The interaction-strength interpolation method for main-group chemistry: benchmarking, limitations, and perspectives
aa r X i v : . [ phy s i c s . c h e m - ph ] S e p The interaction-strength interpolation method formain-group chemistry: benchmarking, limitations,and perspectives
Eduardo Fabiano, ∗ , † , ‡ Paola Gori-Giorgi, ¶ Michael Seidl, ¶ and Fabio Della Sala † , ‡ Istituto Nanoscienze-CNR, Euromediterranean Center for Nanomaterial Modelling andTechnology (ECMT), Via per Arnesano 16, 73100 Lecce, Italy., Istituto Italiano di Tecnologia(IIT), Center for Biomolecular Nanotechnologies@UNILE, Via Barsanti, 73010 Arnesano, Italy.,and Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW,Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
E-mail: [email protected]
Abstract
We have tested the original interaction-strength-interpolation (ISI) exchange-correlation functionalfor main group chemistry. The ISI functionalis based on an interpolation between the weakand strong coupling limits and includes exact-exchange as well as the Görling-Levy second-order energy. We have analyzed in detail the basis-set dependence of the ISI functional, its depen-dence on the ground-state orbitals, and the influ-ence of the size-consistency problem. We showand explain some of the expected limitations ofthe ISI functional (i.e. for atomization energies),but also unexpected results, such as the good per-formance for the interaction energy of dispersion-bonded complexes when the ISI correlation is usedas a correction to Hartree-Fock. ∗ To whom correspondence should be addressed † Istituto Nanoscienze-CNR, Euromediterranean Centerfor Nanomaterial Modelling and Technology (ECMT), Viaper Arnesano 16, 73100 Lecce, Italy. ‡ Istituto Italiano di Tecnologia (IIT), Center forBiomolecular Nanotechnologies@UNILE, Via Barsanti,73010 Arnesano, Italy. ¶ Department of Theoretical Chemistry and AmsterdamCenter for Multiscale Modeling, FEW, Vrije Universiteit, DeBoelelaan 1083, 1081HV Amsterdam, The Netherlands
Current approximations for the exchange-correlation functional of Kohn-Sham (KS) den-sity functional theory (DFT) work for systems thatare weakly or moderately correlated, as they arebased on information (exact or approximate) fromthe weakly correlated regime, when the physi-cal system is not too different from the KS one.The idea of including information from the op-posite limit of infinite correlation dates back toWigner, who approximated the correlation en-ergy of the uniform electron gas by interpolatingbetween the limits of zero and infinite interac-tion strength. Seidl and coworkers importedthis idea in the framework of KS DFT. They ana-lyzed the structure of the DFT limit of infinitecoupling strength, proposed a semilocal approxi-mation for it, and built an exchange-correlation(xc) functional by interpolating along the adiabaticconnection between zero and infinite interactionstrength (“interaction-strength interpolation,” orISI). The original ISI functional interpolates be-tween exact ingredients at weak coupling (exactexchange and second-order perturbation theory)and approximate ingredients at infinite couplingstrength, given by the semilocal “point-chargeplus continuum” (PC) model. In the recent years, the exact solution for thelimit of infinite interaction strength in DFT has1een derived: it is given by a highly non-localfunctional of the density, and can be mappedinto a mathematical problem appearing in mass-transportation theory.
Comparison againstthese exact results showed that the PC model(with a minor readjustment on the next leadingterm ) is a rather accurate approximation for thexc energy at infinite coupling strength, whileits functional derivative misses the non-local fea-tures of this limit needed to describe many strong-correlation phenomena in DFT in a spin restricted framework. Another approximation for thestrong-coupling limit that retains some of its non-locality (the “non-local radius” model, or NLR)has been recently proposed in Ref., and used byZhou, Bahmann, and Ernzerhof to construct newxc functionals that use the information at infinitecoupling strength.A formal drawback of the original ISI functionalis that it is size consistent only when a systemdissociates into equal fragments. This problemis shared by different non-local methods in DFT(see e.g. Refs. ) and in particular by the ap-proximations based on a global interpolation (i.e.,performed on quantities integrated over all space)along the adiabatic connection, like the one ofRef. For the latter, a possible way to restore size-consistency in the usual DFT sense is to turnto models based on local interpolations, performedin each point of space, a route that is beingpresently explored by different authors. Anefficient implementation of the ingredients neededfor a local interpolation along the adiabatic con-nection in the ISI spirit is not yet available, andit is the object of ongoing work.While a considerable amount of theoretical workon xc functionals that include in an approxi-mate or exact way the strong-interaction limit hasbeen done, benchmarking has been restricted sofar to atomization energies, ionization poten-tials, or to simple paradigmatic physical and chemical models only. Very little isknown about the performance of such function-als for bigger systems and for other chemical andphysical properties, and about technical aspectssuch as their sensitivity to reference orbitals andtheir basis set dependence.The purpose of the present work is to fill thisgap, by starting from a systematic study of the ISI functional in its original formulation, for which allthe ingredients are readily available. This allowsus to start to analyze quantitatively which effectsare well captured by a functional that includesthe strong-coupling limit, together with the practi-cal consequences of the size-consistency error forheterolytic dissociation, as well as to examine re-stricted versus unrestricted calculations, and otheraspects such as sensitivity to the reference orbitals.Our main aim is to provide valuable informationfor a future generation of functionals based on lo-cal interpolations along the adiabatic connectionthat can include the strong-coupling limit withoutviolating size consistency. As we shall see, our results show some of theexpected limitations of the original ISI functional,but also unexpected results, like an excellent per-formance for the interaction energy of dispersion-bonded complexes that definitely deserves furtherstudy.
The ISI xc functional is built by modeling thestandard density-fixed linear adiabatic connectionintegrand W l [ r ] , W l [ r ] = h Y l [ r ] | ˆ V ee | Y l [ r ] i − U [ r ] , (1)where Y l [ r ] is the wavefunction yielding the den-sity r and minimizing h Y | ˆ T + l ˆ V ee | Y i , and U [ r ] is the Hartree (or Coulomb) energy, with a func-tional form W ISI l [ r ] that has the exact weak- andstrong-coupling asymptotic behavior, W l → [ r ] = E x [ r ] + l E GL2 [ r ] + ..., (2) W l → ¥ [ r ] = W ¥ [ r ] + W ′ ¥ [ r ] √ l + .... (3)Its final form for the xc energy is obtained as E ISI xc [ r ] = Z W ISI l [ r ] d l , (4)and reads E ISI xc = W ¥ + (5) + XY (cid:20) √ + Y − − Z ln (cid:18) √ + Y + Z + Z (cid:19)(cid:21) , X = xy z , Y = x y z , Z = xy z − , (6)and x = − E GL2 , y = W ′ ¥ , and z = E x − W ¥ . TheISI functional is thus based on four ingredients:two come from the limit of weak interaction ofEq. (2) expressed in terms of orbital and orbitalenergies, namely the exact exchange energy E x = − (cid:229) i , j Z d r Z d r ′ f ∗ i ( r ) f ∗ j ( r ′ ) f j ( r ) f i ( r ′ ) | r − r ′ | (7)and the Görling-Levy second-order energy E GL2 = − (cid:229) abi j |h f i f j || f a f b i| e a + e b − e i − e j −− (cid:229) ia |h f i | ˆ v KSx − ˆ v HFx | f a i| e a − e i , (8)where h· · || · ·i denotes an antisymmetrized two-electron integral; two are derived from the limitof strong coupling of Eq. (3): W ¥ [ r ] is the in-direct part of the minimum possible expectationvalue of the electron-electron repulsion in a givendensity, and W ′ ¥ [ r ] is the potential energy ofcoupled zero-point oscillations of localized elec-trons. They are both highly non-local densityfunctionals that are presently expensive to com-pute exactly.
They are well approxi-mated by the semilocal PC model, which weuse in this work, W ¥ [ r ] = Z (cid:20) A r ( r ) / + B | (cid:209) r ( r ) | r ( r ) / (cid:21) d r (9) W ′ ¥ [ r ] = Z (cid:20) C r ( r ) / + D | (cid:209) r ( r ) | r / ( r ) (cid:21) d r . (10)The parameters A = − . B = . × − ,and C = . while the parameter D cannotbe derived in the same way, and different choicesare possible. For example, we can fix D by re-quiring that W ′ ¥ [ r ] be self-interaction free for theH atom density. Another possible choice, whichwas adopted when the ISI functional was first pro-posed and tested for atomization energies, is tofix D by requiring that W ′ ¥ [ r ] be exact for the He atom density. At the time, however, the exact so-lution for W ′ ¥ [ r ] was not available, and the accu-rate W ′ ¥ [ r ] for He was estimated from a metaGGAfunctional. Few years later, when the exact W ′ ¥ [ r ] has been evaluated for several atomic densities,it has been found that the metaGGA values werenot accurate enough. The parameter D has thenbeen changed and fixed by using the exact W ′ ¥ [ r ] for the He atom. This choice, corresponding to D = − . × − , improves significantly theagreement between the PC model for W ′ ¥ [ r ] andthe exact values for several atomic densities andit is the one we use in this work.To see how the limits of Eqs. (2)-(3) are includedin the ISI functional of Eq. (5), we can expand E ISI xc [ r ] in a series for small E GL2 , E ISI xc (cid:12)(cid:12)(cid:12) E GL2 → = E x + E GL2 + ( E x − W ¥ ) E + ... (11)showing that ISI includes the exact-exchange andrecovers second-order perturbation theory.The opposite limit of strong correlation is nor-mally signaled by the closing of the energy gapbetween the highest occupied molecular orbital(HOMO) and the lowest unoccupied molecular or-bital (LUMO), which usually makes appear a bro-ken symmetry solution with lower energy. If wedo not allow symmetry breaking, the gap closes,implying that E GL2 → − ¥ and E ISI xc (cid:12)(cid:12)(cid:12) E GL2 →− ¥ = W ¥ + W ′ ¥ − (12) − W ′ ¥ E x − W ¥ ln (cid:18) + E x − W ¥ W ′ ¥ (cid:19) . The first two terms, W ¥ [ r ] + W ′ ¥ [ r ] , give the xcenergy in the limit of strong coupling, which is thesum of a purely electrostatic indirect part ( W ¥ [ r ] )and electronic zero point oscillations (the factortwo in front of W ′ ¥ [ r ] accounts for the zero pointkinetic energy, and comes from the integration ofthe term ∼ l − / in Eq. (3)). The last term inEq. (12) is dependent on the interpolating function,and can change if we choose different forms (see,e.g., the ones of Refs. and ).If the four ingredients E x , E GL2 , W ¥ and W ′ ¥ are size consistent, then the ISI xc functional issize consistent only when a system dissociates into3qual fragments, as it can be easily derived fromEq. (5). A detailed and quantitative analysis of theproblem is reported in Sec. 5.1.We should notice, however, that within the lessusual restricted framework for open shell frag-ments, which seems crucial to capture strong cor-relation without introducing artificial magnetic or-der and it is the present focus of a large theoret-ical effort, size-consistency of the E x and E GL2 is lost, and usually E GL2 → − ¥ at disso-ciation. In this case, the ISI xc functional stays fi-nite and tends to the expression of Eq. (12). In thiswork we have tested the ISI functional followingthe standard procedure of allowing spin-symmetrybreaking (for a very recent review on spin symme-try breaking in DFT see Ref. ), and we discussonly briefly paradigmatic calculations (the H andN dissociation curves) in a spin-restricted formal-ism. It is however clear from Eq. (12) that the ISIxc functional is not able to dissociate a single ormultiple bond properly in a spin-restricted frame-work, since Eq. (12) will not provide the right en-ergy in this limit. The ISI accuracy in the usualunrestricted KS (or Hartree Fock) formalism areless easy to predict, and its analysis is the mainobject of this work. The calculations with the ISI xc functional de-fined by Eqs. (5)-(10) have been performed in apost-self-consistent-field (post-SCF) fashion, us-ing reference orbitals and densities obtained fromdifferent methods; namely, DFT calculations us-ing the Perdew-Burke-Ernzerhof (PBE ), the hy-brid PBE (PBE0 ), and the hybrid Becke-half-and-half (BHLYP ) exchange-correlationfunctionals, the localized Hartree-Fock (LHF) ef-fective exact exchange method and the Hartree-Fock (HF) method.In different parts of the paper we consider the ISIcorrelation energy, which is defined, as usual in theDFT framework, as E ISIc = E ISIxc − E x , where E ISIxc and E x are the ISI xc energy [Eq. (5)] and the exactexchange energy [Eq. ( ?? )], respectively. Notethat this definition of the ISI correlation energy iswell justified since the ISI xc functional includesthe full exact exchange. Unless otherwise stated, all energies have beenextrapolated to the complete basis set limit as de-scribed in subsection 3.1, using data from calcu-lations performed with the Dunning basis set fam-ily cc-pV n Z ( n = , . . . , For spin-polarizedsystems, an UHF formalism has been employedin the self-consistent calculations. All calculationshave been performed using a development versionof the TURBOMOLE program package.
To assess the performance of the ISI xc func-tional in practical applications we considered thefollowing set of tests:
Thermochemistry dataset . It contains atom-ization energies (AE6,
G2/97 ), ionizationpotentials (IP13 ), electron and proton affinities(EA13 and PA12 ), barrier heights (BH76 and K9 ), and reaction energies (BH76RC and K9 ) of small main-group molecules. Non-covalent interactions dataset . It containsinteraction energies of non-covalent complexeshaving hydrogen bond (HB6 ), dipole-dipole(DI6 ), charge-transfer (CT7 ), dihydrogen-bond (DHB23 ), and various (S22 ) char-acter. The ISI correlation energy formula contains theGL2 correlation energy of Eq. (8). The latter iswell known to exhibit a relevant basis set depen-dence as well as a slow convergence to the com-plete basis set (CBS) limit. Thus, a similar be-havior can be expected also for the ISI correlationenergy. Nevertheless, because the ISI energy alsoincludes other input quantities, whose basis set de-pendence is different from that of GL2, and be-cause all the input quantities enter non-linearly inthe ISI formula, it is not simple to derive analyti-cally the ISI basis set dependence. This situationis depicted in Fig. 1, where we report, for the Fatom, the basis set evolution of the different inputquantities of the ISI energy as well as of the ISIcorrelation energy itself.For this reason it is not convenient trying to de-rive the ISI basis set behavior starting from theassumed behavior of GL2 and other input quan-tities, as given by popular basis set interpolation-extrapolation formulas.
Instead, it is morepractical to consider the basis set evolution of the4 -0.12-0.09-0.06-0.030.000.03 V a r i a t i o n w i t h b a s i s s e t WW’EXXGL2ISIc
Figure 1: Variation with the basis set (cc-pV n Z) ofthe various input quantities (in Ha) used to com-pute the ISI correlation energy, here for the F atom.The black square at 1 / n = ?? ).ISI correlation energy as a whole. To this end, inanalogy with previous works on basis set extrapo-lation, we consider the following ansatz E ISI c [ n ] = E ISI c [ ¥ ] + An − a , (13)where the notation [ n ] indicates that the energy iscomputed with an n -zeta quality basis set (herespecifically the cc-pV n Z basis set), A is a system-dependent constant, and a is an exponent deter-mining the strength of the basis set dependence.Equation ( ?? ) provides an accurate fit for the ISIcorrelation energies of different systems as weshow in Fig. 2 where we report, for some exam-ple systems, the ISI correlation energies computedwith several basis sets and the corresponding fitobtained from Eq. ( ?? ). Note also that, as shownin Fig. 1, Eq. ( ?? ) reproduces correctly the CBSextrapolated value of the ISI correlation energy ascomputed using the extrapolated values of all inputingredients.Use of Eq. ( ?? ) allows to obtain accurate CBS-ISI energies. However, a more practical approachis to use Eq. ( ?? ) into a two-point scheme, to have the extrapolation formula E ISI c [ ¥ ] = E ISI c [ n ] n a − E ISI c [ m ] m a n a − m a , (14)where n and m label two selected basis sets. In thiswork we considered n = m = -0.11-0.10-0.09 C -0.39-0.36-0.33 E c I S I ( H a ) Ne -0.38-0.36-0.34 H O cc-pVTZ cc-pVQZ cc-pV5Z cc-pV6Z Basis set -0.81-0.78-0.75-0.72 F Figure 2: Evolution of the ISI correlation energyof different test systems with basis set. The reddashed lines denote interpolations obtained usingEq. ( ?? ).ical noise in some cases) and fixed the parameter a = . ?? ) to the full set of data correspond-ing to n = , . . . ,
6. The calculations have been per-formed in a post-SCF fashion using LHF orbitals(almost identical results have been obtained usingHartree-Fock orbitals). Note that the optimizedvalue of a is a bit larger than the correspondingones obtained in Ref. for MP2 and CCSD (1.91and 1.94, respectively). This indicates that the ISIcorrelation converges slightly faster than the MP2and CCSD ones to the CBS limit, possibly becauseit benefits from the fast convergence of the puredensity-dependent contributions.A test of Eq. ( ?? ) is reported in Fig. 3 where weshow the errors on ISI absolute correlation ener-gies (upper panel) and atomization correlation en-ergies (lower-panel) computed with different ba-sis sets, as compared to CBS reference ones, i.e. E ISI c [ ¥ ] of Eq. ( ?? ) fitted to the data with n = , . . . , ?? ). The results obtained usingEq. ( ?? ) are labeled as E-45 in the figure. For theabsolute correlation energies we see that even atthe cc-pV6Z level errors of about 10 mHa can beexpected, while only energies obtained via the ex-trapolation formula of Eq. ( ?? ) show accuraciesof about 1 mHa. For the atomization correlationenergies, we deal with energy differences. There-fore, error compensation effects are quite relevant,especially for the smallest basis sets. Thus, the er-5 N H C H H O F H H
CN C O N C H H C O HOOH F N e S i H P H H C l C l F S O System | E c - a t I S I [ n ] - E c - a t I S I [ ¥ ] | | E c I S I [ n ]- E c I S I [ ¥ ] | TZQZ5Z6ZE-45
Figure 3: Deviations of the ISI correlation ener-gies (in mHa) computed with various basis setsfrom the benchmark CBS ones. Absolute corre-lation energies (upper panel) and atomization cor-relation energies (lower panel).rors are close or lower than 10 mHa even for thecc-pVTZ basis set. Nevertheless, accurate results(about 1 mHa) can be obtained systematically onlyusing at least a cc-pV5Z basis set or, even better,via the extrapolation formula of Eq. ( ?? ). The ISI correlation functional is a complicatedorbital-dependent non-linear functional. Thus, astable self-consistent implementation is a compli-cated task going beyond the scope of this paper.Here the ISI correlation is employed in a post-SCFscheme, where the ground-state orbitals and den-sity are computed using a simpler approach andthen used to evaluate the ISI correlation (and alsothe exact exchange contribution).The relevance of the reference density and or-bitals for different DFT calculations has beenpointed out in several works in literature.
Therefore, it appears important to assess the re-liability of different reference orbitals for the cal-culation of the ISI correlation. Furthermore, be-cause the ISI functional is including the GL2 cor-relation energy as input ingredient, the orbitalenergies, and in particular the HOMO-LUMOgap, can be expected to play a major role (see discussion in subsection 5.2). Hence, we takeinto account reference ground-state orbitals com-puted with the generalized gradient xc approxima-tion of Perdew-Burke-Ernzerhof (PBE), with thehybrid functionals PBE0 and BH-LYP, which include 25% and 50% of exact exchange re-spectively, with the optimized effective potentialnamed localized Hartree-Fock (LHF), and withthe Hartree-Fock (HF) method. Note that the in-clusion of larger fractions of non-local Hartree-Fock exchange yields increasingly large HOMO-LUMO gaps, which are also effectively used indouble-hybrid functionals. We remark that theLHF method is instead a de facto exact exchangeKohn-Sham approach. As such it gives signifi-cantly smaller values of the HOMO-LUMO gapthan Hartree-Fock. Moreover, it may providea better approximation of the self-consistent ISIground-state density and orbitals than approximatefunctionals or the Hartree-Fock method (we re-call that in general correlation contributions to thedensity and orbitals are rather small ). Any-way, we cannot exclude that the self-consistent ISIpotential may display non-negligible differenceswith respect to the LHF (or the exact exchange)one. These differences might concern especially areduction of the HOMO-LUMO gap that will in-duce a lowering of the total xc energy (note, how-ever, that for the ISI functional a complete collapseof the HOMO-LUMO gap is not likely because,unlike in the GL2 case, the large increase of ki-netic energy associated to it cannot be compen-sated by the divergence of the correlation energy,which is bounded from below in ISI) and caseswhere static correlation is rather important.In Table 1 we report the ISI correlation energies(in absolute value) obtained using different refer-ence ground-state densities and orbitals (see alsoFig. 4). The corresponding GL2 energies are alsolisted in order to compare to the ISI ones (a staris appended to the mean absolute errors reportedin Table 1 to indicate, for each choice of the ref-erence orbitals, the best method between ISI andGL2). We recall that the GL2 and ISI results areextrapolated to CBS limit, as described in Eq. ( ?? )(for GL2 we used Eq. ( ?? ) with cc-pVQZ and cc-pV5Z results and the optimized value a = . /
10 mHa (this explains the6act that for a few cases, e.g. Si, S, SiH usingPBE0 orbitals, we have | E ISI c | > | E GL2 c | , whereasby construction it holds | E ISI c | ≤ | E GL2 c | ). Table 1also shows the correlation energies computed withsome popular semilocal generalized-gradient ap-proximation (GGA) functionals (namely the Lee-Yang-Parr (LYP), the PBE, and the PBE withlocalization (PBEloc) functionals), in order toprovide a comparison for the expected accuracyof standard DFT calculations. The correlationenergies for GGA functionals have been com-puted using the cc-pV5Z basis set and the PBEself-consistent orbitals. Note that GGA correla-tion functionals include only dynamical correla-tion, whereas the ISI method includes bothdynamical- and static-correlation.We see that the results depend rather importantlyon the used reference ground-state orbitals. Thisindicates that any non-self-consistent use of the ISIfunctional must be considered with the due cau-tion, while only self-consistent calculations couldgive definitive information on the real quality ofthe ISI energy. However, the self-consistent im-plementation of the ISI functional is an extremelyhard task. On the other hand, using reference or-bitals which are simpler to compute, in order toevaluate ISI functional non-self-consistently mayoffer a more pragmatic approach that can still pro-vide interesting information on this method. Forthis reason we consider this analysis in the follow-ing.A first inspection of the overall results, i.e. theMAE in the overall statistics at the bottom of thetable, shows that the best ISI results are found us-ing PBE0 and LHF orbitals (overall mean abso-lute errors (MAEs) of 26.0 and 31.8 mHa, respec-tively). We remark that these results are of simi-lar quality as those of the semilocal DFT function-als: the MAE of the best GGA functional (LYP)is 28 mHa. On the other hand, the use of PBEorbital leads to overestimated absolute ISI corre-lation energies, while the use of HF or BH-LYPorbitals yields largely underestimated absolute en-ergies. Similar trends are obtained for the under-lying GL2 (MP2 in the case of HF) correlationenergies. It is interesting to see that ISI stronglyimproves over GL2 for PBE, PBE0 and LHF; anopposite trend is found for HF, while no relevantdifferences are found for BH-LYP. C N O F N e S i P S C l A r H N H C H H O F H H
CN C O N C H H C O HOOH F S i H P H S O C l F H C l System -300-200-1000100200300 | E c I S I | - | E c R e f | ( m H a ) Atoms Molecules H e PBEPBE0BHLYPLHFHF
Figure 4: Errors on absolute ISI correlation en-ergies (mHa) calculated using different referenceground-state densities and orbitals.A more detailed analysis of the different sys-tems can be obtained by inspecting the statisticsreported for different classes of systems as wellas inspecting Fig. 4 which reports the errors onthe absolute ISI correlation energies for all thesystems. The plot clearly shows that the use ofHartree-Fock orbitals leads to an underestimationof the absolute correlation for all systems. Instead,when LHF and PBE orbitals are considered atomiccorrelation energies are computed with quite goodaccuracy but molecular correlation energies aresignificantly overestimated. This finding has animportant effect on the calculation of atomization-correlation energies as shown in Table 2. In thiscase the smaller errors are found for HF-based cal-culations, which benefit from a large error cancel-lation effect: indeed, as shown in Fig. 4 moleculesand atoms are underestimated by about the same quantity. On the contrary, for all other referenceorbitals an important overestimation of the abso-lute ISI correlation energy is observed. Note that,in any case, the ISI correlation atomization ener-gies computed for the present test set are alwaysbetter than the corresponding GL2 correlation at-omization energies, yielding MAEs of 138.5, 92.5,62.4, 116.5, and 23.7 mHa for PBE, PBE0, BH-LYP, LHF, and HF orbitals respectively. Moreover,the HF-ISI results are also almost three times bet-ter than those obtainable by semilocal DFT func-tionals (the best being PBEloc with a MAE of 36mHa).7able 1:
Total correlation energies (in mHa, with opposite sign) from semilocal DFT functionals (LYP,PBE, PBEloc, all using PBE orbitals), ISI, and GL2 methods calculated using different reference ground-stateorbitals. Reference data are taken from Ref. The last lines report the mean error (ME) and the mean absolute error (MAE) for each case; a star is appended to the MAEs to indicate, for each choice of the reference orbitals,the best method between ISI and GL2.
Correlation: LYP PBE PBEloc ISI GL2Orbitals: PBE PBE PBE PBE PBE0 BHLYP LHF HF PBE PBE0 BHLYP LHF HF Ref.Closed-shell atomsHe 43.7 41.1 33.8 44.3 40.9 38.3 41.8 34.5 52.3 46.9 42.8 48.4 37.4 42Ne 383.1 347.1 358.3 433.6 400 375.8 406.5 338 500.3 452.9 419.8 462.6 370.1 391Ar 751.5 704.4 757.4 709.4 658.3 618.8 696.2 556.5 723.1 663.1 619.8 708 558.2 723ME 7.4 -21.1 -2.2 10.4 -18.9 -41.0 -3.8 -75.7 39.9 2.3 -24.5 21.0 -63.4MAE 12.7 21.1 25.1 19.5* 24.9* 41.0* 14.2* 75.7 39.9 42.2 44.3 31.0 63.4*Open-shell atomsC 158.3 144.3 139.8 145.8 129.7 118.2 138 102.1 178.6 152.4 135 166.3 112.7 156N 191.9 179.9 176.4 181.9 166.2 154.4 172.5 136.6 217.3 192.9 175.6 203.2 151.3 188O 256.6 235.2 234.8 251.1 230.5 215.4 237 191.7 303.2 270.4 247.7 281.4 214.2 255F 321 292.6 297.4 328.4 303.1 284.6 309.6 255.4 398.4 357.7 329.5 368.7 287.6 323Si 529.2 484.2 516.8 515.3 474.5 444.4 498.9 395.2 518.3 470.3 446.3 508.8 393.5 505P 566.4 526.5 564 544.4 504.1 473.6 531.6 424.5 553.3 505.5 480.6 543.3 426.7 540S 627.7 584.1 626.1 592.1 547.5 514.4 575.4 459.8 600.9 546.9 517.9 585.8 456.6 603Cl 689.7 644.5 691.6 640.2 592.8 557 623.8 499.3 658.5 600.1 559.3 638.6 501.3 664ME 13.4 -17.8 1.6 -4.4 -35.7 -59.0 -18.4 -96.2 24.3 -17.2 -42.8 7.8 -86.3MAE 13.9 17.8 20.0 9.4* 35.7 59.0 18.4 96.2 26.2 31.0* 44.4* 18.4 86.3*Closed-shell moleculesH
318 314.2 310.8 371.2 337.2 312.1 354.7 271.9 463.3 406.7 367.6 436.3 309.4 340CH
295 300 292.5 315.8 287.3 265.1 304.5 230.2 391.5 344.9 310.8 373.5 261.1 299H O 340.4 324.8 325.5 416.1 378.1 350.7 393.8 307 514.7 452.5 410.2 478.7 347.6 371FH 362.2 335.1 340.4 439.6 400.8 373 412.1 329.6 528.9 469.1 428.2 486.8 368.2 389HCN 464.8 439.7 437.8 604.4 536.8 488 561 414.3 773.8 655.2 577.1 714.7 470.1 515CO 485.2 448.4 451 627.4 558.1 508 586.3 434.1 787.5 670.7 593 718.2 487.9 535N H CO 540.7 514.4 514.6 673.8 602.9 550.9 632.4 473.6 844.5 725.1 644.6 775.3 534.3 586HOOH 636.7 598.5 604.7 818.2 736.2 677.4 775.1 586.3 1023.1 885.6 794.1 951.3 663.2 711F able 2: Correlation atomization energies (mHa, with opposite sign) from semilocal DFT functionals(using PBE orbitals) as well as ISI and GL2 methods calculated using different reference ground-stateorbitals. Reference data are taken from Ref. The last lines report the mean error (ME) the mean absoluteerror (MAE) for each case; a star is appended to the MAEs to indicate, for each choice of the referenceorbitals, the best method between ISI and GL2.
Correlation: LYP PBE PBEloc ISI GL2Orbitals: PBE PBE PBE PBE PBE0 BHLYP LHF HF PBE PBE0 BHLYP LHF HF Ref.H O 83.8 89.6 90.7 165.0 147.6 135.3 156.8 115.3 211.5 182.1 162.5 197.3 133.4 116.0FH 41.2 42.5 43.0 111.2 97.7 88.4 102.5 74.2 130.5 111.4 98.7 118.1 80.6 66.0HCN 114.6 115.5 121.6 276.7 240.9 215.4 250.5 175.6 377.9 309.9 266.5 345.2 206.1 171.0CO 70.3 68.9 76.4 230.5 197.9 174.4 211.3 140.3 305.7 247.9 210.3 270.5 161.0 124.0N H CO 125.8 134.9 140.0 276.9 242.7 217.3 257.4 179.8 362.7 302.3 261.9 327.6 207.4 175.0HOOH 123.5 128.1 135.1 316.0 275.2 246.6 301.1 202.9 416.7 344.8 298.7 388.5 234.8 201.0F In Table 3 we report the total atomization ener-gies. We compare the ISI results to HF+GGAcorrelation approaches. Note that in the lattermethods no error cancellation between exchangeand correlation occurs and static-correlation is notconsidered.
Thus HF+GGA calculations givemuch worse results than conventional GGA xc ap-proaches. However, here they can be used to assesthe quality of the ISI results. ISI-HF has a MAEof only 11.7 mHa which is 4 times better thanHF+GGA. Conversely, ISI-LHF largely overesti-mates atomization energies, yielding an absoluteaccuracy close to HF+GGA (which, on the otherhand, underestimate the atomization energies. )We note that the present results for ISI atomiza-tion energies are slightly different from the onesreported in the original ISI publication. This isdue to the different choice of the parameter D inEq. (10), which has been fixed here by using theexact value of W ′ ¥ [ r ] for the He atom density in-stead of the one estimated from a metaGGA func-tional used in Ref., and due to the different basis-set used (recall that in the present work we usedextrapolation towards the complete basis set limit). To assess the practical applicability of the ISIfunctional to main-group chemistry, we have per-formed a series of tests involving different prop-erties of interest for computational chemistry. Wehave restricted our study to ISI calculations em-ploying HF and LHF reference orbitals (hereafterdenotes as ISI-HF and ISI-LHF, respectively).This choice was based on the fact that, as ex-plained in subsection 4.1, ISI-HF is expected toyield the best performance for these tests (accord-ing to the results of Table 2) while ISI-LHF pro-vides the best approximation for the performanceof self-consistent ISI calculations. For compari-son, we report also the MP2 and B2PLYP re-sults, which are based on GL2 energies, as wellas the performance of calculations using the pop-ular PBE functional and of the Hartree-Fock ex-change coupled with the semilocal PBEloc cor-relation (HF+PBEloc). The latter is a sim-ple approach adding semilocal dynamical correla-tion to Hartree-Fock and can give information onthe possible accuracy of “standard” DFT methodswhen used together with exact exchange; we re-9able 3: Total atomization energies (mHa, with opposite sign) form semilocal DFT functionals (usingPBE orbitals) as well as from ISI methods calculated using different reference ground-state densities andorbitals. Reference data are taken from Ref. The last lines report the mean error (ME), the mean absoluteerror (MAE), and the mean absolute relative error (MARE) for each case. 1mHa=0.62751 kcal/molMethod: HF+LYP HF+PBE HF+PBEloc ISIorbitals: PBE PBE PBE PBE PBE0 BHLYP LHF HF Ref.H
660 655 661.8 689.3 678.7 668.9 687.4 651.4 626H O 332.1 325.9 331.9 411 395 383 404 363.6 371HF 195.6 190.9 193.9 264.2 251.6 242.6 256.6 228.7 216.4HCN 431.8 426.7 435.2 586.1 554 530.4 459.3 492.7 496.9CO 348.5 347 354.6 500.9 471.8 450.9 486 418.5 413.8N H CO 536 533.1 543.1 676 646.9 625 662.4 590 596.7HOOH 333.9 326.7 338.4 516.7 481 454.8 505 413.4 428.9F In the upper part of Table 4 we report the meanabsolute errors (MAEs) for several standard testsconcerning thermochemical properties. In the lastline of Table 4 we report, for each method X , therelative mean absolute error (RMAE) with respectMP2, i.e. RMAE X = (cid:229) i MAE Xi MAE MP i , (15)where i indicates the different tests.The results clearly show that ISI-LHF oftengives the largest MAEs, with a RMAE of 4.1.Significantly better results are obtained by ISI-HFcalculations (RMAE=1.7). However, the perfor-mance for barrier heights (BH76 and K9) is quitepoor and even worse than that obtained by adding asimple semilocal correlation to Hartree-Fock. We note also that for this property Hartree-Fockand LHF based ISI calculations yield a quite simi-lar performance. On the other hand, ISI-HF yieldsthe best results for the PA13 test and the S22 test.When the focus is on non-covalent interac-tions (bottom part of Table 4), ISI-HF performsquite well for both hydrogen bond (HB6) anddipole-dipole (DI6) interactions having a compa-rable accuracy as MP2 and B2PLYP. The ISI-HF functional outperforms other approximationsfor the S22 test, which contains different kindsof biology relevant non-covalent complexes hav-ing hydrogen-bond, dipole-dipole, and dispersioncharacter.
The small error for the S22 test set suggeststhat ISI-HF may be more accurate than other ap-proaches (e.g. B2PLYP) in the description of dis-persion complexes. As further evidence, we re-port in Fig. 5 the signed error obtained from ISI-HF, MP2, and B2PLYP in the calculation of theinteraction energy of a collection of different dis-persion complexes, which includes the dispersion-dominated S22 cases as well as additional test
Complexes -6-4-2024 E rr o r ( k c a l/ m o l ) ISI-HFMP2B2PLYP
Figure 5: Signed errors (kcal/mol) in thecalculation of the interaction energy of dif-ferent dispersion complexes (1:He-Ne, 2:Ne-Ne, 3:CH -Ne, 4:CH -F , 5:CH -CH , 6:C H -Ne, 7:CH -CH , 8:C H -C H , 9:C H -CH ,10:C H sandwich dimer, 11:C H T-shapeddimer, 12:C H displaced dimer, 13:Pyrazine-dimer, 14:Uracil stacked dimer, 15:Adenine-Thymine stacked dimer, 16: Indole-Benzenestacked dimer).cases from the literature. It can be seen that,indeed, ISI-HF results are always very accurate( / H -C H , which showagain that ISI-HF accurately captures dispersioninteractions. Further analysis and discussion ofthese results are reported in subsection 5.3. One of the purposes of including the strong-coupling limit into approximate functionals is thehope to capture static correlation without resort-ing to symmetry breaking. However, it is alreadyclear from Eq. (12) that the ISI functional will notdissociate correctly a single or multiple bond ina restricted framework. In fact, as the bond isstretched, the ISI xc energy of Eq. (12) will bequite different than the one for the two equal openshell fragments. The problem is that only the elec-trons involved in the bonds should be strongly cor-related. The rest of the fragment should be in theusual weak or intermediate correlation regime, butthe global interpolation makes the whole fragmentbe in the strong-coupling regime. A local interpo-lation might fix this issue, but it needs to be con-11able 4: Mean absolute errors (kcal/mol) on several tests as obtained from ISI calculations using LHF andHF orbitals. PBE, HF+PBEloc, MP2 and B2PLYP results are reported for comparison. The best (worst)result for each test is in boldface (underlined). The last line reports the relative MAE with respect MP2(see Eq. ( ?? )). Test PBE HF+PBEloc MP2 B2PLYP ISI-HF ISI-LHFThermochemistryAE6 13.3 24.0 9.6 distance (au) -0.20-0.15-0.10-0.050.000.05 I n t er a c t i o n e n er gy ( k c a l/ m o l ) MP2B2PLYPISI-HFISI-LHFRef.
Distance (au) -2-101 I n t er a c t i o n e n er gy ( k c a l/ m o l ) Ne-Ne C H -C H Figure 6: Interaction energy curves for Ne-Ne andC H -C H . All energies have been corrected forthe basis-set superposition error. Reference valuesfor Ne-Ne and C H -C H have been taken fromRefs. and, respectively.structed carefully. An exception is the H molecule for which allthe electrons are involved in the bond. Indeed,Teale, Coriani and Helgaker had found a verygood agreement between the ISI model for theadiabatic connection curve (in a restricted frame- work) and their accurate results in the case ofthe H molecule dissociation, when the bond isstretched up to 10 bohr. Their study used fullconfiguration-interaction (FCI) densities and thecorresponding KS orbitals and orbital energiesfrom the Lieb maximization procedure as inputquantities. They have also tested how the choiceof the parameter D in Eq. (10) affects the shape ofthe adiabatic connection curve. They found thatthe original metaGGA choice used in Ref. doesnot yield accurate results. whereas the parameter D used here was found to yield rather accurate re-sults up to 10 bohr.In Fig. 7 we report the dissociation curves ofthe H and N molecules in a spin-restricted for-malism for different methods. Our ISI resultsare not very accurate if compared to the refer-ence CCSD(T) results, but qualitatively better thanMP2 and B2PLYP which diverge for large dis-tance. The inaccuracy of our ISI results origi-nates form the approximated LHF (or HF) densi-ties, orbitals and orbital energies: in fact, the spin-restricted ISI turns out to be very sensitive to the12 E n er gy ( H a ) H ISI-HFISI-LHFB2PLYPMP2CCSD(T)
Distance (au) -110.5-110.0-109.5-109.0 N Figure 7: Dissociation energy curves for the H and N molecules in a spin-restricted formalism.In this case second-order perturbation theory di-verges as the molecule is stretched, and the ISIfunctional tends to Eq. (12). Note also that ISI-LHF and ISI-HF will coincide at infinite distance.input ingredients. The ISI results in Ref. aremuch more accurate due to the fact that FCI in-put density, orbitals and orbital energies have beenused. Moreover for H , we recall that the ISI re-sults will be exact at infinite distance only if theparameter D is self-interaction free for the H atomdensity. The results reported in Section 4 show that the per-formance of ISI-HF is quite good when comparedwith HF+GGA methods (eg. HF+PBEloc), sincethe former describes dynamical and static correla-tion without any error cancellation while the latterdo not. On the other hand ISI-HF is much less ap-pealing, if compared to MP2 which yields in manycases better results at similar computational cost.One important exception are dispersion interac-tions, for which ISI-HF outperforms MP2. Insteadwhen ISI is applied to DFT orbitals (i.e. LHF) theresults are rather bad. In the following subsectionswe try to analyze and rationalize this performance,in order to provide useful information which canbe used to improve functionals based on interpola-tions between the weak and the strong interactinglimits.
Being a non-linear function of exact exchange andGL2 total energies, the ISI xc energy functional isformally not size consistent. This means that com-puting the (spin-unrestricted or spin-restricted) ISIxc energy of two systems separated by a distancelarge enough (eventually infinite) to make the in-teraction between them negligible, yields a resultwhich is different form the sum of the ISI xc ener-gies of the two isolated systems.One exception is the case of a set of identicalsystems, e.g. a homonuclear dimer A − A , where A is closed-shell or the spin-unrestricted formalismis used: under these conditions E x , E GL2 , W ¥ , and W ′ ¥ are all size-consistent, thus X [ A − A ] = X [ A ] ,while Y [ A − A ] = Y [ A ] and Z [ A − A ] = Z [ A ] . Sincethe ISI xc energy (see Eq. (5)) is linear in X and W ¥ is a size-consistent quantity, the the whole re-sult is size-consistent.The issue of size-inconsistency may, of course,affect the results when atomization or interactionenergies are calculated. To investigate the rele-vance of this problem, we perform a numericalstudy on the magnitude of this effect. Consider asystem M (e.g., a molecule) composed of differentfragments A i (e.g. atoms) with i = , . . . , N . Thetotal xc interaction energy in this system is E intxc ( M ) = E xc ( M ) − N (cid:229) i = E xc ( A i ) . (16)Here, E xc ( M ) denotes the xc energy of M and E xc ( A i ) the xc energy of the isolated fragment A i .Consequently, if we denote with M ∗ the systemobtained by bringing all fragments A i at large dis-tance from each other (such that their mutual in-teraction is negligible), this interaction energy canalso be written as E intxc ( M ) = E xc ( M ) − E xc ( M ∗ ) . (17)For any size-consistent method, Eqs. ( ?? ) and ( ?? )give the same result. However, for a non-size-consistent method such as ISI, their difference D xc ( M ) = E xc ( M ∗ ) − N (cid:229) i = E xc ( A i ) (18)13an provide a measure for the size-consistencyproblem (clearly, D xc = E ISI xc ( M ∗ ) = (19) = f ISI (cid:16) E x ( M ∗ ) , E GL2 ( M ∗ ) , W ¥ ( M ∗ ) , W ′ ¥ ( M ∗ ) (cid:17) , where f ISI ( w , w , w , w ) is the non-linear func-tion of four variables defined in Eqs. (5) and( ?? ). Assuming that all four ingredients are size-consistent, we can further write E ISI xc ( M ∗ ) = f ISI N (cid:229) i = E x ( A i ) , N (cid:229) i = E GL2 ( A i ) , N (cid:229) i = W ¥ ( A i ) , N (cid:229) i = W ′ ¥ ( A i ) ! . (20)Even then, we typically have D ISI xc ( M ) =
0, since f ISI is not linear, i.e. E ISI xc ( M ∗ ) = N (cid:229) i = E ISI xc ( A i ) . (21)As previously mentioned, an exception arises incases with identical fragments, A i = A (all i ), sincethe function f ISI has the property f ISI ( Nw , Nw , Nw , Nw ) = N f
ISI ( w , w , w , w ) . Using Eq. (19) and the corresponding expres-sion for E ISI xc ( A i ) , it is possible to evaluate the ef-fect of the size-consistency violation of ISI for dif-ferent systems. The results of these calculationsare reported, for a selected test set of molecules,in Table 5. In these calculations we have con-sidered a spin-unrestricted formalism for HF andGL2 calculations on open-shell atoms, assumingthat the corresponding results are properly size-consistent (whether this is formally correct is stillunder debate in literature; however, numericalresults suggest that our approximation is quite ac-curate in the considered cases). Inspection ofthe Table shows that for molecules composed offirst row elements (plus hydrogen) the values of D xc are negligible. Thus, the ISI functional be-haves, in practice, as a size-consistent method. Onthe other hand, for molecules including both first Table 5: Values of D xc per bond (in mHa), calcu-lated using Eqs. ( ?? ), (19), and (20), for a selec-tion of molecular systems. Note that since E x is asize-consistent quantity D xc = D c .Molecule D xc Molecule D xc CH CO 0.01NH O 0.04 SiH -2.44CO -0.02 HCl 0.00C H -1.33and second row elements, larger values are found.We remark that these values are, anyway, oftensmaller than few mHa per bond, so that the size-inconsistency problem is not too large also in thesecases.The difference between the two kinds of behav-iors observed in Table 5 traces back to the fact thatwhen only first row elements are present, all atomsdisplay quite similar values of exchange and GL2correlation; thus, the ISI behavior is rather simi-lar to the ideal case of identical systems and thesize-consistency violation is small. On the con-trary, when both first and second row atoms arepresent, the atomic properties are significantly dif-ferent and the non-linear nature of the ISI formulaleads to a non-negligible size-inconsistency. Fur-ther evidence of this fact is given in Fig. 8, wherewe report the values of D xc for the atomization ofa N molecule into two atomic fragments having 7electrons each (as the N atom) but nuclear charges Z = + D Z and Z = − D Z , for various valuesof D Z . Indeed, the plot clearly shows that the size-consistency problem grows with the difference be-tween the two atomic fragments. The fact that similar trends are observed in Tables1 and 2 for ISI and GL2 correlation energies fordifferent reference orbitals and different systems,suggests that the energy gap between occupied and14 D x c D z -0.5-0.4-0.3-0.2-0.10.0 D E x , D E c G L D E x / 10 D E cGL2 x 100 Figure 8: Upper panel: Values of D xc as functionsof D Z for the dissociation of N into two atomicfragments having 7 electrons and nuclear charges Z = + D Z and Z = − D Z . Lower panel: Val-ues of D E x = E x [ f ragment ] − E x [ f ragment ] and D E GL2 c = E GL2 c [ f ragment ] − E GL2 c [ f ragment ] asfunctions of D Z . The values of D E x and D E GL2 c have been scaled only for graphical reasons.unoccupied molecular orbitals may play a majorrole in determining the accuracy of the ISI corre-lation energy. This difference is in fact smaller forsemilocal DFT (PBE) and larger for HF, havingintermediate values for hybrid and the LHF meth-ods. Similarly, the energy gap is larger for closedshell atoms and smaller for open-shell atoms andmolecules. These observations fit well with the be-havior reported in Table 1 and Fig. 4.To investigate this feature, we have consideredfor all the systems in Table 1 the application tothe LHF ground-state orbitals of a scissor operator to rigidly move all the unoccupied orbitals up inenergy by D E = a ( E g [ HF ] − E g [ LHF ]) , (22)where E g [ HF ] and E g [ LHF ] are the HOMO-LUMO gaps for HF and LHF, respectively, while a is a parameter used to tune the effect. Thus, for a = a = molecule and twice the N atom,as functions of the a parameter of Eq. ( ?? ). Theatomization correlation energy error is thus the dif-ference between these two curves. For simplicitywe considered here results with the 5Z basis set M A E ( m H a ) open-shell atomsoverallmolecules AbsoluteAtomization a -100-50050 | E c I S I - L H F | - | E c R e f . | ( m H a ) N Figure 9: Top panel: Mean absolute errors(MAEs) for the ISI correlation and atomizationcorrelation energies of the systems of Table 1 asfunctions of the a parameter of Eq. ( ?? ). Bottompanel: Deviations from reference values of the ISIcorrelation energies of the N molecule and two Natoms as functions of the a parameter of Eq. ( ?? ).All results were computed using a cc-pV5Z basisset.and not the extrapolated CBS ones. Hence, evenat a = are slightly differentfrom the ones reported in Tables 1 and 2. At a = a is increased the absolutecorrelation energies decreases due to an increasedenergy in the denominator. However, the slopesof the lines are different. At a ≈ a for the ISI correlation energies ofopen-shell atoms, molecules and both, as well asthe MAE of the correlation atomization energies,are reported in the upper panel of Fig. 9. Theplot shows that the application of a shift for theunoccupied orbitals generally leads to a worsen-ing of the ISI correlation energies. This is partic-ularly true for atoms, which already suffer for anunderestimation (in absolute value) of the correla-tion energy, thus increasing the energy differencebetween occupied and virtual orbitals adds a fur-ther underestimation. For molecules instead at low15alues of a a moderate improvement of the corre-lation energy is observed, since in most moleculesLHF-ISI overestimates (in absolute value) correla-tion energies that are thus improved by the appli-cation of a shift. Nevertheless, for larger values of a in all molecules an underestimation of the corre-lation is found, so the results rapidly worsen withincreasing shift. We note that the rate of worseningfor molecular correlation energies is quite fasterthat that observed for atoms. The good performance of the ISI-HF for disper-sion interactions is surprising and deserves furtherthoughts. First of all, we notice that the functional E ISI xc [ r ] defined by Eqs. (5)-(10) inherits (at leastfor the case of equal fragments) the long-range ∼ R − dispersion interaction energy dependencefrom its E GL2 component (MP2 in the case of HFreference orbitals considered here). Yet, it sys-tematically outperforms MP2, suggesting that itadds a sensible correction to it. The analysis ofStrømsheim et al. shows that the adiabatic con-nection curve for the interaction energy of disper-sion complexes deviates significantly from the lin-ear behavior, requiring a considerable amount of“non-dynamical” correlation, which seems to bewell accounted for by the ISI functional (althoughthe picture may be different with HF orbitals).A possible explanation may be derived by look-ing at the functional W ′ ¥ [ r ] , which describes thephysics of coupled oscillations of localized elec-trons. Its PC semilocal approximation of Eq. (10)is a quantitatively good approximation of this en-ergy. The physics of dispersion interactions is ac-tually very similar, describing oscillations of cou-pled charge fluctuations on the two fragments. Wesuspect that when looking at the interaction en-ergy, the physics introduced by W ′ ¥ [ r ] (when sub-tracting the internal part of each fragment) is actu-ally correct. However a more detailed study of thisaspect is required and it will be the object of ourfuture work. We have reported the first detailed study of theperformances of a functional that includes (inan approximate way) the strong-coupling limit,analysing its dependence on basis-set, referenceorbitals and other aspects such as the size consis-tency error. Overall, the ISI functional has seriouslimitations, which could have been expected fromsome of its formal deficiencies. We have ratio-nalized our findings, providing useful informationfor functionals that can retain the information fromthe strong-coupling limit while remedying to thesedeficiencies. In future work, we plan to extendour analysis to functionals based on local interpo-lations along the adiabatic connection, im-plementing the needed input quantities.An unexpected finding that emerged from ourstudy is a very good performance of the ISI func-tional (when used as a correction to Hartree Fock)for dispersion interactions, yielding a mean abso-lute error of only 0.4 kcal/mol on the S22 set, andconsistently improving over MP2 in a significativeway for dispersion complexes (see Figs. 5-6). Wesuspect that the functional W ′ ¥ [ r ] , which describescoupled oscillations of localized electrons, is ableto capture the physics of interaction energy in dis-persion complexes. This is an interesting perspec-tive for the ISI functional, which we will investi-gate in detail in a future work. Acknowledgments
We thank Evert Jan Baerends, Oleg Gritsenkoand Stefan Vuckovic for insightful discussions andTURBOMOLE GmbH for providing the TUR-BOMOLE program. PG-G and MS acknowledgethe European Research Council under H2020/ERCConsolidator Grant “corr-DFT” [grant number648932] for financial support.
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