An empirically scaled version of the explicitly correlated F12 correction to second-order Møller–Plesset perturbation theory (MP2-F12) is introduced. The scaling eliminates the need for many of the most costly terms of the F12 correction while reproducing the unscaled explicitly correlated F12 interaction energy correction to a high degree of accuracy. The method requires a single, basis set dependent scaling factor that is determined by fitting to a set of test molecules. We present factors for the cc-pVXZ-F12 (X = D, T, Q) basis set family obtained by minimizing interaction energies of the S66 set of small- to medium-sized molecular complexes and show that our new method can be applied to accurately describe a wide range of systems. Remarkably good explicitly correlated corrections to the interaction energy are obtained for the S22 and L7 test sets, with mean percentage errors for the double-zeta basis of 0.60% for the F12 correction to the interaction energy, 0.05% for the total electron correlation interaction energy, and 0.03% for the total interaction energy, respectively. Additionally, mean interaction energy errors introduced by our new approach are below 0.01 kcal mol−1 for each test set and are thus negligible for second-order perturbation theory based methods. The efficiency of the new method compared to the unscaled F12 correction is shown for all considered systems, with distinct speedups for medium- to large-sized structures.

It is well-known that the computation and evaluation of electron correlation effects are critical for the accurate and quantitative description of chemical systems. One of the simplest ab initio wave function based correlation methods is second-order Møller–Plesset perturbation theory (MP2).1 Since its first formulation in 1934, several extensions and simplifications have been proposed to improve its accuracy and decrease its computational cost: The resolution of the identity (RI) approximation2–4 and the related pseudospectral approach5 can be used to significantly reduce the computational prefactor of the method, while local orbital or atomic orbital (AO) based formulations (see, e.g., Refs. 6 and 7) are able to lower the asymptotic computational scaling with the system size to as low as linear. In addition, complete basis set (CBS) extrapolation8,9 and explicitly correlated R12/F12 methods10–15 have been introduced to overcome the basis set incompleteness error (BSIE). The latter incorporates explicitly coupled two-electron terms (geminals) into the wave function to better describe short-ranged correlation and satisfy electronic cusp conditions,16 leading to much faster convergence with respect to the size of the one-electron basis set.

To improve accuracy for non-covalent interactions (NCIs), some authors have introduced a scaling of the correlation energy with an empirically determined factor.17,18 In 2003, Grimme19 established the spin-component-scaled MP2 (SCS-MP2) method, which employs separate scaling factors for the same-spin (SS) and the opposite-spin (OS) energy contributions,

(1)

where cOS=65 and cSS=13 are the fixed OS and SS scaling factors, respectively. In particular, the description of stacked unsaturated complexes and the computation of thermochemical properties benefit by using SCS-MP2 instead of MP2, where for general NCIs, no superior results are obtained.20 However, several variations of this ansatz21–25 could improve its accuracy and have been applied to other correlation methods within the context of configuration interaction26,27 and coupled cluster28,29 theories. One popular variation of the SCS-MP2 method is Jung et al.’s30 scaled-opposite-spin MP2 (SOS-MP2) method, which focuses on the opposite-spin MP2 energy contribution and completely neglects the same-spin part, leading to the energy expression

(2)

The choice of cSOS = 1.3 as a scaling factor results in small differences to SCS-MP2 and improved reaction and atomization energies compared to standard MP2. The use of SOS-MP2 is beneficial in terms of efficiency since it allows for a reduction in the computational scaling of the method with the system size M from O(M5) for standard MP2 and SCS-MP2 to O(M4) when SOS-MP2 is combined with Laplace-transform methods.30,31

Besides SCS-MP2 and SOS-MP2, the hybrid supermolecular MP2 coupled (MP2C) approach by Pitoňák and Heßelmann32 has become popular in the last decade for the calculation of NCIs. It focuses on correcting the MP2 dispersion interactions of complexes by neglecting and replacing the uncoupled Hartree–Fock (UCHF) dispersion energy with a more accurate time-dependent density-functional theory (TDDFT) based quantity, resulting in

(3)

as interaction energy expression, and a notably improved description of all kinds of NCIs. In general, MP2, SCS-MP2, and MP2C suffer from large BSIEs and are improved when an explicitly correlated F12 correction is applied, allowing for the use of smaller double-zeta basis sets.33 However, the F12 correction introduces significant overhead, especially for MP2 theory and its variation, and its computation is by far the most expensive step in the correlation calculation.

In this paper, we address this issue of the computationally demanding F12 correction by using the general SOS idea in the context of explicitly correlated second-order Møller–Plesset perturbation theory. Our new method does not require the computationally expensive geminal–geminal exchange-type integrals and thus drastically reduces the prefactor of MP2-F12 calculations, especially for small basis sets. We have determined scaling factors designed to reproduce the standard explicitly correlated MP2-F12 correction to the interaction energy as accurately as possible for a wide range of organic and biologically relevant systems.

Consider the closed-shell, spin-adapted explicitly correlated second-order F12 correction to the correlation energy,

(4)

where eijs is the singlet (s = 0) or triplet (s = 1) energy contribution for a pair of spatial orbitals (ϕiϕj). In the popular diagonal, orbital-invariant formulation of Ten-no,15,34,35 which satisfies Kato’s cusp condition16 without the need for geminal amplitude optimization, the pair contributions can be written as

(5)

where eij,V¯s represents the contributions of all terms involving one explicitly correlated geminal (orbital-geminal) and eij,B¯s contains all terms involving two geminals (geminal–geminal). The explicitly correlated pair energies are

(6)
(7)

with Kronecker delta δij,

(8)
(9)

and

(10)
(11)
(12)
(13)
(14)
(15)

Here, f^1 and f^2 are the one-electron Fock-operators, ϵi and ϵa are occupied and virtual orbital energies, respectively, F^12 is the correlation factor, and Q^12 is the strong orthogonality operator,

(16)

The exchange-type expressions for B̃ijji and Ṽijji are simply obtained by switching |ϕiϕj⟩ to |ϕjϕi⟩ in Eqs. (10)(14). More details on the computation and derivation of these intermediates can be found in the literature.36–38 The singlet and triplet F12 energy corrections are

(17)
(18)

and the contribution from one pair of spatial orbitals is thus

(19)

We aim to avoid the need to calculate the O(M5) scaling exchange-type term B̃ijji since it is by far the most costly37 and also contributes relatively little to the final F12 correction. Therefore, we introduce a scaling of the geminal–geminal triplet energy by a factor of 4/3 and an empirically determined scaling factor cSF12 used to fit the resulting energy to the standard MP2-F12 interaction energy values. The resulting explicitly correlated total pair energy correction then becomes

(20)

where summation over all pairs of spatial orbitals in combination with the MP2 energy leads to the definition of our scaled explicitly correlated second-order Møller–Plesset perturbation energy denoted as MP2-SF12.

Because triplet correlation energies converge faster than singlet energies,39 the F12 triplet correction vanishes earlier in the complete basis set limit than the singlet correction. For this reason, the initial scaling of the triplet geminal–geminal correction by 4/3 becomes increasingly negligible as the size of the basis set increases, and the ideal cSF12 factor changes accordingly. Thus, it is necessary to determine an optimal factor for each basis set. This is not a problem in practice since the determination of good factors can be performed quite cheaply by fitting to a set of small molecules (see below). In any case, the F12 correction is designed to deliver accurate energies with smaller basis sets, which become necessary for calculations of larger systems where the SF12 approximation allows for sizable speedups.

In explicitly correlated second-order F12 theory, the use of RI in the form of Valeev’s CABS approach13 leads to MP2 like expressions involving products of two-electron integrals, which scale as O(M5) with the size of the system M. Through the use of density fitting (DF) techniques,40,41 the scaling of direct-type terms (B̃ijij,Ṽijij) can be reduced to fourth-order, while the fifth-order scaling remains for the exchange-type terms, albeit with a significantly reduced prefactor. Both RI and DF require auxiliary basis sets μ′ and μ″, and such sets have been specifically designed for both cases.42,43 Although the computational scaling is not worse than for MP2 itself, the sheer number of terms involved in the F12 correction, along with the appearance of the very large CABS basis set in the most expensive terms, means that it is significantly more costly than an MP2 calculation when using the same one-electron basis set. The SF12 approximation reduces the number of O(M) terms substantially, leading to significant cost savings for larger systems.

During testing, we noted that the size of the one-electron basis set used has an effect on the computational cost-saving, with slightly smaller speedups seen for larger basis sets (see below). Since the expensive steps of the F12 calculation (integral calculation, integral transformation, and integral contraction) all scale as O(N3) with the size of the one-electron basis sets N for a given system, one could expect that speedups should stay roughly the same as the size of these basis sets increases. The decrease in computational advantage for larger basis sets occurs due to the fact that the computational steps saved through the SF12 approximation involve only orthogonalized basis sets in which linear dependencies within the given atomic orbitals have been removed. The number of linear dependencies can become quite large in the CABS method so that expensive O(M4) scaling steps involving overdetermined atomic orbitals (integral calculation and integral transformation), which are largely unaffected by our SF12 approximation, gain more importance with increasing atomic orbital basis sets. This, however, is not a problem in practice since the intended goal of the SF12 approximation is to reduce the computational overhead for calculation of large systems with relatively small basis sets.

All reported MP2-SF12 and MP2-F12 energy calculations were performed in our program package FermiONs++.44–48 Besides the necessary use of Ten-no’s fixed amplitude ansatz,15 we applied the extended Brillouin condition (EBC)11 and thus neglect the last term in Eqs. (8) and (9). A fixed Slater type geminal (STG) correlation factor14,49 of the form F^12=1γ1exp(γr12), with γ = 1.3, was utilized in the 3*C variant36 of the explicitly correlated F12 correction. Furthermore, we employed the cc-pVXZ-F1250–52 basis set family, with the corresponding RI cc-pVXZ-F12/OptRI+42 and DF cc-pVXZ-F12/MP2fit43 basis sets.

In order to obtain reliable, well-balanced cSF12 factors for each member of the cc-pVXZ-F12 (X = D, T, Q) basis set family, which are capable of reproducing the F12 correction to the interaction energy to a high degree of accuracy, we decided to employ the S66 complexes53 for parameterization. The small- to medium-sized S66 dimers cover a broad range of organic and biologically relevant systems and incorporate different kinds of non-covalent interactions (NCIs). With 23 systems representing frequently occurring hydrogen bond donor and acceptor groups, 23 structures dominated by dispersion effects, and 20 with mixed dispersion and electrostatic interactions, respectively, we consider the S66 complexes as suitable reference for fitting SF12 energies to NCIs.

First, the MP2-F12 interaction energies for each basis set and S66 complex were once evaluated via Boys and Bernardi counterpoise correction,54 and the total and intermediate energy results of these calculations were saved. These data were subsequently used in a minimization procedure to reduce the mean percentage error (MPE) between the SF12 and F12 corrections of the S66 complexes. Starting with initial guesses of 0.9 and 1.0 for cSF12 to avoid potential local minima, the factors were stepwise increased and decreased with a systematic reduction in the stepsize close to the minimum value. Figure 1 visualizes the minimization procedure for the MPE and the corresponding mean absolute errors (MAE), demonstrating very small errors for all basis sets. We determined for the double-, triple-, and quadruple-zeta basis sets factors of 0.935, 0.95, and 0.96, respectively.

FIG. 1.

Visualization of the minimization procedure for the cc-pVXZ-F12 (X = D, T, Q) basis set family using S66 complexes. (a) MPE minimization between the MP2-F12 and MP2-SF12 corrections to the interaction energy and (b) corresponding MAEs (kcal mol−1).

FIG. 1.

Visualization of the minimization procedure for the cc-pVXZ-F12 (X = D, T, Q) basis set family using S66 complexes. (a) MPE minimization between the MP2-F12 and MP2-SF12 corrections to the interaction energy and (b) corresponding MAEs (kcal mol−1).

Close modal

1. Accuracy

The main results of the F12 and SF12 interaction energy corrections using our cSF12 factors are reported in Table I, showing MAEs, maximum errors (MAX), and MPEs for the S66 fitting set and two validation sets S228 and L7.55 Together, S22 and L7 cover a broad range of small- to medium-sized (S22) as well as large-sized structures (L7) with up to 112 atoms, including single, double, and triple bonds. In combination, a scope of NCIs such as dispersion interactions like ππ stacking, hydrogen bonds, mixed electrostatic-dispersion effects, and interaction of aliphatic hydrocarbons is represented. In general, excellent MAEs for all test and basis sets were obtained, with values of 0.004 kcal mol−1, 0.002 kcal mol−1, and 0.007 kcal mol−1 (S66, S22, L7) for a cc-pVDZ-F12 basis. Even for the large L7 complexes, highly accurate results were obtained, which indicates no size limitation of our SF12 method using the determined cSF12 factors. Only a few small aliphatic S66 hydrocarbons lead to outlying larger deviations to F12, which are visualized in Fig. 2 alongside with the errors for each system basis set combination. In total, almost no error variations are observed supporting the general applicability of SF12.

TABLE I.

MAEs, MAXs, and MPEs of the S66, S22, and L7 explicitly correlated F12 correction to the interaction energy using cc-pVXZ-F12 (X = D, T, Q) basis sets.

TestBasisMAEMAXMPE
setsetcSF12(kcal mol−1)(kcal mol−1) (%)
S66 cc-pVDZ-F12 0.935 0.004 12 0.082 22 0.627 31 
S22 cc-pVDZ-F12 0.935 0.002 42 0.007 79 0.342 08 
L7 cc-pVDZ-F12 0.935 0.007 29 0.011 29 0.475 16 
S66 cc-pVTZ-F12 0.95 0.001 86 0.023 41 1.036 43 
S22 cc-pVTZ-F12 0.95 0.001 92 0.010 77 0.535 26 
S66 cc-pVQZ-F12 0.96 0.000 95 0.011 36 0.924 76 
S22 cc-pVQZ-F12 0.96 0.001 18 0.005 33 0.692 09 
TestBasisMAEMAXMPE
setsetcSF12(kcal mol−1)(kcal mol−1) (%)
S66 cc-pVDZ-F12 0.935 0.004 12 0.082 22 0.627 31 
S22 cc-pVDZ-F12 0.935 0.002 42 0.007 79 0.342 08 
L7 cc-pVDZ-F12 0.935 0.007 29 0.011 29 0.475 16 
S66 cc-pVTZ-F12 0.95 0.001 86 0.023 41 1.036 43 
S22 cc-pVTZ-F12 0.95 0.001 92 0.010 77 0.535 26 
S66 cc-pVQZ-F12 0.96 0.000 95 0.011 36 0.924 76 
S22 cc-pVQZ-F12 0.96 0.001 18 0.005 33 0.692 09 
FIG. 2.

SF12 Errors with respect to F12 for the S66, S22, and L7 complexes using cc-pVXZ-F12 (X = D, T, Q) basis sets and our cSF12 factors. Dashed guidelines are at ±0.01 kcal mol−1, where negative values note overbinding (OB) and positive values note underbinding (UB).

FIG. 2.

SF12 Errors with respect to F12 for the S66, S22, and L7 complexes using cc-pVXZ-F12 (X = D, T, Q) basis sets and our cSF12 factors. Dashed guidelines are at ±0.01 kcal mol−1, where negative values note overbinding (OB) and positive values note underbinding (UB).

Close modal

The MPEs are considerably below 1% for the cc-pVDZ-F12 basis set and show almost no error. For larger basis sets, only S66 and S22 were employed (since some L7 complexes were too large for their computation). Here, absolute errors decrease and MPEs moderately increase. Again, SF12 introduces insignificant errors, which leads to negligible percentage errors in the correlation and total interaction energy of 0.05% and 0.03% for a cc-pVDZ-F12 basis.

Overall, the additional mean absolute errors introduced by SF12 are two orders of magnitude smaller than MP2-F12, SCS-MP2-F12, and MP2C-F12 errors compared to the CCSD(T)/aug-cc-pVTQZ (gold standard) for the S22 complexes. The most accurate of these stated methods for the S22 complexes is the MP2C-F12 approach, which results in MAEs of 0.18 kcal mol−1, 0.15 kcal mol−1, and 0.16 kcal mol−1 for the comparable aug-cc-pVXZ (X = D, T, Q) basis sets.33 Our SF12 approach introduces additional errors for all these methods of 0.002 kcal mol−1, 0.002 kcal mol−1, and 0.001 kcal mol−1, which are thus completely negligible.

2. Speedup

For comparison of the computational costs for the second-order SF12 and F12 corrections and associated therewith the applicability of our SF12 approach, the required computational times for both methods were measured for the adenine–thymine Watson–Crick complex from the S22 set, the L7 complexes, and additionally for a set of linear alkanes and amylose chains. The adenine–thymine Watson–Crick complex was calculated on 2 × Intel© Xeon© processor E5-2620 CPUs (12 cores, 2.00 GHz), while the L7 complexes as well as the linear alkanes and amylose chains were computed on 2 × Intel Xeon CPU E5-2667 v4 (16 cores, 3.20 GHz).

Table II reports the results of the S22 adenine–thymine Watson–Crick and the L7 complexes for a cc-pVDZ-F12 basis. The SF12 calculation of the explicitly correlated terms for the medium-sized adenine–thymine complex is 1.79 times faster than F12, which reduces to 1.53 and 1.14 times, for a cc-pVTZ-F12 and cc-pVQZ-F12 basis, respectively. As expected, the computation of larger systems like the L7 systems experiences a substantial speedup employing SF12. On average, the calculations of the explicitly correlated terms for these structures are by a factor of 2.41 times faster using in MP2-SF12 instead of MP2-F12, which has considerable effects on the total post Hartree–Fock computational time, e.g., the total MP2-SF12 correlation calculation with additional evaluation of the CABS+ singles correction requires for the L7 octadecane-dimer only 60% of the time MP2-F12 needs. In general, the SF12 approach combines distinct speedups with accurate explicitly correlated corrections to the interaction energy. Thus, its usage becomes highly attractive for medium- to large-sized structures.

TABLE II.

Ratios of the MP2-F12 and MP2-SF12 timings of the explicitly correlated F12 terms for the S22 adenine–thymine Watson–Crick and L7 complexes using a cc-pVDZ-F12 basis.

ComplextF12 (h)tSF12 (h)tF12tSF12
Adenine–thymin Watson–Crick 0.19 0.11 1.79 
Octadecane-dimer 3.20 1.03 3.10 
Guanine-trimer 0.56 0.25 2.19 
Circumcoronene-adenine 7.94 2.74 2.90 
Circumcoronene-guanine-cytosine 11.06 4.35 2.54 
Phenylalanineresidues-trimer 2.40 1.07 2.24 
Coronene-dimer 1.74 0.81 2.15 
Guanine-cytosin–guanine-cytosin stack 0.85 0.48 1.76 
ComplextF12 (h)tSF12 (h)tF12tSF12
Adenine–thymin Watson–Crick 0.19 0.11 1.79 
Octadecane-dimer 3.20 1.03 3.10 
Guanine-trimer 0.56 0.25 2.19 
Circumcoronene-adenine 7.94 2.74 2.90 
Circumcoronene-guanine-cytosine 11.06 4.35 2.54 
Phenylalanineresidues-trimer 2.40 1.07 2.24 
Coronene-dimer 1.74 0.81 2.15 
Guanine-cytosin–guanine-cytosin stack 0.85 0.48 1.76 

To make quantitative scaling assessments and investigate the properties of SF12 and F12 for a series of systematically increasing structures, the timings for the explicitly correlated terms for linear alkanes and amylose chains were measured for a cc-pVDZ-F12 basis, showing good speedups that increase with the molecule size. For the largest members of both sets, C60H122 and an amylose chain out of four d-glucose monomers, SF12 performs 3.60 and 2.83 times faster than F12. This has, in practice, distinct effects on the computational cost, e.g., the F12 calculation of C60H122 requires 34.78 h, which drops to 9.66 h for SF12. Since our cSF12 are fitted to interaction energy corrections, larger deviations to F12 absolute energy corrections are expected. Regardless, our factors have proven to accurately treat all kinds of NCIs, and going to chemically related complexes of these polymers such as L7 octadecan–dimer complex, SF12 results in a negligible interaction energy correction error of 0.000 65 kcal mol−1.

All evaluated F12 and SF12 timings of the alkane and amyloses are visualized in Figs. 3 and 4 as linear and log–log plots, and the scaling behavior with the system size M was determined by linear regression of the wall times between two neighboring members of the test sets. SF12 shows a noticeably decreased scaling behavior with O(M4.24) and O(N3.60) for the largest systems, where F12 leads to O(M4.67) and O(M4.10), respectively. The observed scaling of the explicitly correlated SF12 terms is smaller than the formal O(M5) scaling of F12 and SF12 in the 3*C variant.37 Lower scaling terms dominate the calculation due to the neglect of the exchange-type B terms even for large systems and thus lower the measured scaling exponent. Considering the beneficial performance enhancement and the achieved precision of SF12, which was demonstrated even for large systems, highly accurate results are expectable, and its usage enables the computation of chemical relevant systems very close to the level of F12 but in remarkably less time.

FIG. 3.

Linear plot (a) and corresponding log–log plot (b) of the wall times for the explicitly correlated F12 and SF12 corrections of linear n-alkanes (CnH2n+2, n ∈ {10, 20, 30, 40, 50, 60}) for a cc-pVDZ-F12 basis. The numbers in brackets correspond to scaling exponents between two neighboring structures.

FIG. 3.

Linear plot (a) and corresponding log–log plot (b) of the wall times for the explicitly correlated F12 and SF12 corrections of linear n-alkanes (CnH2n+2, n ∈ {10, 20, 30, 40, 50, 60}) for a cc-pVDZ-F12 basis. The numbers in brackets correspond to scaling exponents between two neighboring structures.

Close modal
FIG. 4.

Linear plot (a) and corresponding log–log plot (b) of the wall times for MP2-F12 and MP2-SF12 explicitly correlated F12 correction calculations of linear amyloses (nd-glucose, n ∈ {1, 2, 4}) for a cc-pVDZ-F12 basis. The numbers in brackets correspond to scaling exponents between two neighboring structures.

FIG. 4.

Linear plot (a) and corresponding log–log plot (b) of the wall times for MP2-F12 and MP2-SF12 explicitly correlated F12 correction calculations of linear amyloses (nd-glucose, n ∈ {1, 2, 4}) for a cc-pVDZ-F12 basis. The numbers in brackets correspond to scaling exponents between two neighboring structures.

Close modal

A simple extension to the 3*C variant of the explicitly correlated F12 correction to the second-order Møller–Plesset perturbation theory is introduced, which noticeably reduces its computational demand, especially for medium- to large-sized molecules, while retaining accurately to the MP2-F12 interaction energy reference. By scaling of the Coulomb type geminal–geminal contributions to the F12 correction and additional neglect of exchange-type integrals, we achieved a substantial computational cost reduction.

In order to determine the necessary cSF12 factors of each cc-pVXZ-F12 (X = D, T, Q) basis set, a minimization procedure was employed to reduce the mean percentage error between S66 MP2-F12 and MP2-SF12 Boys and Bernardi counterpoise corrected F12 interaction energy corrections. The obtained general cSF12 has shown some remarkably good results for the S66, S22, and L7 test set with a very low MAE of 0.004 kcal mol−1, 0.002 kcal mol−1, and 0.007 kcal mol−1 for a cc-pVDZ-F12 basis. Supported by small MPE for the explicitly correlated interaction correction, the correlation interaction energy, and the total interaction energy with values of 0.60%, 0.05%, and 0.03% for a cc-pVDZ-F12 basis, MP2-SF12 allows the computation of non-covalent interactions for all kinds of chemically relevant systems close to the level of MP2-F12. Besides the good accuracy, a noticeable computational cost reduction was observed as demonstrated for L7 complexes, linear alkanes, and amylose chains. For medium- to large-sized systems, a speedup of the explicitly correlated F12 correction to the correlation calculation by a factor of 2 and more compared to MP2-F12 are expectable using a cc-pVDZ-F12 basis.

In total, our SF12 correction is recommended for large systems and molecular complexes with a cc-pVDZ-F12 basis to incorporate the full potential of F12 theory. Future work will focus on applying our SF12 approach to dynamical studies to investigate novel chemical problems. Furthermore, we are planning to combine SF12 with different theories and fitting references to reproduce high quality, explicitly correlated quantum chemical interaction energies, e.g., the reproduction of CCSD(T)-F12 interaction energies.

See the supplementary material for energy data and timings of all test set calculations.

Financial support is gratefully acknowledged from the DFG Excellence Cluster, Grant No. EXC 2111-390814868 (Munich Center for Quantum Science and Technology—MCQST). C.O. further acknowledges financial support as Max-Planck-Fellow of the MPI-FKF Stuttgart.

The data that support the findings of this study are available within the article and its supplementary material.

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