On the Use of the Pseudo-Diatomic Model for Experimental Evaluation of Stabilization Energies of Weakly Bound Molecular Complexes

Energias de estabilização para complexos moleculares fracamente ligados, relativas aos monomeros isolados, tem sido determinadas experimentalmente através do modelo pseudo-diatômico (PD) o qual faz uso de um potencial do tipo Lennard-Jones 6-12 (LJ), aqui denominado modelo PD-LJ. Este método utiliza da dos espectroscópicos para avaliar a profundidade do poço de potencial para sistemas fracamente ligados. Neste trabalho é mostrada uma comparação sistemática en tre valores experimentais de energias de estabilização disponíveis na literatura e calculados à nível ab in itio para uma série de complexos binários, objetivando avaliar a eficácia do modelo PD-LJ.


In tro duc tion
The sub ject of weakly bound mo lec u lar com plexes has at tracted the at ten tion of many experimentalists and the o reti cians and, con se quently, this re search field has grown con sid er ably over the past years.Ex per i men tal stud ies on such mo lec u lar sys tems, mainly dimeric and trimeric species, have been con ducted in the gas phase with struc tural and en ergy data be ing ob tained through the aid of in fra red (IR) and mi cro wave spectroscopies.The sam ple is usu ally pro duced in a su per sonic ex pan sion of the mix ture of the mono mers where the for ma tion of small mo lec u lar as so ci ations, mainly bi nary and ter nary com plexes, are de tected.The ex per i ments are con ducted at low tem per a ture and pres sure con di tions.Ex per i men tal in ves ti ga tions on a num ber of gas phase dimers have been ac com plished us ing pulsed-nozzle Fou rier-transform mi cro wave spec tros copy.
In many cases the ge om e try of the com plex is unambiguosly de ter mined by the experimentalists.The exper i men tal de ter mi na tion of mo lec u lar prop er ties for weakly bound com plexes through spec tro scopic techniques has been well doc u mented.For a more re cent review see Ref. On the other side the o ret i cal stud ies of weakly bound spe cies have been in creas ingly re ported in the lit er a ture.Re views on the the o ret i cal ap proaches and pro gresses achieved in this field are given in Refs.6-10.For a more recent re view of the pres ent sta tus of the o ret i cal stud ies of mo lec u lar as so ci a tions see Ref. 1. One-electron prop er ties like elec tric field gra di ent, have been found to be very sat isfac to rily pre dicted by ab in itio cal cu la tions, com pared to ex per i men tal re ported val ues, as it was found for the se ries Ar ti cle of B...HCl dimers, where B is a neu tral monomeric species 11 .Also the vi bra tional spec tra of weakly bound species 12,13 , and the me chan i cal and elec tri cal anharmonicities con tri bu tions to the vi bra tional fre quen cies and in ten si ties of the nor mal modes 14,15 , have been suc cess fully ad dressed by ab in itio stud ies.This has given a sub stan tial con tri bution to the as sign ment of ex per i men tal gas phase spec tra.
Re gard ing the mo lec u lar ge om e try the agree ment between the ory and ex per i ment has been re mark able.As an ex am ple both lin ear and cy clic iso mers of the (HCN) 3 trimer were de tected ex per i men tally 16,17 and the o ret i cally pre dicted to ex ist 18,19 .Good agree ment be tween the o ret i cal and ex per i men tal pre dic tions, re gard ing isomerism in weakly bound spe cies has been found.An ex am ple is the CO 2 ...HCN com plex 20,21 , where both lin ear and T-shaped struc ture were the o ret i cally pre dicted to co ex ist and the ab in itio cal cu la tions 22 antecipated the ex per i men tal de tec tion of the lin ear struc ture.Cal cu la tions on the par ent com plex CO 2 ...C 2 H 2 were also re al ized and the agree ment be tween the o ret i cal 23 and ex per i men tal 24,25 re sults was very good.The o ret i cal stud ies, car ried out at the ab in itio level of theory in clud ing elec tron cor re la tion ef fects, have been of great help in sit u a tions where the ex per i ment alone is unable to pro vide pre cise in for ma tion on both ge om e try and sta bi li za tion en ergy of the com plex.A good ex am ple is the (Cl 2 ) 2 homodimer, where the ab in itio cal cu la tions 26 were cru cial for the cor rect assignement of the global min i mum en ergy struc ture on the po ten tial en ergy sur face (PES).In other sit u a tions the existency of dimeric spe cies has been pre dicted for spe cies ((PCCP) 2 ) where ex per i men tal data are still un avail able 27 .More re cently the ab in itio stud ies [28] have also proved to be able to pro vide an un am big u ous in ter pre ta tion of gas phase ex per i men tal re sults for tetrameric spe cies, like the (CO 2 ) 3 ...HCN tetramer 29 .
An other phe nom e non that has been suc cess fully treated by the o ret i cal ap proaches is the tun nel ing ef fect.The mech a nism of tun nel ing in the C 2 H 4 ...SO 2 30 and (C 2 H 2 ) 2 31 dimers has been stud ied re cently at the ab in itio level of the ory, and the the o ret i cal treat ment pro vided a sub stan tial help for the in ter pre ta tion of the ex per i men tal data for the eth yl ene-sulfur di ox ide 32 and acet y lene dimer [33][34][35][36][37][38][39][40] .There are some other in ter est ing sit u a tions where a weakly bound com plex play a key role for undertanding phe nom ena like for ex am ple mech a nisms of or ganic re ac tions, and where the ab in itio cal cu la tions, includ ing elec tron cor re la tion, were of fun da men tal im portance.The H 2 O...CCl 2 com plex, where a carbene spe cies is in volved, is a good ex am ple of the non for ma tion of a ylide type struc ture, con firmed by the the o ret i cal study 41 and wrongly pro posed ex per i men tally.
In the pre vi ous para graphs the agree ment be tween theory and ex per i ment re gard ing the de ter mi na tion of mo lec ular prop er ties for weakly bound spe cies has been dis cussed.How ever the ex per i men tal de ter mi na tion of sta bi li za tion en er gies still de serve more in ves ti ga tion.Im prove ment on the the o ret i cal de ter mi na tion of vdW en er gies has been recently achieved by us ing more ex tended and mid-bond func tions within the Moller-Plesset sec ond or der per tur bation the ory (MP2) treat ment of weakly bound spe cies, as it has been re ported for the HF...ClF dimer 42 .How ever the in ter ac tion en ergy be tween two or more mono mer units is not usu ally mea sured di rectly for com par i son with high level ab intio re sults.What is most com monly done is to use a pseudo di atomic (PD) model em ploy ing a Lennard-Jones (LJ) 6-12 type po ten tial, here de nom i nated PD-LJ model, for the eval u a tion of sta bi li za tion en er gies.This model makes use of some as sump tions and this ar ti cle aims to assess the qual ity of the ex per i men tal en ergy data ob tained from this model.It is be lieved that a com par i son be tween high level ab in itio cal cu lated bind ing en er gies and ex per imen tal re ported val ues may shed some light about the re liabil ity of the pseudo-diatomic ap prox i ma tion for the eval u a tion of sta bi li za tion en er gies of gas phase mo lec u lar com plexes.

Meth od ol ogy
The ab in itio cal cu la tions of sta bi li za tion en er gies make use of the supermolecule ap proach, where the in terac tion en ergy is eval u ated as the dif fer ence be tween the ab in itio to tal en ergy of the com plex and the iso lated mono mer en er gies.By do ing such type of cal cu la tion it has to take into ac count the ba sis set su per po si tion er ror (BSSE) effects, which cause an un de sir able over es ti ma tion of the com plex sta bi li za tion en ergy.This has been dis cussed exten sively in the lit er a ture.For a re cent re view see Ref. 43.The full coun ter poise (CP) method of Boys and Bernardi 44 is widely used for BSSE cor rec tions.In this ap proach the mono mer en er gies are cal cu lated in the pres ence of ba sis set of the other sub unit, placed on ghost at oms.That is, the sec ond mono mer is not ac tu ally in cluded in the en ergy calcu la tion, only its ba sis set.It is well known that elec tron cor re la tion ef fects play an im por tant role on com plex forma tion and this has to be taken into ac count in the ab in itio stud ies, which is more fre quently done at the MP2 level of the ory.
The ex per i men tal meth od ol ogy for sta bi li za tion en ergy de ter mi na tion usu ally re lies on the PD-LJ model.In this ap proach it is as sumed that the cen trif u gal dis tor tion effects arises from the stretch ing of the intermolecular van der Waals (vdW) bond only and so the cen trif u gal dis tortion con stant, D J , is re lated to the vdW stretch ing force constants, ks.It is also as sumed that the ra dial in ter ac tion be tween the mono mer sub units can be de scribed by the Lennard-Jones 6-12 po ten tial which ex panded as a Tay lor se ries about the equi lib rium intermolecular dis tance yields an ex pres sion for the depth of the intermolecular po ten tial well, i.e., Ta ble 1.A com par i son be tween a Pseudo-diatomic (PD) model, us ing a Lennard-Jones 6-12 po ten tial (LJ), and ab in itio supermolecule cal cu lations of BSSE cor rected sta bi li za tion en er gies (in units of cm -1 ) for some gas phase mo lec u lar com plexes.

72
where R eq is the cen ter of mass equi lib rium intermolecular dis tance.It is there fore im plicit that the an gu lar de pendence of the in ter ac tion en ergy is com pletely ne glected.This model was ap plied early for the Kr...HCl dimer 45 and subse quently to the Xe...HBr dimer 46 .Millen 47 has ob tained a re la tion ship be tween the ex per i men tally ob served D J values and k s for dis tinct types of weakly bound dimers.For triatomic lin ear dimers B...H-A, the fol low ing re la tion ship was found, where µ D is the dimer re duced mass and B D and B HA respec tively the dimer (D) and mono mer (HA) ro ta tional con stants.Ex pres sions for other types of dimers than lin ear are given in Ref. 47. So, once the struc tural pa ram e ters, rota tional con stants and D J val ues are avail able from ro tational spec tros copy an es ti mate of the ex per i men tal sta bi li za tion en er gies can be made for gas phase complexes.
The ab in itio cal cu la tions car ried out in the pres ent work were done with the GAMESS pack age 48 , as im plemented on a SUN UL TRA-1 work sta tion, at the Laboratório de Química Computacional e Modelagem Molec u lar (LQC-MM), Departamento de Química, Universidade Fed eral de Minas Gerais (UFMG).

Re sults and Dis cus sions
A com par i son be tween the PD-LJ model and ab in itio supermolecule cal cu la tions of sta bi li za tion en er gies for a se ries of mo lec u lar com plexes is re ported in Ta ble 1.It can be seen from Ta ble 1 that the av er age agree ment be tween high level ab in itio cor re lated sta bi li za tion en er gies for weakly bound com plexes and the PD-LJ model ex per i mental ↔ val ues is very sat is fac tory for some dimeric spe cies, the max i mum dis crep ancy be ing ca.10%.One of the se rious dis agree ment hap pened for the (CO 2 ) 2 dimer (79%).The car bon di ox ide dimer has been in ves ti gated at the MP2 level of the ory by Bone and Handy 49 and their MP2/DZP (DZP stand for dou ble zeta plus po lar iza tion func tion ba sis set) value for the intermolecular stretch ing har monic frequency was in poor agree ment with the ex per i men tal value.Since the D 2h sta ble form of the (CO 2 ) 2 dimer does not possess a per ma nent di pole mo ment mi cro wave stud ies are unable to mea sure a D J value.Con se quently the k s was ob tained di rectly from the ex per i men tally re ported intermolecular stretch ing fre quency us ing the well known ex pres sion for a lin ear di atomic sys tem, which is no lon ger the case of the (CO 2 ) 2 dimer.So the very poor agree ment found for the car bon di ox ide dimer may be re lated to the im pre ci sion in the eval u a tion of the stretch ing force constant (some how in agree ment with Bone and Handy findings 49 ) rather than to the fail ure of the PD-LJ model.The PD-LJ en er gies of more strongly bound dimers exibited an ac cen tu ated de vi a tion from ab in itio and ex per i men tal mea sured val ues.This re sult shows that the PD-LJ model should be used with care.From the re sults quoted in Ta ble 1 it seems that as the in ter ac tion en ergy value in creases the model loses its efficaciousness.The dis trib uted multipole anal y sis (DMA) model per formed better than the point quadrupole model for the pre dic tion of the elec tro static inter ac tion en ergy, but the agree ment with ex per i ment is not so good as for the MP2 BSSE cor rected supermolecule approach.
It should also be said that the PD-LJ model re lies strongly on the knowl edge of ac cu rate val ues for the intermolecular cen tre of mass dis tance (R CM ) and stretching force con stant (k s ).This means that the cur va ture of the LJ curve near the min i mum is quite cru cial.So it is not unex pected at all that the use of ab in itio k s val ues within the PD-LJ ap proach may no give so good re sults as the supermolecule ap proach, as shown for the N 2 ...O 2 heterodimer 50 .The rea son is that, as pointed out by Simandiras et al. 51 , very large ba sis sets in clud ing f functions are needed in or der to re pro duce ac cu rately har monic fre quen cies.So the cal cu la tion of very re li able force constants be comes a hard com pu ta tional task.On the other hand the ab in itio MP2 BSSE CP cor rected supermolecule sta bi li za tion en er gies are not fun da men tally based on the pre cise de ter mi na tion of the cur va ture (sec ond en ergy deriv a tive) of the PES.Dif fer ently from har monic fre quency cal cu la tions, re li able en ergy val ues may be ob tained us ing a not so so phis ti cated level of the ory.It is seen that the ab in itio and PD-LJ treat ments are based on dis tinct grounds but the pre dicted sta bi li za tion en er gies are com pa ra ble.
By an a lyz ing care fully the re sults re ported in Ta ble 1, it can be seen that the MP2 treat ment of elec tron cor re la tion is in suf fi cient for the pre dic tion of sta bi li za tion en er gies for the very weakly bound dimers (Kr...HCl and Ar...HCl).In this case the dis per sion con tri bu tion to the in ter ac tion en ergy is dom i nant and so a higher or der of per tur ba tion the ory is re quired for the eval u a tion of the cor re la tion energy.As it has been shown re cently for the (PCCP) 2 dimer 27 , a cor rect de scrip tion of the at trac tive dis per sion forces is cru cial in the o ret i cal in ves ti ga tion of weakly bound mo lec u lar com plexes.
The re sults of the pres ent study gives strong sup port for the use of the ab in itio supermolecule cal cu la tions em ploying per tur ba tion the ory for treat ing elec tron cor re la tion, with ex tended-polarized ba sis set plus BSSE cor rec tion by the CP ap proach, for pre dict ing the mo lec u lar struc ture and energetics of gas phase mo lec u lar com plexes.The PD-LJ model was shown to yield sta bi li za tion en er gies in good agree ment with high level ab in itio cal cu la tions for the follow ing mo lec u lar com plexes: Ar...ClF, HF...ClF, CO 2 ...HCN, C 2 H 4 ...SO 2 and (CH 3 ) 3 ...HCl.This re sult is an in di ca tion that the ani so tropy of the po ten tial func tions, absent in the LJ 6-12 po ten tial, may not be so im por tant for the eval u a tion of the sta bil ity of these gas phase weakly bound com plexes.How ever for the other com plexes listed in Table 1 the agree ment be tween the PD-LJ re sults and the ab in itio ones is very poor, re veal ing that the pseudodiatomic model does not pro duce con sis tent re sults and so may not be the most ad e quate for ac cu rate pre dic tions of sta bi li zation en er gies.It can also be seen from ta ble 1 that the PD-LJ sta bi li za tion en er gies seem to be con sis tently un der es timated when com pared to the high level ab in itio val ues.Nev er the less the PD-LJ model can still be used to make an es ti mate of the bind ing en er gies of weakly bound spe cies.

Con clu sion
In the pres ent work an as sess ment of the efficaciousness of the pseudo-diatomic ap proach (the PD-LJ model) for eval u at ing sta bi li za tion en er gies has been at tempted.A com par i son be tween ex per i men tal and ab in itio cal cu lated en er gies for a se ries of mo lec u lar complexes has been made.For some dimers in the se ries of molec u lar com plexes stud ied here it has been found that, de spite ne glect ing the an gu lar de pend ence of the in ter action en ergy, the PD-LJ model ∈ val ues ex hibit a very sat isfac tory agree ment with the ab in itio supermolecule sta bi li za tion en er gies, cal cu lated in clud ing elec tron cor rela tion ef fects through the Moller-Plesset per tur ba tion theory treat ment.How ever, the over all agree ment for the whole se ries of mo lec u lar sys tems treated in the pres ent work is not con sis tent and so the pseudodiatomic ap proach can not be rec om mended as a suit able pro ce dure for ob taining re li able sta bi li za tion en ergy val ues for weakly bound spe cies.

Ac knowl edg ments
The au thor would like to thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and the Fundação de Amparo a Pesquisa no Estado de Minas Gerais (FAPEMIG) for sup port.The au thor also thanks all the other mem bers of the Grupo de Química Computacional e Modelagem Mo lec u lar, par tic u larly Josefredo R. Pliego Jr. and Stella M. Resende, for the very helpul dis cus sions dur ing the prep a ra tion of this manuscript.
1 and ref er ences there cited.The re views re ported in Refs.2-5 are also rec om mended.
Eval u ated us ing struc tural and spec tro scopic data from Ref. 45; b The pres ent work.c Eval u ated us ing struc tural and spec tro scopic data from Ref. 52. d See Ref. 53; e Eval u ated us ing struc tural and spec tro scopic data from Ref. 54; f Eval u ated us ing struc tural and spec tro scopic data from Ref. 55; g See Ref. 56; h See Ref. 42; i Eval u ated us ing struc tural and spec tro scopic data from Ref. 57; j Ex per i men tal value mea sured from ab so lute inten si ties of ro ta tional tran si tions, Ref. 4; k See Ref. 58; l Eval u ated us ing struc tural and spec tro scopic data from Ref. 59; m See Ref. 60; a n Eval u ated us ing struc tural and spec tro scopic data from Ref. 61; o See Ref. 22; p Eval u ated us ing struc tural and spec tro scopic data from Ref. 20; q Eval u ated us ing struc tural and spec tro scopic data from Ref. 34; r See Ref. 49; s Eval u ated us ing struc tural and spec tro scopic data from Ref. 62; t Eval u ated us ing the stretch ing force con stant from Ref. 63; u Eval u ated us ing struc tural and spec tro scopic data from Ref. 64; v Eval u ated us ing struc tural and spec tro scopic data from refs.32,65; x See Ref. 66; y Eval u ated us ing struc tural and spec tro scopic data from Ref. 67; z Eval u ated us ing struc tural and spec tro scopic data from Ref. 68; w See Ref. 69; + Point quadrupole model, Ref. 49; ++ Dis trib uted multipole anal y sis (DMA) model, Ref. 49; +++ Elec tro static model us ing the DMA ap proach from Ref. 32; ++++ DMA model from Ref. 67.