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noncovalent interactions of organic molecules play a fundamental role in biochemistry , solvation , surface science , and supermolecular chemistry . an accurate description of the noncovalent interactions is thus very important for many applications @xcite . actually , the `` golden standard '' for the simulation of noncovalently interacting complexes is the coupled cluster single and double with perturbative triple ( ccsd(t ) ) approach @xcite . however , the ccsd(t ) method has a high computational cost , scaling as @xmath0 , and it is not easily applicable in general , especially when numerous single  point calculations are required as in the determination of a potential energy surface ( pes ) or for geometry optimizations .    in various applications is thus necessary to recover to lower level computational methods . however , most approaches ( including density functional theory ( dft ) and standard mller plesset second  order perturbation theory ( mp2 ) , are not fully satisfactory for an accurate description of noncovalent interactions and the corresponding pess @xcite . therefore , in the last years different computational schemes , mainly based on variants of mp2 , have been proposed to treat noncovalent interactions with sufficient accuracy and relative small computational effort @xcite . one of such methods is the so called spin  component  scaled mp2 ( scs  mp2 ) method @xcite which is based on the spin resolved mp2 formula for the correlation energy @xmath1 with @xmath2 } { \epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b},\end{aligned}\ ] ] where @xmath3 denote occupied orbitals , @xmath4 denote virtual orbitals , @xmath5 denotes a two  electron integral in the mulliken notation , @xmath6 is the energy of the @xmath7th orbital , and @xmath8 and @xmath9 are the scaling coefficients for the opposite spin ( os ) and same spin ( ss ) correlation , respectively . this approach was shown to yield quite improved results with respect to the standard mp2 , when a proper choice of the scaling parameters is performed @xcite . alternatively , a good performance was also obtained by limiting the method to consider only the opposite spin part ( i.e. setting @xmath10 and optimizing only @xmath8 ) , resulting in the scaled  opposite spin mp2 ( sos  mp2 ) method @xcite , which allows , when properly implemented , to reduce the computational scaling to @xmath11 . however , the choice of the optimal scaling parameters in scs and sos mp2 is not trivial and different proposals have been made @xcite , either based on theoretical considerations or on empirical fittings , showing that the `` best '' scaling parameters are somehow system and basis  set dependent . this issue may be not particularly relevant for covalent bonds , where the considered binding energies and improvements over mp2 are much larger that the inaccuracies due to a non optimal choice of the parameters . it becomes anyway relevant for noncovalently bonded complexes where the energies to be computed are much smaller . for these latter cases , very good scs parameters were proposed by distasio and head  gordon ( scs(mi)mp2 method ) @xcite by optimizing the scaling coefficients against benchmark noncovalent interaction energy data . the same authors , as well as grant hill and platts in a separate paper , also showed that accurate results can even be achieved by just same  spin scaling ( sss  mp2 ; @xmath12 , @xmath131.75 ) @xcite methods . additional investigations concerned the optimization of the sos mp2 for long  range interaction by the introduction in the sos mp2 method of a distance dependent scaling coefficient and a modified distance  dependent two  electron operator @xcite . the effort spent to optimize the scaling parameters and the fact that the results of these works are not in an agreement with `` standard '' scs mp2 ( @xmath14 , @xmath15 ) or sos  mp2 ( @xmath16 ) values nor with the values suggested by theoretical arguments within wave function theory @xcite , indicates the difficulty to fix optimal values of the scaling for the proper description of noncovalent interactions . moreover , all these approaches pay the prize of introducing a high level of empiricism . therefore , a simple non  empirical procedure to fix the value of the various scaling parameters appears highly desirable .    in this paper we address this issue and we show a simple non  empirical scheme to fix the scaling factor in calculations of the dissociation curve of noncovalent bonded complexes . here , non  empirical denotes the fact that scaling factors will be fixed by direct use of information from high  level _ ab initio _ calculations and not from some empirical fitting procedure . of course , some empiricism is still implied in the use of spin resolved mp2 formulas . our approach focuses on scaled mp2 calculations with a single parameter . to this end , we introduce the sos(r)mp2 , sss(r)mp2 , and scaled mp2 ( s(r)mp2 , i.e. with @xmath17 ) methods . in acronym ( r ) stands for `` calculated from reference non  empirical values '' , to avoid confusion and distinguish from empirically scaled versions of standard scs  mp2 methods . our procedure is based on the observation that there exist a well defined proportionality between scaled(same / opposite)mp2 energies and the correlation energy computed by high  level methods ( e.g. ccsd(t ) ) , which is almost independent on the intermolecular distance . thus , the scaling parameter can be fixed once by using the information from only one expensive high  level calculations and successively the whole pes can be computed with high accuracy by performing only relatively cheap scaled(same / opposite)mp2 calculations . in particular , in this way the efficiency of the sos mp2 ( sos(r)mp2 ) method ( @xmath11 scaling @xcite ) can be fully exploited for large scale explorations of dissociation potential energy surfaces ( pes ) without introducing empirical parameters and achieving almost the ccsd(t ) accuracy . we acknowledge that a similar approach was already recently used in ref . @xcite , to compute accurate interaction energies of an ar monolayer adsorbed on an mgo substrate . however , in ref . @xcite only local mp2 calculations were performed while spin - resolved mp2 calculations , which are the main topic of the present work , were not considered . in addition , in the present work , a systematic assessment of the non - empirical procedure for fixing the scaling parameters is carried out for different systems and interactions . the main quantity of interest in the present paper is the inter fragment correlation ( ifc ) energy , defined as : @xmath18 where in our calculations @xmath19=ccsd(t ) , ccsd , mp2 , sos(r)mp2 , sss(r)mp2 or s(r)mp2 . the ifc is the difference between the correlation energy ( @xmath20 ) of a complex ( @xmath21 ) and that of its constituting fragments ( a and b ) , computed with a method @xmath19 . the ifc for two exemplary cases ( h@xmath22o dimer and ne@xmath22 ) is reported in fig . [ fig_ifc ] as a function of the intermolecular distance ( @xmath23 ) for different methods . the ifc is always negative and its absolute value rapidly decreases with @xmath23 . inter  fragment correlation ( ifc ) energy of ne@xmath22 and ( h@xmath22o)@xmath22 as computed with different methods . ] it can be observed that for noncovalent interactions , with good approximation , the ifc energies computed at the mp2 ( or equivalently mp2(ss ) or mp2(os ) ) level and at the ccsd(t ) level are proportional to each other over a wide range of inter  fragment distances @xmath23 . this proportionality can be verified by direct comparison of the ifc energies calculated at different levels of theory for various noncovalent interacting complexes ( see fig . [ fig_ratio ] ) . ratio between different spin  resolved mp2 ifc energies and ccsd(t ) ones , for various systems at several distances . @xmath24 denotes the equilibrium distance of each complex . @xmath19 = mp2 [ continuous line ] , mp2(os ) [ dashed line ] , and mp2(ss ) [ dotted line ] . ]    for mp2 the proportionality is very well satisfied for all systems and distances . mp2(ss ) and mp2(os ) show instead some larger deviations at higher distances . however such deviations are not very relevant , because the ifc energy is rapidly approaching zero for large distances . in fact , as shown in fig . [ fig_ifc ] , for @xmath251.5 the ifc is already close or below subchemical accuracy . the proportionality of the ( spin  resolved ) mp2 and coupled cluster energies can be rationalized considering that in noncovalent complexes the variation of ifc with distance is mainly due to relaxation of intramolecular contributions , thus both mp2 and ccsd(t ) ifc energies display a very similar behavior with @xmath23 and their ratio stays almost constant . this proportionality can be used to define single parameter scaled ( spin  resolved ) mp2 approaches suitable for the description of the dissociation of noncovalent complexes . to do that , we define @xmath26 with @xmath27 where @xmath28 is some reference distance . with this definition we fix ( at the cost of a single expensive ccsd(t ) calculation ) the proportionality between ( spin  resolved ) mp2 and ccsd(t ) results , by simply imposing that at some point , the two ifc energies , or analogously , the respective total binding energies ( @xmath29 ) are the same . the choice of the reference point @xmath28 in eqs . ( [ e9])([e11 ] ) is not a major concern , since the value of @xmath30 is almost independent on it ( see fig . [ fig_ratio ] and subsection [ sect_dist ] ) . in this work we have chosen for the the reference point the equilibrium distance @xmath24 , computed at the ccsd(t ) level . this choice is not mandatory , nevertheless it appears the best compromise between the need to avoid too short distances ( where the proportionality relation may be less accurate ) and the necessity to avoid the use of too small energies ( as would result for large @xmath23 values ) in the ratio of eqs . ( [ e9])([e11 ] ) to minimize numerical noise(see subsection [ sect_dist ] for a further discussion ) . each of the equations ( [ e9 ] ) , ( [ e10 ] ) , ( [ e11 ] ) provides a simple non  empirical procedure for fixing the ( system  dependent ) scaling factor in one  parameter ( scs)mp2 calculations when dissociation energies of noncovalent systems are of interest . thus , a whole dissociation curve , comprising many single  point results , can be simulated by performing only a single ccsd(t ) calculation and without the introduction of any empirical parameter . the main question is if the accuracy of the such procedure is high enough . in section [ sec : res ] we will show that deviations from the ccsd(t ) reference results are indeed well below subchemical accuracy ( i.e. 0.1 kcal / mol ) .      to test our scaling approach we considered a representative set of small noncovalently interacting systems : he@xmath22 , ne@xmath22 , he ne , ar@xmath22 ( dispersion interaction ) , ( h@xmath22o)@xmath22 , ( hf)@xmath22 ( hydrogen bond ) , ( h@xmath22s)@xmath22 , ( hcl)@xmath22 ( dipole  dipole interaction ) . additionally , benzene hcn , the stacked benzene dimer , and be@xmath22 have been also considered , as particular cases . the former two are in fact relatively large systems with the first one also displaying a mixed electrostatic  dispersion character , which changes with the bond distance @xcite . the latter is a system where mp2 and ccsd completely fail even qualitatively @xcite . the sizes of the molecules included in our test set were limited by the need to compute in all cases the full ccsd(t ) dissociation curves for reference purposes , except for benzene  hcn and the stacked benzene dimer in which case the reference data were taken from ref . @xcite . calculations were performed with the aces ii @xcite and turbomole @xcite program packages . for all systems an aug  cc pvqz basis set @xcite was used , except for ne@xmath22 ( uncontracted aug  cc pvtz ) , be@xmath22 ( cc  pv5z ) , and ar@xmath22 ( aug  cc pv5z ) .    to assess the performance of the different methods to reproduce the full dissociation curves we considered the mean absolute error : @xmath31 where @xmath19=ccsd , mp2 , sos(r)mp2 , ... , and @xmath32 note that because all the methods considered here use the hartree  fock exchange , the difference between the ifc energies of different methods in eq . ( [ edelta ] ) are exactly the same as the differences in the total interaction energies . the mae@xmath33 defined in eq . ( [ emae ] ) measures the average deviation of the results from the reference ccsd(t ) data over a given interval . in our work we decided to remove from this evaluation the smallest values of @xmath23 , because our approach might be not fully justified at such small inter  fragment distances . hence , the lower bound for the integration was fixed at @xmath24 . the upper bound of the integral was fixed instead to the value of @xmath23 beyond which the ccsd(t ) ifc energy is lower than 10@xmath34 hartree , in order to keep the normalization factor finite in eq . ( [ emae ] ) . finally , we note that in all calculations no correction for the basis set superposition error ( bsse ) was included , for computational simplicity and because it is readsorbed in the scaling coefficient . for the water dimer , e.g. , we check that , including a bsse correction , the mae of eq . ( [ emae ] ) is 36.7 , 36.6 , 5.5 @xmath35 kcal / mol , for sos(r)-mp2 , sss(r)-mp2 and s(r)-mp2 respectively . these values are in good agreement with the ones ( without bsse ) reported in tab . the values of the @xmath36 , @xmath8 , and @xmath9 coefficients , obtained by applying eqs . ( [ e9 ] ) , ( [ e10 ] ) , and ( [ e11 ] ) to the systems considered in this work , are reported in table [ tablecos ] , together with the reference inter  fragment separation @xmath24 . .[tablecos ] values of the @xmath8 , @xmath9 , and @xmath36 coefficients obtained via eqs . ( [ e9 ] ) , ( [ e10 ] ) , ( [ e11 ] ) calculated at distance @xmath24 for all the systems presented in this paper . [ cols="<,^,^,^,^,^,^,^",options="header " , ]     the tabulated results show that scaling coefficients change very little , as can be already inferred from fig . [ fig_ratio ] . this provides a validation of the working hypothesis and indicates the robustness of our simple non  empirical scaling procedure . by inspecting the relative maes reported in tab . [ tab3 ] we see that the accuracy of the methods is not only preserved but in many cases is also improved . the improvement is anyway rather small and can be considered to lay within the numerical uncertainty related to the definition of the mae ( eq . ( [ emae ] ) ) . this result is important because it remarks the possibility to utilize the present non empirical scaling procedure also in all those cases where the equilibrium distance is not known , by simply performing a single ccsd(t ) calculations at any `` reasonable '' bond distance .      in this subsection we consider the application of our new scaling method to larger systems , namely the benzene hcn t  shaped complex and the stacked benzene dimer . the first of these systems is particularly interesting because the interaction has a mixed character involving both electrostatic and dispersion contributions in a variable proportion along the dissociation curve @xcite . its dissociation curves computed at the ( spin  resolved ) mp2 level as well as the corresponding scaled results are reported in fig . [ fig_benz ] . reference ccsd(t ) data from ref . @xcite are also reported .     binding energy for the benzene hcn complex as a function of the inter  atomic distance @xmath23 , for different methods . ]    the scaling coefficients have been computed at the equilibrium distance of 2.34  and are @xmath37 , @xmath38 , and @xmath39 . consistently with the results discussed in the main body of section [ sec : res ] , the non  empirical scaling methods all perform remarkably well , substantially improving over `` bare '' mp2 , mp2(os ) , and mp2(ss ) calculations and especially over empirically scaling methods ( e.g. scs(mi)mp2 ) . s(r)mp2 , sos(r)mp2 and sss(r)mp2 show in fact deviations from the reference data below 0.5 kcal / mol along the whole range of distances considered . similar results are found also by using scaling coefficients obtained considering a different reference distance @xmath28 . for @xmath40 , indeed , we found @xmath8=1.15 , @xmath9=1.14 , @xmath41=0.57 , and essentially the same performance for the binding energy ( or equivalently the ifc energy ) . in fact , the scaling coefficients computed at the new distance are in good agreement , although slightly smaller , with the ones computed at the equilibrium distance . we remark that the small difference between the coefficients computed at different reference distances shall , in this special case , be partially traced back to the computational noise originating from the fact that the reference ccsd(t ) values were not computed with the same set up as the mp2 ones , but rather extracted from the total binding energies ( hf+ccsd(t ) at the cbs limit ) reported in ref . @xcite . despite this small issue , the non  empirically scaled methods display an impressive robustness .    as a further test we consider a typical @xmath42-stacking complex . in fig . [ benzdimer ] we thus report the dissociation curve of the stacked benzene dimer as computed at the ( spin  resolved ) mp2 level and with the various scaled approaches . reference ccsd(t ) data from ref . @xcite are also reported . the scaling coefficients have been computed at the equilibrium distance of 3.765  and are @xmath43 , @xmath44 , and @xmath45 .     binding energy for the stacked benzene dimer as a function of the inter atomic distance @xmath23 , for different methods . ] inspection of the figure shows that , similarly with the case of the benzene - hcn complex , all the scaled methods perform remarkably well yielding deviations within 0.2 kcal / mol from the reference ccsd(t ) data . in particular , the non empirical scaling procedure proposed here demonstrates to be able to correct well both the overbinding of the mp2 method and the strong underbinding of the mp2(os ) and mp2(ss ) approaches . overall , the dissociation curves computed with our scaling approach agree very well with reference data and show that the method is very effective even for such `` large '' and `` difficult '' systems as the present ones . we note that this performance is remarkable especially for sos(r) and sss(r)mp2 , because the mp2(os ) and mp2(ss ) binding energies are almost degenerate , so that the corresponding scaling coefficient are almost identical . thus , the same scaling is obtained for the two methods . however , this has only a minor effect on the sss(r)mp2 results at relatively small distances , while a good performance is observed for bond distances larger than the equilibrium one . this shows once more that the here presented scaling procedure is rather robust in many different situations . finally , we consider in fig . [ fig_be2 ] the beryllium dimer , which is a very difficult system where mp2 and ccsd fail to give even an approximate quantitative description of the dissociation curve @xcite . for this system even ccsd(t ) shows some limitations @xcite . however , this high - level method can at least capture most of the features of the dissociation curve . therefore , it will be used anyway as reference in the present example , although caution must be paid to the quantitative analysis of the results .    in this case , the computed scaling coefficients are @xmath46 , @xmath47 , and @xmath48 . however , an accurate description of the dissociation curve is found only using the sos(r)mp2 method . on the contrary , methods based on the same  spin correlation fail quite evidently . this may be due to the fact that in this system , where static correlation is important , the stretching of the bond is not only promoting polarization and induction phenomena in each atom , but is also changing the actual `` multireference '' description of the system , influencing the same  spin correlation which includes antisymmetrized integrals . this fact indicates that the use of mp2(ss ) correlation energy in scaling procedures has strong limitations for this peculiar system . due to this limitations also the s(r)mp2 method , which includes important contributions of same  spin correlation , can not be accurate . nevertheless , we remark that s(r)mp2 and even sss(r)mp2 can compensate the limitations of the underlying mp2(ss ) energy relatively well , thanks to appropriate values of the scaling factor granted by eqs . ( [ e9 ] ) , ( [ e10 ] ) , ( [ e11 ] ) . however , the independence of the scaling factor from the reference separation is partially lost and only a limited portion of the dissociation curve can be properly described when same  spin correlation is considered . the sos(r)mp2 method with coefficients calculated in our scaled procedure provides for the be@xmath22 the best dissociation curve ( as compared to ccsd(t ) ) significantly outperforming mp2 , ccsd or sos(mi)mp2 methods .     binding energy and ifc energy for be@xmath22 as a function of the inter atomic distance @xmath23 , for different methods . @xmath24 denotes the equilibrium distance . ] we have proposed a simple non  empirical procedure which can be used to calculate optimal scaling coefficients for ( spin  resolved ) mp2 calculations of the dissociation of noncovalent complexes . the applicability of the proposed method was demonstrated for a series of test systems , considering equilibrium and non  equilibrium reference distances , as well as for a notoriously difficult case as be@xmath49 . the obtained results show that the presented method works well in most cases , confirming the robustness of our hypothesis . the proposed method is especially attractive because it allows to obtain a full dissociation curve of almost ccsd(t ) quality at the cost of just a single ccsd(t ) calculation . moreover , it is conceivable to replace the costly ccsd(t ) calculation with a focal  point analysis @xcite ( the @xmath50ccsd(t ) procedure ) to obtain an estimate of the ccsd(t ) correlation energy from a cheaper calculation . in this way the computational cost is much reduced , losing only little on accuracy . thus , future applications on large systems can be foreseen . moreover , it might be also possible to replace the ccsd(t ) reference with an alternative accurate treatment of the correlation , e.g. the ccsd[t ] @xcite or the fno ccsd(t ) @xcite method , to optimize the accuracy or the computational effort . finally we remark that , although in this paper we focused only on single  parameter scaled mp2 methods , the methodology presented here can be extended in the future to treat the more general case of spin  component  scaled ( scs ) mp2 calculations with two parameters ( @xmath8 and @xmath9 ) . to this end eqs . ( [ e6])-([e8 ] ) will be generalized to @xmath51 and an additional constraint will be needed for the scaling parameters ( e.g. fixing the sum or the ratio of the parameters from theoretical considerations ) . we also have to stress that this approach can be useful for the verification of the quality of scs coefficients used in different scs  mp2 methods . * acknowledgments * this work was supported by the polish committee for scientific research mnisw under grant no . n n204 560839 and by the erc stg fp7 project dedom ( no . 207441 ) . we thank turbomole gmbh for providing the turbomole program package and m. margarito for technical support .
we present a simple and non  empirical method to determine optimal scaling coefficients , within the ( spin  component)scaled mp2 approach , for calculating intermolecular potential energies of noncovalently  interacting systems . 
 the method is based on an observed proportionality between ( spin  component ) mp2 and ccsd(t ) energies for a wide range of intermolecular distances and allows to compute with high accuracy a large portion of the dissociation curve at the cost of a single ccsd(t ) calculation . 
 the accuracy of the present procedure is assessed for a series of noncovalently  interacting test systems : the obtained results reproduce ccsd(t ) quality in all cases and definitely outperform conventional mp2 , ccsd and scs 
 mp2 results . 
 the difficult case of the beryllium dimer is also considered .