In this section the available features of the Mrcc code are summarized. We also specify what type of reference states (orbitals) can be used, and if a particular feature requires one of the interfaces or is available with Mrcc in standalone mode. We also give the corresponding references which describe the underlying methodological developments.
conventional and density-fitting (resolution-of-identity) Hartree–Fock SCF (Ref. 138): restricted HF (RHF), unrestricted HF (UHF), and restricted open-shell HF (ROHF)
conventional and density-fitting (resolution-of-identity) Kohn–Sham (KS) density functional theory (DFT) (Ref. 84): restricted KS (RKS) and unrestricted KS (UKS); local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA (depending on kinetic-energy density and/or the Laplacian of the density), hybrid, and double-hybrid (DH) functionals (for the available functionals see the description of keyword dft); dispersion corrections
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 (Ref. )
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. 84)
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), and approximate RPA with exchange (RPAX2); currently the dRPA, SOSEX, and RPAX2 methods are available for RHF/RKS and UHF/UKS references, while the RPA method is only implemented for RHF/RKS (Refs. 84 and 85)
density-fitting (resolution-of-identity) second-order coupled-cluster singles and doubles (CC2), iterative doubles correction to configuration interaction singles [CIS(D${}_{\infty}$)], and second-order algebraic diagrammatic construction [ADC(2)] approaches (Refs. 96 and )
arbitrary single-reference coupled-cluster methods (Ref. 82): CCSD, CCSDT, CCSDTQ, CCSDTQP, …, CC($n$)
arbitrary single-reference configuration-interaction methods (Ref. 82): CIS, CISD, CISDT, CISDTQ, CISDTQP, …, CI($n$), …, full CI
arbitrary perturbative coupled-cluster models (Refs. 77, 15, and 79):
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)${}_{\Lambda}$, CCSDT(Q)${}_{\Lambda}$, CCSDTQ(P)${}_{\Lambda}$, …, CC($n-1$)($n$)${}_{\Lambda}$
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
multi-reference CI approaches (Ref. 83)
multi-reference CC approaches using a state-selective ansatz (Ref. 83)
arbitrary single-reference linear-response (equation-of-motion, EOM) CC methods (Ref. 76): LR-CCSD (EOM-CCSD), LR-CCSDT (EOM-CCSDT), LR-CCSDTQ (EOM-CCSDTQ), LR-CCSDTQP (EOM-CCSDTQP), …, LR-CC($n$) [EOM-CC($n$)]
linear-response (equation-of-motion) MRCC schemes (Ref. 76)
DFT/WFT embedding (Ref. 62): DFT-in-DFT, WFT-in-DFT, WFT-in-WFT, and WFT-in-WFT-in-DFT embedding (where WFT stands for wave function theory); currently embedding is only available for closed-shell systems using density-fitting; 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.
The CI and CC approaches listed above are available with RHF, UHF, and standard/semi-canonical ROHF orbitals. Density fitting with CI or 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. 79).
The CI and CC approaches listed above are also available with the following interfaces and references.
Cfour, Columbus, and Molpro
Cfour, Columbus, and Molpro
Cfour
Cfour, and Molpro
Columbus and Molpro
QM/MM calculations can be performed by the Amber interface using any method implemented in Mrcc for the QM region.
Single-point calculations are also possible with several types of relativistic Hamiltonians and reference functions, see Sect. 6.7 for more details.
CC$n$ calculations with ROHF orbitals are not possible for theoretical reasons, see Ref. 79 for explanation.
Single-point CI and CC calculations are, in principle, possible with RKS and UKS orbitals.
Geometry optimizations and first-order property calculations can be performed using analytic gradient techniques with the following methods.
conventional and DF (RI) HF-SCF (Ref. 138): RHF and UHF
conventional and DF (RI) DFT (Ref. 84): RKS and UKS with LDA, GGA, meta-GGA (depending only on kinetic-energy density), and hybrid functionals as well as dispersion corrections
double hybrid density functional methods, 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)
DF-MP2 (RI-MP2), currently only for RHF references (Ref. 84)
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, and intrinsic atomic orbital (IAO) 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
– to select an algorithm for the optimization
– maximum number of iterations allowed
– convergence criterion for energy change
– convergence criterion for the gradient change
– 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.
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.
Cfour and Columbus
Cfour and Columbus
Cfour
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.
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.
Geometry optimizations and first-order property calculations can also be performed via numerical differentiation for all methods available in Mrcc using the Cfour interface.
Analytic gradients are are also available with several types of relativistic Hamiltonians and reference functions, see Sect. 6.7 for more details.
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.
In addition to harmonic vibrational frequencies [75], the analytic Hessian code has been tested for NMR chemical shifts [75], static and frequency-dependent electric dipole polarizabilities [78], magnetizabilities and rotational $g$-tensors [40], electronic $g$-tensors [39], spin-spin coupling constants, and spin rotation constants. These properties are available via the Cfour interface.
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.
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.
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.
The analytic third derivative code has been tested for static and frequency-dependent electric-dipole first (general, second-harmonic-generation, optical-rectification) hyperpolarizabilities [113] and Raman intensities [112]. Please note that the orbital relaxation effects are not considered for the electric-field. These properties are available via the Cfour interface.
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.
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.
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. 96 and ).
time-dependent HF (TD-HF), Tamm–Dancoff approximation (TDA), time-dependent DFT (TD-DFT)
second-order coupled-cluster singles and doubles (CC2) method
iterative doubles correction to configuration interaction singles [CIS(D${}_{\infty}$)] method
second-order algebraic diagrammatic construction [ADC(2)] approach
Ground to excited-state transition moments are available for TD-HF, TDA, TD-DFT, and ADC(2). For the CIS, TD-HF, TDA, TD-DFT, CC2, CIS(D${}_{\infty}$), and ADC(2) methods an efficient reduced-cost approach is 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.
Cfour, Columbus, and Molpro (only excitation energy)
Cfour, Columbus, and Molpro (only excitation energy)
Cfour and Molpro (only excitation energy)
Columbus and Molpro (only excitation energy)
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.
With CI methods excited to excited-state transition moments can also be evaluated.
Excited-state harmonic frequencies can be evaluated for the above methods with the help of numerical differentiation of analytical gradients, see Sect. 6.3.
Excited-state harmonic frequencies can also be calculated for the above methods via numerical differentiation using the Cfour or Columbus interface.
Excited-state harmonic frequencies and second-order properties can be evaluated for CI methods using analytic second derivatives and the Cfour interface.
Treatment of special relativity in single-point energy calculations is possible for all the CC and CI methods listed in Sect. 6.1 using various relativistic Hamiltonians with the following interfaces.
With Molpro relativistic calculations can be performed with Douglas-Kroll-Hess Hamiltonians using RHF, UHF, ROHF, and MCSCF orbitals. The interface also enables the use of effective core potentials (see Molpro’s manual for the specification of the Hamiltonian and effective core potentials).
With Cfour exact two-component (X2C) and spin-free Dirac–Coulomb (SF-DC) calculations can be performed. The evaluation of mass-velocity and Darwin corrections is also possible using analytic gradients for all the methods and reference functions listed in Sect. 6.2. (See the description of the RELATIVISTIC keyword in the Cfour manual for the specification of the Hamiltonian.)
Treatment of special relativity in analytic gradient calculations is possible for all the CC and CI methods listed in Sect. 6.2 using various relativistic Hamiltonians with the following interfaces.
With Cfour analytic gradient calculations can be performed with the exact two-component (X2C) treatment.
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 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. See the description of keywords ovirt, eps, and ovosnorb for more details.
The cost of density-fitting methods can be reduced using natural auxiliary functions (NAFs) introduced in Ref. 84. The approach is very efficient for dRPA, but considerable speedups can also be achieved for MP2 and CC2. See the description of keywords naf_cor and naf_scf for more details.
The computational expenses of CIS, TD-HF, TDA, TD-DFT, CIS(D${}_{\infty}$), CC2, and ADC(2) excited-state calculations can be efficiently reduced using state-averaged NOs and NAFs (Refs. 96 and ). See the description of keyword redcost_exc for more details.
The cost of MP2, dRPA, SOSEX 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, 85, 105, 104, and 106). This method combines ideas from the cluster-in-molecule approach of Li and co-workers [88], the incremental approach of Stoll et al. [147], domain- and pair approximations introduced first by Pulay et al. (see, e.g., Ref. 131) with frozen natural orbital, natural auxiliary function, and Laplace transform techniques. It is currently available only for closed-shell molecules using RHF (RKS) orbitals. See the description of keywords localcc, lnoepso, lnoepsv, domrad, lmp2dens, dendec, nchol, osveps, spairtol, wpairtol, laptol, lccrest, and lcorthr for further details.
Utilizing the above local correlation techniques a multi-level scheme is defined in which the LMOs are classified as active or environment (Ref. 62). 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.
The optimization of basis set’s exponents and contraction coefficients can be performed with any method for which single-point energy calculations are available (see Sect. 6.1). The implementation is presented in Ref. 95. The related keywords are
– to turn on/off basis set optimization
– to select an algorithm for the optimization
– maximum number of iterations allowed
– convergence criterion for energy change
– convergence criterion for parameter (exponent, contraction coefficient) change
For their detailed description see Sect. 12.
For the optimization of basis sets it is important to know the format for the storage of the basis set parameters. In Mrcc the format used by the Cfour package is adapted. The format is communicated by the following example.
actual lines | description | ||||
C:6-31G | $\hookleftarrow$ Carbon atom:basis name | ||||
Pople's Gaussian basis set | $\hookleftarrow$ comment line | ||||
$\hookleftarrow$ blank line | |||||
2 | $\hookleftarrow$ number of angular momentum types | ||||
0 | 1 | $\hookleftarrow$ 0$\rightarrow$s , 1$\rightarrow$p | |||
3 | 2 | $\hookleftarrow$ number of contracted functions | |||
10 | 4 | $\hookleftarrow$ number of primitives | |||
$\hookleftarrow$ blank line | |||||
3047.5249 | 457.36952 | … | $\hookleftarrow$ exponents for s functions | ||
$\hookleftarrow$ blank line | |||||
0.0018347 | 0.0000000 | 0.0000000 | $\hookleftarrow$ contraction coefficients | ||
0.0140373 | 0.0000000 | 0.0000000 | for s functions | ||
0.0688426 | 0.0000000 | 0.0000000 | |||
0.2321844 | 0.0000000 | 0.0000000 | |||
0.4679413 | 0.0000000 | 0.0000000 | |||
0.3623120 | 0.0000000 | 0.0000000 | |||
0.0000000 | 0.1193324 | 0.0000000 | |||
0.0000000 | 0.1608542 | 0.0000000 | |||
0.0000000 | 1.1434564 | 0.0000000 | |||
0.0000000 | 0.0000000 | 1.0000000 | |||
$\hookleftarrow$ blank line | |||||
7.8682724 | 1.8812885 | … | $\hookleftarrow$ exponents for p functions | ||
$\hookleftarrow$ blank line | |||||
0.0689991 | 0.0000000 | $\hookleftarrow$ contraction coefficients | |||
0.3164240 | 0.0000000 | for p functions | |||
0.7443083 | 0.0000000 | ||||
0.0000000 | 1.0000000 |
In a basis set optimization process you need two files in the working directory: the appropriate MINP file with the basopt keyword set and a user supplied GENBAS file that contains the basis set information in the above format. You do not need to write the GENBAS file from scratch, you can use the files in the BASIS directory of Mrcc to generate one or you can use the Environmental Molecular Sciences Laboratory (EMSL) Basis Set Library [30, 32, 142] to download a basis in the appropriate form (AcesII format). Note that you can optimize several basis sets at a time: all the basis sets which are added to the GENBAS file will be optimized.
You can perform unconstrained optimization when all the exponents and contraction coefficients are optimized except the ones which are exactly 0.0 or 1.0. Alternatively, you can run constrained optimizations when particular exponents/coefficients or all exponents and coefficients for a given angular momentum quantum number are kept fixed during the optimization. The parameters to be optimized can be specified in the GENBAS file as follows.
Unconstrained optimization: no modifications are needed—by default all exponents and contraction coefficients will be optimized except the ones which are exactly 0.0 or 1.0.
Constrained optimization: by default all the exponents and coefficients will be optimized just as for the unconstrained optimization. To optimize/freeze particular exponents or coefficients special marks should be used:
use the “--” mark (without quotes) if you want to keep an exponent or coefficient fixed during the optimization. You should put this mark right after the fixed parameter (no blank space is allowed). If this mark is attached to an angular momentum quantum number, none of the exponents/coefficients of the functions in the given shell will be optimized except the ones which are marked by “++”.
use the “++” mark (without quotes) if you want a parameter to be optimized. Then you should put this mark right after it (no blanks are allowed). You might wonder why this is needed if the default behavior is optimization. Well, this makes life easier. If you want to optimize just a few parameters, it is easier to constrain all parameters first then mark those, which are needed to be optimized (see the example below).
Examples:
To reoptimize all parameters in the above basis set but the exponents
and coefficients of s-type functions you should copy the basis set to
the GENBAS file and put mark “--” after the angular
momentum quantum number of 0. The first lines of the GENBAS file:
C:6-31G | |||||
---|---|---|---|---|---|
Pople's Gaussian basis set | |||||
2 | |||||
0-- | 1 | ||||
3 | 2 | ||||
10 | 4 | ||||
3047.5249 | 457.36952 | … |
Both s- and p-type functions are fixed but the first s-exponent:
C:6-31G | |||||
Pople's Gaussian basis set | |||||
2 | |||||
0-- | 1-- | ||||
3 | 2 | ||||
10 | 4 | ||||
3047.5249++ | 457.36952 | … |
During the optimization the GENBAS file is continuously updated, and if the optimization terminated successfully, it will contain the optimized values (in this case it is equivalent to the GENBAS.opt file, see below, the only difference is that the file GENBAS.opt may contain the special marks, i.e., “++”, “--”). Further files generated in the optimization are:
GENBAS.init – the initial GENBAS file saved
GENBAS.tmp – temporary file, updated after each iteration, can be used to restart conveniently a failed optimization process
GENBAS.opt – this file contains the optimized parameters after a successful optimization.