference determinant, for theoretical reasons, semi-canonical orbitals are used (see Ref. 65).

1. 1.

   The CI and CC approaches listed above are also available with the following interfaces and references.

   RHF:

   Cfour, Columbus, and Molpro

   ROHF, standard orbitals:

   Cfour, Columbus, and Molpro

   ROHF, semi-canonical orbitals:

   Cfour

   UHF:

   Cfour, and Molpro

   MCSCF:

   Columbus and Molpro

2. 2.

   QM/MM calculations can be performed by the Amber interface using any method implemented in Mrcc for the QM region.

3. 3.

   Semi-empirical quantum mechanical calculations are available through the Mopac and xTB interfaces in the ONIOM framework with mechanical embedding using automated link atom handling. The electrostatic embedding version of the ONIOM scheme is only available with the xTB program.

1. 1.

   Single-point calculations are also possible with several types of relativistic Hamiltonians and reference functions, see Sect. 6.7 for more details.

2. 2.

   Reduced-scaling approaches for the above CC and CI methods are available (Ref. 136). See Sect. 6.8 for details.

3. 3.

   Local CC approaches for arbitrary single-reference and perturbative coupled-cluster models, local MP2 approaches, and local dRPA are available (Refs. 135, 138, 72, 107, 105, and 108). See Sect. 6.8 for details.

4. 4.

   CC$n$ calculations with ROHF orbitals are not possible for theoretical reasons, see Ref. 65 for explanation.

5. 5.

   Single-point CI and CC calculations are, in principle, possible with RKS, UKS, ROKS orbitals.

6. 6.

   Solvation effects can be modeled by the polarizable continuum model (PCM) [147] via an interface to the PCMSolver library [16, 17, 1]. The PCM treatment is self-consistent for HF and KS SCF calculations, while, in the post-SCF steps, the potential of the solvent optimized at the SCF level is frozen. See the descriptions of keywords pcm*. An improved treatment of solvent effects is possible using the embedded cluster reference interaction site model (EC-RISM) [RISM], which is available via the Amber interface, see Sect. 5.5 and keyword rism for further details. Note that the implicit solvation models of the Mopac and xTB programs are also available in the ONIOM framework.

Geometry optimizations and first-order property calculations can be performed using analytic gradient techniques with the following methods. See also Tables LABEL:GrTable and LABEL:ExTable for the availability of analytic gradients for ground- and excited-state models, respectively.

1. 1.

   conventional and DF (RI) HF-SCF (Ref. 138): RHF and UHF

2. 2.

   conventional and DF (RI) DFT (Ref. 71): RKS and UKS with LDA, GGA, meta-GGA (depending only on kinetic-energy density), and hybrid functionals as well as dispersion corrections

3. 3.

   double hybrid density functional methods (Ref. 71), such as B2PLYP, B2PLYP-D3, B2GPPLYP, etc. (current limitations: only MP2 correlation, closed shell RKS, no spin-component scaling, no meta-GGA functionals, only DH functionals for which the DFT contribution to the energy is stationary with respect to the variation of the MO coefficients)

4. 4.

   DF-MP2 (RI-MP2), currently only for RHF references (Ref. 71)

5. 5.

   arbitrary single-reference coupled-cluster methods (Refs. 69 and 60): CCSD, CCSDT, CCSDTQ, CCSDTQP, …, CC($n$)

6. 6.

   arbitrary single-reference configuration-interaction methods (Refs. 69 and 60): CIS, CISD, CISDT, CISDTQ, CISDTQP, …, CI($n$), …, full CI

7. 7.

   multi-reference CI approaches (Refs. 70 and 60)

8. 8.

   multi-reference CC approaches using a state-selective ansatz (Refs. 70 and 60)

9. 9.

   CIS and TD-HF methods using RHF references

10. 10.

    arbitrary single-reference linear-response (equation-of-motion, EOM) CC methods (Refs. 60 and 62): LR-CCSD (EOM-CCSD), LR-CCSDT (EOM-CCSDT), LR-CCSDTQ (EOM-CCSDTQ), LR-CCSDTQP (EOM-CCSDTQP), …, LR-CC($n$) [EOM-CC($n$)]

11. 11.

    linear-response (equation-of-motion) MRCC schemes (Refs. 60 and 62)

12. 12.

    The ONIOM approach with arbitrary number of layers [86]. Currently, only the IMOMO [57] part of ONIOM is available in the standalone version of Mrcc with mechanical embedding and automated link atom handling. ONIOM gradients with electronic embedding are also available with perturbation independent point charges.

Currently only unrestricted geometry optimizations are possible, and electric dipole, quadrupole, and octapole moments as well as the electric field at the atomic centers can be evaluated. In addition, Mulliken, Löwdin, intrinsic atomic orbital (IAO), CHELPG, and Merz–Singh–Kollman atomic charges, and Mayer bond orders can be computed using the SCF wave functions. Analytic gradients for the CI and CC methods listed above are available with RHF, UHF, and standard ROHF orbitals without density fitting.

The following keywords are available to control the optimization process

optalg

– to select an algorithm for the optimization

optmaxit

– maximum number of iterations allowed

optetol

– convergence criterion for energy change

optgtol

– convergence criterion for the gradient change

optstol

– convergence criterion for the step-size

The optimization will be terminated and regarded as successful when the maximum gradient component becomes less than optgtol and either an energy change from the previous step is less than optetol or the maximum displacement from the previous step is less than optstol. For their detailed description see Sect. 12.

1. 1.

   The implemented analytic gradients for the CI and CC approaches listed above can also be utilized via the Cfour and Columbus interfaces with the following references.

   RHF:

   Cfour and Columbus

   ROHF, standard orbitals:

   Cfour and Columbus

   UHF:

   Cfour

   MCSCF:

   Columbus

   In addition to geometries, most of the first-order properties (dipole moments, quadrupole moments, electric field gradients, relativistic contributions, etc.) implemented in Cfour and Columbus can be calculated with Mrcc.

2. 2.

   QM/MM geometry optimizations and MD calculations can be performed by the Amber interface using any method implemented in Mrcc for which analytic gradients are available.

3. 3.

   Analytic gradients are also supported for the semi-empirical quantum mechanics methods of the Mopac and xTB programs with the mechanical embedding of the ONIOM framework. The electronic embedding with perturbation independent point charges is also supported for the DFTB methods of the xTB program.

1. 1.

   Geometry optimizations and first-order property calculations can also be performed via numerical differentiation for all methods available in Mrcc using the Cfour interface.

2. 2.

   Analytic gradients are are also available with several types of relativistic Hamiltonians and reference functions, see Sect. 6.7 for more details.

3. 3.

   Analytic gradients are available with the PCM via an interface to the PCMSolver library [17, 1] at the HF and KS SCF levels. See the descriptions of keywords pcm*.

4. 4.

   The implicit solvation models of the xTB and Mopac programs and their analytic gradient are available in the ONIOM implementation. See the descriptions of keywords pcm* and oniom_pcm.

Harmonic vibrational frequencies, infrared (IR) intensities, and ideal gas thermodynamic properties can be evaluated using numerically differentiated analytic gradients for all the methods listed in Sect. 6.2.

CC and CI harmonic frequency and second-order property calculations for RHF and UHF references can also be performed using analytic second derivatives (linear response functions) with the aid of the Cfour interface. Analytic Hessians (LR functions) are available for the following approaches.

1. 1.

   arbitrary single-reference coupled-cluster methods (Refs. 69, 60, 61, 64, 32, and 31): CCSD, CCSDT, CCSDTQ, CCSDTQP, …, CC($n$)

2. 2.

   arbitrary single-reference configuration-interaction methods (Refs. 69, 60, and 61): CIS, CISD, CISDT, CISDTQ, CISDTQP, …, CI($n$), …, full CI

3. 3.

   multi-reference CI approaches (Refs. 70, 60, and 61)

4. 4.

   multi-reference CC approaches using a state-selective ansatz (Refs. 70, 60, 61, and 64)

In addition to harmonic vibrational frequencies [61], the analytic Hessian code has been tested for NMR chemical shifts [61], static and frequency-dependent electric dipole polarizabilities [64], magnetizabilities and rotational $g$-tensors [32], electronic $g$-tensors [31], spin-spin coupling constants, and spin rotation constants. These properties are available via the Cfour interface.

1. 1.

   Using the Cfour interface, harmonic frequency calculations are also possible via numerical differentiation of energies for all implemented methods with RHF, ROHF, and UHF orbitals.

2. 2.

   Using the Cfour or the Columbus interface, harmonic frequency calculations are also possible via numerical differentiation of analytic gradients for all implemented methods for which analytic gradients are available (see Sect. 6.2 for a list of these methods). With Cfour, the calculation of static polarizabilities is also possible using numerical differentiation.

3. 3.

   NMR chemical shifts can be computed for closed-shell molecules using gauge-including atomic orbitals and RHF reference function.

Third-order property calculations can be performed using analytic third derivative techniques (quadratic response functions) invoking the Cfour interface for the following methods with RHF and UHF orbitals.

1. 1.

   arbitrary single-reference coupled-cluster methods (Refs. 69, 60, 61, 114, and 113): CCSD, CCSDT, CCSDTQ, CCSDTQP, …, CC($n$)

2. 2.

   multi-reference CC approaches using a state-selective ansatz (Refs. 70, 60, 61, 114, and 113)

1. 1.

   The analytic third derivative code has been tested for static and frequency-dependent electric-dipole first (general, second-harmonic-generation, optical-rectification) hyperpolarizabilities [114] and Raman intensities [113]. Please note that the orbital relaxation effects are not considered for the electric-field. These properties are available via the Cfour interface.

2. 2.

   Using the Cfour interface, anharmonic force fields and the corresponding spectroscopic properties can be computed using numerical differentiation techniques together with analytic first and/or analytic second derivatives at all computational levels for which these derivatives are available (see Sect. 6.2 and 6.3 for a list of these methods).

Diagonal Born–Oppenheimer correction (DBOC) calculations can be performed using analytic second derivative techniques via the Cfour interface for the following methods with RHF and UHF references.

1. 1.

   arbitrary single-reference coupled-cluster methods (Refs. 69, 60, 61, and 33): CCSD, CCSDT, CCSDTQ, CCSDTQP, …, CC($n$)

2. 2.

   arbitrary single-reference configuration-interaction methods (Refs. 69, 60, 61, and 33): CIS, CISD, CISDT, CISDTQ, CISDTQP, …, CI($n$), …, full CI

3. 3.

   multi-reference CI approaches (Refs. 70, 60, 61, and 33)

4. 4.

   multi-reference CC approaches using a state-selective ansatz (Refs. 70, 60, 61, and 33)

Excitation energies, first-order excited-state properties, and ground to excited-state transition moments can be computed as well as excited-state geometry optimizations can be performed using linear response theory and analytic gradients with the following methods.

1. 1.

   arbitrary single-reference linear-response CC methods (Refs. 69, 60, and 62): LR-CCSD, LR-CCSDT, LR-CCSDTQ, LR-CCSDTQP, …, LR-CC($n$)

2. 2.

   linear-response MRCC schemes (Refs. 70, 60, and 62)

3. 3.

   arbitrary single-reference configuration-interaction methods (Refs. 69, 60, and 62): CIS, CISD, CISDT, CISDTQ, CISDTQP, …, CI($n$), …, full CI

4. 4.

   multi-reference CI approaches (Refs. 70, 60, and 62)

Excitation energy and property calculations for the aforementioned methods are available with RHF, UHF, and standard ROHF orbitals. Density-fitting is only possible for RHF-based single point calculations. So far electric and magnetic dipole transition moments, both in the length and the velocity gauge, as well as the corresponding oscillator and rotator strengths have been implemented. For the list of implemented first-order properties see Sect. 6.2.

Excitation energies can also be computed for closed-shell systems using the density-fitting approximation and RHF (RKS) orbitals with the following methods (Refs. 93, 94, 91, 90, RSDH, SCS_RSDH, and ADC_RSDH).

1. 1.

   CIS, time-dependent HF (TD-HF), Tamm–Dancoff approximation (TDA), time-dependent DFT (TD-DFT)

2. 2.

   configuration interaction singles with perturbative correction for double excitations [CIS(D)]

3. 3.

   second-order coupled-cluster singles and doubles (CC2) method

4. 4.

   iterative doubles correction to configuration interaction singles [CIS(D${}_{\mathrm{\infty }}$)] method

5. 5.

   second-order algebraic diagrammatic construction [ADC(2)] approach

6. 6.

   spin-scaled versions of the latter approaches: SCS-CIS(D), SCS-CC2, SCS-CIS(D${}_{\mathrm{\infty }}$), SCS-ADC(2), SOS-CIS(D), SOS-CC2, SOS-CIS(D${}_{\mathrm{\infty }}$), and SOS-ADC(2)

7. 7.

   CIS(D)- and ADC(2)-based double hybrid TD-DFT methods even with range separation

Analytic gradients are implemented for CIS and TD-HF. Ground to excited-state transition moments are available for TD-HF, TDA, TD-DFT, hybrid and double-hybrid TD-DFT even with range separation, CIS(D), SCS-CIS(D), SOS-CIS(D), ADC(2), SCS-ADC(2), and SOS-ADC(2). For the CIS, TD-HF, hybrid TDA and TD-DFT, CC2, CIS(D${}_{\mathrm{\infty }}$), ADC(2), SCS-CC2, SCS-CIS(D${}_{\mathrm{\infty }}$), SCS-ADC(2), SOS-CC2, SOS-CIS(D${}_{\mathrm{\infty }}$), and SOS-ADC(2) methods efficient reduced-cost approaches are also implemented, see Sect. 6.8 for details.

Excitation energies, first-order excited-state properties, and ground to excited-state transition moments can also be calculated as well as excited-state geometry optimizations can also be carried out using the following interfaces and reference states.

RHF:

Cfour, Columbus, and Molpro (only excitation energy)

ROHF, standard orbitals:

Cfour, Columbus, and Molpro (only excitation energy)

UHF:

Cfour and Molpro (only excitation energy)

MCSCF:

Columbus and Molpro (only excitation energy)

1. 1.

   Please note that for excitation energies and geometries LR-CC methods are equivalent to the corresponding EOM-CC models. It is not true for first-order properties and transition moments.

2. 2.

   With CI methods, excited to excited-state transition moments can also be evaluated.

3. 3.

   Excited-state harmonic frequencies can be evaluated for the above methods with the help of numerical differentiation of analytical gradients, see Sect. 6.3.

4. 4.

   Excited-state harmonic frequencies can also be calculated for the above methods via numerical differentiation using the Cfour or Columbus interface.

5. 5.

   Excited-state harmonic frequencies and second-order properties can be evaluated for CI methods using analytic second derivatives and the Cfour interface.

6. 6.

   Natural transition orbitals (NTOs) can be computed for the CIS, TDHF, TDA, TD-DFT, ADC(2), CC2, and CIS(D${}_{\mathrm{\infty }}$) methods, while state-averaged natural orbitals (NOs) can be computed for the second-order methods. See the description of keyword nto for details.

The different types of SCF calculations can be sped up utilizing various tricks. In the dual basis set methods [81, 49], the SCF equations are solved with a smaller basis set, the resulting density matrix is projected into a large basis set, and then, the Fock matrix is constructed and diagonalized in the large basis (see keyword dual). In a similar manner, dual auxiliary bases, dual grids, or dual fitting metrics can also be used [DualDF] (see keywords dual_df, grid_sm, and fitting).

For the reduction of the computational expenses of the exchange contribution to the Fock matrix, several techniques have been implemented. The evaluation of three-center Coulomb integrals and their contribution can be accelerated with the multipole approximation [Multipole] (see keywords fmm, fmmord). The exchange contribution can also be computed with semi-numerical techniques exploiting pseudo-spectral approximations [PSSPFMM, 110, DualDF], for which also a dual grid approach is available [DualDF] (see keywords pssp, agrid_pssp, agrid_pssp_sm). Probably the most efficient method to speed up SCF calculations is the local density fitting approach, where products of orbitals are expanded in local domains of fitting functions [127, 138, 107] (see keywords scfalg, excrad, excrad_fin). The efficiency of the latter can further be improved with the OCC-RI-K algorithm [occri, DualDF] algorithm, which can also be used together with non-local direct DF-SCF calculations (see keyword occri).

The computational expenses of the CC and CI methods listed in Sect. 6.1 can be reduced via orbital transformation techniques (Ref. 136). In this framework, to reduce the computation time, the dimension of the properly transformed occupied and/or virtual one-particle space is truncated. Currently optimized virtual orbitals (OVOs) or MP2 natural orbitals (NOs) can be chosen. This technique is recommended for small to medium-size molecules. This scaling reduction approach is available using RHF or UHF orbitals for virtual orbitals. Occupied NOs are only implemented for RHF orbitals and with density fitting. Spatial symmetry for the frozen NO (FNO) methods can also be utilized with the general-order CC and CI methods of the mrcc program. For the correction of the dropped orbitals, besides the conventional $\mathrm{\Delta }$MP2 approach, novel PPL+ and (T+) corrections [FNOExt] can also be used. See the description of keywords ovirt, eps, lnoepsv, and ovosnorb for more details. Reduced-cost ground-state correlation methods can also be activated by adding the FNO- prefix to the acronym of the method [e.g., calc=FNO-CCSD(T)], see notes for keyword calc.

The cost of density-fitting methods can be reduced using natural auxiliary functions (NAFs) introduced in Ref. 71. The approach is very efficient for dRPA, but considerable speedups can also be achieved for MP2, CC2, and ADC(2). See the description of keywords naf_cor and naf_scf for more details.

The computational expenses of CIS, TD-HF, hybrid TDA and TD-DFT, CC2, CIS(D${}_{\mathrm{\infty }}$), ADC(2), SCS-CC2, SCS-CIS(D${}_{\mathrm{\infty }}$), SCS-ADC(2), SOS-CC2, SOS-CIS(D${}_{\mathrm{\infty }}$), and SOS-ADC(2) excited-state calculations can be efficiently reduced using local fitting domains as well as state-averaged NOs and NAFs (Refs. 93, 94, and 91). The scaling of CC2 and ADC(2) calculations can also be decreased by local correlation approaches (Ref. 90). See the description of keywords redcost_exc and redcost_tddft for more details.

The cost of MP2, dRPA, SOSEX, corresponding double- and dual-hybrid DFT, as well as single-reference iterative and perturbative coupled-cluster calculations can be reduced for large molecules by the local natural orbital CC (LNO-CC) approach (Refs. 135, 138, 72, 107, ROLocalMP2, 105, 108, and 106). This method combines ideas from the cluster-in-molecule approach of Li and co-workers [80], the incremental approach of Stoll et al. [144], domain- and pair approximations introduced first by Pulay et al. (see, e.g., Ref. 129) with frozen natural orbital, natural auxiliary function, and Laplace transform techniques. All of the above methods are available for closed-shell molecules using RHF (RKS) orbitals, while currently only LMP2 [ROLocalMP2] is implemented using ROHF (ROKS) orbitals or QROs formed from unrestricted orbitals. The LMP2 and LNO-CCSD(T) implementations are highly competitive, applicable up to a thousand atoms or about 50,000 atomic orbitals [108, 106] with accurate settings and basis sets. Highly-converged LNO settings and quadruple- or quintuple-$\zeta$ basis sets are feasible up to a few hundred atoms [L7CCDMC, 26, MeLysine] depending on the system type.

Further unique features of our local correlation methods include the exceptionally small memory and disk use, frequent checkpointing and restarting capability (see lccrest), support for general-order LNO-CC approaches, and treatments of point group symmetry (see localcorrsymm), and near-linear dependent AO basis sets [108, 106]. See the description of keywords localcc, lnoepso, lnoepsv, domrad, lmp2dens, dendec, nchol, osveps, spairtol, wpairtol, laptol, lccrest, and lcorthr for further details. Extensive benchmarks regarding the accuracy and efficiency of the local correlation methods are also provided in Refs. 72, 107, ROLocalMP2, 108, and 106 and Section II.G of Ref. 66. The definition, performance analysis, and the illustration of the systematic convergence of the lcorthr settings, as well as of the corresponding LNO extrapolation scheme is documented in Ref. 106. Extensive independent LNO-CCSD(T) benchmarks and comparisons to alternative local CCSD(T) approaches are also available [MobiusCC, LCCcompRuMartin, 117, LCC_MOBH35re, LCCcomp_anionbind].

Utilizing the above local correlation techniques a multi-level scheme is defined in which the LMOs are classified as active or environment (Ref. 48, 49). The contributions of these LMOs to the total correlation energy are evaluated using different models for the two subsystems, for instance, one can choose a LNO-CC model for the active subsystem and LMP2 for the environment. See the description of keyword corembed for further details.