Magnetic susceptibility of alkali-TCNQ salts and extended Hubbard models with bond order and charge density wave phases
Manoranjan Kumar, Benjamin J. Topham, Rui Hui Yu, Quoc Binh Dang Ha, Zolt'an G. Soos
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Magnetic susceptibility of alkali-TCNQ salts and extended Hubbard models withbond order and charge density wave phases
Manoranjan Kumar, Benjamin J. Topham, Rui Hui Yu, Quoc Binh Dang Ha ∗ , Zolt´an G. Soos † Department of Chemistry,Princeton University, Princeton NJ 08544 (Dated: November 11, 2018)The molar spin susceptibilities χ ( T ) of Na-TCNQ, K-TCNQ and Rb-TCNQ(II) are fit quantita-tively to 450 K in terms of half-filled bands of three one-dimensional Hubbard models with extendedinteractions using exact results for finite systems. All three models have bond order wave (BOW)and charge density wave (CDW) phases with boundary V = V c ( U ) for nearest-neighbor interaction V and on-site repulsion U . At high T , all three salts have regular stacks of TCNQ − anion radicals.The χ ( T ) fits place Na and K in the CDW phase and Rb(II) in the BOW phase with V ≈ V c . TheNa and K salts have dimerized stacks at T < T d while Rb(II) has regular stacks at 100K. The χ ( T )analysis extends to dimerized stacks and to dimerization fluctuations in Rb(II). The three modelsyield consistent values of U , V and transfer integrals t for closely related TCNQ − stacks. Modelparameters based on χ ( T ) are smaller than those from optical data that in turn are considerablyreduced by electronic polarization from quantum chemical calculation of U , V and t on adjacentTCNQ − ions. The χ ( T ) analysis shows that fully relaxed states have reduced model parameterscompared to optical or vibration spectra of dimerized or regular TCNQ − stacks. I. INTRODUCTION
The strong π -acceptor A = TCNQ (tetracyano-quinodimethane) forms an extensive series of ion-radicalsalts [1–3] with closed-shell inorganic ions as well ascharge-transfer (CT) complexes with π -donors such asD = TTF (tetrathiafulvalene). The high conductivityand phase transitions of TTF-TCNQ on cooling werethoroughly investigated as an important step towardsthe realization of organic superconductivity [4]. TCNQsalts crystallize in face-to-face stacks that immediatelyrationalize their quasi-one-dimensional (1D) electronicstructure. Endres has reviewed the many structuralmotifs of 1D stacks [5]. We consider in this paper themagnetic properties of “simple” 1:1 alkali-TCNQ saltswith half-filled stacks of A − radical ions. Complexsalts with stoichiometry such as 1:2 or 2:3 have lessthan half-filled stacks; they are semiconductors withhigher conductivity than simple salts. Hubbard andrelated models are the standard approach to TCNQsalts or CT complexes [1–4]. Each molecule in a stackis a site with a single frontier orbital, the lowest un-occupied orbital of A or the highest occupied orbital of D.Heisenberg spin chains were initially applied to themagnetic properties of TCNQ salts [6], especially todimerized stacks whose elementary excitations aretriplet spin excitons. Subsequently, Hubbard models[1–4] opened the way to discuss optical and electrical aswell as magnetic properties. Limited understanding of ∗ Present address: University College Dublin, Dublin Ireland 01 7167777 † Electronic mail:[email protected]
1D models hampered early treatments. Theoretical andnumerical advances now make it possible to treat thespin susceptibility of 1D models almost quantitatively.Alkali-TCNQ salts offer the possibility of joint modelingof magnetic, optical and vibrational properties. Aninteresting consequence reported below is that differentmodel parameters are needed for magnetic and opticalproperties.There is considerable literature on K-TCNQ or Na-TCNQ, recently in connection with photo-induced phasetransitions [7, 8]. They form [5] regular stacks with A − at inversion centers at high T , dimerized stacks withtwo A − per repeat unit at low T . The transitions are[9] at T d = 348 K and 395 K, respectively, for Na andK-TCNQ. Torrance [10] and others [11] sought to model T d as a spin-Peierls transition, as discussed in the reviewof Bray et al. [12] who noted that such high T d requiresunacceptably large exchange constants. The transitionshave some 3D character since the cations also dimerize[13, 14]. We model the molar spin susceptibility χ ( T )of the Na and K salts at T > T d using regular stacks.We also consider χ ( T ) of dimerized stacks for T < T d without, however, treating the transition. There aretwo Rb salts: Rb-TCNQ(I) is strongly dimerized [15] at300 K while Rb-TCNQ(II) has regular stacks [16, 17]with A − at inversion centers at both 100 and 295 K.The recent 100 K structure [17] rules out a dimerizationtransition around 220 K that was inferred from magneticsusceptibility [18] and infrared [19] data. We reinterpretthese observations. Regular stacks make Rb-TCNQ(II)the best target for modeling χ ( T ).Fig. 1 shows the molar spin susceptibilities of Na, Kand Rb-TCNQ(II). The K and Na data are integratedelectron spin resonance (esr) of Vegter and Kommandeur[18], who identified the transitions. Dimerization opensa magnetic gap E m > χ ( T )that vanishes at T = 0, whether or not χ ( T ) can bemodeled. Crystal data [13, 14] at T > T d indicateeclipsed (ring over ring, Fig. 1) stacks with TCNQ − at inversion centers and interplanar separation R(Na)= 3.385 ˚ A at 353 K, R(K) = 3.479 ˚ A at 413 K. Thesolid line for Rb(II) is esr intensity [18]. The dotted lineis static susceptibility [17] corrected for diamagnetism.The measurements agree at 300 K and both have a kneearound 220 K, less prominently in static susceptibility.The structure has slipped stacks [17] (ring over externalbond, Fig. 1) of TCNQ − at inversion centers with R= 3.174 and 3.241 ˚ A at 100 and 295 K, respectively.These regular stacks clearly have large E m . They arenot compatible with finite χ (0) at T = 0 and E m = 0 inregular Heisenberg [20] or Hubbard [21] chains.The 1D extended Hubbard model [22] (EHM, Eq. 1 be-low) has nearest-neighbor interaction V in addition to on-site repulsion U >
0. Increasing V in a half-filled EHMinduces a transition to a charge density wave (CDW)phase [22]. The CDW boundary is at V c ( t = 0) = U/α M in the atomic limit of U >> t , where t is electron transferbetween adjacent sites and α M = 2 for the EHM is theMadelung constant of the lattice. The CDW transitionis closely related to the neutral-ionic transition of CTsalts from largely neutral DADA stacks to largely ionicD + A − D + A − stacks [23–25]. In either case, the groundstate (gs) undergoes a first-order quantum transition atsmall t/U or a continuous transition when t/U exceedsa critical value, or when U < U ∗ . Nakamura [26] rec-ognized that the EHM with U < U ∗ has a narrow bondorder wave (BOW) phase between the CDW phase at V c ( U ) > U/ E m = 0 at V s ( U ) < U/
2. The BOW phase has finite E m in a regularstack and broken inversion symmetry C i at sites. Sub-sequent studies [27–29] confirmed the BOW phase of theEHM and sought accurate values of V s ( U ), V c ( U ) and U ∗ . We recently characterized the BOW phase of theEHM and related broken C i symmetry to electronic soli-tons [30]. Finite E m in regular stacks is an attractiveway to rationalize χ ( T ) in Fig. 1, and we have proposedthat Rb-TCNQ(II) is a BOW phase system [17, 31].In this paper, we model the spin susceptibility of Na,K and Rb-TCNQ(II) quantitatively with the EHM andrelated models with more realistic Coulomb interactions.We find the Na and K salts to be in the CDW phase with V slightly greater than V c ( U ) and the Rb(II) salt to bejust on the BOW side of V c ( U ). Modeling χ ( T ) is bothchallenging and decisive for several reasons. First, thefull electronic spectrum is required, not just the groundstate. Second, comparison with experiment is absolutesince the magnitude of χ ( T ) follows without scalingin π -radicals with weak spin-orbit coupling. Third,all three salts have 1D stacks of A − = TCNQ − withsimilar U and other parameters on physical grounds.To the best of our knowledge, Hubbard models have RR FIG. 1: Solid lines: molar magnetic susceptibility of alkali-TCNQ salts based on electron spin resonance intensity fromref. [18]; dotted line, static susceptibility from ref. [17].The Na and K salts dimerize at T d and have regular face-to-face stacks for T > T d with separation R between molec-ular planes. The Rb salt has regular ring-over-external-bondstacks down to 100 K and a knee around 220 K. not been applied quantitatively to both magnetic andoptical/vibronic properties of the same system. 1:1alkali-TCNQ salts provide such an opportunity.The paper is organized as follows. We present inSection II the spin susceptibility of Hubbard-type modelsnear the boundary V c ( U ) between the BOW and CDWphases. The magnetic gap E m ( V ) to the lowest tripletstate increases rapidly at V ≈ V c . In Section III wemodel the χ ( T ) data in Fig. 1 with similar param-eters for TCNQ − stacks in related but not identicalcrystals. We compute model parameters in Section IVfor individual TCNQ − or for adjacent TCNQ − . Theseparameters are reduced substantially in crystals, more sofor magnetic than for optical or vibrational properties.The Discussion briefly addresses the parameters ofH¨uckel, Hubbard or other semiempirical models. II. BOW/CDW BOUNDARY
We consider a half-filled extended Hubbard model [22](EHM) in 1D and extend it to second-neighbor interac-tions V = γV . The EEHM with γ > H ( γ ) = N X p =1 ,σ − t ( a † p,σ a p +1 ,σ + h.c )+ N X p =1 U n p ( n p − / N X p =1 V n p ( n p +1 + γn p +2 ) (1)The first term describes electron transfer betweenadjacent sites with retention of spin σ . Regular stacksin this Section have equal t s taken as t = 1. The numberoperator is n p . The last two terms are on-site repulsion U >
0, nearest-neighbor interaction V and second-neighbor interaction γV . The spin fluid phase with n p = 1 at all sites is the gs for small V while the chargedensity wave (CDW) with two electrons per site on onesublattice is the gs for large V . The CDW boundary is V c ( t = 0) = U/α M ( γ ), where α M ( γ ) = 2(1 − γ ) is the 1DMadelung constant. As recognized from the beginning[6, 32], electrostatic interactions are 3D and ion-radicalorganic salts have α M ≈ .
5. Point charges in 1D lead to α M = 2ln2. Physical considerations set α M = 1 . α M ( EHM ) = 2.Finite t in a regular stack leads to a narrow BOWphase [26] between V s < U/α M and V c > U/α M for U < U ∗ , with[29] U ∗ ≈ t for the EHM. Smaller α M gives a less cooperative CDW transition and extendsthe BOW phase to higher U ∗ . The point charge model(PCM) with long-range Coulomb interactions V n = V /n in Eq. 1 has [33] U ∗ (PCM) ≈ t . By the sameanalysis, we estimate that the EEHM with γ = 0 . U ∗ ≈ t . Quantum chemical evaluation[34] of U and V places alkali-TCNQ salts at the CDWboundary and imposes the constraint V ≈ V c in Eq. 1.The symmetry properties of H ( γ ) are the same forspin-independent interactions. Total spin S is conservedand E m is from the singlet gs to the lowest tripletstate. The half-filled band has electron-hole symmetry J = ±
1. We define E J as the excitation energy to thelowest singlet with opposite J from the gs. A regularstack has inversion symmetry C i at sites that we labelas σ = ± E σ as excitation to lowest singletwith opposite σ from the gs. The energy thresholds E m , E J and E σ of extended stacks are not known exactly for V > V ≈ V c ( N ) and use valence bond meth-ods [35] to solve H ( γ ) exactly for N = 4 n or 4 n + 2sites with periodic or antiperiodic boundary conditions,respectively. Low-energy excitations are accessible upto N = 16, and the full spectrum to N = 10. Atconstant U and t , the condition E m ( V ) = E σ ( V ) gives V s ( N ) while E σ ( V ) = E J ( V ) gives V c ( N ). We also TABLE I: Boundaries V s and V c of the BOW phase of H ( γ ),Eq. 1, with N sites, t = 1, U = 8, γ = 0 .
2, and periodic(antiperiodic) boundary conditions for N = 4 n (4 n + 2) basedon excitation thresholds E m , E σ , E J . N V s ( E σ = E m ) V ( E σ = 0) V c ( E σ = E J )8 4.834 5.105 5.19910 4.908 5.139 5.20112 4.908 5.139 5.20114 4.932 5.149 5.20216 4.952 5.157 5.203TABLE II: Magnetic gap E m ( V ) to the lowest triplet of H ( γ ),Eq. 1, with N sites, t = 1, U = 8, γ = 0 .
2, and periodic(antiperiodic) boundary conditions for N = 4 n (4 n + 2). N E m ( V ) E m ( V c ) E m ( V c + 0 . define V ( N ) where E σ ( V ) = 0 and the degenerate gs inthe BOW phase can be explicitly constructed as linearcombinations of σ = ± V s in units of t for γ = 0 . α M = 1 .
6) and U = 8 t inEq. 1 up to N = 16. The V s cluster as expected about U/α M = 5 .
0. Their weak N dependence makes it possi-ble to extrapolate to the extended system as discussed[36, 37] in connection with a frustrated spin chain. Wehave computed V s ( N ), V ( N ) and V c ( N ) of all threemodels (EHM, EEHM, PCM) as functions of U < U ∗ and have previously reported [30] EHM values at U = 4 t .The magnetic gap E m dominates χ ( T ) as T → V s , remains small at V and grows rapidly on crossing the CDW boundary at V c . The size dependence of E m in Table II is for theEEHM at U = 8 t . Decreasing E m ( N ) is found inspin or Hubbard chains with E m = 0 in the extendedsystem. Instead, E m increases with N in all threemodels when V slightly exceeds V c ( N ). A densitymatrix renormalization group (DMRG) calculation [33]for the EHM at U = 4 t shows increasing E m ( N, V )for
N >
30 at V = V c ( N ). Hence E m of the extendedsystem may exceed the N = 8 gaps that we use below for V ≈ V c in the BOW phase for V > V c in the CDW phase.We compute the full spectrum of H ( γ ) for N = 8 (10)sites with periodic (antiperiodic) boundary conditions[30]. Standard methods give the partition function Q and molar spin susceptibility χ ( T ). Fig. 2 shows χ ( x ) asa function of reduced temperature x = k B T /t , where k B is the Boltzmann constant. Since g (TCNQ − ) ≈ . χ ( x ) is directly related to Avo-gadro’s number N A and the Bohr magneton µ B . J¨uttneret al. [38] obtained quantitative χ ( x ) for the Hubbard k B T/t χ t/ ( N A µ Β g ) N=8N=10Ref. 38 PCM, U=8tEHM, U=5t,EEHM, U=8t
Hubbard ModelU=6tU=5tV=V c N=8
FIG. 2: Molar magnetic susceptibility χ of Hubbard-typemodels, Eq. 1, with finite N . The U = 5 t and 6 t , V = 0curves for N = 8 and 10 match the extended results (opensymbols) from ref. [38] for k B T > . t . The other curveshave V = V c . The EHM has U = 5 t , the EEHM has U = 8 t and χ = 0 . U = 8 t and Coulombinteractions V n = V /n . model with V = 0 in Eq. 1 and finite χ (0); their re-sults for U = 5 t and 6 t are shown by open symbols inFig. 2. The lines are exact N = 8 and 10 results thatfor x > . χ ( x ) curves in Fig. 2are for N = 8 with periodic boundary conditions and V = V c . We find similar χ ( x ) for the EHM with U = 5 t ,the EEHM with U = 8 t , γ = 0 . U = 8 t . Small V is conveniently approximated as aHubbard model with an effective U e = U − V . Thisrationalizes reduced χ ( x ) with increasing V , but not thequalitative change of χ (0) = 0 due to finite E m in theBOW or CDW phase. Large t ≈ K and limitedthermal stability of ion-radical oganic solids limits χ ( x )to x < . III. MAGNETIC SUSCEPTIBILITY
In this Section, we model χ ( T ) data in Fig. 1 usingregular stacks for Rb(II) and for K and Na at T > T d .The first terms of Eq. 1 for dimerized stacks at T < T d has transfer integrals t p = − (1 + ( − p δ ) (2)along the stack. We did not change V in dimerizedstacks. Since all three salts have TCNQ − stacks, similar U is expected on physical grounds, and we have soughtsimilar U < U ∗ without strictly enforcing the constraint. T ( K ) χ ( T ) ( - e m u / m o l e ) EHM EEHMPCM δ =0.2 δ =0 T d Na - TCNQ
FIG. 3: Three χ M ( T ) fits shifted by 50 K for clarity of theNa-TCNQ data (open symbols) in Fig. 1 for Hubbard-typemodels in Eq. 1 with N = 8 and parameters in Table III. Thestacks are regular ( δ = 0) for T > T d , dimerized ( δ = 0 .
20) atlow T and interpolated using Eq. 3 in between. It soon became apparent that χ ( T ) for T > T d requires V > V c . We studied the PCM with V n = V /n andEEHM with γ = 0 . γ = 0) in part to search for a fit with V ≤ V c andin part to probe the dependence of t and U on themodel. The following χ ( T ) calculations are all for N = 8sites with periodic boundary conditions in Eq. 1. Theexperimental data in Fig. 1 are now shown as opensymbols.We start with χ ( T ) for Na-TCNQ in Fig. 3 and obtaingood fits for T > T d for the EHM with U/t = 4, t =0 .
097 eV and
V /t = V c + 0 .
19. The χ ( T ) results forthe EEHM and PCM are displaced by 50 and 100 K,respectively, for clarity. They are equally good for the t , U and V parameters listed in Table III. All threemodels return t ≈ .
10 eV and V slightly larger than V c .Good χ ( T ) fits to T = 310 K in the dimerized phase areshown in Fig. 3 with δ = 0 .
20 in Eq. 2 and the same t , U and V. Konno and Saito [13] followed the temperaturedependence of the Na-TCNQ crystal structure and founda coexistence region. The regular phase for T > T d =345 K appears already at T = 332 K and grows at theexpense of the dimerized phase that disappears at T d .The χ ( T ) fits in Fig. 3 between T = 310 K and T d arelinear interpolations according to χ ( T ) = T − T T d − T χ ( T, δ = 0) + T d − TT d − T χ ( T, δ = 0 .
2) (3)The coexistence region is 10 K wider in the fit. Ter-auchi [9] studied the intensity of selected superlattice
TABLE III: Parameters for the spin susceptibility of Na, Kand Rb-TCNQ(II) in Figs. 3,4,5Salt Model t ( eV ) U ( eV ) V ( eV )EHM 0.0956 0.383 0.214Na EEHM 0.0969 0.630 0.434PCM 0.0965 0.627 0.492EHM 0.0780 0.370 0.211K EEHM 0.0801 0.601 0.406PCM 0.0758 0.569 0.440EHM 0.0745 0.373 0.199Rb EEHM 0.0767 0.614 0.399PCM 0.0707 0.601 0.440 reflections for T < T d in both Na and K-TCNQ. Theintensity is proportional to δ ( T ) and decreases linearlyas T d − T near T d . The susceptibility between 320-345K can also be modeled as variable δ ( T ).Fig. 4 shows χ ( T ) fits for K-TCNQ, again displacedby 50 K for clarity and again in the CDW phase with V > V c for T > T d = 398 K. The K-TCNQ parameters t , U and V are in Table III. The same parameters and δ = 0 .
40 agree with experiment up to 350 K. Thereis no evidence of coexisting phases. The intensity ofsuperlattice reflections decreases over an 80 K intervaland changes discontinuously from δ/ χ ( T d , δ/ χ ( T d ) indicates that χ ( T ) between 350 Kand T d can be fit with variable δ ( T ) in these models.Smaller t ( K ) ≈ . eV is consistent with larger R inK-TCNQ.Figure 5 shows χ ( T ) fits for Rb-TCNQ(II) for thethree models displaced by 50 K. We took V = V c (8)at the upper limit of the BOW phase and set E σ = 0.The δ = 0 fit for regular stacks is markedly improved byslightly increasing E m beyond E m (8) /t in Table II, by0 . t for the EHM and by 0 . t for PCM and EEHM.Finite-size effects are critical in view of other evidence[17, 30] for broken C i symmetry in Rb-TCNQ(II), whichimplies V ≤ V c . By contrast, finite-size effects for theNa or K salts are absorbed in V > V c . Good δ = 0fits are obtained down to T ≈
250 K with the t and U parameters in Table III. The esr intensity in Fig. 1 hasa pronounced knee around T kn ≈
220 K. The knee isless prominent in the static susceptibility. Dimerizationis ruled out by the 100 K structure, which has regularstacks and the 300 K space group [17].An adiabatic (Born-Oppenheimer) approximation forthe lattice is typically invoked to model the Peierls [39] orspin-Peierls [12, 40] instability of 1D systems, althoughquantum fluctuations[41] are important for small δ (0)at T = 0. The BOW phase has finite δ (0) for linearelectron-phonon (e-ph) coupling α to a harmonic lattice T ( K ) χ ( T ) ( - e m u / m o l e ) δ =0.4 δ =0 EHM EEHMPCMT d K-TCNQ
FIG. 4: Three χ M ( T ) fits shifted by 50 K for clarity of theK-TCNQ data (open symbols) in Fig. 1 for Hubbard-typemodels in Eq. 1 with N = 8 and parameters in Table III. Thestacks are regular ( δ = 0) for T > T d , dimerized ( δ = 0 .
40) atlow T and jump to δ/ T d . [30], where α = ( dt/du ) is the first term of the Taylorexpansion of t ( R + u ). The electronic gs energy per sitein units of t has a cusp [30] ǫ ( δ ) − e (0) = − B ( V ) | δ | + O ( δ ) (4)where B ( V ) is the order parameter of the BOW phase. B ( V ) ≈ . V = V ( U )in Table I for all three models for the U s in Table III.For comparison, a half-filled band of free electrons with δ = ± . B ( δ ) for partial double andsingle bonds.The BOW phase has long-range order that cannotpersist for T > T has regions with reversed δ (0) that areseparated by topological solitons whose width 2 ξ goesas 1 /δ (0). Spin solitons are also found numerically inthe BOW phase [30] of the EHM or in the magneticproperties of organic ion-radical salts [42].We consider dimerization fluctuations in the BOWphase. This regime has equal densities ρ ( T ) of spinsolitons and dimerized segments with successively ± δ (0)in Eq. 4. We approximate each S = 1 / ξ sites in an otherwise dimerizedstack. Since E m is not degenerate, E m ( V, δ (0)) /t ini-tially increases as B ( V ) | δ (0) | N due to the cusp in Eq. 4,as found directly [30] up to N = 16 at V = V ( N ) where E σ = 0. Such size dependence cannot go on indefinitely.It suffices for our purposes to note that δ (0) < . T ( K ) χ ( T ) ( - e m u / m o l e ) Static Susceptibility esr Intensity
PCMEEHMEHMRb - TCNQ δ =0 Eq. 5
FIG. 5: Three χ ( T ) fits shifted by 50 K for clarity of the Rb-TCNQ(II) data (open symbols) in Fig. 1 for Hubbard-typemodels in Eq. 1 with N = 8 and parameters in Table III. Thestacks are regular ( δ = 0). Spin solitons with width 2 ξ = 60in Eq. 5 are used at low T . generates large E m with negligible χ ( T ) at low T in dimerized regions between solitons. Such regionsdecrease with increasing T and vanish at 2 ξρ ( T ′ ) = 1when the stack is regular everywhere.In this approximation, dimerization fluctuations re-duce χ ( T ) for T < T ′ . The soliton density ρ ( T ) = χ ( T ) T /C follows directly from the molar Curie constant C = N A g µ B of noninteracting spins. The knee regionin Fig. 5 up to T is modeled as χ ( T < T ′ , ξ ) = 2 ξρ ( T ) χ ( T ) (5)with 2 ξ = 60. The fit is adequate for the simpletreatment of fluctuations. The choice of 2 ξ = 60 gives T ′ ≈
250 K, somewhat higher than experiment. Thesame soliton width accounts for the T dependence ofthe infrared intensity of a totally symmetric TCNQ − vibration [30]. Such IR data is decisive evidence [43, 44]for broken C i symmetry, whether due to B ( δ ) in well-characterized K-TCNQ stacks [45] at 300 K or to finite B ( V ) in a BOW phase.It remains to reconcile dimerization fluctuations atlow T with the X-ray data for regular stacks at 100K and thermal ellipsoids that conservatively limit [46]R + − R − = 2 u < . A . To be detectable, u mustexceed zero-point motions. The stack at 100 K has small ρ ( T ) that prevents long range order. Soliton motionmodulates R as δ (0) = αu/t , and the magnitude of α/t determines whether δ (0) is consistent with X-ray data. We conclude this Section by assessing the parame-ters in Table III. Three models (EHM, EEHM, PCM)with narrow BOW phases have been applied to threeTCNQ salts (Na, K, Rb(II)). The CT integral t of reg-ular stacks depends on overlap, as sketched in Fig. 1,and on separation between TCNQ − planes. It is reas-suring that the models return identical t to better than10 % with t ( N a ) > t ( K ) > t ( Rb ) in an unconstrainedfit. We sought similar U in TCNQ − stacks. The U sin Table III are identical within 5% for each model. TheBOW/CDW boundary V c of the EHM with α M = 2 leadsto U ( EHM ) < U ( EEHM ) ≈ U ( P CM ) ≈ .
61 eV thatwe prefer on the basis of α M ≈ .
5. The Na and K saltsare in the CDW phase with
V > V c while χ ( T ) of Rb-TCNQ(II) is consistent with a BOW phase with V closeto V c . The χ ( T ) fit of Na leads to δ = 0 . T ≈ T d . The K-TCNQ fit has larger δ = 0 . T . Theknee region of Rb-TCNQ(II) is fit by Eq. 5 with 2 ξ = 60,the soliton width used previously [30] for IR data. IV. MODEL PARAMETERS
The parameters in Table III are for three models withBOW and CDW phases. They are internally consistent,but considerably smaller than expected from opticaldata. Typical values [1–4] are t ≈ . − . U ≈ U e = U − V ≈ t /U e ≈ . U , V and t for individual or adjacentTCNQ − . The results are based on density functionaltheory (B3LYP) with the 6-311**G(p,d) basis in theGaussian 03 package [47]. An eclipsed (TCNQ) − dimerat R = 3.2 or 3.4 ˚ A is correctly found to have singlet gs,while smaller basis sets [48] yield a triplet gs. Smallerbasis sets are adequate for model parameters, however,as discussed [49] for t .The disproportionation reaction 2 A − → A − + A relates U to the gs energies E ( A − ) , E ( A ) and E ( A − ).The optimized TCNQ − structure leads to U (vertical) =4.413 eV. Optimization of TCNQ and TCNQ − returns U (adiabatic) = 4.192 eV. The relaxation energy of 0.22eV for electron transfer is in excellent agreement with0.1 eV per TCNQ − deduced [43] from Raman and IRspectra. The interaction V depends on adjacent TCNQ − and can be estimated several ways: (1) electrostaticrepulsion between the atomic charges q i of the two ions;(2) repulsion between q i obtained in a dimer calculation;(3) energy difference E − E ( A − ) between the tripletgs of the dimer, which precludes the formation of a π − π bond, and two radical ions. The same values are obtained[48] to better than 5%, and V s in Table IV are based onthe triplet. The listed V ( N a ) and V ( K ) are for eclipsed TABLE IV: Calculated model parameters for adjacentTCNQ − . Parameter Na-TCNQ K-TCNQ Rb-TCNQ(II) V (eV) 2.713 2.671 2.594 t (eV) 0.345 0.299 0.182 t /t (eV) 0.299 / 0.468 0.254 / 0.444 - t /t (eV) 0.266 / 0.451 0.265 / 0.429 - α (eV/˚ A ) 0.59 0.55 0.34 (TCNQ) − with R = 3.385 and 3.479˚ A , respectively.The regular Rb-TCNQ(II) stack has R = 3.241 ˚ A anda 2.0 ˚ A displacement along the long axis shown in Fig. 1.The ratio U ( ad ) /V ≈ . α M = 1 . γ = 0 . α M (8) = 17 /
12 for an 8-sitePCM. The 1D stack is close to the CDW boundary of theEEHM or PCM. The magnitude of U is strongly reducedin the solid state by electronic polarization P ≈ P is approximately quadratic incharge, we have P ( A − ) − P ( A − ) ≈ A − reduces V by a smaller amount.The t s in Table IV are for the 300 K structure ofRb and the T > T d structures of Na and K. We find t ( N a ) > t ( K ) > t (Rb) as expected but for calculated t s that exceed the magnetic parameters in Table III bya factor of 2.5 for Rb and 3.5 for Na or K. The reasonfor such large reduction is not understood. There are twodimerized stacks [5] in Na or K-TCNQ at 300 K. Table IVlists the calculated t , t and the larger, smaller separa-tion R + , R − . We obtain δ ( N a ) = ( t − t ) / ( t + t ) = 0 . δ = 0 .
20 at low T in Fig.3. The corresponding δ ( K ) are 0.27 and 0.25, smallerthan δ = 0 .
40 in Fig. 4. But the K salt has substan-tially larger ( t + t ) / . t in the dimerized phasethat leads to an equally good χ ( T ) for δ = 0 .
30 when themean value of the transfer integral is used. Overall, thecalculated and fitted δ are reasonably consistent.The two-point derivative dt/d R = α is an estimatefor the e-ph coupling constant. The two stacks of Naor K-TCNQ at T < T d have almost the same α , whoseaverage value is reported in Table IV. The 300 and100 K structures of Rb-TCNQ(II) return a smaller α ≈ . eV / ˚ A . The structural constraint of 2 u < . A discussed above leads to δ (0) < αu/t = 0 . t in the crystal.Dimerization fluctuations of such small amplitude wouldbe difficult to detect.Direct evaluation of t has been discussed before[49, 51]. Eclipsed TCNQ − gives the largest t (R,0) thatdecreases with increasing separation R. Displacing theion by L along the long axis leads to tilted stacks in Fig.1. The nodes of the singly occupied orbital of TCNQ − generate t (R , L) = 0 at L=1.3˚ A and to secondary maxima at other L [49, 52]. The first maximum at L= 2.1 ˚ A is close to the Rb-TCNQ(II) or TTF-TCNQstructures. A series of substituted perylenes illustrateswider variations of t with displacements along both thelong and short molecular axes [51].We consider next parameters derived from nonmag-netic data. Simple TCNQ salts have a broad CTabsorption ~ ω CT ≈ eV polarized along the stack.The optical conductivity of K-TCNQ has a shoulder athigher energy that has been variously associated withdimerization [53], with a band edge [54] or with a localexcited state [45] of TCNQ − . Meneghetti [55] modeledK-TCNQ with special attention to totally symmetricmid-IR modes that are coupled in dimerized stacks tothe CT absorption. Polarized spectra yield the couplingconstants g n . Meneghetti [55] used an EHM with N = 4sites, periodic boundary conditions, and adjustable t , t and V , V at T < T d . Comparison with experimentalso entails lifetime or broadening parameters. Nearlyquantitative fits are shown in Fig. 8 of ref. [55] forthe optical conductivity at 300 K with coupled mid-IRmodes and in Fig. 7 for polarized spectra at 27, 300 and413 K.The EHM parameters of ref. [55] for a regular K-TCNQ stack are t = 0 .
19 eV, U = 1 .
20 eV and V = 0 . V for a moment, we have a Hubbardmodel with χ (0) ≈ . × − emu/mole. Including V = 0 .
02 eV in a N = 8 calculation leads to χ ( T ) with abroad maximum at 1.6 x 10 − emu/mole at T max ≈ χ ( T ) slope for T > T d is muchsteeper, however, and finite χ (0) is not consistent withRb-TCNQ(II). We note that V = 0 .
02 eV is a finite-sizeeffect since N = 4 confines an e-h excitation to be closetogether. The CT absorption shifts to lower energy withincreasing N and optical spectra of longer regular stacksreturn different parameters. The 300 K parameters ofref. [55] are again U = 1 .
20 eV and alternating t = 0 . t = 0.37 eV (or δ = 0 . / .
47 = 0 . V = 0 . V = 0 .
31 eV. The strong CT absorption at N = 4hardly shifts to the red at N = 8 or 12. But δ = 0 . E m . The calculated χ ( T ) for N = 8 withthese parameters is very small ( < − emu/mole) up to500 K, completely incompatible with the magnetic datain Fig. 1.Quantitative treatment of e-mv coupling in dimerizedstacks such as K-TCNQ is based on linear responsetheory and force fields for molecular vibrations [56, 57].The coupling constants g n depend on just one electronicparameter, the zero-frequency optical conductivity.While the CT band is of central importance, its precisemodeling is not. Dimerized stacks with broken C i symmetry are required for coupling to mid-IR modes.Coupling to the same mid-IR modes in Rb-TCNQ(II)in regular stacks is strong evidence for a BOW phasewith broken C i symmetry. The same modes appear[58] with slightly higher intensity in powder spectra ofRb-TCNQ(I), which is dimerized [15] at 300 K. The T dependence of the intensities I IR ( T ) of coupled modes ischaracteristic of a BOW phase, and spin solitons with2 ξ = 60 account [30] for I IR ( T ).The optical spectrum in the narrow BOW phaseis dominated by t due to competition between thelarger U and V terms. The CT absorption peak isaround 3 t for a regular stack of rigid molecules [34]and shifts to higher energy by U (vert) - U (ad) = 0.22eV. Dimerization also shifts ~ ω CT to higher energy.Preliminary modeling with all eigenstates of N = 8 or10 indicates that the t s in Table III have to be doubledfor optical spectra and that δ ≈ . t s have been assumed all along for optical spectra. V. DISCUSSION
We have modeled the molar spin susceptibility χ ( T ) ofalkali-TCNQ salts in Fig. 1 that were previously beyondquantitative treatment. We have not treated the phasetransitions of Na or K-TCNQ, but relied on crystal datafor T d and coexisting phases or diffuse scattering. Wefound consistent parameters in Table III for 1D Hubbardmodels with point charges or second-neighbor V thatreduced the Madelung constant to α M ≈ .
5. The χ ( T )fits in Fig. 3,4,5 have t , U , V parameters in Table IIIthat are about half as large as parameters from opticaldata.It should perhaps be no surprise that quantitativeanalysis of magnetic and optical data within the samemodel leads to different parameters. Hubbard modelsmake the zero-differential-overlap (ZDO) approxima-tion of H¨uckel theory for conjugated molecules orof tight-binding theory in solids. The PCM with V n = V /n is a special case of the Pariser-Parr-Pople(PPP) model [59, 60]. Salem [60] has summarizedthe merits and limitations of ZDO, which does notconcern us here. But his discussion of t , the H¨uckel β parameter, bears directly on different magnetic andoptical parameters. Systematic variations are illustratedby many conjugated hydrocarbons with sp hybridizedC atoms. H¨uckel theory provided a convenient approachto analyze variations prior to modern digital computers.Thermochemical data were successfully fit with a β th that is roughly half of β op inferred from optical spectra[60]. The correlated PPP model with β op is definedby the geometry of planar conjugated molecules andhas considerable predictive power [57, 61], includingtwo-photon spectra and nonlinear optical properties.More recently, INDO (intermediate neglect of differentialoverlap) and its spectroscopic version INDO/S havedifferent β parameters [62].Instead of closely related hydrocarbons, Hubbardmodels are used to study electron-electron correlationsolids in general. Quantitative application is rare and soare homologous series. Moreover, magnetic and opticalor other properties are typically modeled separately anda single half-filled Hubbard band is rarely thought tobe quantitative. Na, K and Rb-TCNQ(II) are closelyrelated quasi-1D systems that nevertheless crystallize indifferent space groups.At least qualitatively, differences between magneticand optical parameters may be rationalized in termsof relaxed states in thermal equilibrium and electronicexcitations that are fast compared to atomic or molec-ular motions. Equilibrium states that contribute to χ ( T ) are fully relaxed with respect to both molecularand lattice modes, and relaxed states have reducedexcitation energies. Hubbard or other approaches toelectronic excitations start with vertical 0 − U and V significantlyin the solid state, but this fast process is fully includedin model parameters for optical spectra. The Holsteinmodel [63] illustrates reduced t due to linear coupling toa molecular vibration. Lattice phonons are consideredin 1D for selected modes such as the Peierls modebut complete 3D relaxation is prohibitively difficult.Yet such relaxation is the most likely explanation forsmall parameters derived from χ ( T ) data. Quantitativemodeling of the spin susceptibility clearly points todifferent magnetic and optical parameters for Na, Kand Rb-TCNQ(II). The magnetism also indicates theHubbard-type models for the Na and K salts at T > T d are in the CDW phase while the Rb(II) salts is in theBOW phase close to the CDW boundary. Acknowledgements.
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