Chemical Physics Letters 479 (2009) 133–136

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Chemical Physics Letters

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Understanding Peierls distortion in one-dimensional infinite V-chain andV–Bz multi-decker complex

Sudipta Dutta a, Tarun K. Mandal b, Ayan Datta c, Swapan K. Pati a,d,*

a Theoretical Sciences Unit and DST Unit on Nanoscience, Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur Campus, Bangalore 560 064, Indiab Department of Biotechnology, Haldia Institute of Technology, Haldia 721657, Indiac Indian Institute of Science Education and Research, Thiruvananthapuram 695016, Indiad New Chemistry Unit, Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur Campus, Bangalore 560 064, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 July 2009In final form 4 August 2009Available online 8 August 2009

0009-2614/$ - see front matter � 2009 Published bydoi:10.1016/j.cplett.2009.08.014

* Corresponding author. Address: Theoretical ScieNanoscience, Jawaharlal Nehru Center for AdvancedCampus, Bangalore 560 064, India.

E-mail address: pati@jncasr.ac.in (S.K. Pati).

We perform first-principles calculations based on density-functional theory to study the stable structuresof one-dimensional (1D) linear infinite vanadium (V) chain. The calculation shows that it prefers todimerize according to the Peierls theorem. However, in 1D infinite neutral V–benzene (V–Bz) multi-decker complex, the dimerization almost disappears because of the screening effect of the interveningbenzene rings. Additionally, we study the effect of electronic correlations on dimerization in 1D chains.Our numerical analysis reveals that, although the strong electron–electron interaction suppresses thedimerization, strong electron–phonon coupling overwhelms it to gain stability through dimerization insuch systems.

� 2009 Published by Elsevier B.V.

The fabrication of nanoscale devices has been of great interestin the recent times due to their fascinating electronic properties,differing significantly from those of the bulk. The formation of sta-ble monatomic gold wires in between two gold tips [1,2] is a majorbreakthrough in nanotechnology. Theoretical studies reveal that,the gold wire is stabilized in a zigzag structure, making an angleof 131� with � 2:9 Å distance within the nearest neighbors [3–8].A comparative theoretical study [4] shows planar zigzag structureswith equilateral triangular geometry for Cu, Ca and K infinitechains. Aluminium with 3s23p1 valence electronic configurationalso forms planar zigzag chains as well as ladder structures [9].

One-dimensional transition metal–benzene (benzimidazole)multi-decker sandwich compounds are also well studied both the-oretically and experimentally [10–23]. These compounds havebeen of great interest for a long time because of their interestingmagnetic and electronic properties with huge possibility in deviceapplications. Generally, it has been observed that, the early transi-tion metals (Sc–V) form multi-decker sandwich compoundswhereas, the late transition metals prefer to form rice-ball struc-tures [13–16]. Another interesting group of multi-decker sandwichcomplexes consists of bridging neutral transition metal (V) or alka-li metal cations between two cyclopentadienyl rings of two succes-sive ferrocene complexes [18,19]. Given all these, the obvious

Elsevier B.V.

nces Unit and DST Unit onScientific Research, Jakkur

question arises, what drives the stability in such one-dimensional(1D) systems.

In this Letter, we focus on the structural stabilities of suchsystems in the light of Peierls theorem [24,25]. According to Pei-erls theorem, a monatomic linear chain with one electron peratom undergoes dimerization leading to a more stable structurewith opening of band gap at the Brillouin zone-boundary bysplitting the degeneracy of the expected metallic band, character-istic of completely undistorted structure. It is analogous to theJahn–Teller distortion in molecules. Polyacetylene is one of theinnumerable examples [26–33], which undergoes Peierlsdistortion.

Here we study stability issues, related to Peierls distortion usingfirst-principles density-functional theory in case of infinite 1D lin-ear vanadium (V) chain and neutral vanadium–benzene (V–Bz)multi-decker sandwich chain, as shown in Fig. 1. There can betwo conformers for the V–Bz multi-decker compounds. One is withD6h symmetry, where all the benzene rings are arranged in eclipsedconformations (Fig. 1b). The other is the staggered conformer withD6d symmetry, where alternate benzene rings are rotated by 30�with respect to each other. In the present study, we have consid-ered the first one because of its higher stability compared to thestaggered conformer, as observed earlier [34]. Although there havebeen reports of V-chain stabilizing in zigzag geometry [35], ourmain focus here is to find the validity of Peierls theorem in termsof distortion in 1D linear V-chain in presence of intervening Bzrings. Additionally, we model the one-dimensional chains withinextended Hubbard Hamiltonian with electron–phonon interac-tions in adiabatic limit to obtain microscopic understanding of

Fig. 1. Systems considered: one-dimensional (a) linear V-chain and (b) V–Bz multi-decker sandwich complex.

134 S. Dutta et al. / Chemical Physics Letters 479 (2009) 133–136

the interplay between electron–electron and electron–phononinteractions in stabilizing such systems.

All the ab initio calculations have been performed using thecommercially available Amsterdam density-functional package[36–42] and BAND graphical user interface for the periodic sys-tems. At the generalized gradient approximation (GGA) level, weuse the PW91 functional (equivalent to PW91x + PW91c) to calcu-late the exchange and correlation energy with double zeta (DZ) ba-sis set and small frozen core. The electronic wave functions aredeveloped on the basis of numerical atomic orbitals (NAO) and Sla-ter-type orbitals (STO).

To obtain the stable equilibrium structures without any dimer-ization, we vary the distance between two successive V atoms inthe linear 1D V-chain. For V–Bz infinite chain, we vary the distancebetween the V atoms and the center of mass of benzene (Bz) rings.In both the cases, we start with completely undistorted structuresand perform single point energy calculations to obtain the undis-torted equilibrium distances. Then, to investigate the Peierls dis-tortion, we vary the bond lengths of two alternate bonds suchthat the sum of two successive bond lengths always remains equalto twice of equilibrium bond length.

Fig. 2 shows the stability of infinite undistorted linear V-chainas a function of the internuclear distance. It can be seen that, thestructure stabilizes at equilibrium bond length of 2.0 Å and the

Fig. 2. Energy of linear one-dimensional infinite V-chain as a function of internu-clear distance without any distortion. Inset shows the stabilization energy as afunction of distortion of alternate bonds of the equilibrium structure.

ground state energy turns out to be �10.298 eV (including nuclearcontribution) for two atoms. We then introduce dimerization intothis stable structure, keeping the total length of the chain as con-stant and in the inset of Fig. 2, we plot the stabilization energieswith respect to the undistorted structure, i.e., the differences be-tween the energies of distorted geometry ðEðdÞÞ and the undis-torted ground state ðEð0ÞÞ as a function of distortion ðdÞ. The plotclearly shows that, the 1D linear V-chain prefers to undergo Peierlsdistortion with most stable structure preferring d ¼ 0:3.

To study the effect of screening on the electron–phonon cou-pling, more specifically, the effect of electron–electron interactionon the Peierls distortion, we now consider 1D infinite V–Bz multi-decker sandwich complex (Fig. 1b). We chose benzene since, theelectron cloud on benzene ring can screen two V atoms on eithersides. We vary the distance between the V atom and the centerof mass of one benzene ring and obtain the equilibrium length of1.686 Å between them as can be seen from Fig. 3. Introduction ofdistortion shows that, although this system undergoes dimeriza-tion, the amount of distortion (0.001 Å) in this case is negligiblecompared to the linear V-chain. So, in presence of screening ben-zene rings in between V atoms, the dimerization gets substantiallyreduced. This observation can be attributed to a number of factslike the reduced bonding connectivity within the V centers dueto the presence of intervening benzene rings, localization ofcharges, weak electron–phonon coupling in multi-decker com-plexes etc. It can also be an interplay among all these factors.

To focus on these key factors controlling Peierls distortion inone dimension, we model the 1D chain within extended HubbardHamiltonian with adiabatic phonons, written as,

H ¼X

i

ðt þ ð�1Þiþ1diÞðayi aiþ1 þ h � cÞ þ UX

i

ni"ni#

þ VX

i

ðni � navÞðniþ1 � navÞ þ1pk

Xi

d2i ð1Þ

where t is the hopping integral, d is the bond alternation parameter,U and V are the on-site and nearest-neighbour Coulomb repulsions,respectively. We set t as the unit of energy. The ayðaÞ is the creation(annihilation) operator and nðnavÞ is the number operator (averageelectron number on every site). We consider di ¼ ½aðui � uiþ1Þ�=t andk as a dimensionless electron–phonon coupling constant defined ask ¼ 2a2=pKt2. ui is the displacement of the ith atom from its meanposition and K and a are the elastic spring constants associated withthe nuclear motion and the electron–phonon coupling constant,

Fig. 3. Energy of one-dimensional infinite V–Bz sandwich complex as a function ofthe distance between V atom and the center of mass of the benzene ring withoutany distortion.

S. Dutta et al. / Chemical Physics Letters 479 (2009) 133–136 135

respectively. The actual distortion of the ith bond from equilibriumis given by di=2a.

We minimize the above Hamiltonian with the constraint thatthe energy changes associated with the net displacement aboutthe mean positions should be zero so that the total length of thewire remains constant, i.e,

Pidi ¼ 0. This way, the relaxed ground

state of the system has the same overall chain length as the unre-laxed state. Imposing the above constraint in terms of Lagrangemultiplier g, we define the functional, F ¼ H þ g

Pidi. Minimization

of this with respect to the di leads to,

di ¼pk2hayi aiþ1 þ h � ci � 1

N

Xi

hayi aiþ1 þ h � ci" #

ð2Þ

where N is the total number of the bonds. The Lagrange multiplier gturns out to be the average bond order of the system, g ¼1N

Pihayi aiþ1 þ h � ci [43].

For a detailed understanding of the electronic correlations onthe Peierls distortions, we consider several values of electron–electron interaction and vary the electron–phonon couplingstrength for all the cases. We solve the Hamiltonian for one-dimen-sional finite chain using exact diagonalization method with an ini-tial guess for ds (to be precise, we start with all di ¼ 0). We thencalculate the bond orders, which essentially gives the d values.With this new ds, we solve the Hamiltonian again self-consistently.The process continues until all the bond orders and the ds converge[43].

Fig. 4 shows the bond alternation parameter as a function ofbond index for different electron–phonon interactions. We con-sider three different systems with electron–electron interactionterms (a) U = 0 and V = 0, (b) U = 2 and V = 0, and (c) U = 2 andV = 0.5. In absence of correlation, the first system undergoes Peierlsdistortion and amount of dimerization increases with increase inelectron–phonon interaction. Case (b) and (c) represents a systemwith spin density wave (U > 2V) ground state. Here the V term triesto make the electrons paired at alternate sites, whereas the U termlocalizes single electron on every site. As can be seen in Fig. 4,although the electron–phonon interaction dimerizes the stronglycorrelated systems, it seems less operative for non-interacting orweakly interacting systems at lower strength. The 1D V–Bz mul-ti-decker sandwich complex, being a weakly interacting or almostnon-interacting system [10,11,17] reduces the dimerization. Theoverlap between the V d-orbitals and the carbon p-orbitals causesthe delocalization of electrons over the whole system resulting in ametallic or half-metallic behavior. Thus the electronic correlation

Fig. 4. Bond alternation parameter as a function of the bond index of a 1D chainwith 10 sites for electron–phonon interaction terms 0.1 (triangle-up), 0.2 (triangle-down), 0.3 (circle) and 0.4 (square) for three different systems, (a) with U = 0 andV = 0, (b) with U = 2 and V = 0 and (c) with U = 2 and V = 0.5. The t value is set as 1.

becomes less significant in this class of systems. However, the 1DV-chain shows significant dimerization because of localization ofelectrons in V d-orbitals which owing to its narrow band naturemake the electronic correlations more significant.

We have additionally investigated infinite benzene stack with-out the V atoms, keeping the benzene rings on top of each other.The stable structure with equilibrium distance of 3.5 Å betweenthe center of masses of two successive benzene rings does not pre-fer dimerization. Actually, the dispersive forces among the benzenerings cannot distort the structure in absence of any explicit bond-ing interaction. Note that, the PW91 functional used for the calcu-lation of the geometry of the infinite benzene stacks is quitesuitable for the systems with dispersive forces and has been exten-sively used for studying weak intermolecular forces [44–49]. Thus,in 1D infinite V–Bz multi-decker sandwich complex, the major roleof benzene is to reduce the dimerzation. Indeed, we conjecture thathigher screening effect can completely remove the dimerization,due to lack of true bonding connectivity and consequently decreasein electron–phonon coupling.

In conclusion, we have studied 1D infinite linear V-chain and V–Bz multi-decker sandwich complex in the light of Peierls distor-tion. We have found that, the significant dimerization in the formeralmost disappears in case of multi-decker system. Our numericalstudies within quantum many body formalism suggest that, the re-duced bonding between V atoms due to the presence of interven-ing Bz rings and due to almost non-interacting nature, thedimerization becomes insignificant in multi-decker complex. How-ever, the strong electron–electron or electron–phonon interactioncan introduce higher distortion in such systems. Although ourmany body calculations do not have direct correspondence withthe density-functional theory results, we believe to obtain acoherent picture of the Peierls instability, both the methodologiesare required hand in hand for reliable predictions.

Acknowledgments

S.D. acknowledges the CSIR, Government of India for researchfellowship and S.K.P. acknowledges the research support fromDST and CSIR, Government of India.

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