Ab-initio modeling of an anion C − 60 pseudopotential for fullerene-based compounds
EEPJ manuscript No. (will be inserted by the editor)
Ab-initio modeling of an anion C − pseudopotential forfullerene-based compounds. I I Vrubel , R G Polozkov , , and V K Ivanov ITMO University, 49 Kronverksky Pr., St. Petersburg, 197101, Russia Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland Peter the Great Saint-Petersburg Polytechnic University, 29 Politekhnicheskaya, St. Petersburg, 195251, RussiaReceived: date / Revised version: date
Abstract.
A pseudopotential of C − has been constructed from ab-initio quantum-mechanical calculations.Since the obtained pseudopotential can be easily fitted by rather simple analytical approximation it canbe effectively used both in classical and quantum molecular dynamics of fullerene-based compounds. Fullerene C is the most studied and widely used amongall fullerenes because of the availability, high symmetryand low price [1]. Some of the most promising fields ofapplication of these novel materials are artificial photo-synthesis, non-linear optics and the preparation of pho-toactive films and nanostructures (see, for example [2]).Because of high electron affinity and small rearrangementenergy fullerenes, in particular C , play a role of electron-acceptors in such systems and produce very stable radi-cal pairs. In particular, it was demonstrated that a π -conjugated polymer was able to efficiently transfer elec-trons to the C core giving rise to long-lived charge-separated states. For example the donor-acceptor com-pound [ C ]PCBM [3] is the most known and effectivelyused in organic solar photoelectric cells for the last time.But, due to the difficulties in the modeling extended andpossible nanostructured materials as [ C ]PCBM an iso-lated molecular limit is preferable, so the investigationsof the isolated anions and radical anions of fullerenes, inparticular C − , seems to be actual.The anions and radical anions of fullerenes have beenan object of intensively investigations during last two decade[1, 4, 5]. The anionic fullerenes have been observed in ioncyclotron resonance traps [6], storage rings [7] and electro-spray mass spectrometry [8]. The theoretical systematicstudy of the stability of highly charged anionic fullereneshas been performed within different levels of theory [9,10].But for the developing efficient quantum simulation meth-ods, which allow us to predict the optimized geometry ofthe fullerene C compounds with reasonable computercost and accuracy, we suggest to construct a pseudopo-tential of C − .The recent calculations showed that the application ofsimple and widely used jellium model doesn’t bring datainto accordance with results of more complicated but ac- curate ab-initio calculations [11]. In this paper the pseu-dopotential of C − has been constructed on the basis ofthe total electrostatic potential of C − calculated withinthe ab-initio approach.The atomic system of units, m = | e | = (cid:126) = 1, is usedthroughout the paper. All ab-initio computations are performed by using theFireFly QC package [12]. For the first the fully optimizedgeometry and the total energy of C − have been obtainedfrom the Hartree-Fock and density functional theory (DFT)calculations by ROHF/6-31G(d) and B3LYP/6-31G(d) lev-els respectively. Then within the optimized geometry the ab-initio calculations of the electronic structure and thetotal charge density of C − have been performed at thesame levels of theory, which is shown to provide reasonableresults for small carbon clusters [13] and fullerenes [10,14],both charged and neutral. Although the inclusion of dif-fuse functions is usually important to obtain accurate ab-solute energies for anions, it has recently been shown thatthe 6-31G(d) and 6-31G+(d) basis sets give similar resultsfor geometries, charge distributions, and relative energiesof anionic C and C fullerenes [15, 16].The important point of the ab-initio calculations ofthe total charge density is to apply the corresponding keyin the input of the FireFly program like AIM P AC = 1to obtain the practical information about molecular or-bital wave functions which are used in the next step ofconstruction of the pseudopotential of C − . a r X i v : . [ phy s i c s . a t m - c l u s ] D ec R G Polozkov et al: Pseudopotential of an anion C − The pseudopotential of C − can be construct on the basisof the total electrostatic potential. The latter is presentedas a sum of two summands: the potential of nuclei U n ( r ),which depends on positions of sixty carbon atoms, and thepotential created by electron density ρ ( r ) of 361 electrons U el ( r ): U tot ( r ) = U n ( r ) + U el ( r ) = − (cid:88) i=1 | r − R i | + (cid:90) ρ ( r (cid:48) ) | r − r (cid:48) | d r (cid:48) . (1)The positions of the carbon atoms within the opti-mized geometry and corresponding charge density havebeen extracted from results of the ab-initio FireFly QCpackage calculations by using of a Multifunctional Wave-function Analyzer (Multiwfn) [17] (see for example a colorfilled map of the electron charge density of the C − ob-tained from ab-initio calculations prepared within the Mul-tiwfn [17] software on the Fig. 1). This software has beenthen used for computation of the corresponding electro-static potentials on a specified grid of the position vector r . After that we have averaged the electrostatic potentialobtained from Multiwfn software over the directions ofthe position vector r to construct the radial dependenceof C − pseudopotential U pseudo ( r ) and to obtain averagedelectron density ρ ( r ): U pseudo ( r ) = U tot ( r ) = U n ( r ) + U el ( r ) ,U i ( r ) = 14 π (cid:90) U i ( r ) dΩ (i = tot , n , el) ,ρ ( r ) = 14 π (cid:90) ρ ( r ) dΩ . (2) For the first we have checked the non-applicability of thejellium model for purpose of construction of pseudopoten-tial of C − . The averaged radial valence electron density ρ ( r ) of C − calculated by ab-initio method has been com-pared with results of the jellium model and ab-initio cal-culations for C [11]. The Hartree-Fock method has beenused for self-consistent calculations in all three cases. Fig.2demonstrates the density profiles of valence electrons of C [11] and C − (present work, ROHF/6-31G(d) level)as a function of radial distance from a center of fullerene.As Fig.2 indicates, the results of ab-initio calculations forfullerene and anion are close, but substantially differ fromthe results of jellium model calculations for C , whichmakes this approach non applicable to solving a problemof determination of C − pseudopotential.The results of the C − pseudopotential calculations byHartree-Fock method (ROHF/6-31G(d)) and within DFT Fig. 1.
A color filled map of the electron charge densityof the C − obtained from ab-initio calculations preparedwithin the Multiwfn [17] software in the plane X,Y. Theorigin of the X,Y,Z is set at the center of C − , lengths unitis Bohr. ab initio, C -60 ab initio, C jellium, C R ad i a l e l e c t r on den s i t y ( a . u . )- Fig. 2.
Radial valence electron density of C − (ROHF/6-31G(d) level, solid line) is compared with the same of C calculated by ab-initio method [11](ROHF/6-31G(d)level, blue dotted line) and with use the jellium model [11](red dashed line).(B3LYP/6-31G(d)) are presented and compared in Fig.3.Note that the usage of the different approaches for theelectronic structure calculations leads to the significantdiscrepancy of the corresponding one-particle energies re-sults but doesn’t lead to the any noticeable differences inthe resulting behavior of pseudopotential (compare blueand red solid line in Fig.3 ).It should be mentioned several important features ofthe pseudopotential obtained. The first one is the correctasymptotic behavior at the large distances as 1 /r , which is G Polozkov et al: Pseudopotential of an anion C − typical for a single negative ion (see Fig.3). The numericalanalysis shows that the pseudopotential behavior beginsto satisfy the 1 /r law at the radial distance about 10 a.u.Secondly the pseudopotential demonstrates two differenttypes of interaction between the fullerene’s anion and anexternal electron: the strong attraction close to a radiusof fullerene’s anion and the weak repulsion outside andinside of fullerene cage. This combination of repulsion andattraction gives rise the weak barriers for an any negativeprojectile particle and can lead to increasing of probabilityfor the projectile to ”getting stuck” on the fullerene cage. P s eudopo t en t i a l ( a . u . ) Fig. 3.
Pseudopotential of C − obtained from ab-initio calculations: ROHF/6-31G(d) (blue solid line), B3LYP/6-31G(d) (red solid line) and compared with an 1/r asymp-totic behavior (black dots).For purposes of the classical and quantum moleculardynamics of fullerene-based compounds it is reasonableto make the analytical approximation of the numericallyobtained pseudopotential. Within the range 0 −
10 a.u.of radial distance the pseudopotential has been approx-imated by sum (3) of constant and Chesler-Cram singlepeak function (see Fig.4). This Chesler-Cram function [18]is applied to approximate experimental results in the pro-cessing of chromatographic data. We use this function be-cause it may describe discontinuity point and consists ofelementary functions. The general view of our function ispresented by the following formula: u ( r ) = y + A [ e − ( r − rc w + Be − k ( | r − r c | +( r − r c )) × (1 − . − tanh( k ( r − r c ))) )] , (3)where r is the radial distance, y , A , r c , w , B , k , r c , k , r c are approximation constants. The array of constantsthat allows to achieve the best result is represented in thetable 1. P s eudopo t en t i a l ( a . u . ) Radial distance (a.u.)
Fig. 4.
Approximation (red solid line) of the pseudopo-tential of C − obtained from ab-initio B3LYP/6-31G(d)calculations (black dots).
Table 1. array of constants symbol valuey r c A -1,60691 w , ∗ − k r c B k r c In this work we have constructed the pseudopotential ofthe fullerene anion C − for molecular dynamic purposes.The method of construction is based on the using of thecharge density obtained by the ab-initio calculations andon the averaging of the corresponding total electrostaticpotential to make the radial dependence of the pseudopo-tential.The pseudopotential of the fullerene anion C − ob-tained has rather simple analytical approximation andthen can be effectively used both in classical and quan-tum molecular dynamics of fullerene-based compounds. RGP and IIV acknowledge support from Russia’s FederalProgram ”Scientific and Educational Manpower for In-novative Russia” (grant no RFMEFI58715X0020), RGPacknowledges support by Rannis Project ”BOFEHYSS”.
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