Journal of Modern Physics, 2013, 4, 522527 http://dx.doi.org/10.4236/jmp.2013.44074 Published Online April 2013 (http://www.scirp.org/journal/jmp) Non Local Corrections to the Electronic Structure of Non Ideal Electron Gases: The Case of Graphene and Tyrosine Yamila García1, John Cuffe1,2*, Francesc Alzina1, Clivia Marfa SotomayorTorres1,3,4 1Catalan Institute of Nanotechnology (CIN2CSIC), Campus UAB, Bellaterra, Spain 2Department of Physics, Tyndall National Institute, University College Cork, Cork, Ireland 3Instituciò Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain 4Department of Physics, Universitat Autonoma de Barcelona, Bellaterra, Spain Email: ygarcia@icn.cat Received January 4, 2013; revised February 6, 2013; accepted February 14, 2013 Copyright © 2013 Yamila García et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We introduce a formal definition of a non local functional and show that the non local exchangecorrelation potential functional, derived within DensityFunctional Theory, is non local in the space of electronic densities. A previously developed non local exchangecorrelation potential term, is introduced to approach the exact densityfunctional poten tial. With this approach, the electronic structure of the graphene surface and the tyrosine amino acid are calculated. Keywords: Nonlocality; Interfaces; ExchangeCorrelation Potential; Graphene 1. Introduction Functional architectures at the nanoscale are natural can didates to overcome the limitations encountered by the conventional road map of micro technologies. The effi cient realization of nanoscale devices demands the deve lopment of theoretical methods with enough precision to predict properties in a regime where physical interfaces play a key role. However, it is precisely the description of interfaces that is the main obstacle to rely in current theoretical methods. This is due to limitations in model ing and numerically assess the required quantum defini tion of electronic states. On the one hand, reduced mod els suffer from the inability to include actual many body potentials while retaining numerical efficiency. On the other hand, abinitio approaches, while exactly introduc ing many body interactions, suffer from the lack of effi ciency in terms of computational resources. Consensually [1], an efficient, elegant, and exact alternative involves abinitio calculations by means of DensityFunctional Theory (DFT). While in principle, DFT can yield exact electronelec tron interactions, its effectiveness relies in a combination of the HohenbergKohn (HK) theorems [2] and the KohnSham (KS) scheme [3]. The HK theorems reduce any property of the manybody interacting system to a functional of the ground state electronic density and then, the KS scheme replaces the original manybody problem by an auxiliary independentparticle problem, whereby all manybody interactions are mapped in an effective singleparticle potential, namely, the exchangecorrela tion (xc) potential, c [4]. This xc potential, which contains all the information regarding electronelectron interactions, is implicitly defined as a functional deriva tive of the xc energy, Exc, in the space of electronic den sities, xc xc Enr vnr nr (1) However, in most general cases, an algebraic form for xc Enr is unknown. Thus, the definition of xc nnr is by far the most important task for an ef fective DFT. Initially, the manageability of DFT arises from the in troduction of the LocalDensity Approximation (LDA) to c [3]. LDA, and further extensions beyond it within local exchange models, such as generalized gradient ap proximations (GGA) [5] and MetaGGA [6], work fairly well for homogeneous and wellbehaved electronic den sities [7]. While this is probably the case of bulk systems, *Present address: Department of Mechanical Engineering, Massachu setts Institute of Technology, Cambridge, Massachusetts, 02139, USA. C opyright © 2013 SciRes. JMP
Y. GARCÍA ET AL. 523 U it is no longer the expected behaviour for confined sys tems, such as nanostructures and interfaces, [8], therefore, failures within DFT can be attributed to the inability of local xc potentials, to retain all many body effects when the many body interacting wave function representation is mapped to a meanfield densitybased representation [9]. Recent attempts to overcome such difficulties and to manage DFT are framed within the context of the con trolled addition of the non local Fock exchange to current local xc potentials [10]. To some extent, such hybrid po tentials correct the electronic structure, even when such improvements are not handled systematically [11,12]. Therefore, the purpose of this work is twofold. First, we provide a means to understand the role that non local ity plays in DFT by introducing a formal definition of a non local functional which, as far as we know, is not presently available in the literature. The use of our defi nition provides a rigorous frame to assess the failure of DFT when describing non homogeneous electron sys tems. Secondly, we use a method developed previously by some of us to devise hybrid functionals in a system atic manner [13]. In this way, we analyze the role that non locality plays in the electronic characterization of two benchmark systems, graphene and tyrosine. The cho ice of these two systems responds to the emergence of these materials as platforms for effective molecular elec tronics [14], and thus pay particular attention to graphene interfaced with biosystems [15,16]. 2. Definition of Non Local Functional by Its Opposite A fundamental issue to address when dealing with a mean field theory is the definition of effective potentials. DFT is a mean field theory within which the selection of the effective c potential is controversial [9,17], with continuous efforts towards improvement in accuracy [11, 12,1820]. It is our point of view that all such efforts suf fer from the lack of a rigorous and explicit definition of one of the analytic properties that a c functional must fulfill, namely the non locality in the position space [9]. Thus, in this section we introduce a definition for a non local functional. This definition should be sufficient to connect DFT with other effective theories and gain in sights in the physical implications that the inclusion of non local effects has on mean field theories. Two complementary definitions are given for a local functional as follows: 1) A local functional from F to : n CR A functional n : CR n UR is local if for any open set and fields and , so that UU UU (2) This means that is determined by the values of in an arbitrarily set . U : 2) A local functional from F to R: R A functional is local if it is given by the integral, d n R x :n (3) of a local functional CR (4) This means that the functionals which do not fulfill either of these two conditions in the fields where they are defined are called non local functionals. These two defi nitions should contribute to elucidate controversies con cerning the inclusion of non local effects in current ex changecorrelation potentials. 3. Non Locality in the ExchangeCorrelation Potential In the fundamentals of DFT by Hohenberg and Kohn [2], the analytical expression for the xc energy functional is determined for two particular cases, the electron gas with almost constant density and the electron gas with slowly varying density. By using the first of these two cases, we will show the non local character of the most general exchangecorrelation functional. The gas with almost constant density is defined in ac cordance with the conditions that follow, 0 nrn nr (5a) with 0 1 nr n (5b) and d0nr r (5c) where nr is the electronic density as a function of the position. HK proposed a formal expansion for Exc pro vided such conditions above are met [2]. It is written as, 0dd ,, ddd xc xc EnEnKrrnrnr rr Lrr rnrnrnrrrr (6) In order to capture the essential physics of the problem in question, and to avoid tedious mathematical treatments, we reduce all the expressions above to their one dimen sional (1D) form. Then, by doing the functional deriva tive we obtain an enlightening expression for the xc po tential. It is then written as, d ,, dd xc vnx Kxxnxx Lxxxnxnxx x (7) Copyright © 2013 SciRes. JMP
Y. GARCÍA ET AL. 524 Noticeably, the spatial integrals of the electronic den sity products are, by themselves, non local mathematical operations in the space in which they are defined, see definition of non locality above. Thus, for the most gen eral case, this analytical expression for c is non local in the xspace provided that ñ is different from zero. 4. Non Local xc Potentials for Interfaces The choice of the xc functional describing the most gen eral physical system must fulfill the following conditions: 1) it has to be non local in the space of electronic densi ties [9]; 2) it should correct selfinteraction errors in ex change functionals [10]; 3) it should contain a fraction of the Fock potential to approach exact exchange [13]; and 4) it must agree with the adiabatic connection theorem [21]. With these conditions in mind, and as an improve ment to the first hybrid approaches introduced by Becke [22], we propose the following analytical equation to evaluate numerically c , Fock 1 xc x vv local local x c vv Fock v local (8) Here x is the Focklike potential contribution, which accounts for non local effects, the , c contribu tions are given by any of the available exchange (x) or correlation (c) local functionals. Finally, α is a single fitting parameter to be determined by exact rules. v HOMO LUMO To estimate α, we follow an exact methodology within meanfield Green’s function schemes [23,24]. The value is chosen so that HOMOLUMO matches the charge gap, GAP. This quantities are defined as HOMOLUMO (9) and 0 11E N 00 2GAPE NE N (10) respectively, where N represents the number of electrons, E0 is the total energy for the selected ground states in the KohnSham system of independent electrons with single particle energies denoted as HOMO and LUMO , for the Highest Occupied Molecular Orbital (HOMO) and Low est Unoccupied Molecular Orbital (LUMO) respectively [4]. The choice of the finite region in which to search for α is constrained by two conditions: 1) it must contain the zone which confines the electron and 2) the HOMO and LUMO orbitals must be well localized in to the cluster model. Finally, we should remark that from the reasoning pre sented here we could not derive an explicit mathematical relation between the non local corrections to c re quired by seminal DFT [2], and our adjustable hybrid approach defined by Equation (8). However, we note that both xc potentials are non local functionals of the elec tronic density. It is this common property that justifies an algebraic connection between both equations. Moreover, the fundamental physical relation between both correc tions is now clear: in both cases non local corrections are included beside the causal effects, when non ideal elec tron gases are described in terms of the electronic den sity. 5. Numerical Results and Discussion In this section the influence of the gradual inclusions of non local effects in c upon the electronic structure of two confined systems, tyrosine and graphene, is discussed. To do this, we analyze the evolution of the OLUMO , when the non local Fock contribution to HOM xc varies [see Equation (8)]. The code GAUSSIAN09 is used for numerical DFT calculations [25]. The local DFT functional introduced to define c in the Equation (8) is PW91 [26], which is used either to account for local exchange and local cor relation effects. In regard to the other two inputs required to manage DFT approaches, i.e., 1) finite cluster models definition and 2) the selection of basis sets, the following reasoning was followed. First, the atomic structure of the amino acid reported in the pdb data base is selected [27], and a finite cluster model for the ideal semiinfinite graphene is chosen. To this end, we calculate the position of the ionization potential 00 1 PEN EN when the size of graphene cluster models is increased [see Figure 1]. Graphene is known to have an IP close to 5 eV [28]. Thus, this reference value was used to estimate the level of certainty of our finite model approach to the semi infinite graphene system [29]. Using these results, see Figure 1, we select a 10 × 10 unit cells cluster to model graphene1. Second, the basis sets are selected such that they are 3  21 g for all the atoms in the amino acid and graphene systems. Higher order basis sets were in tro duced to prove the robustness of our results for all the calculations in this work2. The convergence criterium was set to the 10 percent for the single particle energy le vels as well as for ground state energies calculated with fully local c potentials. Figures 2 and 3 show HOMOLUMO and GAP energies calculated for graphene and tyrosine respectively, using a 1Bigger cluster sizes will certainly increase the accuracy of the calcula tions but will also increase the computational time by orders of magni tude. e.g. Using 3  21 g basis sets [25], and going from a cluster with 1 carbon atom [0 unit cells] to clusters with 240 carbon atoms [10 × 10 unit cells], the time consumption is increased by 3 orders of magnitude. In addition, improvements to the accuracy of selfconsistent energies do not affect the main message of this work. 2The electronic structure has been optimized using localizedatomic orbitals asis sets for all the atoms in the structure. The basis selected where sto3 g, 3  21 g, 6  31 g and ccpvtz. http://www.emsl.pnl.gov/forms/basisform.html Copyright © 2013 SciRes. JMP
Y. GARCÍA ET AL. 525 Figure 1. Ground state energy calculations of the ionization potentials, IP, as a function of the number of unit cells used in graphene. Numerical results are represented by filled dots and the solid line is a guide to the eye. Figure 2. Evolution of ΔHOMOLUMO gap for a 10 × 10 unit cell model of graphene [see inset], when non local Fock ex change is included in the abinitio approach to xc . The dots represent numerical calculations for the ΔHOMOLUMO gap when the non local Fock exchange contribution varies in the abinitio approach to xc . These points are interpo lated with a thin solid line as a guide of the eye. The hori zontal solid line represents the GAP value calculated using ground state energies and a local approach to xc . As ex plained in the main text, the intersection between both con tinuous lines, when ΔHOMOLUMO = GAP, determines the proportion of non local exchange required to improve mod els to xc . wide range of c Figure 3. Evolution of ΔHOMOLUMO gap for a tyrosine ami noacid [see inset] when non local Fock exchange is included in the abinitio approach to xc . The dots represent nu merical calculations for the ΔHOMOLUMO gap when the non local Fock exchange contribution varies in the abinitio ap proach to xc . These points are interpolated with a thin solid line as a guide for the eye. The horizontal solid line represents the GAP value calculated using ground state energies and a local approach to xc . As explained in the main text, the intersection between both continuous lines, when ΔHOMOLUMO = GAP, determines the proportion of non local exchange required to improve models to xc . By adjusting this percentage, with the aid of the method previously developed [13], we conclude that the best approach to the electronic structure of the graphene rib bon is ~28% Focklike, and that the best approach to the electronic structure of the amino acid should contain ~58% of Fock exchange contribution. As expected, the value of α decreases as we move to the infinite systems, i.e. as we move from a graphene ribbon to the semi infi nite graphene model, where local (~0% Focklike) ap proaches are expected to perform better. By contrast, the non local exchange is more important to describe elec tronic density regions were the confinement is more stronger, e.g. a 3D confinement as occurring in an iso lated molecule exemplified by the tyrosine amino acid. 6. Conclusion In this paper, a formal definition for a non local function al has been introduced. This definition should open new doors to DFT approaches, while distinguishing whether c proposals actually fulfill the requirement of non locality in the space of electronic densities, as dictated by the original DFT theory [1,9]. In conjunction with previ ously developed non local hybrid approach to c functionals, while a clear indication on the amount of non local effects included is made. Here, it is initially shown that the inclusion of non local Fock exchange has a strong influence on the charge gap for both systems, which rapidly increases with the per centage of Focklike exchange in the hybrid functional. [13], we show how including non local effects in c affects the electronic structure calculations on two benchmark systems: graphene and tyrosine. It is shown that the pro portion of nonlocal Fock exchange required to repro Copyright © 2013 SciRes. JMP
Y. GARCÍA ET AL. 526 duce correctly the gap energy increases with increasing the degree of confinement in the system. The discrepan cies between our non local corrected calculations and the local calculation results, suggest that care should be taken when forthcoming local densityfunctional appro aches in favour of those results obtained by means of hybrid densityfunctional approaches. However, in light of the results discussed here, it is still difficult to assess to what extent current hybrids c potentials can de scribe the electronic states in a more general frame of non ideal electron gases, e.g. nanostructures and inter faces, where the amount of non local corrections is ex pected to vary. Therefore, further developments of hybrid functionals should also include the spatial dependence of non local contributions while exactly approaching the xc potentials. 7. Acknowledgements We are indebted with E. 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