arXiv:0909.0776v1 [cond-mat.mtrl-sci] 3 Sep 2009 nld ihrprn n ooeeo n ie rnaph- or side, one on coronene that and pyrene the progression either explore include two-dimensional will we hydrocar- and benzene, aromatic one-dimensional from polycyclic Starting (PAH). in both bons for carbon shifts and chemical hydrogen calculate our to for systematically principle here of molecules proof simple a approach, for provided results hydrocarbons early and by While circumvented is framework construction. linear-response a for need ic eoeoeo h otwdl sdmtosin methods used widely most the chemistry. of structural one 1930’s, the become since known since been has (NMR) nance uaeceia hfsi eidcsystems, periodic in shifts chemical culate implement. to difficult and complex fairly approaches them these makes all to central framework linear-response uhcluain eefis eeoe ntequantum- the in developed community; first chemistry un- were real- in spectra. calculations NMR soon aid Such from greatly was structures could it determining calculations ambiguously decades, ab-initio past that several ized the over ments to lc ymty hssotoigwsovercome was Mauri shortcoming This de- by therefore symmetry. and Bloch potential stroy vector non-periodic which a fields, the require by magnetic clus- macroscopic hindered and including is of molecules systems difficulty extended as to such Application systems, ters. finite to only plied nteapiaino nifiieia nfr magnetic uniform infinitesimal an based of field. approach application linear-response the alternative on representation an periodic Wannier derive a the to used of Parrinello field. and limit magnetic bastiani applied long-wavelength the the crys- of periodic modulation on in based shieldings tals NMR calculating for proach ln-aepedptnilcodes. pseudopotential plane-wave refined, and vanced h xeietltcnqeo ula antcreso- magnetic nuclear of technique experimental The ehv eetydvlpdanvlmto ocal- to method novel a developed recently have We 5 nrcn er hs prahshv enad- been have approaches these years recent In 12 M hfsfrplcci rmtchdoabn rmfirst- from hydrocarbons aromatic polycyclic for shifts NMR tal. et 2 letlmtdt yrgnsit,w pl it apply we shifts, hydrogen to limited albeit ASnmes 11.m 11.b 52.g 76.60.Cq 75.20.-g, 71.15.Mb, size. 71.15.-m, system of numbers: function PACS experimen a with as agreement shifts chemical good the very for show trends results grap Our in finally ribbon. and pentacene, and naphthacene, anthracene, esuyo qa otn the footing equal orbi on local microscopic, ar of study a shifts we to theory chemical respect developed where with magnetization calculations recently orbital she NMR the to and approach of ribbons novel combination graphene a of using limits infinite two-dimensional oyylcaoai yrcros trigwt ezn a benzene with starting hydrocarbons, aromatic polycyclic eateto aeil cec n niern,MT Camb MIT, Engineering, and Science Materials of Department 1 eateto hsc,Wk oetUiest,Winston-Sa University, Forest Wake Physics, of Department epeetfis-rnils est-ucinlter ca theory density-functional first-principles, present We .INTRODUCTION I. h eeoe ierrsos ap- linear-response a developed who , 6–8 2 3 ln iheprmna advance- experimental with Along edn oagoiguei modern in use growing a to leading oee,teedvlpet ap- developments these however, .Thonhauser, T. 9–11 eetees the Nevertheless, 4 1 iial,Se- Similarly, 12 and H 1 Dtd coe 3 2018) 23, October (Dated: hr the where 1 aieCeresoli, Davide n has and 13 hfsi ezn,prn,crnn,i naphthalene, in coronene, pyrene, benzene, in shifts C etdi e.I .W ilcnld n umrz in summarize in and conclude col- presented will are are We systems V. PAHs B. Sec. periodic IV finite Sec. for in for results lected whereas Results A, IV found Sec. III. be can practi- Sec. method the converse Details in the about of shifts. and implementation sum- NMR calculations cal theory numerical to short approach our a converse about II our Sec. of in mary present we completeness in PAHs air. detecting or for soil devices e.g. precise to and lead efficient might more techniques experimental with information idcssesi oapyamgei edadcalcu- as linear-response approach and a this using to field refer nucleus magnetic will the We at a framework. field apply local the to late is systems riodic eetyage hti satal osbet aclt the calculate to possible actually is it that argued recently fPH norenvironment. NMR content our years the in PAHs recent determine of in to that developed surprising been have not techniques thus terato- is and It mutagenic, genic. can carcinogenic, pollutants toxic, these tobacco, extremely structure, fat, be byprod- on wood, a Depending are e.g. incense. they in or and combustion coal, incomplete and their of oil uct occur as and PAHs such materials fuels issues. fossil health related in and energy environmental on in impact interest of surge or ribbons. graphite), extension, graphene by limits (and, infinite graphene by the represented reach we until etc., anthracene, thalene, hoyi h ii fln-aeeghperturbations. long-wavelength of limit the linear-response use in to theory been has community solid-state with by in- the compatible developed approach and not the is applied conditions, field constant the periodic-boundary a from Since directly fields. duced shifts the computes it hsppri raie ntefloigwy For way: following the in organized is paper This h sa praht acltn M hfsi pe- in shifts NMR calculating to approach usual The efcsti td nPH eas fterecent the of because PAHs on study this focus We 2 zdmgei ioe sn hs methods these Using dipole. magnetic ized cltoso h M hmclsit for shifts chemical NMR the of lculations n ioaMarzari Nicola and ee rpie n nifiiegraphene infinite an and graphite, hene, dicesn ie pt h n-and one- the to up sizes increasing nd t.Orcluain r performed are calculations Our ets. bandfo h eiaieo the of derivative the from obtained e a antzto nsld,ada and solids, in magnetization tal e,NrhCrln 70,USA. 27109, Carolina North lem, sadalwu ocaatrz the characterize to us allow and ts ig,Msahsts019 USA. 02139, Massachusetts ridge, I THEORY II. 2 13 olcmiigab-initio combining tool A principles direct since , 4 We 2 shifts in periodic systems without using a linear-response distribution.15 We added an extra term to the Hamilto- framework. In our converse approach we circumvent the nian taking into account the electron orbital interaction linear-response framework in that we relate the shifts with a nuclear magnetic dipole ms sitting at nucleus rs, to the macroscopic magnetization induced by magnetic by the usual substitution for the momentum operator e point dipoles placed at the nuclear sites of interest. We p → p + c A, where A is the vector potential of a peri- report here the essential result and refer the reader to odic array of nuclear dipoles. This is done conveniently Ref. [12] for details. in reciprocal space and requires only a few dozen lines We define E as the energy of a virtual magnetic dipole of additional code. The calculation and implementation 16–20 ms at one nuclear center rs in the field B for a finite sys- of the orbital magnetization necessary to evaluate tem, or as the energy per cell of a periodic lattice of such Eq. (1) is somewhat more involved. However, a growing dipoles. Then, writing the macroscopic magnetization as number of codes has the calculation of the orbital mag- −1 Mβ = −Ω ∂E/∂Bβ (where Ω is the cell volume), netization already implemented so that the NMR shifts can be calculated readily. Also, if one is only interested ∂ ∂E ∂ ∂E ∂Mβ in finite systems, one can calculate the induced magnetic δαβ − σs,αβ = − = − =Ω . ∂Bβ ∂ms,α ∂ms,α ∂Bβ ∂ms,α moment instead of the orbital magnetization, which is (1) much simpler (the moment can be calculated from the ↔ It follows that σ s accounts for the shielding contribution quantum-mechanical current, which requires only knowl- to the macroscopic magnetization induced by a magnetic edge of the wave functions). Since we are interested in point dipole ms sitting at nucleus rs and all of its pe- shifts of atoms other than hydrogen, we also implemented riodic replicas. Instead of applying a constant (or long- a GIPAW reconstruction, as mentioned above. wavelength) field Bext to an infinite periodic system and calculating the induced field at all equivalent s nuclei, we It might appear that the converse method is computa- apply an infinite array of magnetic dipoles to all equiv- tionally demanding, since we need to perform 3N cal- alent sites s, and calculate the change in magnetization. culations to obtain the shielding tensor for N atoms. Since the perturbation is now periodic, it can simply be However, note that in practice a single full self-consistent computed using finite differences of ground-state calcu- ground-state calculation is performed once, and then the ind dipole perturbations according to each shielding are ap- lations. Note that M = ms/Ω+ M , where the first term is present merely because we have included mag- plied to this ground state. Convergence after the pertur- netic dipoles by hand. The shielding is related to the true bation is much faster than the ground-state calculation, ind and we find that NMR shifts for systems with hundreds induced magnetization via σs,αβ = −Ω ∂Mβ /∂ms,α. NMR is a technique that probes for properties of the of atoms can be easily calculated. electronic structure near the nuclei. Thus, a good de- For our calculations in the following sections we have scription of the wave functions near the nuclei is vital. used a PBE exchange-correlation functional and norm- The formalism described above can directly be imple- conserving pseudopotentials of the Troullier-Martins mented into all-electron programs such as LAPW codes. type21 with a cutoff of 80 Rydberg. The structural pa- However, many codes use plane waves as a basis and the rameters of all systems have been optimized within this ionic potential is replaced by a pseudopotential to keep framework. For the dipole perturbation we chose a value the computational cost low. In these cases, extra care of |m | of 1µ , although the results are independent of has to be taken when calculating NMR shifts. While hy- s B the exact value over a wide range. drogen shifts can be described correctly using Coulombic (pseudo)potentials, a reconstruction of the pseudovalence In order to compare the accuracy of our converse wave functions in the core region has to be performed 4 method to the direct method, we performed test calcu- for elements beyond the first row. This can be achieved lations on benzene and diamond. For the finite system with the so called gauge-including projector augmented- benzene using the converse method we find an absolute wave (GIPAW) method, which has been developed by hydrogen shielding of 22.97 ppm and carbon shielding of Pickard and Mauri in 2001 and proved very efficient for 7 −165.2 ppm. These results compare well with the re- NMR calculations using pseudopotentials. We have de- sults obtained using the direct method: 22.69 ppm and rived the corresponding GIPAW formalism for the con- −163.4 ppm. For periodic diamond we find a shielding of verse method, but the lengthy, mathematical details will 14 −63.89 ppm compared to the direct method which gives be presented in a forthcoming article. −65.85 ppm.6 Note that in all cases the carbon shield- ing includes only the shielding from the valence electrons and does not include the core shielding, which we calcu- III. IMPLEMENTATION AND late to be +200.3 ppm. The small deviations between CALCULATIONAL DETAILS the converse and the direct method are most likely due to the fact that the direct method uses a linear-response We have implemented the converse method outlined framework in which a finite q-vector is used to make the above into the plane-wave density functional theory code magnetic field periodic and modulate it over space, in- PWSCF, which is part of the Quantum-ESPRESSO stead of using a truly constant magnetic field. 3

5 A 1 2 3 B 1 4 3

A C 1 2 A FIG. 1: Schematic model of benzene (left) and pyrene (right). In order to refer to certain atoms and their NMR shift, we use the labeling shown. In general, hydrogen atoms are label by letters, whereas carbon atoms are labeled by numerals. FIG. 2: Schematic model of coronene.

IV. RESULTS TABLE I: Hydrogen and carbon NMR chemical shifts in ppm for pyrene and coronene. For a definition of atomic positions see Figs. 1 and 2. Note that we have used the shifts of benzene NMR experiments usually report the isotropic shield- as a reference. 1 ↔ ing σs = Tr[ σ s ] via the chemical shift δs = −(σs −σref). 3 Molecule atom experiment calculation Here σref is the isotropic shielding of a reference com- a pound such as tetramethylsilane (TMS). Note that we Benzene 1 128.6 — have not calculated the shielding of TMS itself, however, A 7.26 a — we can still report shifts with respect to TMS by using Pyrene 1 125.5 a 122.2 our calculations for e.g. benzene as an intermediate ref- 2 124.6 a 122.1 benzene benzene benzene a erence: δs = −(σs − σcalc − δTMS ). For δTMS 3 130.9 130.4 we have used the experimental values of 128.6 ppm for a 22 4 127.0 126.0 carbon and 7.26 ppm for hydrogen. Henceforth, we will a 5 124.6 123.4 use the notation δmolecule to describe the shift relative to s A 7.60–8.15 b 8.61 TMS of atom s in a certain molecule. We will use letters b s = A, B, C, . . . to refer to hydrogen shifts and numerals B 7.60–8.15 8.57 b s =1, 2, 3,... for carbon shifts. C 7.60–8.15 8.47 Coronene 1 126.2 a 123.5 2 128.7 a 129.3 A. Finite systems 3 122.6 a 122.6 A 8.89 c 9.78 1 13 We start out by calculating the H and C NMR shifts a Reference [22]. of several small PAHs. We start with benzene, and move b Reference [24]. towards the one-dimensional limit by adding one ring at c Reference [25]. a time, to study naphthalene, anthracene, naphthacene, and pentacene. The two-dimensional limit is considered via pyrene and coronene, i.e. surrounding benzene by of the functional used on the NMR shifts.26 In principle, more rings in the plane. For all these calculations we since we are calculating the shift through the orbital mag- used a large supercell with at least 16 Bohr of vacuum netization, which is closely related to electrical currents, between periodic replicas. Note that the magnetic sus- it might be more appropriate to use a current-density ceptibility of these systems effectively vanishes and our functional rather than one of the standard density-only computed shifts can be directly compared to the experi- functionals. Unfortunately, appropriate current-density mental ones without any shape correction.23 functionals (e.g. Ref. [27]) are not yet available in most Our results for pyrene and coronene are collected in Ta- DFT codes. Note that the relative shifts (that is, the ble I. For a definition of their atomic positions see Figs. 1 difference in shift between different atoms) are closely and 2. In general, we find very good agreement between reproduced. This is important, as relative shifts play the the experimental results and our calculations. The small key role in linking NMR data to a structure. deviations are most likely due to the approximative na- As one would expect, all shifts of pyrene and coronene ture of DFT with respect to the exchange and correlation are significantly different from the pure benzene shifts. functional. Using e.g. the local-density approximation for However, it is interesting to see that the hydrogen shifts the exchange-correlation functional, we get slightly dif- increase, while all carbon shifts decrease—except one. It pyrene coronene ferent results. However, a detailed and comprehensive is δ3 and δ2 that rise above the 128.6 ppm of study is necessary to adequately investigate the effect benzene. These shifts both correspond to carbon atoms 4

C 4

3 3 B 2 B 2 D 6 1 1 A A 5 5 C 4 C 4 FIG. 3: (left) Naphthalene. (right) Anthracene. 3 3 B 2 B 2 TABLE II: Hydrogen and carbon NMR chemical shifts in ppm for naphthalene and anthracene. For a definition of atomic 1 1 positions see Fig. 3. A A Molecule atom experiment calculation a Benzene 1 128.6 — FIG. 4: (left) Naphthacene. (right) Pentacene. A 7.26 a — Naphthalene 1 125.7 c 123.0 2 127.8 c 126.2 TABLE III: Hydrogen and carbon NMR chemical shifts in c ppm for naphthacene and pentacene. For a definition of 3 133.4 133.1 atomic positions see Fig. 4. A 7.41 b 8.08 b Naphthacene Pentacene B 7.79 8.13 c atom calc. atom calc. Anthracene 1 126.1 125.0 2 128.1 c 127.8 1 122.3 1 123.3 3 131.6 c 132.0 2 126.5 2 127.4 4 125.3 c 121.6 3 130.5 3 130.6 A 7.41 b 7.95 4 125.5 4 125.0 B 7.95 b 8.24 5 128.1 5 128.5 C 8.40 b 8.44 A 8.31 6 125.4 a B 8.68 A 8.37 Reference [22]. b Reference [24]. C 9.07 B 8.74 c Reference [25]. C 9.21 D 9.46 that have no hydrogens attached. Our results for the one-dimensional chains from naph- of NMR cuvettes. thalene to pentacene are given in Tables II and III. For a definition of the corresponding atomic positions see Figs. 3 and 4. As before, we find very good agreement be- B. Periodic systems tween the experimental results and our calculations. The exact same statements concerning the differences com- The systems discussed above can systematically be ex- pared to benzene apply: hydrogen shifts increase, while panded until they eventually become infinite. In a one- carbon shifts decrease—expect one. In these chains it is dimensional way, we extend the chains naphthalene, an- the carbon shift of atom 3 that increases, which again thracene, naphthacene, pentacene, . . . until we reach an corresponds to a carbon that is not connected to a hy- infinite ribbon. On the other hand, if we extend pyrene drogen. and coronene in a two-dimensional way, we arrive at Note that we have not included experimental results graphene. Stacking graphene sheets on top of each other for naphthacene and pentacene. Although there are some yields graphite. results reported (see e.g. Ref. [28]) these molecules are We have calculated the hydrogen and carbon shifts insoluble in most of the solvents used for the preparation for these extended systems. These infinite systems show 5

teresting in and of themselves. However, we now want to focus on the change in shift as the systems grow larger and eventually become infinite. One would expect to find a correlation between the shifts in finite and infi- A 1 nite systems. Indeed, looking at the carbon shifts we 2 benzene pyrene coronene graphene find δ1 → δ5 → δ3 → δ with the results 128.6 → 123.4 → 122.6 → 118.0 ppm. Al- though NMR is a local probe, it turns out that the size of coronene is not yet big enough for its bulk atoms to behave like graphene. FIG. 5: Section of an infinite ribbon consisting of benzene If we start at benzene again and add rings in a liner rings. fashion, we can grow a chain of arbitrary length. Looking anthracene at the carbon shifts we find the sequence δ4 → pentacene ribbon δ6 → δ1 with the corresponding results metallic behavior and therefore require a dense k-point 121.6 → 125.4 → 128.0 ppm. As with the graphene mesh. For our calculations we have used up to 16 k- sheet, this result suggests that a finite chain would need points in the directions of periodicity and a gaussian to be longer than pentacene so that its carbon atoms be- smearing of 0.001 Rydberg to obtain converged results. have like an infinite ribbon. Looking at sequences that As another result of the metallic behavior, Knight shifts end at δribbon or δribbon, no clear trend is visible. might influence the NMR spectrum. While schemes do A 2 exist to calculate Knight shifts within plane-wave DFT 29 approaches, we did not include them in our calcula- V. CONCLUSIONS tions. For graphene we find a carbon shift of 118.0 ppm. The two in-equivalent carbon shifts in graphite give We have used an alternative first-principles method to 124.3 ppm and 134.9 ppm. For the infinite rib- calculate NMR chemical shifts of a variety of polycyclic ribbon aromatic hydrocarbons and related infinite systems. Our bon depicted in Fig. 5 we find δA =8.56 ppm, ribbon ribbon results are in good agreement with experiment. By going δ1 =128.0 ppm, and δ2 =132.2 ppm. Experimen- tally, solid state NMR spectra of graphite show a broad from finite to periodic systems, we observe trends in shifts peak in the range 155÷179 ppm, depending on sample which suggest that neither coronene nor pentacene are preparation. This broadening is due to a fairly long lat- large enough to model their infinite counterparts. In future work we would like to include the shifts tice relaxation time T1 and the presence of conduction electrons.30 of other PAHs as well as study the Knight shifts of In order to compare our results for graphite with ex- graphene. perimental data, in principle one would have to include a shape correction involving the susceptibility.23 This cor- rection is estimated to be of the order of 1÷2 ppm in VI. ACKNOWLEDGMENTS most forms of graphite sample such as powder samples. The only exception is highly oriented pyrolytic graphite, We are grateful for extensive discussions with F. Mauri. which has a very large and anisotropic magnetic suscep- This work was supported by NSF grant DMR-0549198, tibility, where the shape correction is expected to be of ONR grant N00014-07-1-1095, and the DOE/SciDAC the order of 10÷20 ppm. project on Quantum Simulation of Materials and Nanos- The calculated shifts for these periodic systems are in- tructures.

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