Magnetic molecules created by hydrogenation of Polycyclic Aromatic Hydrocarbons
J. A. Verges, G. Chiappe, E. Louis, L. Pastor-Abia, E. SanFabian
MMagnetic molecules created by hydrogenation of Polycyclic Aromatic Hydrocarbons
J. A. Verg´es
Departamento de Teor´ıa de la Materia Condensada, Instituto de Cienciade Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain. ∗ G. Chiappe and E. Louis
Departamento de F´ısica Aplicada, Unidad Asociada del CSIC and Instituto Universitario de Materiales,Universidad de Alicante, San Vicente del Raspeig, 03690 Alicante, Spain.
L. Pastor-Abia and E. SanFabi´an
Departamento de Qu´ımica F´ısica, Unidad Asociada del CSIC and Instituto Universitario de Materiales,Universidad de Alicante, San Vicente del Raspeig, 03690 Alicante, Spain. (Dated: July 30, 2008)Present routes to produce magnetic organic-based materials adopt a common strategy: the use of magneticspecies (atoms, polyradicals, etc.) as building blocks. We explore an alternative approach which consists ofselective hydrogenation of Polycyclic Aromatic Hydrocarbons. Self-Consistent-Field (SCF) (Hartree–Fock andDFT) and multi-configurational (CISD and MCSCF) calculations on coronene and corannulene, both hexa-hydrogenated, show that the formation of stable high spin species is possible. The spin of the ground statesis discussed in terms of the Hund rule and Lieb’s theorem for bipartite lattices (alternant hydrocarbons in thiscase). This proposal opens a new door to magnetism in the organic world.
PACS numbers: 31.10.+z, 33.15.Kr, 31.15.aq, 75.50.Xx
I. INTRODUCTION
Two successful routes that are being actually followed toproduce magnetic organic materials are the addition of mag-netic atoms and the use of polyradicals . In particular,carbon-based nickel compounds that show spontaneous field-dependent magnetization and hysteresis at room temperature,have been recently synthesized . Moreover, the combinationof two radical modules with different spins has allowed theobtaining of organic polymers with ferro- or antiferromag-netic ordering . Research on molecules containing polyrad-icals goes back to the early nineties and has produced avariety of results as, for example, the synthesis of high spinorganic molecules. In some of these molecules the failureof Hund’s rule has been demonstrated . On the other hand,experimental and theoretical evidence has been recently pre-sented indicating that 5-dehydro-m-xylylene or DMX was thefirst example of an organic tri-radical with an open-shell dou-blet ground-state . Both methods share a common strategy:the use of ingredients (either radicals or atoms) that provide afinite spin.In this work we follow a different approach. Specifically,we predict the existence of spin polarized organic moleculesderived from non magnetic π -conjugated Polycyclic AromaticHydrocarbons (PAHs) by selective hydrogenation of their pe-ripheral C atoms. High hydrogenation of PAHs has been pro-posed as a method for hydrogen storage . More recently, thefeasibility of double hydrogenation of those compounds hasbeen investigated theoretically .Our work is inspired upon Lieb’s theorem for bipartite lat-tices that shows the appearance of magnetism whenever theyare unbalanced . According to Lieb, if a nearest neigh-bor model with a local on-site interaction is applicable to abipartite lattice, the spin multiplicity of the ground state is | N A − N B | +1 , where N A and N B are the number of atoms ineach sublattice. Most PAHs are alternant hydrocarbons wherecarbon atoms can be separated into two disjoint subsets so thatan atom in one set only has neighbors in the other set (Figs. 1and 2 show a colored version of the partition). The same the-orem has been used to support the existence of magnetism ingraphene ribbons and islands . All work we know is based onsingle-determinantal methods, i.e., on a more or less sophis-ticated form of Self-Consistent-Field (SCF) calculation. Letus remark that being π -orbital magnetism a direct result of thestrong correlation among π -electrons, only methods designedexplicitly to catch these effects (like CISD and MCSCF, usedin our work) can help to resolve the doubts regarding the ap-pearance of magnetism in graphite-derived systems.The rest of the paper is organized as follows. The ab ini-tio methods (both mono- and multi-determinantal) used in thiswork are discussed in some detail in section II, while the re-sults obtained with those methods are reported and discussedin section III. Section IV in turn is devoted to the analysis ofthe ab initio results by means of model Hamiltonians, in par-ticular the Hubbard and the Pariser-Parr-Pople Hamiltonians.Finally, the conclusions of our work are gathered in section V. II.
AB INITIO
CALCULATIONS: METHODS ANDNUMERICAL PROCEDURES
Calculations of the spin states of the molecules of Figs.1 and 2 were done using the following basis functions sets:MIDI , cc-pVDZ and cc-pVTZ . Although the latter setguarantees a sufficient precision, varying the dimension ofthe variational space allowed to check the reliability of ourresults. SCF calculations were carried out at the Restricted-Hartree-Fock (RHF) level and by means of the hybrid den- a r X i v : . [ phy s i c s . a t m - c l u s ] F e b FIG. 1: (color online) 1,4,5,8,9,12-hexahydrocoronene (A, hereafterreferred to as D h according to its symmetry group), 1,3,5,7,9,11-hexahydrocoronene (B, hereafter referred to as C h ) and planar1,4,5,6,7,10-hexahydrocorannulene (C, hereafter referred to as C v ).Saturated carbon atoms are represented by black symbols while darkgray (magenta) and light gray symbols are used to distinguish car-bon atoms belonging to different sublattices. Corannulene is a non-alternant hydrocarbon, that is, a frustrated cluster of carbon atoms(note the fully magenta bond between two magenta atoms).FIG. 2: (color online) Two views of curved 1,4,5,6,7,10-hexahydrocorannulene in the calculated stable geometry (hereafterreferred to as C ). As in Fig. 1, black symbols indicate carbon atomsforming only single bonds while dark gray (magenta) and light graysymbols denote each of the two sublattices in which carbon atomscan be separated. sity functional RB3LYP . In both cases the Restricted-Open-Shell variant was used in order to get well-defined to-tal spin values . In order to check the accuracy of the de-scription of the correlation energy of partially filled π -shells,multi-configurational wave-functions calculations were alsoperformed. Configuration Interaction with Single and Dou- TABLE I: Total energies (in Hartrees) for atomic hydrogen, molecu-lar hydrogen, coronene C H , corannulene C H , two moleculesobtained from hexahydrogenation of coronene and one derived fromhexahydrogenation of corannulene (in the latter case the results cor-respond to the planar geometry shown in Fig. 1C). The results wereobtained using three basis sets (MIDI, cc-pVDZ and cc-pVTZ), twoSCF methods (RHF and RB3LYP) and one multi-configurationalmethod CISD. The number of occupied (m) and empty (n) π Molecu-lar Orbitals included in the CISD calculations as well as the numberof electrons that fill them (N) is indicated as (m+n,N). Small starsemphasize the spin multiplicity of the more stable state.MOLECULE METHOD BASIS SETMIDI cc-pVDZ cc-pVTZH RHF -0.4970 -0.4993 -0.4998RB3LYP -0.4953 -0.4979 -0.4988H RHF -1.1217 -1.1287 -1.1330RB3LYP -1.1623 -1.1668 -1.1733Coronene RHF -910.4869 -916.0197 -916.2293C H RB3LYP -915.9341 -921.3874 -921.6253Corannulene RHF -758.6127 -763.2326 -763.4078C H RB3LYP -763.1633 -767.7138 -767.9092C H RHF -913.6339 -919.2239 -919.4337 (cid:63) D h (S=3) (cid:63) RB3LYP -919.1814 -924.6701 -924.9159CISD(11+3,16) -913.7112 -919.2905 -919.4939C H RHF -913.3714 -918.9826 -919.2040 D h (S=0) RB3LYP -919.0751 -924.5639 -924.8175CISD(8+6,16) -913.4988 -919.0286 -919.2431C H RHF -913.5690 -919.1638 -919.3742 C h (S=3) RB3LYP -919.1228 -924.6146 -924.8620CISD(11+3,16) -913.6153 -919.2027 -919.4078C H RHF -913.7732 -919.3607 -919.5746 (cid:63) C h (S=0) (cid:63) RB3LYP -919.3423 -924.8230 -925.0730CISD(8+6,16) -913.8433 -919.4158 -919.6110C H RHF -761.9008 -766.5625 -766.7408 (cid:63) C v (S=2) (cid:63) RB3LYP -766.5503 -771.1250 -771.3348CISD(8+4,12) -761.9652 -766.6291 -766.8014C H RHF -761.7591 -766.4418 -766.6266 C v (S=0) RB3LYP -766.4913 -771.0760 -771.2842CISD(6+6,12) -761.8264 -766.4643 -766.6588 ble excitations (CISD) calculations were carried out in allcases, while some checks were also made by means of theMulti-Configurational SCF (MCSCF) on the fully optimizedset in the active space version . The active space was gen-erated within the following windows (m+n,N) of m occupiedand n empty π Molecular Orbitals (MO) filled with N elec-trons: hexahydrogenated coronene S=0, (8+6,16) and S=3,(11+3,16), and planar hexahydrogenated corannulene S=0,(6+6,12) and S=2, (8+4,12). Other π -MO lie excessively farfrom the HOMO-LUMO gap to give a sizable contribution.Geometries were only optimized at the SCF (RB3LYP) level.The geometry of 6H-corannulene was optimized for both itsplanar metastable form and its curved stable form (see Figs. FIG. 3: (color online) Total spin densities of 1,4,5,8,9,12-hexahydrocoronene and 1,3,5,7,9,11-hexahydrocoronene (bothcorresponding to septuplets, S=3) and planar 1,4,5,6,7,10-hexahydrocorannulene (S=2 state).
1C and 2). However, in order to allow a discussion in terms of π -orbital models, the results for the energies of its spin statesdiscussed hereafter correspond to the planar geometry. Any-how, energy differences between the spin states of the twoallotropes are very small (fragmentation energies for both pla-nar and curved geometries are reported below). All quan-tum chemistry calculations were done using the GAMESSprogram . III.
AB INITIO
CALCULATIONS: RESULTS
Total energies for the singlet and the relevant multiplet ofhydrogenated coronene D h , C h and planar hydrogenatedcorannulene C v (A, B and C in Fig. 1) are reported in Ta-ble I. It is first noted that whereas the energies obtained with TABLE II: Fragmentation energies (in Hartrees) of molecules de-rived from hexahydrogenation of coronene and of corannulene. To-tal energy differences are given both for atomic and molecular formsof hydrogen. Data of Table I have been used and again small starsemphasize the spin multiplicity of the more stable state.Atomic H Molecular H MOLECULE METHOD MIDI cc-pVTZ MIDI cc-pVTZC H RHF -0.1651 -0.2056 0.2181 0.1946 (cid:63) D h (S=3) (cid:63) RB3LYP -0.2755 -0.2980 0.2396 0.2292C H RHF 0.0974 0.0242 0.4805 0.4243 D h (S=0) RB3LYP -0.1692 -0.1996 0.3459 0.3276C H RHF -0.1002 -0.1460 0.2830 0.2542 C h (S=3) RB3LYP -0.2170 -0.2441 0.2982 0.2831C H RHF -0.3044 -0.3465 0.0788 0.0537 (cid:63) C h (S=0) (cid:63) RB3LYP -0.4365 -0.4551 0.0787 0.0720C H RHF -0.3061 -0.3342 0.0770 0.0660 (cid:63) C v (S=2) (cid:63) RB3LYP -0.4152 -0.4290 0.0999 0.0981C H RHF -0.1644 -0.2200 0.2187 0.1802 C v (S=0) RB3LYP -0.3562 -0.3784 0.1589 0.1487C H RHF -0.3024 -0.3259 0.0807 0.0743 (cid:63) C (S=2) (cid:63) RB3LYP -0.4098 -0.4198 0.1054 0.1073C H RHF -0.1607 -0.2072 0.2224 0.1930 C (S=0) RB3LYP -0.3471 -0.3643 0.1680 0.1628 the small basis set MIDI and those obtained with the alreadylarge cc-pVDZ, differ in 4-6 Hartrees (approximately 0.6%),the difference is reduced to 0.1-0.3 Hartrees (approximately0.02%) when cc-pVDZ is replaced by the largest basis usedin this work, namely, the cc-pVTZ basis set. This indicatesthat convergence, as far as the basis set is concerned, is ratheracceptable. In the case of hexahydrogenated coronene (briefly6H-coronene), results clearly show that, no matter the methodor the basis set used, the ground state of molecule D h is aseptuplet and that of molecule C h a singlet. We have checkedthat other spin states lie between those two. In molecule D h the largest energy difference between the high spin groundstate and the singlet occurs for RHF (0.23-0.26 Hartrees).This difference is reduced to approximately 0.1 Hartrees forRB3LYP, increasing again using the CISD method. On theother hand, all results for C h conformation show that the sin-glet is below the septuplet by more than 0.2 Hartrees. Simi-lar results are obtained for 6H-corannulene, although energydifferences are slightly smaller. Table I also reports total en-ergy results for atomic and molecular hydrogen, coronene andcorannulene that allow the calculation of fragmentation ener-gies (Table II analysis). These are negative relative to atomichydrogen but not relative to the molecular form. Therefore,actual synthesis of the hydrogenated molecules would needsophisticated reaction paths . We also note that the sin-glet ground state of C h hydrogenated coronene is more sta-ble than that of the molecule having a septuplet ground state( D h ). Presumably, other forms of 6H-corannulene wouldalso show deeper ground state energies than that of the stud-ied magnetic conformation. Note also that hydrogenation ofthe curved (stable) geometry of corannulene (see Fig. 2) isslightly less favorable than that of its planar geometry (com-pare results for C v and C in Table II). Anyhow, as in theplanar geometry, the quintuplet has a lower energy than thesinglet.Fig. 3 depicts the total spin densities of the septuplet states(S=3) of 1,4,5,8,9,12-hexahydrocoronene and 1,3,5,7,9,11-hexahydrocoronene (A and B) and the quintuplet (S=2)of planar 1,4,5,6,7,10-hexahydrocorannulene. Concerning1,4,5,8,9,12-hexahydrocoronene, the most appealing result isthat the spin density is finite only on the carbon atoms of onesublattice. More precisely, spin density is located in the sub-lattice to which no additional H atoms were attached. Thisresult is highly illustrative allowing some intuition on the rea-sons for a magnetic ground state: electron-electron repulsionis minimized because each electron avoids sitting at nearest-neighbors distances from the others. However, in 1,3,5,7,9,11-hexahydrocoronene, a molecule with a singlet ground state,the spin is equally spread over the two sublattices implyinglarger electronic repulsions at the central hexagon. The caseof 1,4,5,6,7,10-hexahydrocorannulene is even more interest-ing as, being a frustrated molecule, at least one bond be-tween atoms of the same sublattice should be present. Thisis clearly visible in Fig. 3 once a sublattice is identified asthe sites showing spin density while the rest belong to theother sublattice (Colors in Fig. 1 have anticipated this fea-ture). We will show later that the model Hamiltonian calcula-tions for 1,4,5,6,7,10-hexahydrocorannulene show frustrationat the same bond than ab initio calculations (compare Figs. 3and 4). Having identified the atoms at each sublattice, it istempting to use the unbalance in the molecule ( N A − N B =4)to predict the total spin of the ground state using Lieb’s for-mula. The result (S=2) is in perfect agreement with numericalresults. This is particularly interesting as in principle Lieb’stheorem should only work on non-frustrated systems.Spin multiplicity of the ground state of a molecule is usu-ally predicted by means of Hund rule applied to MO energiesobtained by an appropriate method. We have checked that thespin of the ground states of the molecules here investigatedis consistent with the degeneracy of the HOMO that H¨uckel’smethod gives for the skeleton of C atoms having an unsatu-rated π orbital. This is true not only for 6H-coronene, but alsofor 6H-corannulene. Although the extended H¨uckel’s methodused by ab initio codes to initialize the self-consistency pro-cess slightly lifts this degeneracy, the HOMO still appears as anarrow bunch containing a number of orbitals compatible withthe spin of the ground states of the three planar molecules de-picted in Fig. 1. Then, as in Hund rule, such a distributionof molecular orbitals favors high spin ground states througha winning competition of interaction energy gains against ki-netic energy losses. IV. MODEL HAMILTONIANS
Let us critically examine the applicability of Lieb’s theo-rem as the predicting tool of the multiplicity of the groundstate of hydrogenated PAHs. The underlying Hubbard model ignores that: (i) transfer integrals in any realistic system arenot limited to nearest neighbors sites, (ii) σ –orbitals appeararound the HOMO-LUMO gap in the same energy intervalas π –orbitals, (iii) interaction among electrons is not limitedto on-site Coulomb repulsion. In our opinion, the success of atheorem or rule based on the simplest interacting model comesfrom its actual capability of describing the correct antiferro-magnetic spin-spin correlations between nearest π electrons.Strong correlation is the basis for the basic correctness of asimplified image in which up and down spins alternate .Even if the spin multiplicity of the ground state is predictedeither by Hund rule or Lieb’s theorem, a deeper understand-ing of underlying correlations calls for a complete numericalsolution of simple interacting models. We have analyzed bothPariser-Parr-Pople (PPP) model Hamiltonian and the localversion of Hubbard Hamiltonian , which actually is a partic-ular case of the former. The PPP Hamiltonian contains a non-interacting part ˆ H and a term that incorporates the electron-electron interactions ˆ H I : ˆ H = ˆ H + ˆ H I (1)The non-interacting term is written as, ˆ H = (cid:15) (cid:88) i =1 ,N ; σ c † iσ c iσ + t (cid:88)
NUMBER OF π ELECTRONS E N E R GY ( e V ) U (eV) -0.6-0.4-0.20 SP I N C O RR EL A T I ON FIG. 4: (color online)
Left:
Total RB3LYP (circles) and Hubbard model (continuous curves) energies of spin S states versus the on-siteCoulomb repulsion U for dodecahydrogenated coronene C H (each peripheral carbon atom saturated with an additional hydrogen), an-thracene and 6H-corannulene C H (Fig. 1C). The sequence from lowest to highest energy is S=0,1, ... for anthracene and C H andS=2,1,0,3,4, ... for C H (in this molecule the Hubbard model gives an almost twofold degenerate ground state). Energies are referred tothe corresponding state of maximum spin, except for C H that were downward shifted by 1.8 eV to improve the overall fit (the state of S=6is largely participated by H orbitals turning invalid the Hubbard model). Molecular geometries were only optimized for the ground state andtaken unchanged for the calculation of excited spin states. Transfer integral was taken equal to -2.71 eV, and, as the results indicate, U = 3.3 eVnicely reproduces the ab initio energies. Right up:
Spin-spin correlations in C H calculated by means of the Hubbard model (the skeletonof C atoms is shown as an inset) on blue pair (two atoms placed on the symmetry axis) and red pair (frustrated horizontal bond). Right down:
Ab initio ground state energies (circles) of the charged states of a benzene molecule that is not allowed to relax. Energy differences are plottedrelative to the neutral case. Results obtained by means of Hubbard (triangles) and Pariser-Parr-Pople (squares) models are also shown. repulsion U . The results depicted in the left panel of Fig. 4 in-dicate that spin states of these molecules can be reasonably fit-ted with U = 3 . eV (benzene, for which the fitting is as goodas for anthracene, is not shown for the sake of clarity). Albeitnoticeable deviations occur in the three lowest lying states of6H-corannulene, the state ordering is the correct one. We havechecked that the failure to correctly separate the lower excita-tions of 6H-corannulene is not exclusive of the simplest in-teracting model: a PPP calculation using Ohno’s interpolationscheme shows a similar weakness. Let us remark once morethat, despite the rather small U ( U/ | t | = 1 . ) resulting fromthe fittings shown in Fig. 4, anti-ferromagnetic correlationsin these molecules, as calculated by means of the interactingmodel, are significant. Particularly attracting is the case ofcorannulene for which there is a bond at which the spin-spincorrelation is significantly smaller than at other bonds of themolecule (top right panel of Fig. 4). Interestingly enough,placing frustration (two adjacent π orbitals of the same sub-lattice as shown in Fig. 1C) at that bond gives a differencebetween carbon atoms in the two sublattices of four atoms,which, using Lieb’s formula, predicts a ground state of totalspin 2, in agreement with our numerical results. Summariz-ing, U = 3.3 eV works satisfactorily describing the spin statesof aromatic molecules and, in particular, the multiplicity ofthe ground state. The same simple Hubbard model fails, however, in describ-ing the charged states of these systems as results for benzeneshow (bottom right panel of Fig. 2). Lanczos results for the in-teracting Hamiltonian are compared with B3LYP results for acharged benzene ideally restricted to a fixed geometric struc-ture. Actual energy differences are much higher than thosepredicted by the model. Our results do also illustrate the lackof electron-hole symmetry that characterizes any realistic self-consistent field calculation as opposed to Lieb’s model. ThePPP model with t = − . eV and values for the Coulombrepulsion integrals from Ref. 24, although greatly improvesthe fitting, still gives a symmetric curve by the well-knownpairing between occupied and unoccupied molecular orbitals.A full fitting of charge and spin states will surely require in-cluding a larger number of parameters . V. CONCLUDING REMARKS
We have proposed a new route to produce magnetic or-ganic molecules that consists of hydrogenating PAHs. In thecase of alternant PAHs the spin multiplicities of the moleculesground states agree with Lieb’s prediction, even though abinitio
Hamiltonians may significantly differ from Hubbard’smodel. A probably related result is that ab initio energies ofthe spin states of these molecules can be very satisfactorilyfitted by means of the simplest version of the Hubbard model.It seems that the molecule topology is enough support for themain result. Energies of charged molecules, instead, cannotbe described by the simple, most popular, interacting models,suggesting a critical examination of their use in, for instance,graphene. Results for total spin densities in molecules havinga magnetic ground state clearly show that the spin is localizedin only one of the two sublattices. On the other hand, ab initio and model Hamiltonian calculations for hydrogenated coran-nulene, place the frustrated bond at the same location. Thisproduces an unbalance in the molecule which, using Lieb’sformula, gives S=2 for the ground state of the molecule, inagreement with the numerical results. It is also worth notingthat in a recent study we have shown that dehydrogenationmay also produce magnetic molecules. Although dehydro-genation is a highly unlikely process, dehydrogenated PAHs have been intensively investigated by astrophysicists whobelieve they to form part of interstellar matter. Although theresults presented here are encouraging, there is still a long wayto go: finding procedures to synthesize these hydrogenatedPAHs and crystallize them into solids that may eventuallyshow magnetic properties. Acknowledgments
The authors are grateful to J. Feliu, M. Yus and A. Gui-jarro for useful suggestions and remarks. Financial support bythe Spanish MCYT (grants FIS200402356, MAT2005-07369-C03-01 and NAN2004-09183-C10-08) and the Universidadde Alicante is gratefully acknowledged. GC is thankful to theSpanish MCYT for a Ram´on y Cajal grant. ∗ Electronic address: [email protected] J.S. Miller, Inorg. Chem. , 4392 (2000). R. Jain, K. Kabir, J.B Gilroy, K.A.R. Mitchell, K.-C. Wong, R.G.Hicks, Nature, , 291 (2007). A. Rajca, in
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