Local correlation methods with pair natural orbitals (PNOs)

In this page single-reference local correlation methods using pair natural orbitals (PNOs) are described. This program is entirely distinct from the older PAO-based methods. It is designed for parallel execution both on one node and across multiple nodes. By default, the program store some data in distributed memory, which means more memory is required than in other programs. The memory required per CPU core for these distributed data is approximately inversely linear in the number of cores used. Therefore, it is normally not recommended using these programs on a single core. Depending on the molecular size, parallelization works well with up to 100-300 cores using multiple nodes, provided that a fast network (Infiniband or similar) is available (requires compiling Molpro from source code). Calculations can also be performed with reasonable efficiency on one node using disk storage instead of distributed memory when fast SSDs are used for scratch.

Appropriate default values are set which normally yield results that are close to the canonical ones. In particular, sub-kJ/mol accuracy of relative energies is usually achieved with PNO-LMP2-F12 (relative to MP2-F12), and sub-kcal/mol accuracy for PNO-LCCSD(T)-F12 relative to CCSD(T)-F12.

We strongly recommend that the user reads the review WIREs Comput. Mol. Sci. 8, e1371 (2018) for the concepts and local approximations used in the PNO program. More details on the PNO methods can be found in bibliography. We kindly ask you to cite our original publications on the corresponding methods in publications resulting from this program. Please read the important notes in getting started before attempting a PNO calculation!

The program can be invoked using one of the following commands:

  • PNO-LMP2-F12, options runs a second order Møller Plesset perturbation theory calculation.
  • PNO-LCCSD-F12, options runs a local coupled cluster calculation with single and double excitations.
  • PNO-LCCSD(T)-F12, options runs a local coupled cluster calculation with single, double, and perturbative triple excitations. (T) can be replaced by (T0) or (T1) for reduced cost.
  • PNO-LDCSD-F12, options runs a distinguishable cluster calculation with single and double excitations.

Available options are described below. F12 in the above commands can be omitted for calculations without explicit correlation. For coupled-cluster calculations the F12 variants are strongly recommended due to the significantly improved accuracy and very little added cost. The PNO program uses a different Ansatz from the default one in the canonical program, and we recommend using the more rigorous F12b approximation instead of F12a for all basis sets.

For Open-shell molecules (supported since Molpro 2020.1), LMP2 calculations use the spin adapted theory described in J. Chem. Theory Comput. 15, 987 (2019), and coupled cluster and distinguishable cluster calculations use the partially spin-restricted theory (similar to the canonical RCCSD in Molpro) by default. The command PNO-RCCSD is equivalent to to PNO-LCCSD. Spin-unrestricted CC or DC calculations can be performed using, for example, PNO-UCCSD. Calculations on close-shell molecules will use the compatible spin-free theories when the PNO-RCCSD or PNO-UCCSD command is given.

It is important to check the following before attempting a PNO calculation:

  • Larger PNO calculations may require pre-allocating GA memory (shared or distributed memory). Please read memory specifications for details.
  • Multi-node calculation over InfiniBand requires compiling Molpro from the source code. Please check GA Installation notes when doing so.
  • Molpro 2020.1 is not compatible with ealier versions by default. Please see versions for more details.

The PNO program requires a preceding Hartree–Fock calculation, and for the F12 varieties the CABS singles correction should be included. The Hartree–Fock calculation can be performed with the density-fitted HF program (DF-HF) or a well-parallelized local variety of it (LDF-HF). If a canonical F12 calculation is done before the PNO calculation, the CABS singles correction is computed by default and is stored in variable EF12_RHFRELAX. If this variable is nonzero, it will be added automatically added to the PNO energies. The variable is remembered across restarts. However, it is cleared whenever a new Hartree-Fock calculation is done. If variable EF12_RHFRELAX is zero or not set, the CABS singles correction can be computed in the PNO program by setting the option CABS_SINGLES=1. Also in this case EF12_RHFRELAX is set and remembered across restarts, so that in a restarted calculation the CABS correction needs not to be computed again.

A typical input including CABS singles correction is

geometry=...
basis=...
df-hf
pno-lccsd(t)-f12,cabs_singles=1     !energies will automatically include the cabs correction

** Note: Computations of the cabs correction are not well suited for multi-node calculations. They may become slow and require too much GA space. It may therefore be advantageous to carry out these calculations separately on a single node. This can be done with

file,2,name.wfu
geometry=...
basis=...
df-hf
df-mp2-f12,cabs_singles=-1

In this case variable EF12_RHFRELAX is set and available after a restart.

By default, the program uses as RI basis the JKFIT basis corresponding to the orbital basis set. It is strongly recommened to use orbital basis sets that include diffuse functions, e.g. aug-cc-pVTZ or cc-pVTZ-F12 (diffuse functions can be omitted on hydrogen atoms). The cc-pVnZ-F12 basis sets (short names: vnz-f12) [see J. Chem. Phys. 128, 084102 (2008), J. Chem. Phys. 132, 054108 (2010), Phys. Chem. Chem. Phys. 12, 10460 (2010)] are particularly well suited. Furthermore, the RI basis should at least have triple-zeta quality, if JKFIT sets are used for this purpose. Errors of several kcal/mol in relative energies can occur if e.g. ri_basis=vdz is used. Thus, with a double-zeta orbital basis, the RI-basis should be specfied using the ri_basis option, e.g.:

geometry=...
basis={
default=vdz-f12
set,jkfit,context=jkfit
default,avtz
set,mp2fit,context=mp2fit
default,avdz
set,ri,context=jkfit
default,avtz
}
explicit,ri_basis=ri,df_basis=mp2fit,df_basis_exch=jkfit

df-hf,basis=jkfit            !Hartree-Fock using the JKFIT density fitting basis
df-mp2-f12,cabs_singles=-1   !compute cabs correction
pno-lccsd(t)-f12

Both F12 calculations use the basis sets specified on the explicit directive, which must be given before the first F12 calculation in the input. Note that specifications on an explicit directive are not remembered in restarts and must therefore be given again after a restart.

An alternative, usually somewhat more expensive choice is to use the optimized RI basis sets of Peterson et al. (see J. Chem. Phys. 141, 094106 (2014) and references therein). In this case the RI basis is generated as then union of the orbital basis and the optri basis. In Molpro, this is done automatically by specifying, e.g. vdz-f12/cabs.

basis={
...
set,ri
default,vdz-f12/cabs
}

Warning: Do not define the RI basis for PNO calculation with the context optri. This will lead to very large errors. The cabs context should be used instead.

Local coupled cluster calculations on large molecules require a significant amount of memory. The memory requirements of the PNO program consist of two parts:

  • Local memory: the memory for each CPU core allocated using the memory card in the input file or the -m option on the molpro command line. This is primarily used for scratch torage of data, and the usage per core is roughly invariant with the number of cores.
  • Distributed memory: the memory used to store large data structure that are shared by all processors. The usage per core is roughly inversely linear in the number of cores. By default it is implemented with the globalarrays (GA) toolkit, but it is also possible to use disk storage (with the implementation=disk option) when the program is executed on one node.

Some typical memory usage can be found in WIREs Comput. Mol. Sci. 8, e1371. Unless the disk storage option is given, one should not allocate all available physical memory using the memory command in an input file, so that the GA toolkit could allocate sufficient memory when needed. In large cases it may be necessary to pass the -G [ga_mem] option in the molpro command line. This allows the allocation of ga_mem megawords of memory (all cores in total) for GA at the beginning of execution. Without doing this, GA may crash when the distributed data structures get large, most likely due to an upstream bug. More information about memory and GA allocation is givem in sections GA Installation notes and memory specifications. Please read these sections carefully before starting large-scale calculations.

When using the implementation=disk option, allocating memory for GA is not necessary. However it only supports calculations on a single node. Also be aware that the program is not specifically optimized for disk operations, and it requires fast SSDs for optimal performance.

In Molpro 2020.1 we have made a major revision to the PNO-LCCSD program to support open-shell molecules. Some changes in the local approximations and default program settings have been applied. The computed energies will not be identical to those obtained with earlier versions of Molpro.

For backwark compatibility, the closed-shell program in Molpro 2019.2 and earlier can be executed with, for example

{pno-lccsd(t)-f12, version=2019.2}

Default and tight settings from Molpro 2019.2 will also be used.

In most cases the recommended default values should be sufficient and provide chemical accuracy for relative energies. In cases of doubt or to benchmark the accuracy of the local approximations, TIGHT presets can be chosen using one of the following options:

  • DOMOPT=TIGHT Use tight domain approximations.
  • PAIROPT=TIGHT Use tight pair approximations.

In most cases, the domain approximations causes the largest errors, in particular in PNO-LCCSD(T)-F12 calculations, and if very high accuracy is required or in cases of doubt DOMOPT=TIGHT should be tried first. This also reduces the errors of the projection approximations, which depend on the domain sizes. Note, however, the calculations with TIGHT settings are much more demanding than with DEFAULT options regarding CPU time and memory.

For a detailed description of all options see the original publications.

Default and tight settings for PNO calculations (in atomic units)
Description threshold default tight
Domain approximations (affected by DOMOPT)
Primary PAO domains (partial charge) THRLMO 0.2 0.2
Domain extension (connectivity) IEXT 2 3
Domain extension (radius) REXT 5 7
OSV domain occupation number threshold THROSV $10^{-9}$ $10^{-10}$
LMP2 PNO domains (occ. number threshold) THRPNO_OCC_LMP2 $10^{-8}$ $10^{-8}$
LMP2 PNO domains (energy threshold) THRPNO_EN_LMP2 0.997 0.997
LCCSD PNO domains (occ. number threshold) THRPNO_OCC_CC $10^{-7}$ $10^{-8}$
LCCSD PNO domains (energy threshold) THRPNO_EN_CC 0.990 0.997
Large domains for (T0) calculation occ. number threshold THRTNO_T0 $10^{-9}$ $10^{-10}$
Small domains for (T) calculation occ. number threshold THRTNO_T $10^{-7}$ $10^{-7}$
Pair approximations (affected by PAIROPT)
Close pair energy threshold THRCLOSE $10^{-4}$ $10^{-5}$
Weak pair energy threshold THRWEAK $10^{-5}$ $10^{-6}$
Distant pair energy threshold THRDIST $10^{-6}$ $10^{-6}$
Very distant pair energy threshold THRVDIST $10^{-7}$ $10^{-7}$
Triples preselection type TRIPTYP 2 2
Preselection of triples list THRCLOSE_T $10^{-4}$ $10^{-5}$
Selection of triples for iterations THRTRIP_IT $10^{-7}$ $10^{-8}$
Local density fitting and RI approximations (affected by DOMOPT)
Connectivity criterion for DF domains IDFDOM 2 3
Distance criterion for DF domains RDFDOM 5 7
Connectivity criterion for RI domains IRIDOM 3 4
Distance criterion for RI domains RRIDOM 7 9

In this section we describe the parameters most relevant to the accuracy and performance of the PNO program. We note that our team has carefully selected the default options through benchmark calculations, and the options only need to be modified for special cases.

  • IMPLEMENTATION=GA|DISK Choose whether large data structures are stored in distributed memory (default) or disk.
  • THRLMO=value Charge threshold for selection of primary PAO domains when using IBOs or NBOs (default 0.2). For open-shell orbitals the threshold is divided by 2. If the threshold is larger than the largest partial charge cmax of an orbital, it is reduced to cmax*0.9 for this orbital.
  • THRLMO_ACT=value Charge threshold for selection of primary PAO domains when using IBOs or NBOs for active (open-shell) orbitals (default THRLMO).
  • IEXT=value Domain extension using connectivity. value corresponds to the number of bonds by which the primary domains are extended.
  • REXT=value Domain extension using distance. value is the radius in $a_0$ from any atom in the primary domain. IEXT and REXT can be combined. If both are given all atoms are included that are selected by one or the other criterion.
  • THROSV=threshold OSV selection threshold based on natural occupation numbers.
  • THRPNO_OCC_LMP2=threshold PNO selection threshold for LMP2 based on natural occupation numbers. It is also possible to specify 3 values for strong+close, weak, and distant pairs as THRPNO_OCC_LMP2=[thrstrong, thrweak, thrdist].
  • THRPNO_EN_LMP2=threshold PNO selection threshold for LMP2 based on the energy criterion.
  • THRPNO_OCC_CC=threshold PNO selection threshold for LCCSD based on occupation numbers.
  • THRPNO_EN_CC=threshold PNO selection threshold for LCCSD based on the energy criterion.

Note that in LCCSD calculations the thresholds THRPNO_OCC_LMP2 and THRPNO_EN_LMP2 only affect the LMP2 domain corrections. The thresholds THRPNO_OCC_CC must not be smaller and THRPNO_EN_CC not be larger than the corresponding LMP2 thresholds. This, the LCCSD domains are always smaller or equal to the LMP2 ones. PNOs are added to the domains until both the occupation number and energy criteria are fulfilled.

In the PNO-LCCSD program the pairs are classified according to the LMP2 pair energies into strong, close, weak, distant and very distant pairs. Close pairs are treated by approximate CCSD, in which terms that cancel at long-range are neglected. Weak pairs are treated with the same approximations as close pairs, but in addition terms that are non-linear in the amplitudes are neglected (CEPA). Distant pairs are approximated by the iterative SCS-LMP2 multipole approximation. Very distant pairs are treated by the semi-canonical SCS-LMP2 (non-iterative) multipole approximation.

  • THRCLOSE=thrclose Pairs with PNO-LMP2 energies $thrclose \ge E_{ij} \gt thrweak$ are treated as close pairs.
  • THRWEAK=thrweak Pairs with PNO-LMP2 energies $thrweak \ge E_{ij} \gt thrdist$ are treated as weak pairs.
  • THRDIST=thrdist Pairs with PNO-LMP2 energies $thrdist \ge E_{ij} \gt thrvdist$ are treated as distant pairs.
  • THRVDIST=thrvdist Pairs with PNO-LMP2 energies $thrvdist \ge E_{ij}$ are treated as very distant pairs.

The domain approximations in (T) calculations are controlled by the following options:

  • THRTNO_T0=value Occupation number threshold for selecting triples domains for the non-iterative (T0) approximation.
  • THRTNO_T=value Occupation number threshold for selecting triples domains for the iterative (T) approximation. Note that decreasing THRTNO_T will lead to significantly increased GA usage.
  • IEXT_T=value, REXT_T=value PAO domain selection criteria similar to IEXT and REXT but affect only (T) calculations. Defaults to IEXT and REXT, respectively.

The selection of the triple list is controlled by the following options:

  • TRIPTYP=value Determines triples list via pair classes (default 2).
  • THRCLOSE_T=value Energy threshold for close pairs in selecting triples (in $E_h$, default 1.d-4).
  • THRWEAK_T=value Energy threshold for weak pairs in selecting triples (in $E_h$, default 0, i.e., considering all nondistant pairs “close” in the triple selection).
  • THRTRIP=value Threshold for additional triple screening using (T0) triple corrections from a small-domain calculation (in $E_h$, default 0, i.e., perform (T0) calculation for all triples selected by TRIPTYP).
  • THRTRIP_IT=value (T0) energy threshold for screening triples before the iterations (in $E_h$, default 1.d-7).
  • THRF12=threshold LMP2 pair energy threshold for selecting pairs for which F12 corrections are computed (default 0, i.e., all nondistant pairs are included).
  • THRVAL=value The MP2 pair energy threshold in determining the LMO domains $[ij]_{\rm LMO}$ used in the F12 strong orthogonality projector (default 1.d-4). This option only applies to the selection of the valence occupied orbitals in the LMO domains.
  • ICOREDOM_F12=value, RCOREDOM_F12=value The connectivity and distance (in $a_0$) criteria in determining the LMO domains $[ij]_{\rm LMO}$ used in the F12 strong orthogonality projector (defaults are 2 and 5.0, respectively). This option only applies to the selection of the core orbitals in the LMO domains.
  • IRIDOM=value RI domain extension using connectivity. The default is 3.
  • RRIDOM=value RI domain extension using distances in $a_0$. The default is 7. If both IRIDOM and RRIDOM are given, the RI basis functions at a center will be included when either criterion is fulfilled.
  • IDFDOM=value Fitting domain extension using connectivity criterion (default 3).
  • RDFDOM=value Fitting domain extension using distances criterion (in $a_0$, default 7). If both IDFDOM and RDFDOM are given, the DF functions at a center will be included when either criterion is fulfilled.
  • IDFDOM_T=value, RDFDOM_T=value Fitting domain extension criteria similar to IDFDOM and RDFDOM but apply only to (T) calculations. The default values are 0 for both, in which case the primary fitting domains are used.
  • FITLMO=threshold In the PNO program LMOs are truncated if the square sum of the coefficients at one center is smaller than threshold (default 1.d-6). The remaining LMO coefficients are fitted to the original LMO.
  • FITPAO=threshold In the PNO program PAOs at one center are truncated when none of the PAOs at the center has a square sum of coefficients greater than threshold (default 1.d-6). The remaining PAO coefficients are fitted to the original PAOs.

In this section we list some advanced options to the PNO program. These options exist for technical or historical reasons and we do not recommend modifying the default values in general.

The following options are available for the orbital localization:

  • LOC_METHOD=method Localization method. method can be IBO (intrinsic bond orbitals, default), PM (Pipek-Mezey), BOYS (Forster-Boys localization), or NBO (natural bond orbitals). IBO is recommended, as it is most efficient and stable.
  • IBTYPE=value Projector type for generation of intrinsic atomic orbitals. value can be 1 or 2 (default).
  • IBOEXP=value Exponent used in the PM-like localization functional. value can be 2 or 4 (default).

Options for PAO domain selection:

  • THRBP=value Boughton-Pulay (BP) completeness criterion for selection of primary PAO domains (default 0, i.e., do not use the BP procedure). The BP criterion takes precedence over the LMO partial charge criterion if a positive THRBP is given.

Options for OSV generation:

  • OSV_AMPL=PAO|CAN|OPT Amplitudes to used to generate OSVs. PAO (default) means to use semi-canonical amplitudes in the PAO domains; CAN means semi-canonical in the full virtual space; OPT means fully optimized LMP2 amplitudes in the full virtual space.

Options for multipole approximations:

  • MLTP_METHOD=value If 1 use non-iterative multipole approximation for distant pairs, if 2 iterative multipole approximation (default 2).
  • MLTP_ORDER=value Expansion level for multipole approximation (default 3).
  • MLTP_SELECT=value Expansion level of multipole approximation used to select distant pairs (default MLTP_ORDER).

Options for PNO generation:

  • PNO_AMPL=OSV|OSV(OPT)|PAO|PAO(OPT) Amplitudes to used to generate PNOs. OSV (default) means to use semi-canonical (non-iterative) OSV amplitudes; PAO means semi-canonical (non-iterative) PAO amplitudes; if (OPT) is appended the amplitudes are iteratively optimized (can be expensive with OSV(OPT) and very expensive with PAO(OPT)!
  • THRDEG (Since Molpro 2020.1) PNOs with occupation number difference less than THRDEG (default $10^{-12}$) are considered degenerate. Degenerate PNOs are either all included or all excluded from the domains.
  • PNO_DIAG (logical). If true, use PNO domain selection criterion also for diagonal pairs. Otherwise OSV domains are used for diagonal pairs. If PNO_DIAG=true the threshold THROSV only affects the distant pair multipole treatment.

The PNO program divides basis functions to blocks for integral screening. The following options are available for defining the block sizes. A smaller block size encourages more efficient integral screening, reduces scratch memory usage, and improves the parallel efficiency. However, a larger block size improves the performance of matrix operations, and reduces the communication and bookkeeping cost.

  • BB_BLOCKS_AO=value Target blocking size in the AO basis (default 32).
  • BB_BLOCKS_DF_F12=value Target blocking size in the DF basis for the F12 calculations (default 32). The option does not affect PNO-LMP2 calculations.
  • BB_BLOCKS_RI=value Target blocking size in the RI basis for the F12 calculations (default 128).

In addition, the following options control the integral screening thresholds:

  • BB_THRESH=value Block screening threshold (default 1.d-5).
  • BB_RADIUS=value Block screening radius (default 4).

Other miscellaneous options:

  • BB_F12_PNO=value If BB_F12_PNO=0, PNO-LMP2-F12 energies with both OSV and PNO projectors will be computed; If BB_F12_PNO=1, only energies using the PNO projector will be computed; If BB_F12_PNO=2, only energies with the OSV projector will be computed (default 1). PNO projectors are always used in the F12 terms in coupled-cluster equations regardless of this option.
  • BB_F12_LOCCORE (logical) If true (default), localize the core orbitals. This may affect the pair approximations in the LMP2-F12 projector.

The following options that have been deprecated in Molpro 2020.1. They can still be used in calculations with version=2019.2.

  • PROJOPT=TIGHT Use tight projection approximations.
  • ALLTIGHT (logical). Use tight domain, pair, and projection approximations.
Projection approximations (affected by PROJOPT)1)
Project K-integrals PROJECT_K true true
Project J-integrals PROJECT_J all weak
Projection of singles amplitudes to doubles domains PROJECT_S all all
Projection of 3-external ${\bf J}({\bf E}^{kj})$ terms PROJECT_JE on on
Level of projection in the 3-external ${\bf K}({\bf E}^{kj})$ terms PROJECT_KE 2 1
  • LOCFIT=0 Disables local fitting (default is LOCFIT=1). Note that this is extremely expensive and memory demanding for large molecules, and local fitting is seldom a noticeable source of error. Please use a large RDFDOM instead.
  • LOCRI=0 Disables local RI (default is LOCRI=1). Please use a large RRIDOM instead.
  • ENERGR Reference energy (including CABS correction if present).
  • ENERGY Last computed total energy including the Hartree–Fock energy and CABS correction (if present). In PNO-LCCSD-F12 and PNO-LCCSD(T)-F12 calculations, ENERGY(1) is the F12a energy and ENERGY(2) the F12b energy (using F12b is recommended).
  • ENERGT0 (T0) energy contribution in PNO-LCCSD(T) and PNO-LCCSD(T)-F12 calculations
  • EMP2 PNO-LMP2 energy without domain correction.
  • EMP2_DC PNO-LMP2 energy including domain correction.
  • EF12 F12 contribution in PNO-LMP2-F12 (only set in F12 calculattions).
  • EMP2_F12 PNO-LMP2-F12 energy (only set in F12 calculations).
  • EMP2_PNO Same as EMP2.
  • EF12_PNO Same as EF12.
  • EMP2_SCS PNO-SCS-LMP2 energy without domain correction.
  • EF12_SCS SCS-F12 contribution (only set in F12 calculattions).
  • EMP2_F12_SCS PNO-SCS-LMP2-F12 energy (only set in F12 calculations)
  • DOMCORR Domain contribution for PNO-LMP2 (should not be added to F12 energies)
  • DOMCORR_CC PNO-LMP2 domain correction for PNO-LCCSD.
  • DOMCORR_F12 PNO-LMP2-F12 domain correction for PNO-LCCSD-F12
  • DOMCORR_PNO Same as DOMCORR.
  • ENERGT (T) energy contribution in PNO-LCCSD(T) and PNO-LCCSD(T)-F12 calculations

Note: The CABS correction has to be computed beforehand and is stored in variable EF12_RHFRELAX. If this is present, it is in F12 calculations added to all total energies, including the PNO-LMP2 one. It is not added in non-F12 calculations, even if EF12_RHFRELAX is set.

The DF-HF and DF-MP2-F12 programs are not designed for execution over multiple computer nodes, and all disk I/O are replaced by GA operations in this case. It is sometimes helpful running these calculations separately on one node. For example, one can run a single-node calculation for DF-HF and CABS singles correction with

 FILE, 2, some_orbitals.wfu
 [specifications of basis, geometries, options, etc]
 {DF-HF}
 {DF-MP2-F12, CABS_SINGLES=-1}

and then a PNO-LCCSD(T)-F12 calculation can be performed on multiple nodes with

 FILE, 2, some_orbitals.wfu
 [specifications of basis, options, etc]
 {PNO-LCCSD(T)-F12}

The CABS singles corrections must be manually added to the final PNO-LCCSD(T)-F12 results when this procedure is used. Also note that the wave function repository of Molpro needs to be on a network file system accessible to all nodes.

If the program crashes with a message ”ARMCI DASSERT fail” most likely more GA memory must be specified using the -G command line option, see section getting started. This is necessary due to a bug in the GA software, which is out of our control. The GA developers recommend to install the GA software with configure option –with-mpi-pr. This avoids the problem and does not require the -G option. Unfortunately also this GA version, which is MPI based, is not stable on all systems, in particular for large cases. It is also somewhat slower since an extra process is needed.

General reviews of the PNO program:

Closed-shell PNO-LCCSD(T)-F12:

  • H.-J. Werner, G. Knizia, C. Krause, M. Schwilk, and M. Dornbach, Scalable electron correlation methods I.: PNO-LMP2 with linear scaling in the molecular size and near inverse-linear scaling in the number of processors, J. Chem. Theory Comput. 11, 484 (2015)
  • Q. Ma, and H.-J. Werner, Scalable electron correlation methods. 2. Parallel PNO-LMP2-F12 with near linear scaling in the molecular size, J. Chem. Theory Comput. 11, 5291 (2015)
  • M. Schwilk, Q. Ma, C. Köppl, and H.-J. Werner, Scalable electron correlation methods. 3. Efficient and accurate parallel local coupled cluster with pair natural orbitals (PNO-LCCSD), J. Chem. Theory Comput. 13, 3650 (2017)
  • Q. Ma, M. Schwilk, C. Köppl, and H.-J. Werner, Scalable electron correlation methods. 4. Parallel explicitly correlated local coupled cluster with pair natural orbitals (PNO-LCCSD-F12), J. Chem. Theory Comput. 13, 4871 (2017)
  • Q. Ma, and H.-J. Werner, Scalable electron correlation methods. 5. Parallel perturbative triples correction for explicitly correlated local coupled cluster with pair natural orbitals, J. Chem. Theory Comput. 14, 198 (2018)

Open-shell PNO-LCCSD(T)-F12:

  • C. Krause, and H.-J. Werner, Scalable electron correlation methods. 6. Local spin-restricted open-shell second-order Møller–Plesset perturbation theory using pair natural orbitals: PNO-RMP2, J. Chem. Theory Comput. 15, 987 (2019)
  • Q. Ma, and H.-J. Werner, Scalable electron correlation methods. 7. Local open-shell coupled-cluster methods using pair natural orbitals: PNO-RCCSD and PNO-UCCSD, J. Chem. Theory Comput. 16, 3135 (2020)

Other topics on local approximations used in the PNO program:

  • M. Schwilk, D. Usvyat, and H.-J. Werner, Communication: Improved pair approximations in local coupled-cluster methods, J. Chem. Phys. 142, 121102 (2015)
  • C. Köppl, and H.-J. Werner, On the use of Abelian point group symmetry in density-fitted local MP2 using various types of virtual orbitals, J. Chem. Phys. 142, 164108 (2015)
  • H.-J. Werner, Communication: Multipole approximations of distant pair energies in local correlation methods with pair natural orbitals, J. Chem. Phys. 145, 201101 (2016)

Application on intermolecular interactions:

  • Q. Ma, and H.-J. Werner, Accurate intermolecular interaction energies using explicitly correlated local coupled cluster methods [PNO-LCCSD(T)-F12], J. Chem. Theory Comput. 15, 1044 (2019)

1)
Our benchmark calculations show that the projection approximations are unlikely to be a major contributor to the local errors. These options will be deprecated in a future release.