Assessment of interaction-strength interpolation formulas for gold and silver clusters
aa r X i v : . [ phy s i c s . c h e m - ph ] J a n Assessment of interaction-strength interpolation formulas for gold and silverclusters
Sara Giarrusso, Paola Gori-Giorgi, Fabio Della Sala,
2, 3 and Eduardo Fabiano
2, 3 Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit,De Boelelaan 1083, 1081HV Amsterdam, The Netherlands Institute for Microelectronics and Microsystems (CNR-IMM), Via Monteroni, Campus Unisalento, 73100 Lecce,Italy Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti,I-73010 Arnesano, Italy (Dated: 24 September 2018)
The performance of functionals based on the idea of interpolating between the weak and the strong-interactionlimits the global adiabatic-connection integrand is carefully studied for the challenging case of noble-metalclusters. Different interpolation formulas are considered and various features of this approach are analyzed.It is found that these functionals, when used as a correlation correction to Hartree-Fock, are quite robust forthe description of atomization energies, while performing less well for ionization potentials. Future directionsthat can be envisaged from this study and a previous one on main group chemistry are discussed.
I. INTRODUCTION AND THEORETICALBACKGROUND
Noble metal clusters, in particular those made of silverand gold, are of high interest for different areas of ma-terials science and chemistry as well as for technologicalapplications.
Noble metals clusters display, in fact,peculiar properties that differ from those of the bulk ma-terials, due to the higher reactivity of the surface atoms.Moreover, these properties can be often tuned by varyingthe size and shape of the clusters.
For thesereasons, the study of the electronic properties of metalclusters is currently a very active research field, withmany available experimental techniques.
Nonethe-less, in most cases information from theoretical calcula-tions is fundamental to provide a better understandingof the results and to aid the correct interpretation of theexperimental data.
Computational studies of noble metal clusters are,however, not straightforward because of the smallsingle-particle energy gap, implying a possible multi-reference character of the electronic states, and due tothe complex correlation effects characterizing such sys-tems. For these reasons, in principle an accurate de-scription of the electronic structure can only be achievedby high-level correlated multi-reference approaches. However, these methods are hardly applicable for thestudy of clusters, due to the very high computationalcost. On the other hand, “conventional” single-referencewave-function methods (e.g. Møller-Plesset perturba-tion theory , configuration interaction , or cou-pled cluster ) often display important basis set and/ortruncation errors, even for relatively small cluster sizes,which prevent the achievement of accurate, reliable, re-sults. Thus, one of the most used computational toolsto study noble metal clusters is Kohn-Sham density-functional theory (DFT).
DFT calculations on noble metal clusters are often per-formed using a semilocal approximation for the exchange- correlation (XC) functional, e.g. the generalized gradientapproximation (GGA) or the meta-GGA’s. This is anefficient approach, but in variouscases it has also shown limited accuracy, especially in thenot so rare case when it is necessary to discriminate be-tween isomers with rather similar energies (for example inthe prediction of the two- to three-dimensional crossoverin gold and silver clusters ). However, unlike in thecase of main group molecular calculations, the use of hy-brid functionals, which include a fraction of exact ex-change, is not able to provide a systematic improvement.Instead, it often leads to a worsening of the results.
The origin of this problem possibly traces back on thetoo simplicistic idea of mixing a fixed fraction of exactexchange with a semilocal approximation.In the hybrid wavefunction-DFT formalism a certainfraction a of the electron-electron interaction is treatedwithin a wave function method, while the remaining en-ergy is captured with a semilocal functional. In a com-pact notation this can be written as E = min Ψ n h Ψ | ˆ T + a ˆ V ee + ˆ V ne | Ψ i + ¯ E a Hxc [ ρ Ψ ] o , (1)where the complementary Hartree-exchange-correlationfunctional ¯ E a Hxc depends on Ψ only through its density ρ Ψ . In Eq. (1), ˆ T is the electronic kinetic energy opera-tor, ˆ V ee the electron-electron repulsion operator and ˆ V ne the external potential due to the nuclei. When the min-imization over Ψ in Eq. (1) is restricted to single Slaterdeterminants Φ, we obtain the usual hybrid functionalapproximation, which mixes a fraction a of Hartree-Fockexchange with a semilocal functional, while using second-order peturbation theory to improve the wavefunction Ψleads to single-parameter double-hybrid functionals. The XC part E xc [ ρ ] of ¯ E a Hxc that needs to be approx-imated in the standard hybrid functionals formalism isusually modeled starting from the adiabatic connectionformula E xc [ ρ ] = Z W λ [ ρ ] dλ (2)where λ is the interaction strength and W λ [ ρ ] = h Ψ λ [ ρ ] | ˆ V ee | Ψ λ [ ρ ] i − U [ ρ ] is the density-fixed linear adi-abatic connection integrand, with Ψ λ [ ρ ] being the wavefunction that minimizes ˆ T + λ ˆ V ee while yielding the den-sity ρ , and U [ ρ ] being the Hartree energy. Most hy-brid functionals then employ a simple ansatz for thedensity-fixed linear adiabatic connection integrand, forexample W λ [ ρ ] = W DFA λ [ ρ ] + (cid:0) E x − E DFA x (cid:1) (1 − λ ) n − , (3)where DFA denotes a density functional approximation(i.e. a semilocal functional), E x denotes the Hartree-Fockexchange functional, and n is a parameter. Substitut-ing Eq. (3) into Eq. (2), yields the usual linear mixingbetween the exact exchange and the density functionalapproximation with a = 1 /n . However, Eq. (3) is a quitearbitrary expression for W λ . It only satisfies the con-straint that W = E x but for λ = 0 it incorporates noexact information and it is not even recovering the correctweak-interaction limit behavior. Thus, most of the accu-racy of hybrids relies on the empiricism included into theparameter n and the DFA. This seems to work well formain-group molecular systems but not for other systemssuch as metal clusters considered here.A possible non-empirical route that allows to overcomethe limitations of a fixed mixing parameter is the origi-nal idea of Seidl and coworkers to build a model forthe adiabatic-connection integrand of Eq. (2) by inter-polating between the known weak- and strong-couplinglimits, W λ → [ ρ ] = W [ ρ ] + λW ′ [ ρ ] + · · · (4) W λ →∞ [ ρ ] = W ∞ [ ρ ] + W ′∞ [ ρ ] √ λ + · · · , (5)where W [ ρ ] = E x [ ρ ] , (6) W ′ [ ρ ] = 2 E GL2 c [ ρ ] , (7)with E GL c being the second-order G¨orling-Levy (GL)correlation energy, whereas W ∞ [ ρ ] is the indi-rect part of the minimum expectation value of theelectron-electron repulsion in a given density, and W ′∞ [ ρ ] is the potential energy of coupled zero-pointoscillations. The idea is that by using a function of λ able to link the result from perturbation theory withthe λ → ∞ expansion of W λ [ ρ ], an approximate resum-mation of the perturbative series is obtained. The exact W ∞ [ ρ ] and W ′∞ [ ρ ] are highly nonlocal den-sity functionals that were approximated in the orig-inal work of Seidl and coworkers by the semilocalpoint-charge-plus-continuum (PC) model (see the ap-pendix). As a result, a series of XC functionals can be derived depending on the chosen interpolating func-tion and on whether the λ → ∞ expansion includes ornot the order 1 / √ λ : ISI and revISI also include W ′∞ [ ρ ], while SPL and LB only include W ∞ [ ρ ]. Theyare briefly described in the appendix. These function-als, which are all based on an adiabatic connection inte-grand interpolation (ACII), will be generally referred toas ACII functionals. They are non-empirical in the sensethat they are approximate perturbation-theory resum-mations, include full exact exchange, and describe cor-rectly correlation in the weak-interaction limit. There-fore, they are well-suited to try to overcome the limita-tions of semilocal and hybrid DFT approaches. Theirmost severe problem could be the lack of size consistencyfor species made of different atoms, an error that is ab-sent in the case of homogeneous clusters. Moreover, thesize consistency issue is actually quite subtle and canbe corrected in many cases.The ACII functionals have been rarely tested on sys-tems of interest for practical applications, with the ex-ception of a recent assessment of the ISI functional formain-group chemistry. This investigation has revealedinteresting features of this functional and suggested pos-sibilities for future applications.In this paper we move away from main group chemistryto assess different ACII functionals for the description ofthe electronic properties of noble metal clusters, made upof gold and silver. As we have mentioned above, these arevery important systems for materials science and chemi-cal applications but their proper computational descrip-tion is still a challenge. Thus, the testing of high-levelDFT methods for this class of systems has a great prac-tical interest. Moreover, the application of non-empiricalXC functionals, constructed on a well defined theoreti-cal framework, to the challenging problem of the simula-tion of electronic properties of noble metal clusters canhelp to highlight new properties and limitations of suchapproaches. In fact, the next step forward could be tomodel the adiabatic connection integrand locally byinterpolating between the exact exchange energy densityand the λ → ∞ one, for which exact results and ap-proximations compatible with the exact exchange energydensity have been recently designed. In order to becompatible with the exact exchange energy density, theseapproximations are non-local and thus more expensivethan the semilocal PC functionals (which suffer from theusual gauge problem that arises when we want to com-bine semilocal functionals with the exact exchange energydensity and thus cannot be used in this framework). Ithas been found that the local interpolations are in generalmore accurate than their global counterpart. Thus, thestudy carried out here provides also a very useful firstidea of what could be achieved with these higher-levelapproaches.
II. COMPUTATIONAL DETAILS
In this work we have tested four ACII XC functionals,which are based on an interpolation of the density-fixedlinear adiabatic connection integrand, namely ISI, revISI, , SPL, , and LB (see the appendix for de-tails). Additionally, for comparison, we have included re-sults from the Perdew-Burke-Ernzerhof (PBE) and thePBE0 functionals, which are among the most usedsemilocal and hybrid functionals, respectively, as well asfrom the B2PLYP double hybrid functional, which alsoincludes a fraction of second-order Møller-Plesset corre-lation energy (MP2). We have also considered a compar-ison with the second- , third- and fourth-order Møller-Plesset perturbation theory (MP2, MP3, MP4) results.This is because, as explained, the ACII functionals canbe seen as an approximate resummation of perturbationtheory, so that it is interesting to compare them with thefirst few lower orders. The reference results used in theassessment are specified below for each test set consid-ered: • Small gold clusters . This set consists of theAu , Au − , Au , Au +3 , Au − , and Au clusters. Forall these systems we have calculated the atomiza-tion energies; for the anions as well as for Au wehave computed the ionization potential (IP) ener-gies. The geometries of all clusters have been takenfrom Ref. 33; they are shown in Fig. 1. Referenceenergies have been calculated at the CCSD(T) levelof theory. • Small silver clusters . This set includes Ag ,Ag +2 , Ag − , Ag , Ag +3 , Ag − , Ag . As for the smallgold clusters case, we have computed the atomiza-tion energies of all the silver clusters and the IP ofthe anions as well as of Ag . The geometries of allsystems have been taken from Ref. 38; they areshown in Fig. 1. Reference values for the energieshave been obtained from CCSD(T) calcula-tions. • Binary gold-silver clusters . This set consid-ers the AuAg, AuAg − , Au Ag, Au Ag − , AuAg ,and AuAg − clusters. Atomization energies havebeen calculated for all system, while IPs have beencomputed for the anions. Note that for the an-ions we considered as atomization energy the av-erage with respect to the two possible dissociationchannels, that is AuAg − → Au+Ag − and AuAg − → Au − +Ag; Au Ag − → Au + Ag − and Au Ag − → Au − + Ag; AuAg − → Au + Ag − and AuAg − → Au − + Ag . The geometries of the binaryclusters have been obtained considering the struc-tures reported in Ref. 69 (see Fig. 1) and opti-mizing them at the revTPPS/def2-QZVP level oftheory. Reference energies have been calcu-lated at the CCSD(T) level of theory. Au Au −2 Au Au Au Au Ag Ag Ag Ag Ag Ag Ag AuAg − AuAg −− Au Ag AuAg Au Ag AuAg FIG. 1. Structures of the small gold, silver and binary gold-silver clusters. Au − I Au − II Au − IIIAu −12 − I Au −12 − IIAu −13 − I Au −13 − II Au −13 − IIIAg +5 − I Ag +5 − IIAg +6 − IIAg +6 − IAg +7 − I Ag +7 − II FIG. 2. Structures of the gold and silver clusters consideredfor the 2D-3D dimensional crossover problem. • Gold 2D-3D crossover . This set includes theAu − , Au − , and Au − clusters, that are involvedin the two- to three-dimensional crossover of goldclusters. The geometries of all systems have beentaken from Ref. 50 and are shown in Fig. 2. • Silver 2D-3D crossover . This set consists of theAg +5 , Ag +6 , and Ag +7 clusters, which are relevantto study the two- to three-dimensional crossoverof silver clusters. Geometries have been obtainedoptimizing at the revTPPS/def2-QZVP level oftheory, the lowest lying structures reportedin Ref. 38. The structures are reported in Fig. 2.All the required calculations have been performed withthe TURBOMOLE program package, employing,unless otherwise stated, the aug-cc-pwCVQZ-PP basis set and a Stuttgart-Koeln MCDHF 60-electron effec-tive core potential. The calculations concerning theISI, revISI, SPL, and LB functionals have been performedin a post-self-consistent-field (post-SCF) fashion, usingHartree-Fock orbitals. This choice is consistent with theresults of Ref. 82, where it has been found that the ISIfunctional yields much better results when used as a cor-relation correction for the HF energy. The PBE andPBE0 calculations have been performed using a full SCFprocedure; B2PLYP calculations have been carried outas described in Ref. 96, considering a SCF tretment ofthe exchange and semilocal correlation part and addingthe second-order MP2 correlation fraction as a post-SCFcorrection.
III. RESULTS
In this section we analyze the performance of the ACIIXC funcionals for the description of the electronic prop-erties of gold, silver and mixed Au/Ag clusters. Theresults are compared to those obtained from other ap-proaches, such as semilocal and hybrid DFT as well aswave-function perturbation theory.
A. Total Energies
To start our investigation we consider, in table I, theerrors on total energies computed with different methodswith respect to the CCSD(T) reference values. Althoughthis quantity is usually not of much interest in practicalapplications (where energy differences are usually con-sidered), the analysis of the errors on total energies willbe useful to understand the performances of the differentfunctionals for more practical properties such as atom-ization or ionization energies.Inspection of the data shows that the ACII functionalsdo not perform very well for the total energy. In fact,they yield the highest mean absolute errors (MAEs), be-ing even slightly worse than the semilocal PBE approachand giving definitely larger errors with respect to pertur-bation theory (MP2, MP3, and MP4) and to the doublehybrid B2PLYP functional. Among the ACII function-als, the SPL and especially the LB approach perform sys-tematically better than ISI and revISI. Thus LB yieldserrors which are often 30% smaller than ISI, even thoughthey are still usually larger than those of the other non-ACII methods. On the other hand, considering the stan-dard deviation of the errors (last line of Table I) we notethat the ACII results display a quite small dispersionaround the average (with LB and SPL again slightly bet-ter than ISI and revISI). This is related to the fact thatthe ACII functionals all give a quite systematic under-estimation (in magnitude) of the energy of all systems.In contrast, PBE, PBE0, and partly B2PLYP give largervalues of the standard deviation. This depends on thefact that these methods describe quite accurately some
TABLE I. Errors on total energies (eV/atom) of small gold, silver, and binary clusters. For each set of clusters the meanabsolute error (MAE) is reported. In the bottom part of the table we report also the statistics for the overall set (mean error(ME), MAE, and standard deviation).PBE PBE0 B2PLYP ISI revISI SPL LB MP2 MP3 MP4Au -4.93 -3.75 -1.73 2.84 2.93 2.65 1.97 -0.33 0.98 -0.27Au+ -4.58 -3.73 -1.64 2.63 2.70 2.50 1.89 -0.10 0.68 -0.15Au- -4.94 -3.47 -1.65 3.26 3.38 3.02 2.25 -0.35 1.44 -0.41Au2 -4.96 -3.66 -1.69 2.89 2.99 2.67 1.94 -0.53 1.22 -0.43Au2- -4.97 -3.58 -1.66 3.06 3.17 2.83 2.08 -0.46 1.40 -0.44Au3 -4.97 -3.65 -1.67 2.89 3.00 2.67 1.94 -0.54 1.29 -0.48Au3+ -4.90 -3.68 -1.66 2.80 2.90 2.59 1.88 -0.50 1.15 -0.43Au3- -4.94 -3.54 -1.66 2.98 3.10 2.73 1.96 -0.65 1.40 -0.55Au4 -4.96 -3.62 -1.67 2.86 2.97 2.63 1.87 -0.66 1.33 -0.54ME -4.91 -3.63 -1.67 2.91 3.01 2.70 1.97 -0.46 1.21 -0.41MAE 4.91 3.63 1.67 2.91 3.01 2.70 1.97 0.46 1.21 0.41Ag -0.89 -0.36 0.21 3.24 3.39 2.91 2.18 -0.26 1.04 -0.27Ag+ -0.44 -0.25 0.38 2.99 3.13 2.71 2.04 -0.19 0.76 -0.20Ag- -0.99 -0.21 0.24 3.68 3.85 3.31 2.53 -0.08 1.39 -0.32Ag2 -0.96 -0.30 0.22 3.32 3.49 2.97 2.19 -0.37 1.18 -0.38Ag2+ -0.79 -0.34 0.28 3.14 3.28 2.82 2.11 -0.26 0.95 -0.27Ag2- -1.00 -0.27 0.24 3.50 3.67 3.14 2.37 -0.22 1.34 -0.33Ag3 -0.95 -0.30 0.25 3.32 3.49 2.96 2.19 -0.39 1.23 -0.40Ag3+ -0.85 -0.30 0.27 3.21 3.37 2.86 2.11 -0.41 1.10 -0.38Ag3- -0.97 -0.23 0.25 3.45 3.63 3.07 2.27 -0.38 1.32 -0.43Ag4 -0.95 -0.27 0.24 3.29 3.47 2.91 2.12 -0.51 1.24 -0.45ME -0.88 -0.28 0.26 3.31 3.48 2.97 2.21 -0.31 1.15 -0.34MAE 0.88 0.28 0.26 3.31 3.48 2.97 2.21 0.31 1.15 0.34AuAg -2.93 -1.95 -0.72 3.10 3.24 2.81 2.06 -0.47 1.20 -0.41AuAg- -2.97 -1.91 -0.70 3.27 3.41 2.97 2.20 -0.37 1.36 -0.39Au Ag -3.58 -2.49 -1.00 3.03 3.16 2.76 2.01 -0.51 1.28 -0.46Au Ag − -3.57 -2.40 -1.00 3.14 3.28 2.84 2.06 -0.57 1.36 -0.50AuAg -2.26 -1.38 -0.37 3.18 3.33 2.86 2.10 -0.46 1.26 -0.43AuAg − -2.32 -1.34 -0.40 3.30 3.46 2.96 2.18 -0.46 1.36 -0.47ME -2.94 -1.91 -0.70 3.17 3.31 2.87 2.10 -0.47 1.30 -0.44MAE 2.94 1.91 0.70 3.17 3.31 2.87 2.10 0.47 1.30 0.44Overall statisticsME -2.82 -1.88 -0.66 3.13 3.27 2.85 2.10 -0.40 1.21 -0.39MAE 2.82 1.88 0.87 3.13 3.27 2.85 2.10 0.40 1.21 0.39Std.Dev. 1.81 1.51 0.87 0.24 0.27 0.18 0.16 0.15 0.20 0.10 systems (e.g. Ag clusters), which are the ones that effec-tively contribute to produce a quite low MAE, but theygive significantly larger errors for other systems. Thisbehavior is a signature of the too simplicistic nature ofthese functionals, which cannot capture equally well thephysics of all systems.The observed standard deviations suggest that, whenenergy differences are considered, the ACII functionals can benefit from a cancellation of the systematic error,such that rather accurate energy differences can be ob-tained. We must remark also that the standard deviationvalues reported in Table I allow only a partial under-standing of the problem because they are obtained fromall the data but, depending on the property of interest,some energy differences may be more relevant than oth-ers, e.g. for atomization energies the difference betweena cluster energy and the energy of the composing atomsis the most relevant. Thus, for example MP methods allyield quite low standard deviations, but a closer look atthe results shows that the errors for atoms are quite dif-ferent than those for the clusters (much more differentthan for ACII methods); hence, we can expect that, de-spite a quite good MAE and a small standard deviation,MP2, MP3, and MP4 atomization energies can display alimited accuracy. A more detailed analysis of the rela-tionship between the data reported in Table I and somerelevant energy difference properties will be given in Sec-tion IV. B. Atomization and Ionization energies
A first example of an important energy difference isthe atomization energy. The atomization energy valuescalculated for the sets of gold, silver, and binary clusterswith all the methods are reported in Table II. Observingthe data it appears that, as anticipated, for atomizationenergies the ACII functionals work fairly well. In par-ticular, SPL and LB, yield mean absolute relative errors(MAREs) of about 2-3% for all kinds of clusters, beingcompetitive with the B2PLYP functional. The ISI andrevISI functionals perform slighlty worse, displaying asystematic underbinding and giving overall MAREs of4% and 6%, respectively. Moreover, unlike for SPL andLB, non-negligible differences exist in the description ofthe different materials with gold clusters described betterthan silver ones. Overall the ISI and revISI functionalsshow a comparable performance as PBE and better thanPBE0. Finally, the MP results show a quite poor per-formance, exhibiting MAREs ranging form 10% to 20%.In addition, we can note that MP2 results are closer toMP4 results than MP3 ones not only from a quantita-tive point of view but also qualitatively (MP2 and MP4always overbind while MP3 always consistently under-binds). This is a clear indication of the difficult con-vergence of the perturbative series for the metal clusterselectronic properties.In Table III, we report the computed ionization po-tential energies, which are other important energy dif-ferences to consider for metal clusters. In this case theACII functionals perform rather poorly, being the worstmethods, if we exclude MP3. As in the case of atomiza-tion energies, SPL and LB (especially the latter) show aslightly better performance than ISI and revISI. Never-theless, the results are definitely worst than for B2PLYP,PBE and even PBE0. A rationalization of this failure willbe given is section IV.
C. 2D-3D crossover
To conclude this section, we consider the problem ofthe two- to three-dimensional (2D-3D) crossover of an-ionic gold clusters and cationic silver clusters. Differ- ent studies have indicated that for anionic gold clustersthe dimensional crossover occurs between Au − (2D) andAu − (3D), with the 2D and 3D Au − structures beingalmost isoenergetic. On the other hand, for cationicsilver clusters it has been suggested that the dimensionaltransition occurs already for Ag +5 , which has a 2D struc-ture with a slightly lower energy than the 3D one, whileAg +6 and Ag +7 display lowest energy 3D structures. Anyway, this is a quite difficult problem because exper-imentally it is not trivial to distinguish clusters of thesame size but different dimensionality. A computationalsupport is thus required.
However, to de-scribe correctly the energy ordering of several noble metalclusters with very similar energies is a hard task for anycomputational method.
For this reason, this isa very interesting problem from the computational pointof view.In Table IV, we report the energies calculated for theanionic gold clusters and cationic silver clusters relevantfor the 2D-3D transition. The Table shows, for compari-son, also the results obtained with the BLOC meta-GGAfunctional , which is expected to be one of the mostaccurate approaches for this kind of problems. Observ-ing the data, one can immediately note that the PBE,PBE0 and even B2PLYP methods are not reliable forthe dimensional crossover of noble metal clusters. In fact,PBE always favors 2D structures, whereas PBE0 predictsthe 2D-3D transition at a too large cluster dimension forgold, Au − (although the 3D geometry with lowest energyis not the same as the one we find with BLOC and allACII functionals), and for silver the energies of the 2Dand 3D clusters differ slightly for both n = 6 and n = 7,not evidencing a clear transition at the expected clus-ter size. A similar behaviour is found for the B2PLYPfunctional, which was instead one of the best for the at-omization energies and IPs of small clusters. The ACIIfunctionals overall perform all quite similarly, predictingfor all clusters the expected ordering and agreeing wellwith BLOC results for the cationic Ag clusters but tend-ing to favor 3D structures in the anionic Au clusters. Wenote that this behavior is somehow inherited from theMP2 method, which however performs much worse thanany of the ACII functionals considered here. IV. DISCUSSION AND ANALYSYS OF THE RESULTS
In the previous Section we saw that the ACII function-als perform rather well for the calculation of atomizationenergies of noble metal clusters. As mentioned above, agood rationalization of the observed results can be ob-tained in terms of the energy errors that the differentmethods display for the total energies of atoms and ofthe clusters. These have been reported in Table I.
TABLE II. Atomization energies (eV) of small gold, silver, and binary clusters. Note that for anionic binary clusters the averagebetween the two possible dissociation paths has been considered (see Section II). For each set of clusters the mean error (ME),the mean absolute error (MAE), the mean absolute relative error (MARE), and the standard deviation are reported. In thebottom part of the table we report also the statistics for the overall set.PBE PBE0 B2PLYP ISI revISI SPL LB MP2 MP3 MP4 CCSD(T)Au − +3 − +2 − +3 − − Ag 3.65 3.26 3.42 3.47 3.41 3.59 3.73 4.28 2.80 4.22 3.65Au Ag − − A. Energy differences
For a better visualization here we additionaly plot, inFig. 3, the quantity δ ∆ E = ∆ E ( M n M − m M + l ) − (8) − X n ∆ E ( M n ) − X m ∆ E ( M − n ) − X l ∆ E ( M + n ) , where ∆ E are the total energy errors (the ∆ E per atomare reported in Table I), M =Au or Ag, and n, m, l are in-tegers such that M n M − m M + l corresponds to a given clus-ter (e.g. for Au +3 we have M =Au, n = 2, m = 0, and TABLE III. Ionization potentials (eV) of small gold, silver, and binary clusters. For each set of clusters the mean error (ME),the mean absolute error (MAE), the mean absolute relative error (MARE), and the standard deviation are reported. In thebottom part of the table we report also the statistics for the overall set.PBE PBE0 B2PLYP ISI revISI SPL LB MP2 MP3 MP4 CCSD(T)Au 9.54 9.22 9.29 9.00 8.97 9.05 9.13 9.42 8.91 9.32 9.20Au − − − − − − − Ag − − l = 1). This quantity provides a measure of how differ-ent is the energy error made on a given cluster from thatof its constituent atoms. Inspection of the plots showsthat the smaller δ ∆ E values are yielded by the ISI andSPL (revISI and LB, not reported, give similar results).These functionals are also among the best performers forthe atomization energies. On the other hand, for PBEwe observe that the δ ∆ E is small for gold clusters, withthe exception of Au +3 , while for silver clusters is larger.Indeed, looking to Table II we can find that PBE per-forms well for gold clusters, with the exception of Au +3 that yields an error of 0.27 eV (more than twice largerthan the MAE), while it performs less well for silver clus- ters. Finally, for MP2 the values of δ ∆ E are generallyvery large. Thus, despite MP2 is on average quite accu-rate in the description of the total energies (see Table I)it fails to produce accurate atomization energies becauseof accumulation of the errors.A similar analysis, can be made to comment the resultsof the ionization potential calculations (reported in TableIII). However, in this case the difference to consider isbetween the neutral and the charged species. Then, adifferent behavior is observed. In fact, while for mostof the considered methods the total energy error is notmuch different between a neutral and a charged speciesof the same cluster, for the ACII functionals we always TABLE IV. Relative energies (eV) with respect to conformer I (see Computational details) of 2D and 3D anionic gold clustersand cationic silver clusters. For the gold clusters the data include the correction terms reported in Table IV of Ref. 50.PBE PBE0 BLOC B2PLYP ISI revISI SPL LB MP2Au − -I 2D – – – – – – – – –Au − -II 3D 0.217 0.224 0.206 0.147 0.083 0.090 0.070 0.054 -0.006Au − -III 3D 0.270 0.179 0.354 0.254 0.265 0.251 0.302 0.344 0.499Au − -I 3D – – – – – – – – –Au − -II 2D -0.450 -0.340 0.008 -0.144 0.710 0.669 0.789 0.882 1.228Au − -I 3D – – – – – – – – –Au − -II 3D -0.027 -0.032 0.037 -0.024 0.497 0.495 0.499 0.527 0.618Au − -III 2D -0.111 0.056 0.386 0.248 0.802 0.894 0.917 0.824 1.069Ag +5 -I 3D – – – – – – – – –Ag +5 -II 2D 0.021 0.025 0.024 0.020 0.021 0.020 0.018 0.017 0.013Ag +6 -I 3D – – – – – – – – –Ag +6 -II 2D -0.005 0.055 0.280 0.007 0.220 0.211 0.241 0.265 0.348Ag +7 -I 3D – – – – – – – – –Ag +7 -II 2D -0.099 – 0.303 -0.059 0.286 0.270 0.318 0.352 0.474 Au Au Au Au Au Au Ag Ag Ag Ag Ag Ag Ag cluster δ ∆ E ( e V ) PBEISISPLMP2
FIG. 3. Difference in the total energy error between a clusterand its constituent atoms (see Eq. (8)). observe an increase of the error with the charge. Thissituation is schematized in Fig. 4, where we plot, forseveral examples, the quantity∆( q ) = ∆ E ( A q ) − ∆ E ( A ) , (9)with A being any of the systems under investigation and q = − , ,
1. The observed trend may trace back to adifferent ability of ACII functionals to describe the high-and low-density regimes. As a consequence, the ACIIfunctionals are generally the worst performers for thecalculation of ionization potentials, while PBE and es-pecially B2PLYP perform well thanks to the more ho-mogeous description of the differently charged species.This analysis shows that, although the quality of thetotal energies produced by a functional is a key elementto understand the performance of the functional, the ba- -1 0 1 charge q -0.20.00.20.4 ∆ E ( q ) [ e V ] PBEB2PLYPISIMP2 -1 0 1 charge q -0.20.00.20.4 ∆ E ( q ) [ e V ] -1 0 1 charge q -0.10-0.050.000.050.10 ∆ E ( q ) [ e V ] -1 0 1 charge q -0.10-0.050.000.050.10 ∆ E ( q ) [ e V ] AuAu AgAg FIG. 4. Variation of the energy error with the total charge ofthe system (Au top left, Au bottom left, Ag top right, Ag bottom right). The values are scaled to the neutral systemvalue (see Eq. (9)). sic property to observe is not the quality of the absoluteenergies, but rather the variance of the errors. Further-more, the contrasting behaviors we have observed for thedescription of the atomization energies and of the ioniza-tion potentials highlights the subtleties inherent to suchcalculations. In particular, the accuracy of the ACIIfunctionals has been shown to be not much dependenton the investigated material (Au or Ag) nor on the sys-tem’s size but to be quite sensitive to the charge stateof the computed system. The first feature is a positiveone. This is related, as we saw, to the computation ofatomization energies, but even more importantly it indi-cates that the idea beyond the construction of the ACIIfunctionals is in general quite robust such that the func-0tionals, although not very accurate in absolute terms (seeTable I) are well transferabe to systems of different sizeand composition. This is not a trivial results since, as wedocumented, other methods (e.g. PBE and PBE0, buteven MP4) do not share this property. On the contrary,the dependence of the ACII functionals on the chargestate of the system indicates a clear limitation of suchapproaches. They are in fact unable to describe with sim-ilar accuracy systems with qualitatively different chargedistributions. As a consequence, the ionization potentialcalculations are problematic for ACII functionals.Note however that, because accurate experimental dataare not available for all the systems, our assessment of theperformances of the ACII functionals on small clusters,see Fig. 1, and Tables I, II, and III, is carried out w.r.t.CCSD(T) values. This allows a more direct and sensiblecomparison of the results, whereas the comparison withexperimental data would require the consideration of fur-ther effects such as thermal/vibronic ones as well as spin-orbit coupling. Of course CCSD(T) results cannotbe considered “exact” for metal clusters. Nevertheless,an accurate comparison with available experimental datafrom literature shows that, for atoms (regarding ioniza-tion energies) and neutral dimers and trimers (re-garding both ionization and atomization energies),
CCSD(T) yields results within 0.04 eV from the experi-mental ones. While for the charged dimers and trimers(regarding both ionization and atomization energies),CCSD(T) results are within 0.2 eV from the ex-perimental ones. This larger discrepancy may be partlyascribable to a diminished accuracy of the CCSD(T) cal-culation per se in these cases, but it may also be possiblydue to the rather large error bars associated to the mea-sures on the experimental side and on the increased im-portance of correcting terms on the computational side.
B. AC curves: gold dimer showcase
To rationalize the origin of the limitations of the ACIIfunctionals as well as to understand in depth the dif-ferences and the similarities between the different inter-polation formulas it would be necessary to inspect insome detail the shape of the density-fixed linear adia-batic connection integrand defining ISI, revISI, SPL, andLB. However, contrary to small atoms and molecules (see,e.g., Refs. 120–122), for noble metal clusters there existsno reference adiabatic connection integrands to compareto. Thus, such a detailed analysis is not really possi-ble. Nevertheless, some useful hints can be obtained by asemi-qualitative comparison of the various adiabatic con-nection curves. As an example, in Fig. 5 we report, forthe Au case (the other systems studied here have verysimilar features), the atomization adiabatic connectionintegrand, defined as W atλ (Au ) = W λ (Au ) − W λ (Au) , (10) FIG. 5. Atomization adiabatic connection integrands (see Eq.(10)) corresponding to ISI, revISI, SPL, and LB for the Au case; the thick curve in gray corresponds to the linear expan-sion for the atomization adiabatic connection integrand (Eq.(11)) for ISI, revISI, SPL, and LB. The integrated value (be-tween 0 and 1) of this quantity corresponds to the XCatomization energy calculated with a given ACII func-tional. For discussion we have plotted also the weak inter-acting limit expansion truncated at linear order in λ forthe atomization adiabatic connection integrand, which isdefined as W atλ,LE (Au ) = W λ,LE (Au ) − W λ,LE (Au) , (11)where the linear expansion (LE) of the AC integrandfor a species X is W λ,LE (X) = E x (X) + 2 λE GL2 c (X) inagreement with Eq. (4) and in the case of HF orbitals E GL2 c (X) = E MP2 c (X). Because of the weak-interactinglimit constraint, all the curves plotted in the figure sharethe same λ = 0 value, which corresponds to the Hartree-Fock exchange atomization energy, as well as the sameslope at this point. The curves remain very similar upto λ ≈ .
2, which is not strictly dictated by the weak-interacting limit constraint but rather by a possible lackof flexibility in the interpolation formulas. For valuesof λ & .
2, the curves associated to the various func-tionals start to differ, due to the different ways they ap-proach the W ∞ value for λ = ∞ . Note that in thiscase ISI and revISI are further constrained to recover the W ′∞ slope, whereas SPL and LB do not have this con-straint. The interpolation towards the strong-interactionlimit is therefore the main feature differentiating the var-ious ACII functionals, even in the range 0 ≤ λ ≤
1. Ingeneral, revISI is the slowest to approach the asymptotic W ∞ value, whereas LB is the fastest. So the former willusually yield the smaller XC energies, whereas the lat-ter will produce the larger XC energies (in magnitude).In fact, turning to the Au example reported in Fig. 5,the inspection of the plot shows that revISI is indeed theslowest to move towards the asymptotic W at ∞ value (forAu W at ∞ = − . atomization energy by 0.13 eV). On the opposite,LB is the fastest to move towards the asymptotic W at ∞ value, thus it gives the larger atomization energy (over-1estimating it by 0.06 eV). In this specific case, the SPLfunctional, which behaves almost intermediately betweenrevISI and LB, yields a very accurate value of the atom-ization energy, underestimating it by only 0.03 eV.Thus, we have seen that there are two main featuresthat can determine the performance of an ACII func-tional. The first one is surely the behavior towardsthe strong-coupling limit, which is able to influence theshape of the adiabatic connection integrand curve for λ ' . / .
3. This behavior is indeed modeled differentlyby the various functionals examined in this work, but itappears that none of them can really capture the correctbehavior in the range of interest 0 . ≤ λ ≤
1. This ispossibly due to the fact that information on the λ = ∞ point is not sufficient to guide correctly the interpolationat the quite small λ values of interest for the calcula-tion of XC energies. A second factor that is relevant forthe functionals’ performance is the small λ behavior. Atvery small λ values this is determined by Eq. (4), butfor larger values of the coupling constant (at least for0 . ≤ λ ≤ .
2) the shape of the curve should depart fromthe slope given by E GL2 c in order to correctly describe thehigher-order correlation effect. Instead, we have observedthat all the ACII functionals provide the same behaviorup to λ ≈ .
2. This indicates that the interpolation for-mulas have not enough flexibility to differentiate from theasymptotic behavior imposed at λ = 0. C. Role of the reference orbitals
The ACII functionals are orbital-dependent nonlin-ear functionals, thus they are usually employed to com-pute the XC energy in a post-SCF fashion (as we didin this work). Then, the results depend on the choiceof the orbitals used for the calculation. Recent work has evidenced that ISI results for main-group chemistryare much improved when Hartree-Fock orbitals are used.This has been basically traced back to the characteris-tics of the Hartree-Fock single-particle energy gap (whichdetermines the magnitute of E GL2 c and thus the weak-interaction behavior of the curves).For gold and silver clusters, after some test calcula-tions, we found a similar result for all the ACII formu-las considered. For this reason, all the results reportedin Section III are based on Hartree-Fock orbitals. Toclarify this aspect, we have reported in Fig. 6 boththe bare and the atomization adiabatic connection in-tegrands computed with the SPL formula (similar re-sults are obtained for the other formulas) for Au andAu using either Hartree-Fock and PBE orbitals. It canbe seen that the adiabatic connection curve of Au , ob-tained from Hartree-Fock orbitals, is very similar to twicethe Au curve. Hence, the atomization adiabatic connec-tion integrand is rather flat, yielding (correctly) a mod-erate atomization XC energy. This behavior dependspartly on the fact that in Hartree-Fock calculations Au has almost twice the exchange energy of Au but, pri- -27-26-25-24 W λ HF orbs.PBE orbs. λ -0.2-0.10.0 w λ a t HF orbs.PBE orbs. λ -24.2-24.0-23.8-23.6 W λ FIG. 6. Top: Adiabatic connection integrands computedwith the SPL formula [Eq. (A8)] for Au (solid line) andAu (dashed line) using Hartree-Fock and PBE orbitals; theAu curve is multiplied by a factor of 2; the inset shows theweak-interaction part of the curves. Bottom: Atomizationadiabatic connection integrands (see Eq. (10)) computed withthe SPL formula for the Au case. marly, it traces back to the fact that the Au MP2 cor-relation energy is almost perfeclty two times larger thanthe Au one (which in turn depends on the fact that thetwo systems have very close single-particle energy gaps– 7.604 eV and 7.707 eV, respectively – and on the size-extensivity of the MP2 method). Thus, the adiabaticconnection integrands for Au and twice the Au have al-most identical slopes at λ = 0 and similar behaviors for λ ≤
1. Instead, when PBE orbitals are used, larger dif-ferences between the Au and twice the Au curves canbe noted. These originate only partially from the factthat, in the case of PBE orbitals, the exact exchangecontributions of Au and twice Au are not much simi-lar (they differ by 0.045 eV). Mostly they depend on therather different GL2 correlation energies for the systems( E GL2 c (Au ) − E GL c (Au) = − . eV ), which in turntrace back to the fact that the single particle energy gapscomputed for Au and Au are very different: 2.014 eVand 0.718 eV, respectively. Consequently, the atomiza-tion adiabatic connection integrand curve calculated withPBE orbitals is steeper than the Hartree-Fock-based oneand therefore it yields significantly larger atomization XCenergies. This results in a strong tendency of PBE-basedACII functionals to overbind the noble metal clusters. D. Further analysis of the ACII’s formulas
We have seen in Sec. III that SPL and LB formulasshow overall better performances than ISI and revISI. Asmentioned, the main difference between the two groupsis that the former use a three-parameters interpolationformula while the latter make use of a fourth ingredient2from the λ → ∞ limit, i.e. the zero-point oscillationterm W ′∞ [ ρ ]. The revISI formula also recovers the exactexpansion at large λ to higher orders. However, we haveto keep in mind that the ingredients coming from thestrong interaction limit are not computed exactly, butapproximated with the semilocal PC model. Comparisonwith the exact W ∞ [ ρ ] and W ′∞ [ ρ ] for light atoms suggests that the PC approximation of the W ∞ [ ρ ] termis more accurate than the one for W ′∞ [ ρ ]. Moreover, theparameters appearing in the PC model for W ∞ [ ρ ] are alldetermined by the electrostatics of the PC cell, while inthe case of W ′∞ [ ρ ] the gradient expansion does not give aphysical result, and one of the parameters has to be fixedin other ways, for example by making the model exactfor the He atom. Another important point to consider is that, as ex-plained in Sec. IV C, we are using the ACII functionalswith Hartree-Fock orbitals, which means that they areused as a correlation functional for the Hartree-Fock en-ergy. In other words, the ACII correlation functionals areused here as an approximate resummation of the Møller-Plesset perturbation series: they recover the exact MP2at weak coupling, and perform much better than MP3and MP4 for atomization energies (see Table II). Thus,a first question that needs to be addressed is whetherthe PC model used here to compute the infinite couplingstrength functionals is accurate also for the Hartree-Fockadiabatic connection, in which the λ -dependent hamilto-nian reads ˆ H λ = ˆ T + ˆ V HF + λ ( ˆ V ee − ˆ V HF ) , (12)with ˆ V HF the Hartree-Fock non local potential operator.When λ → ∞ , the problem defined by ˆ H λ of Eq. (12)is not the same as the one of the density-fixed adiabaticconnection arising in DFT. The results of this study maysuggest that the PC model can provide a decent approx-imation of the leading λ → ∞ term in the HF adiabaticconnection integrand, at least when dealing with isoelec-tronic energy differences. A careful study of the problemis the object of on-going work.Keeping in mind that the information from W ′∞ [ ρ ] isless accurate (and maybe less relevant in the HF con-text), it can be interesting to consider a variant of ISIand revISI, in which we replace W ′ PC ∞ [ ρ ] with the curva-ture at λ = 0 (obtained from MP3) as input ingredient.In this way, the modified AC integrand expressions re-cover the first three terms of Eq. (4) for small λ , andonly the first term of Eq. (5) for large λ . However, theresulting XC approximations show several drawbacks. Infact, the results for atomization energies are significantlyworse than for the original ISI and revISI functionals (theMAEs are 0.72 and 0.65 eV for Au clusters, 0.40 and 0.37eV for Ag clusters and 0.55 and 0.51 eV for binary clus-ters), despite they are close to MP2 ones and better thanMP3 ones. More importantly, the modified ISI and revISIformulas, with the input ingredients for the Au and Agclusters, result in adiabatic connection integrands thatbecome imaginary at some λ >
1, with revISI breaking down at much larger λ values than ISI. This fact mightbe ascribed to the oscillatory behavior of the MP series,which gives a curvature that is too large, or to the lack offlexibility of the revISI and ISI formulas. This is furtherillustrated in the appendix. V. CONCLUSIONS AND PERSPECTIVES
We have assessed the performance of functionals basedon the idea of interpolating between the weak and thestrong-interaction limits the global adiabatic-connectionintegrand (ACII functionals) for noble-metal clusters, an-alyzing and rationalizing different features of this ap-proach. The study presented here extends a previouspreliminary assessment on main group chemistry, andexplores different interpolation formulas.We have found that the ACII functionals, although notspectacularly accurate, are quite robust for the descrip-tion of atomization energies, as their performance tendsto be the same for different species and different clustersizes, which is a positive feature. We should also stressthat this good performance is achieved by using 100% ofHartree-Fock exchange, and thus avoiding to rely on errorcancellation between exchange and correlation. Rather,as clearly shown in fig. 3, this is achieved by performingin a very similar way for the description of a cluster andits constituent atoms. On the other hand, the ACII func-tionals are found to be inaccurate for ionization energies,as they are not capable to describe differently chargedstates of the same system with the same accuracy, asshown in fig. 4.As in the case of main-group chemistry, we havefound that the ACII functionals perform much betterwhen used with Hartree-Fock orbitals, which means thatthey are used as a correlation functional for the Hartree-Fock energy. In other words, the ACII correlation func-tionals are used here as an approximate resummation ofthe Møller-Plesset perturbation series: they recover theexact MP2 at weak coupling, and perform much betterthan MP3 and MP4 for atomization energies (see TableII). Thus, a first question that needs to be addressed iswhether the PC model used here to compute the infi-nite coupling strength functionals is accurate also for theHartree-Fock adiabatic connection of Eq. (12), which isthe object of a current investigation. The results of thisstudy and of Ref. 82 suggest that the PC model can pro-vide a decent approximation of the λ → ∞ HF adiabaticconnection integrand, at least when dealing with isoelec-tronic energy differences.Another promising future direction is the developmentof ACII functionals in which the interpolation is donein each point of space, on energy densities.
Theselocal interpolations are more amenable to construct size-consistent approximations, but need energy densities alldefined in the same gauge (the one of the electrostatic po-tential of the exchange-correlation hole seems so far to bethe most suitable for this purpose ). In this framework,3the simple PC model, which performs globally quite well,does not provide accurate approximations pointwise, and needs to be replaced with models based on integralsof the spherically averaged density, which, in turn,needs a careful implementation, which is the focus of on-going efforts. Finally, recent models for λ = 1 could bealso used in this framework, both locally and globally. ACKNOWLEDGMENTS
Financial support was provided by the European Re-search Council under H2020/ERC Consolidator Grantcorr-DFT [Grant Number 648932]. We thank TURBO-MOLE GmbH for providing the TURBOMOLE programpackage.
Appendix A: Adiabatic connection integrand interpolationformulas
Several interpolation formulas have been developed torecover the weak- and strong-coupling limit behavioursof Eqs. (4) and (5). For the sake of simplicity, we willnot specify in the following that the expressions of theAC integrand as well as of the XC correlation energy are(explicit or implicit) functionals of the density as well aseach of their fundamental ingredients, W , W ′ , W ′′ , W ∞ , and W ′∞ . Interaction Strength Interpolation (ISI)formula W ISI λ = W ∞ + X √ λY + Z , (A1)with X = xy z , Y = x y z , Z = xy z − x = − W ′ , y = W ′∞ , z = W − W ∞ . (A3)After integration in Eq. (2), it gives E ISI xc = W ∞ + 2 XY (cid:20) √ Y − − Z ln (cid:18) √ Y + Z Z (cid:19)(cid:21) . (A4) Revised ISI (revISI) formula W revISI λ = W ∞ + b (cid:0) cλ + 2 d √ cλ (cid:1) √ cλ (cid:0) d + √ cλ (cid:1) , (A5)where b = − W ′ ( W ′∞ ) ( W − W ∞ ) , c = 2( W ′ W ′∞ ) ( W − W ∞ ) ,d = − − W ′ ( W ′∞ ) ( W − W ∞ ) . (A6) The corresponding XC functional is E revISI xc = W ∞ + b √ c + d . (A7) Seidl-Perdew-Levy (SPL) formula W SPL λ = W ∞ + W − W ∞ √ λχ , (A8)with χ = W ′ W ∞ − W . (A9)The SPL XC functional reads E SPL xc = ( W − W ∞ ) (cid:20) √ χ − − χχ (cid:21) + W . (A10)Note that this functional does not make use on informa-tion on W ′∞ . Liu-Burke (LB) formula W LB λ = W ∞ + β ( y + y ) , (A11)where y = 1 √ γλ , β = W − W ∞ , γ = 4 W ′ W ∞ − W ) . (A12)Using Eq. (2), the LB XC functional is found to be E LB xc = 2 β (cid:20) γ (cid:18) √ c − c/
21 + c (cid:19) − (cid:21) . (A13)Also the LB functional does not use information on W ′∞ . Point-Charge-plus-continuum (PC) model
In all cases, the highly non-local functionals W ∞ and W ′∞ (when used) are approximated by the semilocal PCmodel W ∞ ≈ W PC ∞ = Z (cid:20) Aρ ( r / ) + B |∇ ρ ( r ) | ρ ( r ) / (cid:21) d r (A14) W ′∞ ≈ W ′∞ PC = Z (cid:20) Cρ ( r / ) + D |∇ ρ ( r ) | ρ ( r ) / (cid:21) d r , (A15)where A = − π/ / / B = 3[3 / (4 π )] / / C = √ π/ D = − . D parameter.
1. ISI and revISI with the exact curvature
The ISI and revISI formulas have four parameters thatneed to be fixed by four equations. In the standard forms(see above) the four equations are obtained by impos-ing that W ISI λ recovers the first two terms of the weak-interacting limit expansion, Eq. (4), and the first two4terms in the strongly-interacting limit expansion, Eq. (5)for large λ . For the first time we have explored an alter-native choice that is to constrain ISI and revISI to recoverthe first three terms of Eq. (4) for small λ , and only thefirst term of Eq. (5).The structure of the interpolation formula is thus for-mally the same, but the parameters are given by X = − W − W ∞ ) + W ′′ ( W ′ ) ( W − W ∞ ) ,Y = − W ′′ W ′ + W ′ ( W − W ∞ ) ,Z = − W ′′ ( W ′ ) ( W − W ∞ ); (A16)for ISI, and b = − W − W ∞ ) + W ′′ ( W − W ∞ ) W ′ ) ,c = − W ′′ W ′ + W ′ ( W − W ∞ ) ,d = − W ′′ ( W − W ∞ )3( W ′ ) ; (A17)for revISI.However, as discussed in Sec. IV D, while in the stan-dard ISI and revISI interpolation formulas the parame-ters, Y [ ρ ] and c [ ρ ], which appear under square root, aregiven by the sum of squared quantities (see Eqs.A2, andA6), in these modified versions this is not true and theycan become negative. In the cases studied here bothparameters turn out to be always negative and smallerthan one, meaning that there is, for each species, a crit-ical lambda, λ c , always larger than one, after which thefunction takes imaginary values. In particular we foundan average ¯ λ ISIc ≈ . .
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