Computational Materials Science 49 (2010) S341–S347 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci Nuclear magnetic resonance in one-dimensional spin chains Hoang Nam Nhat *, Pham The Tan Faculty of Technical Physics and Nanotechnology, UET, Vietnam National University Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam a r t i c l e i n f o Article history: Received 11 October 2009 Received in revised form May 2010 Accepted May 2010 Available online June 2010 Keywords: NMR Spin chain Ab initio Quantum Computer a b s t r a c t This paper shows how different lengths, terminations and electronic spin states of one-dimensional chains of Cu–O affected the shieldings in copper nuclear magnetic resonance signals The calculation was performed for both periodic structure and model clusters of different size and shape The obtained results showed that there was relatively large splitting of 63Cu resonance for each copper position in the chain Ó 2010 Elsevier B.V All rights reserved Introduction There are several solid-state compounds in which the copperto-oxygen (Cu–O) bonding exhibits only one-dimensional (1D) or quasi 1D geometry In the compounds of general form A2CuO3 (A = Sr, Ca) there was observed a strong antiferromagnetic coupling between Cu 3d9 electrons in the 1D Cu–O chains Numerous studies have been presented for the magnetic properties of these compounds [1–3], including the experimental nuclear magnetic resonance (NMR) studies [4–6] In modern science and technology, the 1D spin systems are important in several aspects due to their close connection to fundamental phenomena such as high Tc superconductivity (e.g in 2D isostructural La2CuO4), Bose–Einstein condensation (observed in 2D and 3D spin systems), spin-charge separation (reported for Sr2CuO3 [7]), and quantum coherence There were also a number of quantum computer models that have been proposed on basis of 1D spin chain systems [8–10] The model quantum computers usually utilize nuclear spin but in some intrigued proposals the electronic spin and its hyperfine interaction with spin of nucleus were also considered [8] The 1D spin chains are also (almost) ideal candidates for quantum bus, as they represent the realistic 1D channel for quantum transportation [9,10] However, for a possible application in quantum devices, it is important for 1D spin chain system (e.g A2CuO3) to possess different resonance frequencies for each copper site, so to provide a targeting manipulation of each nuclear spin by separate RF pulse Unfortunately, the existing studies showed only a broaden feature * Corresponding author E-mail address: namnhat@gmail.com (H.N Nhat) 0927-0256/$ - see front matter Ó 2010 Elsevier B.V All rights reserved doi:10.1016/j.commatsci.2010.05.014 which has even disappeared below 12 K (near TN $ K) [4–6] For A2CuO3, the development of nuclear spin–lattice relaxation rate 1/ T1 and Gaussian spin-echo decay rate 1/T2G followed the field thep ory prediction, i.e 1/T1 = const and 1/T2G / 1/ T (for a quantum critical region T ( J, J = 1300 K for Ca2CuO3 and 2300 K for Sr2CuO3) The peak broadening may be considered for Cu site due to large electric field gradients for 63,65Cu nuclei and anisotropy of magnetic field at long data collection time but the positiondependent splitting at Cu sites is not expected in a highly symmetric cubic space group Immm (to which A2CuO3 belongs) In this group the copper atoms occupy two crystallographically equivalent positions (0, 0, 0) and (1/2, 1/2, 1/2), which are also electronically equivalent, so the calculation performed for the periodic structure (discussed in the next section) really predicted a single resonance line Apart from the single crystals, the real nanocrystallites on the other hand exhibit features that are not usual for the large periodic structures, i.e they are exposed to effects of edge topology, chain termination and surface deviation of electronic structure In 2D spin 1/2 Heisenberg antiferromagnets, for examples, the edge effects induced a smaller magnetic response of an edge in comparison with the bulk susceptibility due to singlet formation at the edge [12] For the 1D spin chains, there are mainly two reasons for systems to be decomposed into non-equivalent chain segments: the geometry defects and the spin fluctuation These effects should break down the symmetry of copper electronic structure along spin chain and induce a co-existence of different (electronic) spin states: a singlet state for segments with even number of Cu sites and a doublet state for segments with odd number of the same (Cu2+ cation in 3d9 configuration pos1 sesses one unpaired electron, 3dx2 Ày2 , with spin ±1/2) The variation S342 H.N Nhat, P.T Tan / Computational Materials Science 49 (2010) S341–S347 of density of electrons over each Cu position should introduce a position dependent shielding and consequently splitting of NMR signals at Cu sites Unfortunately, the experimental resolution as obtained for the powder Ca2CuO3 [5] inhibit detailed investigation of fine structure of NMR shielding The purpose of this paper is to demonstrate, on computational basis (with largest wave function basis sets), the existence of such splitting in a series of nanocrystallites and chain segment models Calculation procedure and strategy A2CuO3 is a typical system with strong electron correlation [11,14] It is a common notion that to explore all aspects of electron correlation, the largest possible basis sets should be used and all electrons, including the core ones, should be treated As this requirement can be fulfilled only for small fragments, we are frequently approached the restriction of computational cost and capability of available softwares Therefore, it is critical for the real system to be properly modeled Usually, to achieve the consistent results, a systematic exploration of model chemistries should be taken into account Basically, the shielding tensor rji(N) for a nucleus N is given as [19]: rji Nị ẳ @2E ; @Bi @mj ðNÞ ð1Þ where E is energy, Bi a component of external magnetic field, mj(N) a component of induced magnetic moment for nucleus N To calculate (1), the field-dependent basis functions should be used The Gauge-Independent Atomic Orbital (GIAO) method implemented in GAUSSIAN 03 [15] utilizes the exponential function to shift the gauge origin The NMR implementation in CASTEP [13], on the other hand, uses the so-called Gauge-Including Projector AugmentedWave (GIPAW) approach of Pickard and Mauri [18] to describe the behavior of wavefunction in magnetic field There is also another possibility to calculate rji(N) through the evaluation of induced first-order electronic current density J(1)(r) [19]: rNji ¼ @2E ¼À @Bi @mNj Bc Z drN " # ð1Þ rN  Ji ðrÞ ; r 3N ð2Þ j where i, j are the components of external magnetic field and induced moment In the Continuous Set of Gauge Transformations (CSGT) method by Keith and Bader [22], which is available in GAUSSIAN 03 [15], J(1)(r) is evaluated by performing the gauge transformation (via a shift in gauge origin d(r)) for each point in space (see Eq (25) in Ref [19]) There are many other methods for computing magnetic shieldings but in the limit of large basis sets they should all give comparable results In this work, the calculation for periodic structure was performed by using CASTEP code [13] on basis of optimized geometry at the same level theory Two symmetry settings were involved, the Immm cell with original dimension and the P1 cell with doubled b-axis The LDA and GGA schemes were investigated separately and the dependence of shielding on applied functional and wave function (i.e plane wave ei(k+G)Ár) cut-offs (2|k + G|2 Ecut) were also examined A limit of CASTEP code in evaluating NMR spectra is that it cannot afford the calculation of shielding with polarized spin and LDA + U functional, therefore the (electronic) antiferromagnetic spin coupling (between Cu 3dx2 Ày2 electrons) could not be properly set for the P1 cell and the corresponding insulating ground state [11] was not correctly reproduced (in fact, CASTEP produced a metallic ground state instead) For cluster models, the calculation was performed using GAUSSIAN 03 software package [15] We examined the single chain models n(O–Cu) up to n = 14 (length = 5.3 nm) with and without oxygen termination (n(O–Cu)–O and n(O–Cu)) Systematically, by insertion of chains between sufficiently large CaO bilayers (carrying both zeroed charge and spin) of different size we investigated the influence of model topology on resulted shieldings We showed that the values of shielding in A2CuO3 system were solely controlled by the electronic structure of 1D spin chain The obtained shieldings varied less than ppm for the models with and without CaO bilayers attached Recall that, except for the part of density of states (DOS) with E > eV above Fermi level, the electronic structure of A2CuO3 has been demonstrated that it depends only on Cu–O bonding in 1D chain [11,20] It is important to mention here that unlike the situation in La2CuO4 [16] where every copper position is symmetric in a square-planar configuration, the 1D bonding of copper in A2CuO3 rules out all considerations of symmetry except for a case of centrosymmetric n(O–Cu)–O chains Theoretically, the equal electronic structure for all copper atoms in 1D Cu–O chain might only be expected for the chains infinitely long, but not for the chains of finite size and different termination Within the size of clusters investigated so far, our results showed that the chain termination and spin state sufficiently influenced the shielding values We have observed that the antiferromagnetic ex1 change of Cu 3dx2 Ày2 electron spins was not perfect in the long chains: the zones with alternating spin densities ±1 were only seen for limited segments, usually shorter than 1–2 nm The longer chains were often split into parts of different spin states which induced different spin densities over copper positions [20] Therefore, a position-dependent splitting of NMR shielding for A2CuO3 is not an accidental but a fundamental aspect of A2CuO3 nanocrystallites In agreement with the results previously reported in Ref [16] and the analysis there given, we adopted the CSGT scheme with diffuse function added basis sets Where not particularly stated, the open-shell model (unpaired electrons) with Beck–Lee–Yang– Parr hybrid functional (UB3LYP) and the largest basis set 6311++G(3df,3pd) were selected For reference purpose, the shieldings for solid CuCl were also computed The consistent results were obtained using PBE functional with DNP basis set and at k =   For an experimental cell (a = 5.406 Å), the isotropic r(CuCl) was 364 ppm (±1 ppm variation on change of cut-off from 350 to 450 eV) but for an optimized cell (a = 5.417 Å), r(CuCl) = 379 ppm This result is almost half of 700 ± 200 ppm as obtained for the CuCl4 cluster used in Ref [16] (for this cluster, our calculation reported 798 ppm) As the experimental values might be given for the referencing CuCl in solution, we have also computed the shielding for [CuCl4]3À cluster in water based solution using largest basis set 6311++g(3df,3pd) after geometry optimization at the same level (optimized Cu–Cl bond length 2.462 Å vs non-optimized 2.341 Å) The final shielding (absolute, unreferenced) was 1129 ppm (the similar calculation for shielding in methanol resulted at 793 ppm for an optimized cell 2.334 Å) It is worth to note that the shieldings over the copper and chlor were quite independent: while the shielding for chlor held within 1113 ± ppm regardless of Cu–Cl bond length and kind of solvent used (CCl4, H2O or methanol), the shielding for copper varied from 793 (methanol) to 1129 (H2O) and 1391 (CCl4) ppm Results and discussion The calculation for the periodic structure of Ca2CuO3 by CASTEP code with PBE functional resulted at r(Cu) = À3550 ± ppm for the cut-off of 500 eV and variation of k-space from 16 to 32 points in Monkhorst–Pack grid The LDA functional also produced a similar result À3558 ppm but the WC (Wu-Cohen) functional arrived at r(Cu) = À3581 ppm These values showed a sufficiently larger H.N Nhat, P.T Tan / Computational Materials Science 49 (2010) S341–S347 S343 shielding for Cu in Ca2CuO3 when compared to that of atomic Cu (2402 ppm) Scaling with respect to water-based [CuCl4]3À solution would yield a relative shift of À4679 ± ppm A calculation for a cell with doubled b-axis showed a negligible variation of shielding for four different positions of Cu ( 0.99) and may be approximated by the equation: rCuị ẳ 5728:2R 10378 S345 singlet state (by removing one electron) The results are summarized in Fig and Table From the beginning of calculation, all model chains were inserted between two CaO bilayers as static potential layers The subsequent investigation, however, revealed that the inclusion of CaO bilayers had a negligible effect on final shieldings as the obtained results with and without CaO bilayers varied less than ppm in many cases The possibility of exclusion of CaO bilayers from full ab initio treatment allowed us to simulate in real-time the longer chains up to n = 14 (length = 5.3 nm) with largest basis set 6-311++G(3df,3pd) Overall, the calculation for type (b) clusters and doublet chains converged more slowly, or might even fail as for n = 9, 11 and 14, in comparison with type (a) clusters and singlet chains At first observation, the dispersion of shielding values is very large, ranging from À14,000 to 12,000 ppm Fig shows that shieldings for doublet chains are left-shifted (more shielding) in comparison with that of singlet chains The same is observed for the shieldings associated with triplet excitation in singlet chains (left-shifted) and with deduced singlet state of doublet chains (right-shifted) The presence of unpaired electron seemed to cause sufficiently large shifts to stronger shielding at copper sites This effect can also be observed in type (b) chains with no oxygen termination In general, the type (b) chains possess exact n different values of shielding, whereas the O-terminated type (a) chains exhibit only n/2 + values (doublet chains) or less (singlet chains) Due to centrosymmetry of type (a) chains, the distribution of resonance lines also follows the symmetry: two symmetrically equivalent positions always possess the same shielding value It is noteworthy that in the doublet chains the equivalent positions also possess the same Mulliken spin den- ð3Þ Particularly, r(Cu) develops from À125 (1.8 Å) to 1027 ppm (2.0 Å) This result demonstrates how sensitive is the shielding for copper on Cu–O distance: the change Dr(Cu) per 0.1 Å is almost 573 ppm We now discuss the shieldings as obtained for the linear chains of form n(O–Cu)AO (O-terminated) (a) and n(O–Cu) (no termination) (b) The chains of type (a) is centrosymmetric and has negative charge (À2) whereas the chain of type (b) is asymmetric with respect to inversion and has zeroed charge The chains 5(O– Cu)–O (5 units Cu–O) is shown in Fig 2a for illustration This model cluster has 141 atoms, of which 17 (7 Cu and 10 O) were treated fully ab initio; the rest 124 atoms (two CaO bilayers) were simulated as static potential layers The 5(O–Cu)–O cluster might also be extended to cover a total of 409 atoms, 31 of which were included in full ab initio treatment (Fig 2b) From analysis of symmetry one may expect a half number of individual resonance lines in the chains of type (a) in comparison with that of the chains of type (b) By spin state, the chains in each type are divided into two groups, one group has singlet spin state (even number of Cu atoms: Cu2+ has 3d9 electronic configuration with one unpaired electron) and one has doublet spin state (odd number of Cu atoms) For several singlet chains, we have also calculated the resonances for triplet excitation, and for some doublet chains the resonances for Fig The statistical distribution of copper resonances (a) and its decomposition into parts (b) S346 H.N Nhat, P.T Tan / Computational Materials Science 49 (2010) S341–S347 Fig Cu and O shieldings for largest cluster (a) and Mulliken spin density distribution at each position (b) sity, i.e the distribution of Mulliken spin density is also centrosymmetric However, the interplay between the spin density and shielding is rather poor There is not any systematic development of shielding upon specific positions such as central, edge and middle positions, although it seemed that in the doublet chains the central positions tended to show excessively large shielding values For type (b) chains, the first copper position after the beginning oxygen often were largely deshielded (r(Cu) > 10,000 ppm) In Fig 4a we show the statistic composition of shielding values at copper sites to simulate the real 63Cu NMR spectrum of A2CuO3 The decomposition of contribution from different parts is shown in Fig 4b As seen, there are two maxima for n(O–Cu) chains, one at À3500 ± 500 ppm and another at 500 ± 500 ppm Smaller maximum may also be observed at around 4500 ppm The characteristic peaks in the spectrum for type (a) clusters (n(O–Cu)ÀO) are sufficiently different These systems showed five main features: a large broaden central peak at À1500 ± 500 ppm and two equidistant side groups, about 4000 ppm apart from the central peak Each of these groups again consists of two bands equally placed about 2000 ppm apart each other The peak positions are: 4500, 2500, À1500, À6500, and À8500 ppm In the top part of Fig 4b we show the distribution of shielding for the chains with singlet and triplet states As mentioned in the first part of this paper, in the spin chain systems there is always possible a spin fluctuation which consequently leads to a fluctuation of chain’s spin states This is a dynamic aspect of spin chain system and may have some impacts on NMR shieldings Four main features may be recognized from a top graph in Fig 4b: 3500, 1500, À1500 and À3500 ppm The peaks are again equally spaced (by 3000 ppm) but their intensities are different, so they not show the characteristics of the side bands In general, we may conclude that the presence of peaks at 2500, À1500 and À3500 ppm is quite essential for the system under investigation (Fig 4a) Recall that the value À1500 ± 10 ppm is the experimentally determined diamagnetic shielding for Cu [21] The value 2402 ± ppm is the isotropic shielding for metallic copper reported by many groups, e.g in Ref [16] The last value, À3500 ppm, agrees very well with our result of isotropic shielding for Cu in crystalline Ca2CuO3, r(Cu) = À3550 ± ppm as discussed above This comparison demonstrates that despite a wide variation of Cu electronic structure along the spin chain, the 1D Cu–O structure preferably created three electronic environments which are similar to that of metallic copper and crystalline Ca2CuO3 A few experimental data available for these systems partly confirm that the values 2500, À1500 and À3500 ppm are not computationally arbitrary At last, we show that the value near diamagnetic shielding À1500 ppm may be obtained on basis of cluster model Using a large cluster, which contains a copper atoms centered in the environment of 30 other atoms (Cu and O) and embedded by 378 point charges (Fig 5), we obtained À1430 ppm Fig 5b shows that the (Mulliken) spin density over copper sites in the central spin chain follows in general the antiferromagnetic arrangement, although the magnitudes of spin densities are not exactly ±1 (may be caused by the problem of wave function projection onto the atomic orbital basis) The antiferromagnetic setting was shown to represent the correct ground state of A2CuO3 system [11] As noted before, there is not clear a dependence between spin density and shielding Conclusion The experimentally observed featureless 63Cu NMR spectrum of nanocrystallites of spin chain system A2CuO3 [5] is indeed composed of rich structures of resonances The calculation showed that there is a large dispersion of shielding values for copper, ranging from low À10,000 to high 10,000 ppm The values are very sensitive to structural differences, geometry of model clusters and spin states of the chains under investigation However, the characteristic signatures are centered at several values À1500, À3500 and 2400 ppm These values can be used to distinguish the A2CuO3 system from other copper oxide based systems Although it is still far from the stage in which we are able to manipulate individual copper nuclear spins, the present results clearly showed that relatively large position-dependent splitting of copper resonances exits This might be of interest for the future development of copper oxide based quantum spintronics Acknowledgments This work is supported in part by the Grant-in-Aid for Scientific Research from Asian Research Center, 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observed for the shieldings associated with triplet excitation in singlet chains (left-shifted)