fast 
 quantum - chemical methods are indispensable for computationally 
 demanding calculations of electronic properties of large molecules . 
 in the 1970s and 1980s , semiempirical quantum - chemical ( sqc ) methods 
 were the workhorse in computational studies of ground - state properties . 
 since the 1990s , they have largely been replaced in such studies by ab initio and density functional theory ( dft ) approaches , 
 which are typically slower by at least 3 orders of magnitude but also 
 generally more accurate and robust . however , even nowadays , 
 sqc methods often remain the only practical 
 choice when treating huge molecules or very large numbers of molecules 
 or when performing extensive molecular dynamics ( md ) simulations , e.g. , in the context of quantum mechanics / molecular mechanics 
 ( qm / mm ) studies on large biomolecular systems .
examples 
 of recent sqc applications include the calculation of electronic properties 
 used in 3d - qsar models ( 3d - quantitative structure  activity 
 relationship ) , qm - based computer - aided 
 drug design , the study of band gaps , uv / vis spectra , and charge - transfer processes 
 in systems relevant for molecular nanoelectronics and organic photovoltaics , the investigation of local properties used to understand electron 
 and hole transport mechanisms in transistors , gas - phase md simulations of electron impact mass spectra , ground - state qm / mm md simulations of enzymes , excited - state 
 nonadiabatic dynamics simulations of organic chromophores , and studies of organic and enzymatic reactions 
 in solution .
most of the widely used sqc methods are variants of the mndo 
 model which is based on the nddo ( neglect of diatomic 
 differential overlap ) 
 integral approximation .
these mndo - type 
 methods include am1 , pm3 , mndo / d , am1 * , rm1 , pddg / mndo and pddg / pm3 , pm6 , and pm7 ( see the cited references for details and the meaning of the acronyms ) . 

a common feature of these approaches is that they attempt to improve 
 the accuracy within the confines of the given electronic structure 
 model , mainly by adding modifications to the core repulsion functions 
 and by performing a more thorough and extensive parametrization .
prime 
 examples of this strategy are the pmx methods ( pm3 , 
 pm6 , and pm7 ) .
all 
 mndo - based methods solve the hartree  fock secular equations 
 as if the basis set were orthogonal , without explicitly accounting 
 for the terms arising in the ab initio treatment 
 when transforming the fock matrix from the original nonorthogonal 
 to an orthogonal basis .
this neglect of 
 orthogonalization terms gives rise to several qualitative deficiencies 
 of standard sqc methods that can not always be eliminated simply by 
 parametrization .
it has been shown that 
 reintroduction of the overlap matrix into the secular equations can 
 indeed significantly improve the accuracy of the mndo method . in the orthogonalization models ( omx ) developed in our group ,
orthogonalization corrections are included 
 into the fock matrix to different extent ( along with other interactions 
 of similar size ) , which leads to the following general - purpose sqc 
 methods : om1 , om2 , and om3 .
the theoretical formalism , 
 the optimized parameters , and the initial validation of the omx methods are described in detail in a companion article .
both the mndo - type and omx methods formally neglect 
 dispersion which causes their poor performance in systems where dispersion 
 interactions play an important role .
in 
 recent years , a number of empirical dispersion corrections have been 
 developed and successfully applied for dft methods , e.g. , the d2 and d3 corrections proposed by grimme .
the same types 
 of dispersion corrections can also be combined with the omx methods , with no change in the omx parameters 
 and only minor adjustment of the d2 and d3 parameters .
the resulting omx - dn methods provide a much improved treatment of 
 noncovalent interactions .
their formalism , parameters , and initial validation are described 
 in detail in ref ( 42 ) . among the mndo - type methods , the latest variant ( pm7 )
incorporates 
 explicit dispersion corrections in its definition , and unlike in the 
 case of the omx - dn methods , its 
 parameters were optimized with these corrections included .
given the severe approximations in the 
 formalism of sqc methods 
 and the presence of empirically determined parameters , it is essential 
 to validate these methods carefully against reliable experimental 
 data and/or accurate high - level theoretical results . some validation 
 studies are available for omx methods , which cover 
 both ground - state and excited - state properties . they indicate that the 
 omx methods outperform other sqc methods in most 
 cases
. some of this validation 
 work also offers comparisons between sqc methods and standard dft 
 approaches showing that the former may approach or sometimes even 
 exceed the accuracy of dft results for ground - state properties of 
 organic molecules . considering the extensive recent benchmark 
 exercises for ab initio and dft methods
, it is obvious 
 that a more comprehensive 
 validation of the omx and omx - dn methods is required to establish their reliability and 
 to allow for a more detailed assessment of their accuracy compared 
 with other sqc methods .
the objective of this article is to provide 
 such validation for ground - state properties .
we cover standard mndo - type 
 sqc methods ( mndo , am1 , pm3 , pm6 , and pm7 ) , omx methods 
 ( om1 , om2 , and om3 ) , and dispersion - corrected omx - dn methods ( om2 and om3 with d2 and d3 corrections 
 as well as d3 t corrections with additional three - body terms ) .
the 
 omx and omx - dn methods 
 are fully specified in ref ( 42 ) .
in addition to our own validation sets , we use benchmark 
 sets mostly 
 taken from the available ab initio and dft literature . 

we include from these sets only those reference molecules that contain 
 the elements h , c , n , o and/or f , for the simple reason that the omx methods have up to now only been parametrized for these 
 elements . we address the following ground - state properties : heats 
 of formation , bond lengths , bond angles , dihedral angles , relative 
 energies , reaction energies , dissociation energies , atomization energies , 
 proton affinities , activation barriers , vertical and adiabatic ionization 
 potentials ( ips ) , adiabatic electron affinities ( eas ) , dipole moments , 
 noncovalent interaction energies , and geometries of noncovalent complexes . 
 overall , the benchmarks contain 13035 original or derived reference 
 data .
the results from the benchmarking will help to establish the 
 accuracy of different sqc methods in different areas and can also 
 serve as a guide for choosing the most appropriate sqc method in specific 
 application projects .
all mndo , am1 , pm3 , omx , and omx - dn calculations were carried out with 
 our locally 
 modified mndo2005 program .
standard conventions were used in all calculations . for the sake of documentation , we specify some 
 of these standard options here . the convergence criterion for the 
 scf energy
was set to 10 ev ; in addition , the 
 diagonal elements of the density matrix were required to be converged 
 to 10 in mndo2005 .
doublet 
 and triplet states were treated with the restricted open - shell half - electron 
 approach , and higher spin states were 
 described using unrestricted hartree 
we did not use molecular mechanics corrections for peptides in pm6 
 or pm7 , and we did not apply any cutoffs for the three - center orthogonalization 
 corrections in the omx and omx - dn methods .
geometry optimizations were considered 
 converged when the gradient 
 norm became smaller than 0.01 kcal/(mol ) . in mndo2005
the bfgs optimization algorithm was used by 
 default ; in difficult cases , we also applied eigenvector following , 
 with the hessian being computed numerically by one - sided finite differences 
 of the gradient ( at the first step and every 10 following steps ) and 
 with a minimum trust radius of 0.00001 . in mopac the eigenvector following algorithm was used throughout ; in most 
 cases the full hessian matrix was constructed and recalculated every 
 10 steps using numerical single - sided derivatives of the gradient . 

frequently mopac stopped when the heat of formation 
 remained essentially constant in subsequent cycles ; in these cases 
 the gradient norm was usually smaller than 0.1 kcal/(mol ) .
it is well - known that sqc methods generally predict the heat of 
 formation of the proton with very large errors .
therefore , in line with common semiempirical practice , we used the experimental value of 367.171 kcal / mol for the heat of formation of the proton at 298 
 k to calculate the proton affinities included in the gmtkn30-chnof 
 and ce345-chnof databases ( see later text ) .
the reference molecules 
 in the various benchmark sets are generally 
 still quite small by semiempirical standards .
therefore , they were 
 computed using single - cpu serial versions of the mndo2005 and mopac programs . for large - scale applications , 

parallel versions of mndo2005 are available 
 using shared - memory and distributed - memory message - passing parallelization , as well as a hybrid version using graphics processing units ( gpus ) .
similarly , there are shared - memory and gpu - parallelized 
 versions of mopac . to illustrate the scope of sqc calculations
, we note that the hybrid 
 gpu version of mndo2005 has been used to compute 
 water clusters of up to 5400 atoms and to optimize the geometries 
 of a large series of representative proteins .
calculations with d3 and d3 t dispersion corrections were done 
 using 
 our interface of the mndo2005 program to the dft - d3 stand - alone program by grimme ( versions 3.0 rev 
 2 and 3.1 rev 0 ) .
default cutoffs ( 95 
 au for two - body terms and 40 au for coordination numbers ) were used 
 with version 3.0 and increased cutoffs ( 95 au for two - body terms , 
 coordination numbers , and three - body terms ) with version 3.1 .
in this section we define the benchmark sets 
 used presently to 
 evaluate different sqc methods .
we divide these sets into two groups : 
 those designed to benchmark ground - state properties in general ( ovs7-chnof , 
 g2g3-chnof , w4 - 11-chnof , gmtkn30-chnof , ce345-chnof , pddg , pm7-chnof , 
 and c7h10o2 ) and those specifically designed to benchmark noncovalent 
 interactions ( a24-chnof , s66 , s66a8 , jsch-2005-chnof , s7l , s30l - chnof , 
 and af6 ) .
we note that this division is somewhat arbitrary as some 
 reference molecules and complexes from the first group also feature 
 noncovalent interactions and may thus appear again in the second group .
all data sets and subsets from the first and second groups are 
 listed in tables 1 and 2 , respectively .
most of them are taken from the 
 literature and described in the same terms as in the published work . 

included are all molecules from the original data sets that contain 
 only the elements c , h , n , o , and/or f , for which omx parameters are available ; all other 
 molecules are ignored .
detailed numerical results on all individual 
 reference molecules 
 from all benchmark sets are compiled in the supporting information ( si ) which also cites the origin of the published 
 reference data . in the following two sections
, we will focus on statistical 
 evaluations of these results for all sets and subsets ( with more than 
 two molecules ) .
data in these databases were normally 
 corrected to represent total energies without any zero - point vibrational 
 energies and thermal corrections ; this is also true for the  expt  
 and  theor  values given for the subsets ;  theor  
 usually means  best theoretical estimate  .
intermolecular 
 distances were obtained 
 from mp2/cc - pvtz geometries by interpolating estimated ccsd(t)/cbs 
 energies along dissociation curves .
reference geometries were obtained 
 by optimizing intermolecular distances at the ccsd(t)/ha - cc - pvdz level 
 using monomer geometries optimized at the b3lyp level with large basis 
 sets . following semiempirical tradition ,
all sqc methods 
 tested presently 
 were parametrized with regard to heats of formation .
their performance 
 can thus be evaluated in a straightforward manner when reliable reference 
 data for heats of formation are available either from experiment or 
 from high - level ab initio calculations .
this is the 
 case for the chno and fluor data sets used during the parametrization 
 of the omx methods and 
 for the ovs7-chnof , g2g3-chnof , pddg , and pm7-chnof sets used in the 
 present benchmarking .
performance evaluation of atomization enthalpies 
 at 298 k is also straightforward for the c7h10o2 set as reference 
 values can be easily obtained from the reference ab initio enthalpies at 298 k. most of the entries in the other databases 
 of tables 1 and 2 represent relative energies , for example isomerization 
 energies , 
 interaction energies , binding energies , reaction energies , barrier 
 heights , proton affinities , ionization potentials , and electron affinities . 

the corresponding ab initio reference data are generally 
 obtained from energy differences , whereas the respective semiempirical 
 values are differences of heats of formation ( i.e. , enthalpies at 
 298 k ) that implicitly include zero - point vibrational energies ( zpves ) 
 and thermal enthalpic corrections ( from 0 to 298 k ) .
these corrections 
 are normally quite similar for related molecules since they tend to 
 be bond - specific and transferable , and 
 hence they will cancel to a large extent when comparing related systems . 
 as in many semiempirical studies over the past decades
, we will thus 
 mostly ignore the distinction between relative energies and heats 
 of formation in the following , and consider the semiempirical values 
 to be directly comparable with ab initio relative 
 energies .
this line of reasoning obviously breaks down when 
 it comes to atomization 
 energies , because there is no longer any cancellation . for a realistic 
 comparison between ab initio and semiempirical atomization 
 energies , the zpves and thermal enthalpic corrections
this is computationally demanding at an accurate ab initio level , and therefore 
 we computed both these corrections at the sqc level using the harmonic - oscillator 
 and rigid - rotor approximations . to illustrate the necessity of including 
 these corrections , we compare the om2 results for propane with experimental 
 and ab initio w4 reference data : the experimental heat of formation at 298 k ( 25.0 
 kcal / mol ) is well - reproduced by a straight om2 calculation ( 24.4 
 kcal / mol at the ab initio reference geometry ) , whereas 
 the w4 atomization energy ( 1007.9 kcal / mol ) is well - reproduced by 
 om2 ( 1005.6 kcal / mol ) only after applying the corrections described 
 previously , which are far from negligible ( 51.8 kcal / mol ) .
the distinction 
 between relative energies and heats of formation is also relevant 
 for the reaction energies in other fragmentations , e.g. , bond dissociation reactions : while the corresponding zpve and thermal 
 enthalpic corrections are much smaller than for atomization energies , 
 they are still non - negligible . in the next section
, we will 
 apply these sqc corrections to all 
 benchmarks for atomization energies ( especially w4 - 11-chnof ) .
we will 
 also investigate their effect on the reaction energies covered by 
 the four subsets of the w4 - 11-chnof set and by those subsets of the 
 gmtkn30-chnof and ce345-chnof sets that address fragmentation reactions . 

the sqc corrections are combined with single - point sqc results obtained 
 at the ab initio reference geometries .
the conversion 
 of semiempirical heats of formation at 298 k to zpve - exclusive relative 
 energies at 0 k is described in detail in the supporting information .
the omx methods have been parametrized using the 
 relatively small chno and fluor training sets ( 140 and 48 molecules , 
 respectively ) . for these sets ,
the omx results are generally found to be superior to the results 
 from other sqc methods , as documented in detail in a companion article .
various omx parametrization 
 runs starting from different initial values were 
 often found to produce different parameter sets that gave results 
 of similar quality for the molecules in the training set ; therefore 
 we performed further tests on several validation sets to help identify 
 the most suitable parameters .
the main 
 validation sets are collected in the ovs7-chnof benchmark which consists 
 of seven subsets .
detailed numerical results are given in the si ( tables s9s14 ) , while statistical 
 evaluations in terms of mean absolute errors ( maes ) are provided in table 3 .
with regard 
 to heats of formation , om2 and om3 outperform all other sqc methods 
 for six of the seven subsets , namely , radicals71
, anions24 , cations41 , bigmol20 , isomers44 , and fluorine91 ; the corresponding maes for om2 and om3 are 5.05.6 , 8.49.6 , 
 ca .
6.9 , ca . 5.0 , 1.11.8 , and 7.27.3 kcal / mol , respectively . 

in the conformers30 subset , om1 gives 
 the lowest mae ( 1.8 kcal / mol ) followed by pm7 ( 2.4 kcal / mol ) and om2 
 and om3 ( ca .
considering relative energies , om2 and 
 om3 again outperform all other sqc methods in the case of the radicals71 
 and cations41 subsets , and they also give very good results for the 
 isomers44 subset ( maes of 0.82.1 kcal / mol ) .
ionization potentials for the radicals71 
 subset ( from koopmans 
 theorem ) are best predicted by the three omx methods , 
 with maes of 0.40.5 ev .
this is also true for the barriers 
 in the conformers30 subset , for which the maes for the omx methods range between 1.3 and 1.5 kcal / mol . in the fluorine91 subset
, 
 bond lengths are well - described by om1 , om2 , pm3 , and pm6 , while bond 
 angles are best reproduced by the omx methods .
finally , we note that the inclusion of dispersion corrections must 
 deteriorate the accuracy of the omx methods for heats 
 of formation , because the omx parameters were optimized 
 without such corrections .
since dispersion effects are always attractive , 
 their subsequent introduction will systematically lower the computed 
 heats of formation and thus increase their errors compared with experiment . 

on the other hand , dispersion corrections have no or very little effect 
 on ionization potentials and barriers ; they improve the relative energies 
 for the isomers44 subset while leaving those for the radicals71 and 
 cations41 subsets almost unaffected ( for further details see the si table s1 ) .
the g2 and g3 sets provide well - established 
 accurate reference data that have often been used to validate the 
 performance of ab initio and dft methods .
they have 
 been published after the development of om1 and om2 . in previous work
we have reported an evaluation of the omx methods for the g2 and g3 sets , and we can thus be brief . 

the statistical results are summarized in table 4 ( for detailed numerical results , see si tables s16s18 ) .
om2 yields the lowest maes for 
 the heats of formations in g2 and g3 ( 3.23.4 kcal / mol ) and 
 for the relative energies in the alkanes28 subset of the g3 set ( 0.6 kcal / mol ) , while om3 gives the lowest mae for 
 the heats of formations in alkanes28 ( 0.7 kcal / mol ) . as pointed 
 out before
, the performance of om2 and 
 om3 for the g2 and g3 sets is respectable , even when compared with 
 standard dft approaches , and there is no systematic error for alkane 
 chains of increasing length that plagues some standard dft functionals 
 ( including b3lyp ) .
again , as in the ovs7-chnof set and for the same reasons , 
 dispersion 
 corrections deteriorate the accuracy of the omx methods 
 for heats of formation in the g2g3-chnof set , while improving relative 
 energies ( si table s2 ) .
the w4 - 11 benchmark set includes the w4 - 08 subset of the gmtkn30 set 
 ( see later discussion ) and , in addition , contains numerous further 
 atomization energies ( collected in the extended subset tae140 ) as 
 well as four other subsets with reaction energies ( bde99 , hat707 , 
 isomer20 , and sn13 ; see table 1 ) .
we reduced the w4 - 11 to the w4 - 11-chnof data set in the 
 usual manner , by eliminating entries for species containing elements 
 other than h , c ,
n , o , and f. the sqc results from single - point calculations 
 at the reference geometries are compared with the reference data obtained 
 from ab initio total energies without zero - point 
 vibrational corrections . for reasons discussed above , we converted 
 the semiempirical heats of formation to atomization energies at 0 
 k by applying zpve and thermal enthalpic corrections .
we also compared 
 the results for the reaction energies in the subsets calculated with 
 and without such corrections ( see table 5 for the statistical evaluation ) .
( corr ) means that energies are obtained 
 by removing zpve and thermal corrections from the sqc results ( see 
 text ) .
the atomization energies 
 in the tae140 subset are computed with 
 errors somewhat larger than typical errors for heats of formation . 

these errors are of similar magnitude in the omx and 
 pm7 methods ( maes of 4.817.02 kcal / mol , om2 lowest ) .
the maes 
 remain roughly of similar size when multireference cases are excluded 
 ( tae_nonmr124 , maes of 4.846.75 kcal / mol , om2 lowest ) .
errors in the corrected and uncorrected reaction energies for heavy - atom 
 transfer ( hat707 subset ) , for isomerization ( isomer20 subset ) , and 
 for transformations in the sn13 subset are very close to each other , 
 without any systematic improvement due to the corrections .
this provides 
 further justification for the usual practice of directly comparing 
 the semiempirical scf energies with ab initio relative 
 energies in such cases .
thus , we will discuss for these sets and similar 
 benchmark sets of this kind only the noncorrected reaction energies 
 calculated from the semiempirical scf energies . on the other 
 hand , removing the zpve and thermal corrections from 
 the semiempirical scf energies systematically improves the computed 
 bond dissociation energies ( bde99 subset , mae lowered by ca .
this indicates that these corrections are generally relevant for fragmentation 
 reactions , and hence we will discuss them for such transformations 
 in the following .
the reference energies for bond dissociation 
 and heavy - atom transfer 
 are best reproduced by om2 ( maes of 6.25 and 8.92 kcal / mol ) followed 
 by om3 ( maes of 7.51 and 9.44 kcal / mol ) .
isomerization energies are 
 computed reasonably well by all sqc methods considered ( maes of 7.659.37 
 kcal / mol , am1 lowest ) .
the reaction energies in the sn13 subset are 
 also well - reproduced , particularly by pm7 , om1 , and om3 ( maes of 3.14 , 
 4.02 , and 4.31 kcal / mol , respectively ) .
the largest outliers 
 in the computed atomization energies are found 
 for c2 ( om1 ) , isocyanic acid hnco ( om2 ) , hnnn ( om3 ) , and 
 h2 and n2 ( pm7 ) , with errors ranging from 21 
 to 35 kcal / mol .
dispersion corrections have only very little 
 effect on any of the 
 statistical results reported presently ( see si table s19 ) .
the grimme group published 
 comprehensive collections of validation sets for general main group 
 thermochemistry , kinetics , and noncovalent interactions , to allow for a thorough assessment 
 of the performance of various dft methods . in previous work , we extracted from the initial gmtkn24 database all species containing only the elements h , 
 c , n , and o. for the resulting gmtkn24-hcno database , we performed 
 calculations using the am1 , pm6 , om1 , om2 , om3 , scc - dftb , b3lyp , and 
 pbe methods and their dispersion - corrected counterparts to compare 
 their accuracy in a comprehensive manner . here
we update our previous evaluation of sqc methods by starting 
 from the more recent gmtkn30 database and extracting 
 all species containing only the elements h , c , n , o , and f. the resulting 
 gmtkn30-chnof database differs from the previous gmtkn24-hcno variant 
 in the following aspects .
( b ) it includes three new subsets isol22 , bsr36 , and adim6 
 from gmtkn30 ; the other three new subsets in gmtkn30 contain elements 
 other than h , c , n , o , and f and are therefore disregarded .
( c ) it 
 also includes the full w4 - 08 subset with 
 multireference ( mr ) cases , in addition to the previously used w4 - 08womr 
 subset without mr cases .
( d ) it utilizes 
 updated , more accurate reference data from the gmtkn30 database whenever 
 available ( in subsets idisp , s22 , bhperi , and pa ) .
specifically , the 
 gmtkn30 reference data for the bhperi subset are now based on w1 rather than cbs - qb3 calculations ; the latter had been employed in gmtkn24-hcno 
 for some reaction barriers .
the gmtkn30 reference values for the proton 
 affinities of small molecules in the pa set are now taken from vibrationally 
 back - corrected w1 data rather than from 
  best estimates  from vibrationally back - corrected experimental 
 values in gmtkn24-hcno .
overall , 
 the gmtkn30-chnof database consists of 24 subsets with 
 480 reference data that are determined from 806 single - point calculations 
 at the reference geometries .
six subsets from the 
 full gmtkn30 database are missing ( al2x , nbprc , alk6 , rg6 , heavy28 , 
 and cyconf ) , because they consist of species containing elements not 
 yet parametrized for the omx methods .
detailed 
 numerical results for the gmtkn30-chnof database are documented 
 in the supporting information .
generally 
 speaking , our present results support and generalize the conclusions 
 on the performance of sqc methods obtained previously for the gmtkn24-hcno 
 set .
overall , the omx and omx - dn methods provide the 
 most accurate results .
the maes for 
 the entire gmtkn30-chnof set are lowest for om3 ( 7.17 kcal / mol ) followed 
 by om2 ( 7.94 kcal / mol ) , but are substantially higher for the pmx methods ( 14.4416.49 kcal / mol ) and am1 ( 16.45 kcal / mol ) . 

inclusion of dispersion corrections for om2 and om3 changes the overall 
 maes only very slightly ( om3-dn , 7.217.26 
 kcal / mol ; om2-dn , 7.767.91 kcal / mol ) .
compared to the previously reported uncorrected values in ref ( 52 ) ( w4 - 08womr subset ) , the 
 semiempirical scf atomization enthalpies at 298 k become much more 
 accurate by correcting to zpve - exclusive atomization energies at 0 
 k ( errors reduced by ca .
the omx methods remain the most accurate sqc methods in the w4 - 08womr subset 
 ( maes of 4.126.00 kcal / mol ) . for the full w4 - 08 subset ,
the 
 maes of the corrected atomization energies are somewhat higher at 
 the omx level ( 4.197.58 kcal / mol ) and still 
 slightly larger for pm6 and pm7 ( 6.537.95 kcal / mol ) .
( corr ) means that 
 energies are obtained 
 by removing zpve and thermal corrections from the sqc results ( see 
 text ) . without mb08 - 165 
 and c20 . upon geometry optimization , 

some 
 of the artificial molecules adopted structures very different from 
 the reference geometries so that the computed corrections may be less 
 accurate in these cases ( see text ) .
the adduct o3 + c2h2 is better described as an open - shell 
 singlet 
 at omx - dn .
subset dc9 without c20 bowl / cage isomerization 
 energy . c20 bowl / cage 
 isomerization 
 energy . for some sqc methods 

reference dissociation energies 
 of four ( h2o)20 clusters were updated with more 
 accurate values from ref ( 132 ) ( see text ) . (
corr ) means that 
 energies are obtained 
 by removing zpve and thermal corrections from the sqc results ( see 
 text ) . without mb08 - 165 
 and c20 . upon geometry optimization , 

some 
 of the artificial molecules adopted structures very different from 
 the reference geometries so that the computed corrections may be less 
 accurate in these cases ( see text ) .
the adduct o3 + c2h2 is better described as an open - shell 
 singlet 
 at omx - dn .
subset dc9 without c20 bowl / cage isomerization 
 energy . c20 bowl / cage 
 isomerization 
 energy . for some sqc methods 

reference dissociation energies 
 of four ( h2o)20 clusters were updated with more 
 accurate values from ref ( 132 ) ( see text ) . in 
 the case of the mb08 - 165 subset ,
the 
 zpve and thermal corrections reduce the errors for all of the 
 methods except for mndo .
this subset is generally most challenging 
 as it was especially designed to test the robustness of computational 
 approaches .
it consists of the dissociation energies of randomly generated 
 species with unusual bonding situations to diatomics and hydrides . 

for some of the artificial molecules in this subset , the geometry 
 optimization ( needed to evaluate the zpve and thermal corrections ) 
 led to structures very different from the reference geometries ( especially 
 with mndo ) , and hence to corrections that may be less accurate ; the 
 corrected values were included in the statistics also in these cases . 

the omx methods have the smallest maes for both the 
 uncorrected ( 19.4622.47 kcal / mol , om1 and om3 lowest ) and 
 the corrected ( 11.0415.03 kcal / mol , om1 lowest ) dissociation 
 energies of the mb08 - 165 subset , while the maes for pm6 and pm7 exceed 
 100 kcal / mol . in the case of the bsr36 subset , going from the 
 scf bond separation 
 enthalpies at 298 k to the corresponding zpve - exclusive energies at 
 0 k reduces the errors only for the om2 , om3 , and omx - dn methods .
the reason for the increased error 
 with the other methods is the way in which this benchmark is constructed : 
 bond separation energies refer to reactions of hydrocarbons with methane 
 to produce ethane molecules ; thus large numbers of ch4 and 
 c2h6 molecules are involved ( up to 22 and 18 , 
 respectively ) , and any errors in the computed zpve and thermal corrections 
 for these two molecules will accumulate .
this happens , for instance , 
 in the case of pm7 where the mae for the bsr36 subset drops from 17.40 
 to 9.08 kcal / mol ( similar to the value of 9.63 kcal / mol without corrections ) 
 when using the zpve energies of methane and ethane calculated at the 
 w4 level instead of the pm7 values .
such 
 problems are not encountered with the om2 , om3 , and omx - dn methods which give realistic zpve and thermal 
 corrections for methane and ethane .
om3 is most accurate for the bsr36 
 subset , both for the uncorrected ( mae of 3.46 kcal / mol ) and the corrected 
 ( mae of 1.90 kcal / mol ) bond separation energies . as noted previously , the cage / bowl 
 isomerization of c20 fullerene is also a very challenging 
 problem for sqc methods : the absolute error in the isomerization energy 
 exceeds 85 kcal / mol for all sqc methods tested here .
c20 is part of the dc9 subset with only seven members ; to avoid misleading 
 impressions because of the c20 outlier , we provide maes 
 not only for the complete dc9 subset but also for dc9woc20 ( dc9 without 
 c20 ) , as well as the errors for c20 . if we remove the most challenging 
 test cases ( mb08 - 165 and c20 ) 
 from the overall statistics , the maes remain lowest for om3 ( 6.30 
 kcal / mol ) and om2 ( 6.95 kcal / mol ) , whereas those for the pmx methods drop considerably ( 9.6011.26 kcal / mol ) ; 
 am1 benefits less ( 14.65 kcal / mol ) .
inclusion of dispersion corrections 
 for om2 and om3 again affects their maes only slightly . according 
 to the statistical evaluations ,
the omx and omx - dn methods are most accurate 
 for barrier heights of hydrogen transfers , heavy - atom transfers , nucleophilic 
 substitutions , unimolecular reactions , and association reactions ( bh76 ) and for the corresponding reaction energies ( bh76rc ) ; and for selected reaction energies in the g2 set ( g2rc ) .
the om2 , om3 , and omx - dn methods also perform best for adiabatic ionization potentials 
 ( g21ip ) and electron affinities ( g21ea ) ; for isomerization energies of large organic 
 molecules ( isol22 ) ; and for relative 
 energies of phenylalanyl  glycyl  glycine tripeptide conformers 
 ( pconf ) and of sugar conformers ( sconf ) .
the lowest maes are provided for proton affinities ( subset 
 pa ) 
 by om1 followed by om3 and om3-dn ; for systems with 
 intramolecular dispersion interactions ( idisp ) by om2 , om3 , and om3-dn ; for interaction energies between n - alkane dimers 
 ( adim6 ) by omx - d3 , omx - d3 t , and pm7 ; and for relative energies of alkane conformers ( aconf ) by omx - dn .
pm6 is statistically most accurate for systems with problems 
 related 
 to the dft self - interaction error ( sie11 ) ; for other molecules that are difficult cases for dft ( dc9 and dc9woc20 ) ; for reaction energies and barriers of ozone addition 
 to c2h4 and c2h2 ( o3add6 ) ; and for energies of diels  alder reactions 
 ( darc ) .
pm7 appears best suited 
 for calculating barrier heights of pericyclic 
 reactions ( bhperi ) ; isomerization energies of small- and medium - sized 
 organic molecules ( iso34 ) ; and binding 
 energies of noncovalently bound dimers ( s22 ) . in the latter case , the omx - dn methods are of similar accuracy ( see section 5 ) .
benchmark energies for four ( h2o)20 clusters 
 used in the water27 subset ( binding energies of water , h(h2o)n , and oh(h2o)n clusters ) were taken from mp2/cbs calculations .
recently more accurate reference energies were 
 obtained at the ccsd(t)/cbs(45 ) level of theory for those four complexes . when including these updated reference data , 
 the pm7 , om2-dn , and om3-d2 methods perform similarly 
 well ( maes of 6.206.57 kcal / mol , om2-d3 lowest ) .
interestingly , 
 radical stabilization energies ( rse43 ) are best predicted by the venerable am1 method 
 closely followed by om1 .
this demonstrates that the most recent scq 
 models and parametrizations may not always be the best for a problem 
 at hand .
hence , if possible , sqc methods should generally be calibrated 
 against available experimental or high - level theoretical data for 
 related systems before using them in actual applications . as 
 already discussed earlier
the largest outliers ( si figure s1 and table s20 ) are often found for 
 species in the mb08 - 165 subset ; this is the case for pmx , om2 , and om2-dn , with especially large errors 
 of more than 100 kcal / mol seen frequently with pmx . in the case of the om3 and om3-dn
methods , the 
 largest errors are encountered for the isomerization energy of c20 , followed by species in the mb08 - 165 subset .
other notable 
 outliers ( figure 1 ) 
 occur for om1 in the water27 subset ( four errors larger than 100 kcal / mol ) ; 
 for om2 in the pa subset ( error of 58 kcal / mol in the proton 
 affinity of molecular hydrogen ) ; and for om3 in the o3add6 subset 
 ( error of 45 kcal / mol in the energy of adduct formation between o3 and c2h2 ) . according to the statistical 
 evaluations , the omx - dn methods 
 have essentially the same overall accuracy 
 as the standard omx methods .
as expected , the dispersion 
 corrections significantly improve the accuracy for systems , in which 
 noncovalent interactions play a significant role .
the barriers of pericyclic 
 reactions ( bhperi subset ) are also slightly improved when using the 
 omx - dn methods , on average by more 
 than 1 kcal / mol . on the other hand , as already noted , inclusion 
 of dispersion corrections 
 ( without any change in the standard omx parameters ) 
 will lower the total energies systematically and thus tend to increase 
 the errors in the calculated heats of formation .
interestingly , the 
 dispersion corrections even deteriorate the results for the idisp 
 subset designed to capture the effects of intramolecular dispersion . 

apparently , intramolecular noncovalent interactions of this kind are 
 already largely taken into account ( on average ) by the general parametrization 
 of omx methods such that adding further dispersion 
 interactions is detrimental .
overall , the positive and negative 
 effects of including empirical 
 dispersion corrections ( while retaining the standard omx parameters ) seem to cancel out in the case of the gmtkn30-chnof 
 benchmark set so that the overall maes are similar for the omx and omx - dn methods .
the chemistry energetic 
 database with 345 entries ( ce345 ) is a collection 
 of accurate reference data used for the evaluation and parametrization 
 of dft functionals .
we note that some 
 of the reference data in the ce345 database are also present in the 
 gmtkn30 database , although the actual reference values may differ 
 slightly because of the use of different methodology for obtaining 
 them .
we reduced the ce345 database 
 to the ce345-chnof set by accepting only species containing exclusively 
 the elements h , c , n , o and/or f. therefore , the srmbe13 subset had 
 to be excluded .
the mrbe10 and dc9 subsets were also excluded from 
 the error analysis because there are only two surviving species in 
 each subset .
the final ce345-chnof set consists of 11 subsets with a 
 total of 187 entries . error distribution of ground - state properties calculated 
 at the 
 omx and pm7 levels for the gmtkn30-chnof benchmark 
 set ( excluding the mb08 - 165 subset and the isomerization energy of 
 c20 ) .
the subsets are marked with alternating gray and 
 white backgrounds , and their numbers correspond to those in table 6 .
overall , the lowest maes for the ce345-chnof set 
 are obtained 
 from the omx and omx - dn methods ( 6.177.91 kcal / mol ) . as expected
, the inclusion 
 of dispersion corrections improves the results for the ncce31/05 subset 
 targeting noncovalent complexation energies ; this is also true for 
 the subsets abde12 , hc7/11 , and tc13 that contain systems in 
 which dispersion plays a prominent role .
( corr ) means that energies are obtained 
 by removing zpve and thermal corrections from the sqc results ( see 
 text ) .
scf calculations 
 of the nh3f2 complex in the 
 ncce31 subset could 
 not be converged with om2 and om3 ; thus we excluded this complex from 
 the statistics .
looking 
 at the statistical evaluation for the individual subsets , 
 atomization energies ( after correcting scf atomization enthalpies 
 at 298 k to zpve - exclusive energies at 0 k , mgae109/11 ) are found to be best reproduced by omx and omx - dn methods ( maes of 4.194.98 
 kcal / mol , om2 and om2-dn lowest ) .
isomerization 
 energies ( isol6/11 ) 
 are best reproduced by om2 and pm7 ( mae of ca .
we note that this subset contains the six smallest pairs 
 of isomers from the isol22 subset of gmtkn30 ; their isomerization 
 energies were calculated at a more accurate level of theory than those 
 in isol22 .
( corr ) means that 
 energies are obtained 
 by removing zpve and thermal corrections from the sqc results ( see 
 text ) .
scf calculations 
 of the nh3f2 complex in the 
 ncce31 subset could 
 not be converged with om2 and om3 ; thus we excluded this complex from 
 the statistics .
the subsets 
 for adiabatic ionization potentials ( ip21 ) and electron affinities ( ea13/03 ) overlap with the corresponding gmtkn30-chnof subsets ( g21ip and 
 g21ea ) . in all cases ,
the om2 and om3 methods perform best ( maes of 
 0.520.57 ev for ip21 and 0.400.42 ev for ea13/03 ) , 
 with substantially larger errors for the other sqc methods .
as expected , 
 dispersion corrections have essentially no effect on the om2 and om3 
 results for these properties .
all entries in the subset for 
 proton affinities ( pa8/06 ) also appear 
 in the pa subset of gmtkn30 ( albeit 
 with slightly different reference values ) . as in the pa case ,
om1 
 has the lowest statistical error ( mae of 4.8 kcal / mol ) .
alkyl bond 
 dissociation energies ( abde12 ) are best reproduced by the omx - dn and omx methods ( maes after accounting for zpve 
 and thermal corrections of 7.3210.52 kcal / mol and above 17 
 kcal / mol for the mndo - type methods ) , while hydrocarbon chemistry ( hc7/11 ) is best described by om3-dn and pm6 ( maes of 3.204.34 and 4.57 kcal / mol , respectively ) . 

the thermochemistry of  systems ( tc13 ) is best treated by the om2 
 and om2-dn methods ( maes of 2.54 and 2.732.74 
 kcal / mol , respectively ; the data set includes some proton affinities 
 of polyenes that are part of the pa subset of gmtkn30 ) .
the 
 entries in the subsets for hydrogen and non - hydrogen transfer 
 reactions ( htbh38/08 and nhtbh38/08 ) are also present in 
 the bh76 subset of gmtkn30 . om2 and
om2-dn perform 
 best for the barrier heights to hydrogen transfer followed by om3 
 and om3-dn ( maes of 4.954.96 and 5.996.61 
 kcal / mol , respectively ) , while the barrier heights to non - hydrogen 
 transfer are described less well ( lowest mae of om1 9.60 kcal / mol ) . as expected , noncovalent complexation energies ( ncce31/05 ) are best reproduced by the omx - dn methods ( maes of 1.081.51 kcal / mol ) followed by pm7 ( maes 
 of 1.58 kcal / mol ) .
the error distributions of the omx and pm7 methods 
 with regard to the ce345-chnof set are shown in figure 2 .
they appear similar overall ; however , there 
 are more outliers for pm7 compared with the omx methods 
 and especially with om2 .
error distribution of ground - state properties 
 calculated at the 
 omx and pm7 levels for the ce345-chnof set .
the subsets 
 are marked with alternating gray and white backgrounds , and their 
 numbers correspond to those in table 8 .
the largest outliers 
 at the omx and omx - dn levels have errors between 40 and 
 63 kcal / mol in the computed energies .
these are the ionization energy 
 of the carbon atom ( om1 ) ; the proton affinity of molecular hydrogen 
 ( om2 and om2-dn ) ; and the barrier of the hcn  
 hnc isomerization ( om3 and om3-dn ) ; see the supporting information .
the pddg set was used in the 
 development and validation of the pddg / mndo and pddg / pm3 methods .
it contains 979 experimental reference data 
 for heats of formation , bond lengths , bond angles , dihedral angles , 
 ionization potentials , and dipole moments of 622 closed - shell molecules 
 ( elements h , c , n , ond o ) .
the pddg set is divided in the original 
 work into ch , chn , cho , and chno subsets .
in addition , we introduce 
 subsets for bond lengths , bond angles , and dihedral angles according 
 to the elements involved . in our evaluations ,
the resulting maes 
 for the different properties are summarized in table 10 for the mndo , am1 , pmx , and omx methods and in si table s3 for the omx - dn methods .
considering heats of formation , the pddg / pm3 method gives the lowest 
 mae for the pddg set ( 3.2 kcal / mol ) , which is not too surprising since 
 it was trained on this set .
slightly 
 higher are the maes for pm7 ( 3.34 kcal / mol ) , om2 ( 3.55 kcal / mol ) , 
 and om3 ( 3.68 kcal / mol ) , while those for the other sqc methods are 
 above 4.0 kcal / mol .
the om2 and om3 methods systematically overestimate 
 the heats of formation in the pddg set ( mean signed errors ( mses ) 
 of 0.92 and 1.08 kcal / mol , respectively ) , more so than pm7 ( mse of 
 0.49 kcal / mol ) .
error histogram 
 of heats of formation calculated at the omx and pm7 
 levels of theory for the pddg benchmark set .
as noted previously , post - scf dispersion corrections deteriorate 
 the accuracy of heats of formation calculated by sqc methods parametrized 
 at the scf - mo level ( compare table 10 and si table s3 ) .
this 
 shortcoming can be avoided by simultaneously fitting the parameters 
 of the sqc method and of the dispersion corrections , as has been done 
 in pm7 ; this may contribute to its good 
 performance for the pddg set .
considering geometries , all tested sqc methods perform 
 reasonably 
 well for bond lengths ( overall maes of 0.0110.018  , 
 lowest for pm3 and pm7 ) . except for mndo ,
this is also true for bond 
 angles ( overall maes of 1.902.15 , lowest for om3 ) .
dihedral 
 angles are described best by the omx methods ( overall 
 maes of 5.78.5 , lowest for om3 ) .
ionization potentials 
 are reproduced best by om2 ( mae of 0.37 ev ) ; 
 the other sqc methods have somewhat larger errors ( 0.480.70 
 ev ) .
the maes in the calculated dipole moments are in the range between 
 0.24 and 0.38 d ; the am1 , pm3 , and omx methods provide 
 the best estimates ( maes of 0.240.27 d ) .
the pm7-chnof benchmark 
 set was assembled from the online database of reference data that have been used for the development and validation 
 of the pm7 method .
it includes all species 
 of the database that consist only of the elements h , c , n , o , and 
 f. it contains 1595 experimental and high - level ab initio reference values for heats of formation , bond lengths , bond angles , 
 ionization potentials , and dipole moments of 1177 neutral and charged , 
 open - shell and closed - shell species .
it is divided into the da , ch , 
 chn , cho , chf , and chno subsets .
the diatomic da subset consists of 
 the h2 , n2 , o2 ( triplet and singlet ) , 
 and f2 molecules . in our evaluations ,
the resulting maes 
 for the different properties are summarized in table 11 for the mndo , am1 , pmx , and omx methods and in si table s4 for the omx - dn methods .
the pm7 method was trained and validated on this benchmark set , 
 and it is thus not surprising that pm7 shows the best overall accuracy 
 for the heats of formation ( mae of 3.78 kcal / mol , mse of 0.18 
 kcal / mol ) ; its error distribution is narrow and symmetric ( figure 4 ) .
the omx methods tend to overestimate the heats of formation for 
 this set , especially om1 ( figure 4 ) . om2 and om3 perform similarly ( maes of 4.834.85 
 kcal / mol ) .
they outperform pm7 in the da and chf subsets and have 
 similar accuracy in the ch subset , while pm7 is superior in the chn , 
 cho , and chno subsets . again , for the same reasons as discussed before , 
 post - scf dispersion corrections deteriorate the accuracy of the heats 
 of formation calculated by the omx methods ( compare table 11 and si table s4 ) .
error histogram for heats of formation calculated 
 at the omx and pm7 levels of theory for the pm7-chnof 
 benchmark set . concerning the other 
 properties , the conclusions for the pm7-chnof 
 set are similar to those for the pddg set ( see preceding discussion ) . 

bond lengths and bond angles are generally reproduced reasonably well 
 by all sqc methods , and there are no great differences in the overall 
 accuracy of the computed ionization potentials and dipole moments . 
 for bond lengths ,
pm7 has the lowest overall mae ( 0.013  ) followed 
 by pm3 , pm6 , om1 , and om2 ( 0.015  in each case ) .
the omx methods have a slight overall advantage over the other 
 sqc methods for bond angles ( maes of 1.481.76 , om1 lowest ) , 
 ionization potentials ( 0.330.53 ev , om2 lowest ) , and dipole 
 moments ( 0.220.26 d , om3 lowest ) .
accurate thermochemical calculations 
 at the g4mp2 level of theory were recently 
 performed for a set of 6095 constitutional isomers c7h10o2 .
they were drawn 
 from the chemical universe database gdb-17 that contains 166.4 billion 
 molecules with up to 17 non - hydrogen atoms .
we derived atomization enthalpies at 
 298 k for these 6095 c7h10o2 isomers 
 from the corresponding g4mp2 enthalpies reported in ref ( 123 ) .
the reported geometries 
 were obtained at the b3lyp/6 - 31g(2df , p ) level of theory , which is not considered accurate enough to 
 be used as reference in our present benchmarking .
the omx and 
 pmx methods perform similarly , with maes ranging 
 between 6.30 and 8.92 kcal / mol ( om2 lowest , closely followed by pm7 ) . 

dispersion corrections have essentially no effect on the atomization 
 enthalpies for the molecules in the c7h10o2 set ( si table s26 ) . upon the request 
 of a reviewer , we have carried out analogous rm1 calculations for 
 all benchmark sets considered in preceding discussion .
the statistical 
 evaluations of the rm1 results are given in the supporting information ( tables s34s41 ) along with 
 a brief assessment .
overall rm1 tends to be generally more accurate 
 than am1 , about as accurate as the pmx methods , and 
 somewhat less accurate than the omx methods .
noncovalent 
 interactions are difficult to describe for standard 
 sqc methods which do not include dispersion in their formalism and 
 often also have well - documented problems with hydrogen bonding . in 
 this section , we will focus on the pm7 and omx - dn methods with explicit dispersion corrections , while presenting 
 the results for the other ( less accurate ) sqc methods only in the supporting information .
we selected seven 
 data sets from the literature to benchmark the 
 accuracy of sqc methods for noncovalent interactions .
as usual , we 
 excluded in these sets the species containing elements for which omx parameters are not yet available .
thus , three of the data 
 sets were reduced : a24 from 24 to 21 
 entries in a24-chnof , jsch-2005 from 
 143 to 134 entries in jsch-2005-chnof , and s30l from 30 to 24 entries in s30l - chnof .
the remaining four 
 sets were not changed ( s66 , s66a8 , s7l , and af6 ) .
we first report results from single - point 
 calculations 
 at the reference geometries for all seven data sets .
thereafter we 
 present results from sqc geometry optimizations for the a24-chnof , s66 , s7l , and af6 sets , 
 for which high - level reference geometries are available .
we note in 
 this context that the s66a8 set is specifically designed for single - point 
 calculations .
the reference geometries in 
 the jsch-2005-chnof set are taken from different sources and include 
 experimental , purely theoretical , and combined experimental / theoretical 
 data , with purely theoretical geometries constituting 
 only a minor part ; many of these reference geometries are not minima 
 in the gas phase , which would make it meaningless to compare them 
 with optimized gas - phase sqc geometries . in the validation studies 
 with full geometry optimization ,
the reason 
 is simple : if a method fails to predict the correct geometry for a 
 noncovalent complex , the computed interaction energy will be meaningless . 

hence , we primarily examine the most important intermolecular distances 
 and angles for each complex ( except for af6 where we check the intramolecular 
 geometry parameters in the folded conformation ) .
the statistical 
 evaluation of the interaction energies obtained from single - point 
 sqc calculations is given in table 13 for the pm7 and omx - dn methods and in si table s5 for all other 
 methods .
the omx - dn methods 
 perform best overall for the a24-chnof set ( maes of 0.560.69 
 kcal / mol , om2 lowest ) , while pm7 has somewhat larger errors ( mae of 
 1.10 kcal / mol ) .
the most problematic case in a24-chnof is the hfhf 
 complex : at the nonoptimized reference geometry , the reference interaction 
 energy is attractive ( 4.57 kcal / mol ) ; om2 and om2-dn give only a rather weak attraction ( 1.47 to 1.15 
 kcal / mol ) , whereas mndo , pm6 , pm7 , om1 , om3 , and om3-dn yield a repulsive interaction energy ( up to 5.81 kcal / mol in pm7 ) ; 
 see the discussion in section 5.2 .
when this complex is disregarded in the statistics , 
 the errors drop further ( maes of omx - dn 0.430.49 kcal / mol and pm7 0.64 kcal / mol ) . considering the 
 subsets ,
the omx - dn methods outperform 
 pm7 for complexes with dispersion - dominated and mixed electrostatic / dispersion 
 interactions , and the om3-dn methods are best for 
 h - bonded complexes . in the s66 set , the pm7 and omx - d3 methods are 
 of similar accuracy ( maes of 0.79 vs 0.82 kcal / mol ) , whereas the latter 
 are superior in the s66a8 set ( maes of 0.78 vs 0.600.62 kcal / mol ) ; 
 the omx - d2 treatments are generally somewhat less 
 accurate . in all three sets ,
the omx - dn methods outperform pm7 for the complexes with dispersion - dominated 
 and mixed interactions . in the jsch-2005-chnof set ( interaction 
 energies of dna base pairs 
 and amino acid pairs ) , the omx - d2 results are again 
 slightly less accurate than the omx - d3 results , and 
 among the latter , om3-d3 has generally the lowest or close - to - lowest 
 errors . om3-d3
also outperforms pm7 for the jsch-2005-chnof set overall 
 ( maes of 1.33 vs 1.93 kcal / mol ) as well as for two of the subsets 
 ( intrastrand and stacked base pairs ) , whereas the maes are similar 
 in the case of h - bonded base pairs ( maes of 2.53 vs 2.62 kcal / mol ) 
 and higher for om3-d3 in the case of amino acid pairs ( maes of 2.56 
 vs 2.19 kcal / mol ) .
the largest outliers generally occur for pairs 
 of oppositely charged amino acids : when these six complexes are excluded 
 from the analysis , the errors drop substantially , as can be seen from 
 the maes for om3-d3 and pm7 for the neutral amino acid pairs ( maes 
 of 0.49 vs 1.17 kcal / mol ) and for the overall set ( maes of 1.10 vs 
 1.83 kcal / mol ) .
evidently , om3-d3 performs best for the jsch-2005-chnof 
 set . without the hfhf 
 complex ,
two complexes correspond to both 
 h - bonded and charged complexes subsets ; see ref ( 49 ) . in a similar vein ,
om3-d3 describes the  
 and  interactions for large complexes ( s7l set ) 
 significantly better than any of the other sqc methods considered . 

the corresponding mae value for om3-d3 ( 0.66 kcal / mol ) is small on 
 an absolute scale and much lower than that for pm7 ( 4.91 kcal / mol ) .
the s30l set is an expansion of the 
 s12l set and consists of very large noncovalent complexes , 
 which are deemed useful to check  against overfitting to too 
 small molecular cases  .
the omx - dn methods perform fairly well ( maes 
 of 3.596.69 kcal / mol , om3-d3 t lowest ) .
they outperform other 
 sqc methods investigated here and in ref ( 49 ) , and especially pm7 , which strongly overestimates 
 most interaction energies ( mae of 15.44 kcal / mol ) .
the omx - dn methods generally describe  
 stacking interactions better than h - bonding interactions ; however , 
 even in the latter case , they are more accurate than pm7 , which appears 
 to suffer from an accumulation of errors in the very large noncovalent 
 complexes of the s30l - chnof set .
the largest outliers in the omx - dn results are found for positively charged 
 complexes , as previously reported for om2-d3 and om3-d3 in the case 
 of the s12l set .
the af6 set provides 
 reference folding energies and enthalpies 
 for a series of alkanes .
the pm7 values 
 should be compared to the enthalpies , since pm7 was parametrized against 
 heats of formation . in the case of the omx - dn methods
, one may argue that the computed values can be 
 considered as enthalpies or as energies , since the parameters for 
 the underlying omx methods and for the dispersion 
 corrections were calibrated against heats of formation and interaction 
 energies , respectively . to be unbiased ,
we provide data for both types 
 of comparisons , and the results differ only very slightly ( see table 13 ) .
the pm7 and omx - dn methods have rather small errors both 
 for the folding enthalpies ( maes of 0.201.18 kcal / mol , om2-d2 
 lowest , pm7 highest ) and energies ( maes of 0.181.05 kcal / mol , 
 om2-d2 lowest , pm7 highest ) .
however , more important than these statistics 
 is a qualitative finding : only om2-d2 , om3-d2 , and om3-d3 correctly 
 predict that the hairpin conformation of alkanes becomes more favorable 
 than the linear conformation for 17  1 carbon atoms , in agreement 
 with experiment , while the hairpin conformation 
 is more favorable already for alkanes with less than 16 carbon atoms 
 in the case of pm7 and om2-d3 .
finally , we briefly compare the 
 effect of the different dispersion 
 corrections applied in the omx - dn calculations . in the large majority of cases ,
the d3 correction 
 turns out to be superior to the d2 correction , albeit often only by 
 a small margin .
the role of three - body corrections is negligible in 
 small complexes ( a24-chnof , s66 , and s66a8 sets ) and still quite small 
 but not entirely negligible in hydrogen - bonded and stacked base pairs 
 ( jsch-2005-chnof ) and in folded alkanes ( af6 ) .
as expected , three - body 
 corrections become significant in stacked systems modeling  
 interactions between graphene layers ( s7l ) and in very large noncovalent 
 complexes ( s30l - chnof ) . since om2-d3 and om3-d3 underestimate the 
 attractive  dispersion interactions , the repulsive 
 three - body corrections increase the corresponding errors ( on average 
 by 0.7 kcal / mol ) . on the other hand , om2-d3 and om3-d3
tend to overestimate 
 the binding energies in most of the very large noncovalent complexes , 
 and hence the repulsive three - body corrections decrease the error 
 in this case by 0.310.77 kcal / mol for the s30l set , as already 
 reported earlier for the s12l set ( om2-d3 
 and om3-d3 ) and for the s30l set ( om2-d3 ) .
full geometry 
 optimizations were carried out for the a24-chnof , s66 , s7l , and af6 
 sets using all sqc methods examined presently .
the ab 
 initio reference geometries were taken from published work 
 at the following levels : ccsd(t)/cbs extrapolation for a24-chnof ; counterpoise - corrected mp2/cc - pvtz for s66 
 with intermolecular distances obtained by interpolating estimated 
 ccsd(t)/cbs energies along dissociation curves ; intermolecular distances optimized at the ccsd(t)/ha - cc - pvdz level 
 with monomer geometries optimized at the b3lyp level with large basis 
 sets for s7l ; and mp2/cc - pvtz for af6 . in the original work on the a24 set , three 
  stacked complexes were not optimized but constrained 
 artificially ; therefore they are not 
 considered here .
the statistical evaluation of selected optimized 
 interatomic distances 
 and bond angles is presented in table 14 for the pm7 and omx - dn methods and in si table s6 
 for the other sqc methods .
the corresponding statistical data for 
 the associated interaction energies are documented in the supporting information ( tables s7 and s8 ) .
we 
 note that some of the noncovalent complexes are present in several 
 data sets , with slightly different reference geometries .
we 
 start with some general remarks before discussing individual 
 data sets and complexes .
the selected distances and angles mostly 
 refer to intermolecular geometrical parameters , e.g. , distances defining the separation between the monomers or the length 
 of hydrogen bonds and angles characterizing the relative orientation 
 of the monomers in a complex .
the associated minima on the potential 
 energy surface are usually rather extended and flat , and hence the 
 deviations of the sqc results for the geometrical parameters from 
 the ab initio reference values are expected to be 
 much larger than in the intramolecular case .
inspection of table 14 confirms that this 
 is indeed the case : the maes are typically of the order of 0.3  
 for the intermolecular distances and 10 for the angles .
the overall statistics in table 14 for the omx - dn methods 
 indicate that the om3-based results are mostly superior to the om2-based 
 results , the d3 correction normally performs better than the d2 correction , 
 and the three - center dispersion correction usually has no significant 
 influence on the geometries of the rather small complexes considered 
 here ( d3 vs d3 t ) .
in an 
 overall assessment , om3-d3 outperforms pm7 for the a24-chnof 
 set ( maes of 0.26 vs 0.38  and 8.7 vs 12.9 ) and the s66 
 set ( maes of 0.32 vs 0.41  and 13.2 vs 15.8 ) , while it 
 shows somewhat larger errors for the s7l set ( maes of 0.39 vs 0.33 
  ) and the af6 set ( maes of 0.17 vs 0.09  and 7.1 vs 6.6 ) . 

om3-d3 works especially well for the dispersion - dominated complexes 
 in the a24-chnof set ( maes of 0.02  and 1.2 ) .
we 
 now address individual sets and complexes . in the case of the 
 a24-chnof set ,
all sqc methods considered give qualitatively reasonable 
 complex geometries in general ( realistic shape and intermolecular 
 distances within 1  of the reference values ) , except for 
 mndo which often fails qualitatively and yields much too large intermolecular 
 separations .
a problematic case is the hf dimer : the reference 
 geometry has 
 a linear hydrogen bond ( f  hf ) , which is also 
 found with pm7 ( but much too long ) whereas the omx and omx - dn methods give cyclic 
 hydrogen - bonded structures ( si figure s2 ) . 
 if the angle f 
hf is constrained to the reference 
 value , the hf molecules drift apart upon sqc geometry optimization : 
 the hydrogen bond is too long by 0.160.18  at om2 and 
 om2-dn , by 0.510.56  at om3 and om3-dn , and by 1.041.05  at pm6 and pm7 .
thus , 
 when calculations are run on the reference geometry , the sqc methods 
 either underestimate the binding energy ( om2 and om2-dn ) or even predict a repulsive interaction ( om3 , om3-dn , and especially pm6 and pm7 ; see section 5.1 ) . as discussed previously ,
the methane dimer is also problematic : om3-d3 
 gives a qualitatively correct geometry , while other sqc methods fail 
 in this regard .
the reference interaction energies are derived 
 in the original 
 work on the a24 set without accounting 
 for the deformation of the monomers ( i.e. , from single - point calculations 
 on monomers at the geometry of the respective complex ) . however , for 
 small monomers such as those in the a24 set , these deformation energies 
 are negligible , and the interaction energies computed with and without 
 them are essentially the same for all tested sqc methods ( si table s27 ) . in the s66 set , the omx - dn methods 
 again give the lowest maes for the distances and angles , but there 
 are some notable outliers , in particular the benzeneacoh 
 ( oh- ) , h2omeoh , c6h6menh2 , peptidepeptide , 
 and t - shaped pyridine dimer complexes ( for their optimized reference 
 and sqc structures see si figures s3s7 ) . 

the pm7 geometries are reasonable for some of these complexes , presumably 
 due to the inclusion of special h - bond correction terms ( see si figures s3s7 ) .
finally , a severe failure 
 of the om3 and om3-dn methods is that they give symmetric 
 structures for carboxylic acid dimers , in which two hydrogen atoms 
 are equidistant to the oxygen atoms in the corresponding hydrogen 
 bonds .
this is found for the acetic acid dimer in the s66 set and 
 for the formic acid dimer in the validation of omx and omx - dn methods .
since the monomer moieties in such dimers are 
 strongly distorted , their interaction energies are strongly overestimated 
 ( see si table s28 for acetic acid dimer ) . in the s7l set ,
pm7 gives overall the best intermolecular distances ; 
 om3-d3 has slightly smaller errors for the hh distances . 

a specific problem is that pm7 optimizes the parallel displaced naphthalene 
 dimer to the sandwich conformation , contrary to omx - dn . in the af6 set ,
the overall statistical 
 errors of the pm7 and omx - dn methods 
 are rather similar , with slight 
 advantages for pm7 .
all sqc methods without explicit dispersion corrections 
 fail qualitatively since mndo , am1 , pm6 , and the omx methods predict the most favorable conformation to be linear for 
 all alkanes while pm3 predicts the opposite .
only the om2-d2 and om3-dn methods give the correct result that the linear conformation 
 dominates for short chain lengths and that the hairpin conformation 
 becomes more favorable for alkanes with 17  1 carbon atoms , 
 in agreement with experiment .
at the end of such a large and diverse 
 benchmark study , it is impossible 
 to provide a single unbiased 
  grand error estimate  for each sqc method .
the chosen 
 benchmark sets strongly differ in their size and in the quality of 
 the reference data , and they address different properties .
thus , instead 
 of attempting to provide such grand error estimates , we summarize 
 the results for all subsets and identify the methods which are best 
 for a given subset / property combination and which are most robust 
 in general .
this information can be used as a guide for selecting 
 the sqc method that performs best for applications of the kind considered 
 here .
tables 15 and 16 present the benchmark results for 
 ground - state 
 properties and noncovalent interactions , respectively ; they list the 
 two top - performing methods with their maes for all data sets and subsets 
 ( as described in tables 1 and 2 ) .
maes for 
 energies in kcal / mol , for bond lengths in  , for angles in degrees , 
 for ionization potentials in ev , and for dipole moments in d. if maes 
 differ by less than ca .
0.1 kcal / mol for energies or 0.01  for 
 bond lengths , the corresponding methods are listed together .
mae of dispersion - corrected counterpart 
 ( om2-dn or om3-dn ) is very close 
 ( within 1.0 kcal / mol for energies and within 1.0 for angles ) 
 and may actually be slightly lower . without the isomerization energy 
 of c20 ; see text .
only the omx - dn and pm7 methods are considered here 
 since the other sqc 
 methods do not adequately describe noncovalent interactions .
if maes 
 differ by less than 0.1 kcal / mol for energies or 0.01  for distances 
 or 1.0 for angles , the corresponding methods are listed together . without the hfhf 
 complex
single - point 
 calculations on reference 
 geometries . deviations 
 upon optimization with 
 sqc methods .
most of the 
 entries in table 15 involve one of the omx or omx - dn methods , which appear to perform best 
 overall for the large variety of ground - state molecules and properties 
 considered presently . as a rule of thumb ,
the omx and omx - dn methods can thus be 
 recommended as default for ground - state sqc studies , especially when 
 a specific validation against experimental data or high - level calculations 
 is not feasible .
common organic molecules are also described with 
 good accuracy by pm6 and pm7 because of their extensive parametrization . 

this indicates that sqc methods with orthogonalization corrections 
 still have a large potential for improvement when larger data sets 
 are used for their parametrization ; this is one target of our current 
 research .
the omx - dn and pm7 
 methods have 
 similar overall accuracy for noncovalent binding energies and geometries 
 of noncovalent complexes .
the omx - dn methods generally outperform pm7 for dispersion - dominated complexes 
 and mixed electrostatic / dispersion interactions , while pm7 is normally 
 superior for hydrogen - bonded complexes ; the latter can be attributed 
 to special hydrogen - bond corrections that are present in pm7 but not 
 in the omx - dn methods .
while it 
 has been shown for water clusters that the description of hydrogen 
 bonds can be improved by a special parametrization of omx methods , it would seem more promising 
 to target a complete reparametrization at the omx - dn level that includes noncovalent interaction 
 energies and geometries as reference data .
 the semiempirical 
 orthogonalization - corrected omx methods ( om1 , om2 , 
 and om3 ) go beyond the standard mndo model by 
 including additional interactions in the electronic structure calculation . 
 when augmented with empirical dispersion corrections , the resulting 
 omx - dn approaches offer a fast and 
 robust treatment of noncovalent interactions . here 
 we evaluate the 
 performance of the omx and omx - dn methods for a variety of ground - state properties using 
 a large and diverse collection of benchmark sets from the literature , 
 with a total of 13035 original and derived reference data . 
 extensive 
 comparisons are made with the results from established semiempirical 
 methods ( mndo , am1 , pm3 , pm6 , and pm7 ) that also use the nddo ( neglect 
 of diatomic differential overlap ) integral approximation . 
 statistical 
 evaluations show that the omx and omx - dn methods outperform the other methods for most 
 of the benchmark sets . 
Introduction
Methods and Computational
Details
Data Sets
General
Benchmark of Ground-State Properties
Benchmarks of Noncovalent Interactions
Conclusions