noncovalent interactions 
 are of fundamental importance to biomolecular 
 systems , as they help determine the structures and functions of proteins 
 and nucleic acids and play a central role in molecular recognition . a reliable representation of noncovalent interactions therefore is 
 critically important to computational modeling of biomolecules , with 
 applications that include rational drug design and protein engineering . in molecular simulations , noncovalent interactions are typically 
 modeled by the nonbonded terms in an empirical force field . these account for electrostatic and van der waals interactions and 
 may also include terms to account for time - varying changes in electronic 
 polarization during the simulation . although 
 the parameters in an empirical force field are typically adjusted 
 to optimize agreement with experimental data , growing computer power 
 and a shortage of suitable experimental data are also driving increased 
 use of quantum mechanical ( qm ) calculations to parametrize and test 
 force fields . in addition , concerns regarding the accuracy of empirical force 
 fields are motivating the direct application of 
 qm methods to the study of noncovalent binding in host  guest and protein  ligand systems . it would be ideal 
 if such applications could take advantage of 
 the highly accurate qm approach often viewed as the gold standard 
 for computing noncovalent interactions , that is , counterpoise - corrected 
 couple - cluster theory with single , double , and perturbative triple 
 excitations extrapolated to the completed basis set limit . however , the computational demands of such ccsd(t)/cbs 
 cp calculations make them too time consuming for routine use in force 
 field parametrization and prohibit direct application to biomolecular 
 systems . as a consequence because all of these methods make approximations for the sake of computational 
 efficiency , it becomes essential to evaluate their accuracy . while 
 there are many studies that rely on high accuracy reference results 
 for relevant molecular systems , there is still a need for broader comparative validation studies 
 that will provide users and developers with a perspective of the strengths , 
 weaknesses , and trade - offs among the various qm approaches and their 
 applicability to specific classes of noncovalent interactions . here , therefore , we contribute a systematic assessment of accuracy 
 and speed for a range of qm methods using a reference data set of 
 over 1200 gas - phase dimers , for which ccsd(t)/cbs cp reference energies 
 are publicly available in the benchmark energy and geometry database 
 ( begdb ) . the categories of the qm method 
 examined are semiempirical , density functional theory ( dft ) with and 
 without dispersion corrections , and symmetry - adapted perturbation 
 theory ( sapt ) . the semiempirical pm6 and 
 pm7 methods , the most computationally 
 efficient methods tested here , rely on empirically adjustable parameters 
 and are often combined with additional interaction terms . we examine 
 the pm6 method , with post hoc corrections for dispersion and hydrogen 
 bonding interactions ( pm6-dh2 , pm6-dh+ ) , and halogen interactions ( pm6-dh2x ) . the pm7 method , which is based on pm6 , is 
 also included without additional corrections , as its parametrization 
 strategy accounts for such interactions . we test the widely used dft 
 functionals b3lyp , b97-d , and m062x , with and without added dispersion 
 corrections , as well as the b97x - d functional , which includes its own correction 
 for dispersion . finally , we test sapt , which is distinct from the pmx and dft approaches in that it is 
 applicable only to the calculation of noncovalent interactions ( e.g. , 
 it can not be applied to geometry optimizations ) and that it provides 
 an informative decomposition of the overall interaction energy into 
 electrostatic , induction , exchange , and dispersion components . in 
 sapt , the interaction energy is computed as an expansion of perturbative 
 terms , and we examine the sapt0 , sapt2 , sapt2 + , sapt2+(3 ) , and sapt2 + 3 
 truncations . we also explore the potential 
 for the fast sapt0 ( fsapt0 ) method to afford accurate results through 
 empirical scaling of its energy terms , much as done previously in 
 a smaller study , and make available the 
 detailed energy decompositions afforded by sapt across all of the 
 test systems , as these can be useful to guide force field parametrization . the present study provides a unique perspective 
 of the reliability and efficiency of a broad range of qm methods and 
 should be a useful guide to their selection and further improvement . a growing collection of benchmark 
 data sets provides high quality geometries and interaction energies 
 for noncovalent complexes . here , we use 
 several data sets ( table 1 ) from the begdb 
 to explore the accuracy of various qm methods for noncovalent interactions 
 spanning a range of system types and sizes . we study a total of 1266 
 dimers , ranging in size from 20 electrons in 4 atoms to 478 electrons 
 in 101 atoms ( figure 1 ) . these various begdb 
 data sets probe different classes of noncovalent interactions . in 
 particular , the s225 and s668 data sets both contain noncovalent complexes 
 categorized as hydrogen bonded ( electrostatics dominated ) , dispersion 
 dominated , or mixed electrostatic and dispersive . the l7 data set contains even larger extended complexes ; 
 all of them containing greater than 200 electrons . several of these 
 data sets , s225 , s668 , x4010 , and ionic , contain geometries generated along 
 a dissociation path relative to the equilibrium geometry and thus 
 include many nonequilibrium conformations . because the aug - cc - pvtz 
 basis set used in the present study is not applicable to iodine , we omit the 
 nine iodine - containing complexes in x4010 and term the reduced 
 data set x3110 . lastly , we also include the a24 data set , which contains small complexes sized to enable 
 comparisons of more accurate approaches that would otherwise be unfeasible 
 for larger complexes . begdb provides counterpoise - corrected ccsd(t)/cbs 
 interaction energies for all of these data sets , except for jsch , 
 which contains energies evaluated using ccsd(t)/cbs and mp2/cbs without 
 counterpoise ( cp ) correction , and l7 , which uses qcisd(t)/cbs cp . it is also worth noting that there are variations within and across 
 these begdb data sets in both the basis sets and extrapolation schemes 
 employed to obtain the cbs results . such details are not trivial and 
 can produce discrepancies as large as 0.7 kcal / mol , as elaborated 
 in the results section . note that the names match those 
 on the begdb web site , which are not necessarily consistent with the 
 corresponding publications . for example , the x40 , x4010 , and 
 ionic data sets have also been referred to as the 
 pm6 methods were examined with corrections 
 for dispersion , hydrogen bonding , and halogen interactions . pm6-dh2 and pm6-dh+ differ 
 in the hydrogen - bonding correction used , with the latter having improved 
 long - range behavior . pm6-dh2x adds an 
 empirical repulsive correction for halogen interactions to the same 
 dispersion and hydrogen - bonding corrections implemented in pm6-dh2 . the more recent pm7 method is parametrized 
 against a larger reference data set than that used for pm6 and includes 
 its own terms to account for dispersion and hydrogen bonding . the dft calculations with cp correction were carried out with the 
 aug - cc - pvtz basis set in revision c.01 of gaussian 09 . where scf calculations failed to converge using 
 default run parameters , the keyword integral = ( acc2e = 12 ) was used 
 to increase the two - electron integral accuracy . b3lyp , b97-d , m062x , and b97x - d functionals were 
 selected based on previously assessed performance for noncovalent 
 interactions . the d3 dispersion correction 
 was applied using dft - d3 , version 3.0 , to b3lyp , b97-d , and m062x using the default parameters . however , we 
 have found that for the s22 data set , aug - cc - pvtz and ( aug-)def2-qzvp , 
 without any dispersion correction , give results that are within 0.18 
 kcal / mol rmse of each other , across all of the methods examined in 
 the present study . we also observed that using the dft - d3 parameters 
 optimized for ( aug-)def2-qzvp reduced the rmse across all data sets 
 by up to 0.12 kcal / mol , compared with using those optimized for ( johnson 
 damping for the d3 correction ( d3bj ) 
 was also tested for b3lyp and b97-d using parameters optimized for 
 ( aug-)def2-qzvp ; there is no such correction available for m062x . 
 the original d3 corrections are so - called 
 because they employ a damping function for which the dispersion energy 
 approaches zero with small internuclear separations . we note that , 
 with the exception of b3lyp , all the functionals already contain some 
 treatment of dispersion . the sapt energy calculations at varying 
 orders ( sapt0 , sapt2 , sapt2 + , sapt2+(3 ) , sapt2 + 3 ) were carried out 
 with psi4 . in sapt , the total interaction 
 is computed as a sum of energy terms that are each classified as resulting 
 from electrostatic , exchange , induction , or dispersion effects . the 
 specific truncations of the sapt expansion are detailed in table s1 
 of the supporting information . due to memory 
 limitations , only lower order sapt calculations were completed for 
 larger systems . jsch was evaluated through sapt2 with the 
 exception of nine amino acid pair geometries ( f30k46 , f30l33 , 
 f30y13 , f30f49 , f30y4 , f49k46 , f49v5 , 
 f49w37 , and f49y4 ) taken from a rubredoxin crystal 
 structure ( pdb 1rb9 ) , for which only 
 sapt0 calculations were completed . on the opposite end of the system 
 size spectrum , all other data sets ( s225 , s668 , ionic , and x3110 ) 
 were evaluated through sapt2+(3 ) . we use 
 the root mean squared error ( rmse ) as the primary metric of error 
 in comparing the various computational methods . however , the mean 
 signed error ( mse ) is also provided to further characterize the performance 
 of each method ; a negative mse indicates that a method overestimates 
 the attractive interactions of a noncovalent dimer . relative error 
 is often reported in the literature , presumably because errors are 
 thought to increase with interaction energy . here , however , we saw 
 no correlation between error and interaction ( r < 0.2 for all methods ) , so relative errors are not reported . timing studies were carried out on eight cpus of a 16-cpu node , 
 which was dedicated entirely to the calculation being timed . the timings 
 of dft and sapt methods were examined by applying them in triplicate 
 to each system in the a24 data set and noting the shortest of the 
 three wall - clock times reported as elapsed this timing approach accounts for the efficiency 
 with which each method uses the eight available cpus . the timings 
 for 
 dft with d3 dispersion correction are recorded without the add - on 
 correction , as it requires negligible resources compared to the main 
 calculation . the sapt0 interaction 
 energy is the sum of seven energy terms , as detailed in table s1 of 
 the supporting information . in the fsapt0 schemes a separate scaling factor is applied to some or all of these 
 terms . the linear scaling factors for the sapt0 energy terms were 
 determined by randomly splitting the systems in all combined data 
 sets , except l7 , into two equal subsets . thus , we applied multiple linear regression 
 of the sapt0 energy terms to the corresponding ccsd(t)/cbs cp reference 
 energies for the training set to obtain a set of fitted coefficients . we then used these coefficients to compute interaction energies for 
 the test set and computed correlation coefficients and rmse for the 
 test set results . this procedure was repeated 1000 times , with different 
 random selections of the training and testing subsets . three different 
 fitting schemes were tested : fsapt0(1 ) scales all sapt0 energy terms ; 
 fsapt0(2 ) scales only the two dispersion terms , edisp(20 ) and eexch disp(20 ) , treated independently ; and fsapt0(3 ) scales 
 only the sum of the two dispersion terms , edisp(20 ) and eexch we also tried applying scaling factors 
 to sapt2 thru sapt2+(3 ) , but this did not lead to significant improvements 
 in accuracy . we tested a spectrum of 
 quantum mechanical methods , spanning semiempirical 
 ( pmx ) , dft , and sapt , by comparing their results with reference interaction 
 energies for a collection of noncovalent complexes in the gas phase . the reference collection , which comprised the a24 , ionic , jsch , l7 , 
 scai , s225 , s668 , and x3110 data sets from begdb , 
 totals 1266 entries and includes a variety of molecules nonpolar , 
 polar , ionized , and halogenated  in equilibrium and nonequilibrium 
 geometries . the supporting information provides 
 the interaction energies , calculated using the various qm methods 
 for all the systems studied , along with the corresponding begdb reference 
 energies . the present quantum mechanical results were compared 
 with the highest - accuracy 
 reference energies available in the begdb for the data sets used . these were generated with ccsd(t)/cbs calculations including counterpoise 
 corrections , except as noted in the methods section . it is worth noting that the reference energies in the s22 and s66 data sets , 
 which are more limited versions of the s225 and s668 data 
 sets used here , were recently revised based on larger basis sets , 
 additional points for the cbs extrapolation , or both . the more rigorous 
 results differ from those originally published by 0.2 and 0.1 kcal / mol 
 rmse , respectively , with maximum unsigned errors of up to 0.7 kcal / mol . given that the reference energies were not computed at such a high 
 level , they also presumably have errors similar in magnitude . this 
 uncertainty in the reference energies implies that the present study 
 can not meaningfully resolve errors less than about 0.2 kcal / mol . the following subsections provide an overview of the results , followed 
 by more detailed discussions of the pmx , dft , and sapt approaches . finally , 
 we show that a simple scaling of sapt0 energy components offers substantially 
 improved accuracy at minimal computational cost . as shown 
 in figure 2 and tabulated in table 2 , the rmse values 
 of the various quantum methods averaged across all data sets range 
 from 0.52 to 3.76 kcal / mol . the methods that yield the lowest overall 
 errors are sapt2+(3 ) and m062x , both with and without its dispersion 
 correction , but a number of methods also yield overall rmse values 
 within 1 kcal / mol . the methods that yield the highest overall errors 
 are the semiempirical ( pmx ) methods , b3lyp without dispersion correction , 
 and sapt0 . the mse values are more informative ; they show that both 
 the pmx and dft methods without dispersion corrections tend to provide 
 interaction energies more positive ( less favorable ) than the reference 
 results , while the sapt methods tend to provide interaction energies 
 more negative ( more favorable ) than the reference results . as expected , 
 supplementing the dft methods with negative dispersion energy terms 
 reduces the tendency to overestimate the interaction energy ; the resulting 
 improvement is particularly striking in going from b3lyp to b3lyp - d3 . sapt2 
 calculations were only evaluated for 134 out of 143 of the jcsh systems . higher orders of sapt calculations were 
 not completed for some data sets due to memory limitations . a24 was 
 evaluated at orders through sapt2 + 3 , scai through sapt2 + , jsch through 
 sapt2 , and all other data sets through sapt2+(3 ) . sapt2 calculations were only evaluated 
 for 134 out of 143 of the jcsh systems . the performance of the quantum approaches varies significantly 
 across the data sets , as shown in figure 2 . perhaps most striking is that the pmx methods provide substantially 
 better relative results for the s225 , s668 , scai , and 
 jcsh data sets and worse results for a24 , ionic , and x3110 . the problems for a24 and x3110 appear to arise largely from 
 errors associated specifically with halogenated molecules . the ionic 
 data set includes no halogens , however , and we speculate that the 
 errors here may trace to the lack of ionized hydrogen - bonded complexes 
 in the data sets used to parametrize the pmx methods . in addition , 
 the minimal basis sets used in the pmx methods may have difficulty 
 accounting for the strong polarization effects present in such ionized 
 complexes . other than b3lyp , lower - order sapt , and pmx , all methods 
 are within 1 kcal / mol rmse of the reference energies for all sets 
 except jsch . for jsch , all approaches yield larger errors ( note the 
 scale of the vertical axis in figure 2c ) . this 
 is perhaps not surprising because the jsch data set contains the largest 
 dimer systems , and one may expect larger systems to effectively include 
 more interactions , each potentially associated with some level of 
 error ( figure 1 ) . ranking the methods according 
 to their overall accuracy and their accuracy on each data set ( table 2 ) shows that although certain methods remain near 
 the top of the rankings across the board , the detailed ordering of 
 the methods varies across data sets . the scatter plots in figure 3 provide further 
 insight into the performance of the various approaches . all the methods 
 tested provide excellent correlation with the reference energies ( r > 0.86 ) , and not surprisingly , those with 
 the largest rmse values ( figure 2 ) also yield 
 the lowest r values ( figure 3 ) . this analysis also allows further characterization 
 of the errors associated with some of the methods . further analysis 
 indicates that each arc corresponds to the dissociation curve of one 
 dimer system , and the dimer systems that generate these arcs are ones 
 for which the pmx method yields idiosyncratically high errors , as 
 discussed below . second , most of the errors of the b3lyp method are 
 associated with dimer systems whose interactions are primarily dispersive , 
 as indicated by the red cluster of off - diagonal points . in contrast , the tendency of the sapt methods to overestimate dimer 
 affinities is largely independent of interaction type , as points of 
 all colors are found below the lines of identity in the sapt scatter 
 plots . third , adding a dispersion correction to the dft methods uniformly 
 improves the correlation , and the d3 correction performs somewhat 
 better than d2 , where the comparison is made ( b97-d versus b97-d3 ) . 
 finally , the dft methods have a weak tendency to overestimate the 
 unfavorable energy of the most repulsive interactions , while the pmx 
 and sapt methods tend to assign overly favorable energies in these 
 cases . these repulsive interactions tend to have intermediate electrostatic 
 dispersive character , as indicated by the cyan color of these points . we present only 
 those data sets to which all methods could be applied , i.e. , a24 , 
 ionic , s225 , s668 , and x3110 . each entry is colored 
 by interaction type character ( see spectrum below ) , which is defined 
 as |edisp / eelst| , where edisp and eelst are the total electrostatic and dispersion energy 
 components taken from the sapt2+(3 ) calculations . it is of interest to examine how the performance of the various 
 methods depends on whether they are applied to equilibrium versus 
 nonequilibrium geometries . data sets s225 , s668 , x3110 , 
 and ionic make such comparisons possible , as they contain dissociation 
 curves for a total of 134 dimer systems ( see methods ) . figure 4 compares the mse and rmse for 
 each method at close , equilibrium , and far separations , defined here 
 as 90% , 100% , and 200% of the equilibrium separations , respectively . the rankings of the methods for equilibrium geometries correlate well 
 with the rankings for the close geometries but poorly with rankings 
 at far separations ( table 3 ) . evaluation of qm methods 
 for equilibrium and nonequilibrium geometries . dashed 
 lines mark rmse levels 
 of 0.5 and 1.0 kcal / mol . given that the long - range interactions are smaller in absolute 
 terms , this observation suggests that a study of equilibrium geometries 
 suffices to determine which methods work best overall . on the other 
 hand , the errors rise at a short distance for all methods , so that 
 none provide excellent accuracies for the close geometries , and only 
 m062x has an rmse below 0.5 kcal / mol . at far separations , all methods 
 are within 1.0 kcal / mol rmse , and several fall under 0.2 kcal / mol 
 rmse , which is comparable to the size of errors associated with basis 
 set choice in computing the ccsd(t ) correction , as discussed above . the semiempirical pmx methods 
 are roughly comparable in accuracy to the dft - d3 and higher - order 
 sapt methods for the s225 , s668 , and jsch data sets but 
 is considerably less accurate for a24 , ionic , and x3110 , much 
 as previously noted . we conjecture that 
 this difference stems in part from the fact that the pmx methods considered 
 here , as well as their corrections ( e.g. , dh+ ) , were parametrized 
 using systems similar in character to those in the s225 , s668 , 
 and jsch data sets . in addition , as suggested by the scatter plots 
 in figure 3 , some of the larger errors of the 
 pmx methods are associated with specific systems for which they give 
 idiosyncratically poor results . in particular , the pmx methods supply 
 problematic results for bromobenzenetrimethylamine 
 and systems containing hf , i.e. , hf dimer , hfmethane , 
 hfmethanol , and hfmethylamine . accordingly , omitting these problematic cases significantly improves 
 the rmse of the pmx methods by 0.81.8 kcal / mol for the a24 
 and x3110 data sets and by 0.20.7 kcal / mol across the 
 full reference collection of data sets , as shown in table 4 . the bromine nitrogen problem , as found 
 here in the bromobenzenetrimethylamine system , is 
 a known issue for the pm6 method and is improved by the halogen (  x  ) 
 correction for pm6 or by going to the pm7 method . however , we have 
 not found previous comments on the issue for hf , and we are not aware 
 of a correction for it . the fact that hf is problematic for all of 
 the pmx methods is evident from the fact that the corresponding rmse 
 values for the a24 data set , which lacks the bromobenzenetrimethylamine 
 system , improve by 0.50.9 kcal / mol when only the hf systems 
 are omitted . when the bromobenzenetrimethylamine 
 and hf - containing systems are omitted from the pmx results , their 
 rmse values across all systems fall to 1.41.6 kcal / mol ( table 4 , bottom row , right ) . however , this improvement 
 does not significantly change their position in the rankings in table 2 . for example , although pm7 has a slightly better rmse across the entire 
 data collection than the corrected pm6 methods , it is not clear how 
 significant this difference is , as its relative performance is quite 
 inconsistent across the separate data sets ( figure 2 ) . the pm6-dh2 and pm6-dh2x methods are equivalent for all 
 systems except those containing halogens and thus produce identical 
 results for s225 , s668 , ionic , scai , and jsch data sets . as expected , using pm7 or applying the  x  correction 
 provides significant improvement in rmse for the halogen - containing 
 x3110 data set . however , removing the specific problem systems 
 mentioned in the prior paragraph essentially eliminates the advantage 
 of these more advanced methods . the utility of the  x  
 correction , which is specifically designed to improve the treatment 
 of halogens , may be examined more closely by comparing the various 
 pmx methods for the full x4010 data set , for which all systems 
 contain at least one halogen atom , and which includes both equilibrium 
 and nonequilibrium distances . pm6-dh2x is more accurate than pm6-dh2 
 at all distances , except that the  x  correction generates 
 a particularly large error ( 22.6 kcal / mol rmse ) for dimers at 80% 
 of their equilibrium distances , as shown table s2 of the supporting information . interestingly , when the 
 iodine - containing systems are omitted , to create the x3110 subset 
 of x4010 , the  x  correction yields improved or 
 equal results at all distances , and the anomalously high error at 
 short range is absent . thus , although the  x  correction 
 yields an overall improvement , it seems problematic for the particular 
 case of short - ranged interactions involving iodine . the pm7 method 
 lacks this short - range anomaly but is somewhat less accurate for the 
 iodinated compounds at longer ranges . the dft 
 methods that incorporate some treatment of dispersion show good overall 
 performance , with rmse values ranging from 0.52 to 0.83 kcal / mol . in contrast , uncorrected b3lyp yields a rather large rmse of 3.76 
 kcal / mol , and its largest errors are associated chiefly with dispersive 
 systems ( red in figure 2 ) , for which the method 
 underestimates the attractive forces . supplementing b3lyp with attractive 
 d3 dispersion corrections markedly improves the overall rmse across 
 all test systems to 0.68 kcal / mol with d3 and to 0.55 kcal / mol with 
 d3bj . for the b3lyp functional , the bj - damped version of the d3 correction 
 typically produces results closer to reference compared with zero - damped 
 d3 . interestingly , there is no marked improvement going from the two - body 
 d2 correction to the three - body d3 correction in the context of the 
 b97-d method , even for the larger systems in the jsch and scai data 
 sets , where three - body contributions are expected to be more important . furthermore , while b97-d3bj has a lower overall rmse across all systems 
 compared with b97-d3 ( 0.65 kcal / mol compared to 0.79 kcal / mol ) , the 
 former produces higher rmse values for the ionic and scai data sets . thus , it is difficult to gauge the benefit of applying bj - damping 
 over zero - damping for b97-d . the uncorrected m062x method performs 
 equally well for electrostatic and dispersive systems ( figure 3 ) , but its accuracy appears to be slightly improved 
 by supplementing it with the d3 dispersion term ( figure 2 ) . the b97x - d method includes its own dispersion correction 
 distinct from d2 or d3 , and this method ranks well across all the 
 data sets . it is perhaps worth noting that the counterpoise corrections 
 for all for dft methods ( b3lyp , b97-d , m062x , and b97x - d ) are 
 small , averaging 0.15 kcal / mol across all methods and systems . the 
 mean correction rises only slightly to 0.21 kcal / mol for nonequilibrium 
 systems at close range ( 90% of equilibrium separation ) . these corrections 
 are small , in the sense that they are similar in magnitude to the 
 uncertainty in the reference energies used here , as discussed above . 
 finally , it is worth noting that the low errors observed for d3-corrected 
 dft functionals in s225 are , perhaps , unsurprising , because 
 the same molecules in similar geometries were included in the data 
 set used to parametrize dft - d3 . the accuracy 
 of the sapt approach tends to increase 
 with order , as expected , and the higher orders are comparable in accuracy 
 to the best dft methods ( figure 2 ) . the trend 
 of increased accuracy with increased order holds for all individual 
 data sets , except ionic , for which sapt2 yields a lower rmse than 
 sapt2 + . because sapt2 + differs from sapt2 by only two dispersion terms , 
 it is interesting that the inclusion of these terms seemingly degrades 
 accuracy here . perhaps the excellent performance of sapt2 for this 
 particular data set results from a fortuitous cancellation of errors . it is worth noting that all orders of sapt tend to overestimate attractive 
 forces , regardless of system character , as evident from the negative 
 mse values in figure 2 and by inspection of 
 the scatter plots in figure 3 . this overestimation 
 is particularly marked for sapt0 , suggesting the presence of a systematic 
 error that might be mitigated by a post - calculation correction . finally , 
 because the sapt energy components can be useful for tuning individual 
 force field terms , we provide in the supporting information the detailed sapt2+(3 ) 
 decompositions for all the dimer systems studied here . we used the a24 data set to compare 
 the computational speeds of the various methods . the pmx methods all 
 finished in less than 0.02s real ( wall clock ) time on a single cpu , 
 making them over 1000 times faster than the dft or sapt methods . figure 5 plots real time against system size , as measured 
 by the number of atoms , while figure 6 plots 
 the trade - off between accuracy and computer time . overall , sapt0 is 
 clearly the fastest approach , sapt2+(3 ) is the slowest , and the dft 
 timings are rather similar to each other and to sapt2 + . the level 
 of accuracy broadly correlates with computer time , except in the case 
 of uncorrected b3lyp . results 
 are presented 
 for the a24 data set , where system size is measured by the number 
 of atoms in each dimer . accuracies are 
 presented as rmse across the a24 , ionic , s225 , s668 , and 
 x3110 data sets , while calculation times are averaged for the 
 a24 data set alone . note that the post hoc d3 dispersion corrections 
 and the sapt0 fitting require negligible calculation time . the scaling of computer time with system size was 
 examined by fitting 
 the timings for each method to a power model of the form t 
 = an , where t is real time and n is the number of atoms or electrons . as shown in figure 5 , all of the dft methods 
 except b97x - d have exponents of about 2.5 and prefactors of 
 about 0.4 . the b97x - d dft method appears to scale rather differently , 
 as its exponent and prefactor are 1.65 and 3.13 , respectively . on 
 the other hand , the r value of its fit 
 to the power model is only 0.67 , so its scaling behavior is not clearly 
 defined by these data . sapt0 scales as the number of atoms to the power 1.49 , 
 while the sapt2+(3 ) time varies as the 2.44 power . analogous trends 
 across the methods are observed when one fits the timings to the number 
 of electrons in the system , rather than the number of atoms . perhaps 
 surprisingly , however , the r values of 
 the fits are much lower , as evident in table s3 of the supporting information . of the methods 
 tested here , sapt0 is faster than all but the semiempirical pmx methods , 
 as detailed above . the fact that it decomposes the total dimer interaction 
 energy into seven contributions , which capture aspects of electrostatics , 
 exchange , induction , and dispersion , provides an opportunity to try 
 generating a fast method with improved accuracy by scaling these contributions , 
 as detailed in methods . table 5 lists the means and standard deviations of the resulting 
 scaling coefficients for the sapt0 terms across the 1000 different 
 training sets and the mean and standard deviations of the rmse and r values when the trained coefficients are applied 
 to the respective test sets . most of the scaling coefficients are 
 near unity ; the term that requires the most scaling is the eexch disp(20 ) term . scaling all terms in fsapt0(1 ) produces 
 the lowest test set rmse , followed by scaling the dispersion terms 
 individually in fsapt0(2 ) , and then by scaling the summed dispersion 
 terms in fsapt0(3 ) . the fact that these results are obtained on test 
 sets not used to set the coefficients means that the improvement in 
 performance for the more highly parametrized models do not reflect 
 overfitting . the accuracy of the three scaling schemes is also compared 
 with the various qm methods in figures 2 and 3 and tables 2 and 3 . across all data sets , except l7 , applying scaling 
 factors to the sapt0 terms reduces the rmse from 1.58 kcal / mol to 
 as low as 0.47 kcal / mol and corrects the tendency of sapt0 to overestimate 
 the attractive nature of the dimer interactions . indeed , the fitted 
 sapt0 results approach the accuracy of the dft methods , with the differences 
 within the estimates of ccsd(t)/cbs basis set choice errors ( above ) . note that this improvement in sapt0 , through the application of simple 
 scaling factors , incurs negligible additional computational cost , 
 so that the fsapt0 scaling methods provide a particularly favorable 
 combination of efficiency and accuracy , as shown in figure 6 . figure s1 of the supporting 
 information furthermore examines the accuracy of fsapt0 , as 
 well as the other qm methods , for the large noncovalent complexes 
 of the l7 data set ; the results are generally consistent with those 
 obtained for the other data sets . the energy components of the fitted 
 sapt0 method still correlate well with the corresponding energy components 
 calculated at the sapt2+(3 ) level , as detailed in table s4 of the supporting information . the good agreement suggests 
 that the energy decomposition derived using the scaled terms is still 
 physically meaningful . three different 
 fitting schemes 
 were tested : fsapt0(1 ) scales all terms ; fsapt0(2 ) scales only the 
 two dispersion terms , edisp(20 ) and eexch disp(20 ) , treated independently ; and fsapt0(3 ) scales only the sum of the 
 two dispersion terms , edisp(20 ) and eexch the scaling factors were determined over 1000 iterations of multiple 
 linear regression on randomly selected training subsets of the dimer 
 systems , while rmse and r were evaluated 
 over the same iterations using test subsets comprising all dimer systems 
 not included in the training subset . the present study systematically evaluates the accuracy and speed 
 of a broad range of electronic structure methods for estimating noncovalent 
 interaction energies . methods spanning pmx , dft , and sapt were applied 
 to over 1200 geometries of gas - phase dimers drawn from the begdb resource , 
 which is tailored to probe a variety of interaction motifs relevant 
 to biomolecules and drug - like compounds . these results offer useful 
 guidance regarding which methods are most suitable for various types 
 of applications where  gold - standard  ccsd(t)/cbs cp 
 calculations are too time consuming or impractical , as now discussed . the pmx methods studied here are dramatically faster than both 
 the dft and sapt approaches , and they are more readily applied to 
 larger molecules . however , they are in general less accurate , particularly 
 for halogenated and ionic molecules , as well as for a few types of 
 systems with idiosyncratic results . perhaps 
 surprisingly , none of the various pmx methods tested here are clearly 
 superior to the others in terms of overall accuracy . the dft methods 
 are slower and more difficult to apply to large systems , but they 
 can achieve high accuracy , so long as dispersion is accounted for , 
 either implicitly , as in m062x , or via an add - on term , as in b97-d3 . the performance of the sapt approach depends strongly on the order 
 of the sapt expansion . the sapt2 + and sapt2+(3 ) orders span the range 
 of accuracy seen for the dispersion - corrected dft methods . however , 
 while the speed of the sapt2 + method is comparable to that of the 
 dft methods , the more accurate sapt2+(3 ) is considerably slower . it 
 is also worth noting that , at least in the current psi4 software , 
 the memory requirements of sapt at orders higher than sapt0 can become 
 problematic for the larger systems examined here . the lowest order 
 of sapt , sapt0 , is similar in accuracy to the pmx methods but significantly 
 slower . however , we find that a simple empirical scaling of one or 
 more sapt0 energy terms leads to accuracy approaching that of the 
 best dft methods , at far less computational cost . with further development , 
 an empirically adjusted sapt0 approach might provide a powerful alternative 
 to dft methods for the study of noncovalent interactions in larger 
 systems . the results of this study have implications for improving 
 the treatment 
 of noncovalent interactions in molecular modeling , as qm calculations 
 are used to guide the development of force fields for simulation and 
 may even replace force fields in some applications . the more accurate 
 dft and dft - d3 methods maybe most suitable for force field parametrization 
 given their reliability and consistency across many types of molecular 
 systems and the fact that their moderate computational cost is not 
 a major liability for this application . despite the high speed of 
 the pmx methods , their lower accuracy , especially for ionic systems 
 and halogens , along with occasional idiosyncratic performance , makes 
 them less suitable for parametrization of force fields . however , continued 
 development of such semiempirical methods , including training on broader 
 data sets , remains promising . in addition , these methods may already 
 be more accurate than typical simulation force fields , so their high 
 speed makes them a reasonable choice for direct modeling of biomolecular 
 systems . the higher order sapt methods are about as accurate as dft 
 but are relatively slow , while sapt0 is fast but inaccurate . interestingly , 
 the scaled sapt0 method offers a promising blend of accuracy and computational 
 speed , especially for larger molecular systems . in addition , the present 
 scaling approach is relatively simple , and more sophisticated schemes 
 that account for geometry and chemistry might be even more accurate 
 at minimal computational cost .
quantum mechanical ( qm ) calculations 
 of noncovalent interactions 
 are uniquely useful as tools to test and improve molecular mechanics 
 force fields and to model the forces involved in biomolecular binding 
 and folding . because the more computationally tractable qm methods 
 necessarily include approximations , which risk degrading accuracy , 
 it is essential to evaluate such methods by comparison with high - level 
 reference calculations . here , we use the extensive benchmark energy 
 and geometry database ( begdb ) of ccsd(t)/cbs reference results to 
 evaluate the accuracy and speed of widely used qm methods for over 
 1200 chemically varied gas - phase dimers . 
 in particular , we study the 
 semiempirical pm6 and pm7 methods ; density functional theory ( dft ) 
 approaches b3lyp , b97-d , m062x , and b97x - d ; and symmetry - adapted 
 perturbation theory ( sapt ) approach . for the pm6 and dft methods , 
 we also examine the effects of post hoc corrections for hydrogen bonding 
 ( pm6-dh+ , pm6-dh2 ) , halogen atoms ( pm6-dh2x ) , and dispersion ( dft - d3 
 with zero and becke  johnson damping ) . 
 several orders of the 
 sapt expansion are also compared , ranging from sapt0 up to sapt2 + 3 , 
 where computationally feasible . 
 we find that all dft methods with 
 dispersion corrections , as well as sapt at orders above sapt2 , consistently 
 provide dimer interaction energies within 1.0 kcal / mol rmse across 
 all systems . 
 we also show that a linear scaling of the perturbative 
 energy terms provided by the fast sapt0 method yields similar high 
 accuracy , at particularly low computational cost . 
 the energies of 
 all the dimer systems from the various qm approaches are included 
 in the supporting information , as are the full sapt2+(3 ) energy decomposition 
 for a subset of over 1000 systems . 
 the latter can be used to guide 
 the parametrization of molecular mechanics force fields on a term - by - term 
 basis .