Proposal of an extended t-J Hamiltonian for high-Tc cuprates from ab initio calculations on embedded clusters

Proposal of an extended t-JHamiltonian for high-Tccuprates from ab initio calculations on embedded clusters Carmen J. Calzado*and Jean-Paul Malrieu Laboratoire de Physique Quantique, IRSAMC, Universite ´Paul Sabatier, 31062 Toulouse, France 共Received 17 October 2000; revised manuscript received 17 November 2000; published 15 May 2001兲 A series of accurate ab initio calculations on CupOqfinite clusters, properly embedded in the Madelung potential of the infinite lattice, have been performed in order to determine the local effective interactions in the CuO2planes of La2⫺xSrxCuO4compounds. The values of the first-neighbor interactions, magnetic coupling (JNN⫽125 meV兲, and hopping integral (tNN⫽⫺555 meV兲have been confirmed. Important additional effects are evidenced, concerning essentially the second-neighbor hopping integral tNNN⫽⫹110 meV, the displacement of a singlet toward an adjacent colinear hole, hSD abc⫽⫺80 meV, a non-negligible hole-hole repulsion VNN⫺VNNN⫽0.8 eV, and a strong anisotropic effect of the presence of an adjacent hole on the values of the first-neighbor interactions. The dependence of JNN and tNN on the position of neighbor hole共s兲has been rationalized from the two-band model and checked from a series of additional ab initio calculations. An extended t-Jmodel Hamiltonian has been proposed on the basis of these results. It is argued that the hereproposed three-body effects may play a role in the charge/spin separation observed in these compounds, that is, in the formation and dynamic of stripes. DOI: 10.1103/PhysRevB.63.214520 PACS number共s兲: 74.72.⫺h, 74.25.Ha, 74.25.Jb I. INTRODUCTION The insulating cuprates, such as La2CuO4, which are the undoped parent compounds of the high-Tcsuperconductor La2⫺xSrxCuO4, are known to present antiferromagnetic couplings between nearest-neighbor 共NN兲copper centered sites in the CuO2planes. Raman and neutron-diffraction experiments evaluate this coupling to be around JNN⫽130 meV 关128⫾6 meV,1,2 134⫾5 meV,3–5 respectively兴. Nevertheless, the corresponding simple Heisenberg Hamiltonian does not reproduce entirely the features of the Raman spectrum6–11 and additional effects such as second-neighbor magnetic coupling JNNN and four-spin cyclic exchange have been invoked.12–15 While an upper bond for JNNN ( 兩 JNNN 兩 ⭐9 meV兲has been given from Raman experiments,5the amplitude of the four-spin operator in this kind of compounds is a matter of discussion. Previous works have shown that this cyclic operator corresponds to a fourth-order term in the perturbative expansion of the Hubbard model Hamiltonian, scaling as ␭tNN 4/U3, where tNN is the nearestneighbor hopping integral and Uis the on-site Coulombic repulsion, with ␭⫽40 共Refs. 16 and 17兲or ␭⫽80,18,19 depending on the formal writing of the Hamiltonian. Regarding the hole-doped material, where the conduction takes place in the CuO2planes, the holes can be seen as centered on copper atoms with large tails on the four neighboring oxygen atoms. They move from on site to an adjacent one through the effect of a hopping operator of amplitude tNN , for which there is no direct experimental evaluation, but values around ⫺0.5 eV are considered as reasonable.20 One of the most widely employed model Hamiltonians for the interpretation of the properties of these materials, through a hole-pairing mechanism, is the so-called t-Jmodel21,22 which combines spin coupling and hole hopping: H⫽兺 ab JNN•共Sa•Sb⫺1/4兲 ⫹tNN•共aa †ab⫹ab †aa⫹s.c.兲 ␦ 共na⫹nb,1兲.共1兲 The adequacy of such a simple Hamiltonian to incorporate the physics of the problem is questionable. Hopping between second-neighbor sites, t⬘, may be non-negligible. When one derives the t-JHamiltonian from the Hubbard Hamiltonian, three-site operators moving a singlet-coupled electron pair toward the hole appear at second order of perturbation theory, scaling as the JNN operator, i.e., as tNN 2/U. The transferability of JNN from the undoped to the doped material is not guaranteed, the presence of a neighboring hole may affect the coupling of two adjacent spins. The hole-hole repulsion Vij is likely to play a role, influencing the mean distance between the holes. Different extensions of the t-Jmodel have been employed in numerical simulations, for instance, t-t⬘-J共Refs. 23–26兲or t-J-V,27–29 but the values given to the parameters are rather arbitrary, varying widely from one author to the other, the main objective being to exhibit qualitative collective effects. Among them the experimental evidence of the occurrence of stripes have focused attention in the recent past.30 The goal of the present paper is to bring useful information regarding the local effective interactions in undoped and hole-doped cuprates. To obtain them, the most accurate tools of ab initio quantum chemistry will be used. The method consists in considering few-site clusters, properly embedded in the field of the periodic environment, and to calculate the low part of the spectrum using the exact Hamiltonian, large basis sets, and extensive configuration interaction 共CI兲expansions. From this spectrum it is possible to fix the amplitudes of the effective interactions. The procedure has been successfully used to calculate the NN magnetic coupling in a series of perovskites,31 among them La2CuO4for which a PHYSICAL REVIEW B, VOLUME 63, 214520 0163-1829/2001/63共21兲/214520共13兲/$20.00 ©2001 The American Physical Society63 214520-1 value of JNN⫽130 meV is obtained. Similar calculations of the hopping integral in the hole-doped system (t⫽⫺0.57 eV兲have been also reported.32,33 These calculations concerned symmetrical two-site clusters, for which the determination of JNN and tNN is straightforward from the two lowest eigenvalues. This is no longer the case when one considers larger clusters to extract additional parameters, concerning next-nearest-neighbor 共NNN兲interactions, neighboring-hole dependence of JNN and tNN , hole-hole repulsions, and fourspin cyclic operators. Their achievement requires some additional mathematical tools such as the Bloch definition34 of effective Hamiltonians and localization procedures 共for instance, the Boys method35兲. The methodology will be explained in Sec. II. Section III will present the results concerning the four-site square plaquette and three-site linear cluster with different number of electrons 共holes兲. Calculations performed on undoped clusters provide the values of the NN, NNN, and next NNN magnetic couplings. Also the amplitude of the four-spin cyclic exchange for this compound has been established. Hole-doped clusters give informations about the hopping integrals 共NN, NNN, and next NNN兲, the singlet-displacement operator and the dependence of the first-neighbor interactions 共hopping integral and magnetic exchange兲on the number and relative position of the adjacent holes. Section V presents a rationalization of the anisotropy of the effect of hole共s兲in the vicinity on the values of JNN and tNN and reports additional exploratory calculations to evaluate the dependence of the bicentric parameters on the hole positions. Finally, Sec. VI summarizes the results, proposing a refined t-Jmodel, and discusses the possible effect of the additional operators on the charge/spin distribution on the lattice, with possible consequences for the stripping phenomena. II. METHOD A. Mapping of a model Hamiltonian on an ab initio effective Hamiltonian For such materials the unpaired electrons are essentially located on Cu dx2⫺y2in-plane atomic orbitals, with nonnegligible delocalization tails on the adjacent oxygen atoms. Such Cu-centered orbitals will be labeled 兵 a,b,c,... 其 .In the doped material the hole has much larger delocalization tails on O 2porbitals, but as shown by Zhang and Rice,21,22 it remains possible to work within a one-band model Hamiltonian, the precise nature of its valence orbitals being implicit. For a finite cluster involving pcenters and n⭐punpaired electrons, the model Hamiltonian acts in the subspace spanned by the localized determinants ␾ i, containing a common frozen core and nactive electrons. The double occupancy of the active orbitals of these localized determinants is prohibited in both the Heisenberg and the t-JHamiltonians. The ab initio calculations handle a large number of atomic orbitals, symmetry-adapted molecular orbitals 共MO’s兲, and expansions of the wave functions on millions of determinants. Nevertheless, it is possible to construct from these calculations ab initio effective Hamiltonians which are in one-to-one correspondence with the model Hamiltonians. For an undoped cluster involving pCu atoms (psites兲,itis possible first to obtain from variational calculations a set of molecular orbitals containing doubly occupied MO’s 共core兲, unoccupied MO’s 共virtual MO’s兲, and pMO’s with essentially single occupation, which define the ab initio oneelectron valence space, 兵 ␸ i 其 in Fig. 1. A unitary localizing transformation of the psymmetry-adapted orbitals will provide equivalent localized orbitals 兵 a⬘,b⬘,c⬘,... 其 which can be seen as in strict one-to-one correspondence with the implicit valence orbitals of the model Hamiltonian. This localizing unitary transformation Umay be evident by symmetry or defined by a localizing functional, for instance, the Boys criterion 共see Sec. IIIA兲, 兵 a⬘,b⬘,c⬘,... 其 ⫽U 兵 ␸ p, ␸ q,... 其 . Fixing a double occupancy of the core orbitals, and putting n electrons in the valence localized orbitals, avoiding their double occupation, localized neutral determinants 兵 ␾ i,loc ⬘ 其 , are obtained which are in correspondence with the n-electron basis of the model Hamiltonian. These localized neutral determinants can be written as ␾ i,loc ⬘⫽ 兩 core 共a⬘b⬘c⬘...兲 兩 *Si,共2兲 where Siis one of the Cp p/2 spin distributions on the various active localized orbitals. These determinants define a model space 共of projector PS,PS⫽兺 兩 ␾ i,loc ⬘ 典具 ␾ i,loc ⬘ 兩 典 ) for the ab initio calculations. Let NSbe the dimension of that space S. The information obtained by the most refined ab initio calculations will be extracted according to the theory of the effective Hamiltonian proposed by Bloch.34 When one knows the NSeigenstates ⌿mhaving the largest projections on the model space 共which constitute the target space, stable subspace of Hexact) and their eigenvalues Em, the effective Hamiltonian is such that its eigenvalues are the exact ones, and its eigenvectors are the projections of the exact eigenvectors onto the model space: Heff 兩 PS⌿m 典 ⫽Em 兩 PS⌿m 典 ,m⫽1,NS.共3兲 The spectral definition of Heff is Heff⫽兺 m 兩 PS⌿m 典 Em 具 PS⌿m ⬜ 兩 ,共4兲 where 兩 PS⌿m ⬜ 典 is the biorthogonal transformation of 兩 PS⌿m 典 . Actually the projections 兩 PS⌿m 典 of the 共orthogonal兲states ⌿mhave no reason 共except for symmetry reasons兲 to be orthogonal; they define an overlap matrix s, smn⫽ 具 PS⌿m 兩 PS⌿n 典 ,共5兲 and the biorthogonal vectors are defined by 兩 PS⌿m ⬜ 典 ⫽s⫺1 兩 PS⌿m 典 .共6兲 The values of the norms of the projections, i.e., the diagonal elements of smatrix, give an indication on the quality of the description of these states by the truncated space S. The model space and the exact eigenstates must be in strong correspondence, i.e., one must choose both spaces so that the vectors ⌿mhave the largest projections on the model space. CARMEN J. CALZADO AND JEAN-PAUL MALRIEU PHYSICAL REVIEW B 63 214520 214520-2 Then one may express this Hamiltonian on the basis of the localized determinants 兵 ␾ i,loc ⬘ 其 written in terms of the orbitals 兵 a⬘,b⬘,c⬘,... 其 and the matrix elements 具 ␾ i,loc ⬘ 兩 Heff 兩 ␾ j,loc ⬘ 典 ⫽兺 m 具 ␾ i,loc ⬘ 兩 PS⌿m 典 Em 具 PS⌿m ⬜ 兩 ␾ j,loc ⬘ 典 共7兲 can be identified to the matrix elements 具 ␾ i 兩 H 兩 ␾ j 典 of the model Hamiltonian. Since the symmetry-adapted delocalized determinants 兵 ␾ i,deloc ⬘ 其 are linear combinations of the localized neutral determinants 兵 ␾ i,loc ⬘ 其 , constituting two basis sets of the same model space, 兩 ␾ i,deloc ⬘ 典 ⫽兺 kdi,k 兩 ␾ k,loc ⬘ 典 ,共8兲 it is not difficult to calculate the overlap: 具 ␾ i,loc ⬘ 兩 PS⌿m 典 ⫽兺 j 具 ␾ i,loc ⬘ 兩 ␾ j,deloc ⬘ 典 cm,j,共9兲 where cm,jis the coefficient of the 兩 ␾ j,deloc ⬘ 典 determinant on the 兩 PS⌿m 典 wave function. In principle the effective Hamiltonians may be non- Hermitian but the hermitization is straightforward.36 The comparison between the ab initio effective Hamiltonian and the model Hamiltonian fixes the amplitudes of the integrals appearing in the latter and allows one to verify whether nonnegligible additional interactions are not present. Figure 1 summarizes the whole process. Changing the size of the cluster, for instance going from a two-center/one-electron problem to a three-center/two-electron one, one may check the consistency of the procedure and the transferability of the effective interactions. B. Computational details The widely used embedded cluster technique has been employed to model the system.37–42 Finite clusters of the type CupOq(Cu3O10 and Cu4O12 , Fig. 2兲have been selected, where the qoxygen atoms are the first in-plane neighbors of the pCu atoms. 共Previous calculations have shown that the explicit involvement of the out-of-plane oxygen atoms does not change the values of the in-plane interactions.32,33兲The first shell of atoms surrounding the cluster have been replaced by formal charges with pseudopotentials, in order to mimic the Coulombic and exclusion effects. The rest of the lattice has been modeled by means of point charges, which values have been fixed according to Evjen’s method,43 and which correctly represent the Madelung potential of the crystal.44 This is a simplified approach in comparison with more elaborate methods proposed in the literature.45 The ten most internal electrons of Cu atoms have been represented by effective core pseudopotentials, the valence electrons being treated explicitly with triple-zeta basis sets.46 A double-zeta basis set has been used for oxygen FIG. 1. Summary of the strategy used to extract effective interactions in La2⫺xSrxCuO4sys- tems. PROPOSAL OF AN EXTENDED t-JHAMILTONIANFOR... PHYSICAL REVIEW B 63 214520 214520-3 atoms,46 preliminary calculations on bicentric clusters have shown that the inclusion of polarization functions on the bridging oxygen atoms has not an important effect either on the magnetic coupling or on the hopping integral. For undoped clusters, the restricted-spin open-shell Hartree-Fock calculations variationaly define the singly occupied magnetic orbitals. These orbitals define a minimal valence complete active space 共CAS兲. From this space it is possible to calculate the spectrum through a difference dedicated configuration interaction 共DDCI兲procedure47 which implies all the simple and double excitations on the top of this CAS, except the double excitations from the core to the virtual orbitals, which do not contribute to the energy difference at second-order of perturbation theory.47 An alternative solution consists in defining an enlarged CAS including the on-bond 2porbital of the bridging oxygen atoms. These ligand-centered orbitals are the most participating on the intersite spin-exchange and electron transfer processes.48 Performing all the single excitations on the top of this extended CAS, which corresponds to the two-band Hubbard model space, one introduces dynamical polarization effects, i.e., screening by the nonactive electrons, at lower computational cost than the preceding computational scheme. For such type of systems, the equivalence of the DDCI procedure on the top of the small CAS and the single excitations on the top of the enlarged CAS, including the most relevant ligand orbitals, has been illustrated on the La2CuO4system 共using a Cu2O7cluster兲33,49 and on oxalatobridged Cu共II兲complexes.49,50 III. AB INITIO CALCULATIONS ON THE PLAQUETTE AND THE LINEAR CLUSTERS As was mentioned above, two different clusters have been used to extract the effective interactions. A four-site square cluster 共plaquette兲of formula Cu4O12 关Fig. 2共a兲兴 has been employed in order to determine the first- 共NN兲and secondneighbor 共NNN兲interactions, and also the four-spin cyclic exchange. Third-neighbor interactions 共nNNN兲has been estimated by means of the calculations carried out in a linear three-site cluster 关Cu3O10 , Fig. 2共b兲兴. Comparing with the results obtained from the plaquette and previously studied binuclear clusters, it is possible to check the dependence of the NN interactions on the size of the fragment involved in the ab initio calculations. Three fillings of the valence shell have been considered in order to evaluate the dependence of these interactions on the hole concentration. Undoped 共four center/four electron and three center/three electron problems兲, one hole-doped (4c/3e and 3c/2e), and two hole-doped (4c/2eand 3c/1e) situations have been analyzed. From the systems with two holes in the valence shell, it is possible to extract the amplitude of the hole-hole repulsions, an important magnitude for the study of the hole pairing mechanism. It is worthwhile to notice that the here-referred hole dopings are not in correspondence with the total doping of the lattice, induced by the replacement of La⫹3by Sr⫹2. A change in the occupation of the valence shell of these small clusters just induces a local hole doping, which provides information about the local modifications of the effective parameters. A. Localization process In the plaquette, the four symmetry-adapted valence orbitals belong to the A1g,Eu, and B1grepresentations in the D4hgroup. The localizing unitary transformation is straightforward since a1g⫽共a⫹b⫹c⫹d兲/2, 共10兲 eu(1)⫽共a⫹b⫺c⫺d兲/2, 共11兲 eu(2)⫽共a⫺b⫺c⫹d兲/2, 共12兲 b1g⫽共a⫺b⫹c⫺d兲/2. 共13兲 Figure 3 pictures one of these four localized valence orbitals for undoped and for the doped plaquettes, showing the strong localization of the magnetic orbitals and the d⫺phybridization occurring in the hole-doped systems.32,33,44,51 FIG. 2. Representation of the clusters employed in the ab initio calculations. 共a兲the plaquette Cu4O12 ;共b兲the linear cluster Cu3O10 . The first-neighbors of the cluster atoms have been also included. They have been modeled by means of the combination of a pseudopotential and a point charge, to mimic both the exclusion and the Coulombic effects. FIG. 3. Localized active orbitals for the plaquette: 共a兲for undoped systems, centered in 3dx2⫺y2, with important tails on the four in-plane neighbor oxygen atoms; 共b兲for hole-doped systems a strong 3d⫺2prehybridization takes places, the 2pcharacter increases sustantially with respect to the undoped situation. CARMEN J. CALZADO AND JEAN-PAUL MALRIEU PHYSICAL REVIEW B 63 214520 214520-4 In the linear cluster the localizing transformation of the three magnetic orbitals is no longer imposed by the symmetry. Two of these orbitals ( ␸ g, ␸ g ⬘) belong to the Agirreducible representation and the other one ( ␸ u) to the B1usym- metry. The rotation U, which transforms the 兵 ␸ g, ␸ g ⬘, ␸ u 其 into the localized set 兵 a⬘,b⬘,c⬘ 其 , has been performed according to the Boys criterion,35 which maximizes the distance between the centroids of the orbitals. An alternative localization criterion, the minimization of the direct exchange integral Kac , leads to the same rotation. B. Magnetic interactions Table I reports the results obtained from the plaquette and the linear clusters for the magnetic coupling, involving different fillings of the valence shell. For the undoped cluster, the first-neighbor magnetic coupling JNN is ⬃125 meV. The calculated value is independent on the size of the considered cluster and it is in agreement with both the previously determined JNN in the binuclear cluster, by using ab initio configuration interaction techniques31–33,40 or the densityfunctional theory,42 and the estimations from Raman and neutron-diffraction experiments 关128⫾6 meV,1,2 134⫾5 meV,3–5 respectively兴. This independence of JNN with respect to the size of the cluster is in accord with the previous work of Illas et al.,41 performed at a level which only recovers 70% of the experimental value of JNN 共see also Ref. 39兲, due to the neglect of the polarization effects of the ligand to metal charge-transfer 共LMCT兲forms in the CI calculations. The coupling between second neighbors JNNN is also antiferromagnetic, with a value of JNNN⫽6.5 meV in accordance with the experimental upper limit 兩 JNNN 兩 ⭐9 meV.5 The ratio of the NNN and NN interactions is JNNN /JNN⫽0.052, in good agreement with the value accepted by Lorenzana et al.13 in numerical simulations, JNNN /JNN⫽0.04. A negligible antiferromagnetic coupling has been found between Cu atoms placed at a 2Rdistance 共third neighbors, next NNN兲:JnNNN⫽1 meV. Finally, the calculations on the plaquette provide a nonnegligible value for the four-spin cyclic exchange of K⫽14 meV. The four-body operator can be written as HK⫽K兺 具 ijkl 典 关共Si•Sj兲共Sk•Sl兲⫹共Si•Sl兲共Sj•Sk兲⫺共Si•Sk兲 ⫻共Sj•Sl兲兴.共14兲 This operator produces the cyclic permutation of the four spins on the plaquette 共Fig. 4兲plus ordinary two-spin exchanges of all the pairs of spins of the plaquette including those of the diagonals. Its value is somewhat smaller than some estimations used in numerical simulations,14,13,52 but larger than the critical value, (K/JNN)c⫽0.05⫾0.04, estimated by Sakai and Hasegawa15 for the appearance of a magnetization plateau at half the saturation value in the S ⫽1 2antiferromagnetic spin ladders. In spite of the smallness of the here-proposed value of the four-spin cyclic exchange, it could affect the fitting of the experimental data, as it happens in the two-leg spin ladders in the La6Ca8Cu24O41 system, where the best fit to the experimental data of the w ⫺Qdispersion is obtained when a value of the cyclic exchange about 10% of the main Cu-O-Cu linear exchange is included into the Hamiltonian.53 When a hole is introduced in the cluster, the NN magnetic coupling is influenced by its presence, but in different directions depending on the relative position of this hole 共Table I兲. Thus a hole in a colinear position to the two NN spins increases the coupling between these two spins 关Fig. 5共a兲兴. However the NN magnetic coupling diminishes if the hole is placed in a position perpendicular to the bond 关Fig. 5共b兲兴. FIG. 4. Cyclic permutation of the four spins on the plaquette, showing the action of the Koperator. TABLE I. Magnetic interactions in undoped and hole-doped clusters 共in meV兲. aA value of JNN⫽130 meV has been previously reported 共Refs. 32 and 33兲, where polarization functions have been included in the basis set of the bridge oxygen atom. bRefs. 1 and 2. cRefs. 3–5. dRef. 5. PROPOSAL OF AN EXTENDED t-JHAMILTONIANFOR... PHYSICAL REVIEW B 63 214520 214520-5 The same trend is observed in the plaquette when a second hole is introduced, JNN being 94 meV, to be compared with JNN⫽104 meV in presence of one hole and JNN⫽125 meV for undoped systems. C. Hopping integrals As in the case of magnetic interactions, the first-neighbor hopping integral (tNN) is also independent on the size of the cluster 共Table II兲, a value of ⫺558 meV has been found in the one-hole-doped linear cluster, ⫺552 meV in the onehole-doped plaquette and ⫺555 meV in a previously reported binuclear cluster.32,33 This value is in accordance with a generally accepted value of ⫺500 meV for these compounds20 共for instance, ⫺570 meV by Wang et al.,54 ⫺650 by Martin,44 ⫺400 meV by Hybertsen et al.55兲. An evaluation of second- and third-neighbor hopping integrals results from our calculations. The NNN hopping integral tNNN⫽⫹112 meV is unexpectedly large. The sign is in agreement with the negative overlap of the active orbitals placed at a 冑 2Rdistance, but its magnitude is large due to through-bond processes, which involve the oxygen atoms. On the basis of the perturbation theory 共quasidegenerated perturbation theory, QDPT兲,34,36,56 considering the neutral valence bond as the model space and H0being the trace of the Hamiltonian in the basis of the single determinants, there are two contributions to the tNNN hopping integral. One corresponds to a third-order contribution,57 scaling as ⫺tpp(tpd)2/⌬ECT 2, where tpd is the hopping integral between the O 2pand the Cu 3dorbitals, tpp is the hopping integral between the O 2porbitals, and ⌬ECT is the O 2p →Cu 3dcharge-transfer excitation energy 共Fig. 6兲. Since tpp ,tpd , and ⌬ECT are negative quantities, the third-order contribution results in a positive magnitude. There exists an alternative pathway corresponding to a fourth-order contribution, scaling as tpd 4/⌬ECT 3, with opposite sign. Since the third-order contribution is expected to be larger than the fourth-order one, the resulting sign of tNNN is positive. A moderate amplitude of the hopping integral between third neighbors (tnNNN , distance 2R) has been extracted from the calculations on the linear cluster. The extension of this coupling is controlled by through-bond interactions, but in contrast with the plaquette, the third order contribution will be negligible (tpp⬃0) and only fourth-order processes can be written here, smaller in magnitude and with an opposite sign with respect to tNNN . When an additional hole is introduced in the system, the NN hopping integral is unchanged when the hole is placed in a perpendicular position to the bond 关tNN⫽⫺558 meV in the two-hole-doped plaquette, Fig. 7共a兲兴, but its absolute value is augmented when the hole is placed in a colinear position to the bond 关tNN⫽⫺600 meV in the two-hole-doped linear cluster, Fig. 7共b兲兴. The presence of the extra hole does not influence significantly either the second-neighbor (tNNN ⫽⫹130 meV to be compared with ⫹112 meV in absence of this additional hole兲or the third-neighbor (tnNNN⫽⫺36 meV versus ⫺47 meV兲hopping integrals. D. Singlet displacement operator Additional information coming from these calculations concerns the singlet-displacement operator 共Table III兲.Itisa three-site/two-electron operator, which moves the pair of electrons, coupled in a singlet, toward a hole placed in a neighbor position. Thus a singlet on sites aand b, the site c containing a hole, is displaced to the positions band c, the hole being in a共Fig. 8兲. It can be written as FIG. 5. Relative position of two coupled electrons with respect to an adjacent hole: 共a兲colinear position; 共b兲perpendicular position. FIG. 6. Pathways showing the thirdorder and fourth-order perturbative contributions to the second-neighbor hopping integral, tNNN . On the corner, a schematic representation illustrating the hopping processes between oxygen and the metal atoms (tpd) and between two adjacent oxygen atoms (tpp), and precising the sign convention. FIG. 7. Relative position of an electron delocalized between two adjacent sites, with respect to an additional hole: 共a兲perpendicular situation; 共b兲colinear situation. CARMEN J. CALZADO AND JEAN-PAUL MALRIEU PHYSICAL REVIEW B 63 214520 214520-6 hSD abc关 兩 ab ¯ ⫺a ¯ b 典具 bc ¯ ⫺b ¯ c 兩 ⫹ 兩 bc ¯ ⫺b ¯ c 典 ⫻ 具 ab ¯ ⫺a ¯ b 兩 兴 ␦ 共na⫹nb⫹nc,2兲,共15兲 where ␦ (na⫹nb⫹nc,2) controls the fact that the three centers bear only two electrons, i.e., that the singlet can only move toward a hole. As for the preceding parameters, the amplitude hSD abc depends on the relative position of the hole. When the singlet moves to a neighbor bond in the plaquette 共i.e., the hole is placed in a perpendicular position to the singlet bond兲, the value is hSD abc⫽⫺41 meV while it goes to hSD abc⫽⫺80 meV when the hole is on the same axis as the singlet. The presence of a second hole in the plaquette does not affect the amplitude of the singlet displacement (hSD abc ⫽⫺37 meV兲. There exists also a small similar operator involving the diagonal of the square (hSD cbd⫽9 meV兲, hSD cbd关 兩 bc ¯ ⫺b ¯ c 典具 bd ¯ ⫺b ¯ d 兩 ⫹ 兩 bd ¯ ⫺b ¯ d 典具 bc ¯ ⫺b ¯ c 兩 兴 ␦ 共nb⫹nc ⫹nd,2兲,共16兲 moving the electrons as shown in Fig. 9. The value of this operator in presence of a second hole is very close to the preceding value: hSD cbd⫽14 meV. E. Hole-hole repulsions The absolute value of the hole-hole repulsion is not accessible, but the effective Hamiltonian gives the difference between two situations 共Table IV兲. From the plaquette, we can extract the relative stability of two holes placed in NNN positions with respect to two holes in NN: VNN⫺VNNN⫽Vab⫺Vac⫽0.98eV. 共17兲 This is significantly larger than the values usually accepted for simulations.27–29 One should stress the fact that this value takes into account the dynamical repolarization effects of all the atoms explicitly treated in the calculation, i.e., the screening by the 12 in-plane oxygen atoms linked to the four Cu atoms of the plaquette. It misses the polarization of the rest of the environment. This effect can be evaluated taking into account the polarization of a large surrounding shell, with a value of the polarizability of the O⫽ion, ␣ O ⫽1.30 Å 3. This value has been obtained from a series of finite-field ab initio calculations on an embedded CuO4clus- ter and it is in agreement with experimental estimates.58 The environment polarization energy corresponding to the situation represented by ␾ i,loc ⬘can be written as Epol共 ␾ i,loc ⬘兲⫽兺 Op苸cluster ⫺1 2 ␣ OFp2共i⬘兲,共18兲 TABLE II. Hopping integrals. Dependence on the presence and the position of an additional hole 共in meV兲. aRefs. 32 and 33. TABLE III. Singlet-displacement operator 共in meV兲. Effect of the presence of a second hole in the neighborhood of the singlet. PROPOSAL OF AN EXTENDED t-JHAMILTONIANFOR... PHYSICAL REVIEW B 63 214520 214520-7 where F ជ p(i⬘) is the electric field created by the holes in ␾ i,loc ⬘on the position of atom p. Including the polarization effects for the two situations 共two holes in NN and in NNN positions兲results in a decrease of the hole-hole repulsion difference to VNN⫺VNNN⫽0.80 eV which remains a rather large value. From the linear cluster, it is possible to estimate the energy gain obtained when placing two holes in nNNN positions 共distance 2R) instead of adjacent positions 共distance R兲:VNN⫺VnNNN⫽Vab⫺Vac⫽1.77 eV, with a final value of VNN⫺VnNNN⫽1.47 eV once the environment polarization effects have been taken into account. These values of differences between hole-hole repulsions may seem very large and most of the calculations introducing this repulsion in a t-J-Vmodel usually take smaller values (J⬍V⬍4J)27–29 when looking for the hole-pairing mechanism. The above calculated values are smaller than the corresponding electrostatic quantities calculated in a pointcharge approximation: VNN⫺VNNN⫽1.12 eV and VNN ⫺VnNNN⫽1.90 eV. The delocalization of the holes on the oxygen atoms should result in larger repulsions, especially for VNN since the two holes share an oxygen atom, and our values, which exhibit a significant screening, do not seem unrealistic. IV. INTERPRETATION OF THE HOLE DEPENDENCE OF THE ONE BOND JNN AND tNN PARAMETERS The two preceding sections indicate that the presence of holes in the immediate vicinity of a bond may affect the values of the spin coupling and of the hopping integral on this bond, and that this effect is anisotropic, i.e., does not only depend on the minimal distance of the hole to the atoms of the bond. Hence it seems important to rationalize these effects if possible. A. Rationalization from the two-band model 1. Hopping integral The rationalization of this anisotropy is possible in terms of the two-band model. The thopping integral results from a second-order effect 共Fig. 10兲,tscaling as t⬃tpd 2 ⌬ECT ,共19兲 where ⌬ECT is the 2pOto3dCu charge-transfer excitation energy, that is, the energy difference between the charge transfer and the model space states. In presence of an additional hole, the energies of the model space determinants ␾ 1 and ␾ 2and of the intermediate charge-transfer state ␾ CT will be modified. The model space determinant energies are no longer degenerate and one shall take their mean energy as the zero-order energy. Let consider now the effect on the bond directed along xof an adjacent hole placed either on the y direction, as occurs in the plaquette, or on the xaxis, as in the linear cluster 关Figs. 7共a兲and 7共b兲, respectively兴.Inthe situation of Fig. 7共a兲, the change of the zero-order mean electrostatic energy due to the hole is ␦ E0⫽( 冑 2⫹1)/2 冑 2R ⫽0.85/R. In the corresponding charge-transfer state the mean electrostatic energy with the hole is ␦ ECT⫽2/ 冑 5R ⫽0.89/R, i.e., the excitation energy ⌬ECT ⬘is increased by a small quantity, ⌬ECT ⬘⫽⌬ECT⫹ ␦ ECT⫺ ␦ E0⫽⌬ECT⫹0.04 R,共20兲 which should diminish slightly the absolute value of the hopping integral. In contrast, for the situation of Fig. 7共b兲, ␦ E0 ⫽3/4Rand ␦ ECT⫽2/3Rhence ⌬ECT ⬘⫽⌬ECT⫺0.08/R which should increase the absolute value of t, in agreement with our calculation. This analysis is quite rudimentary, it neglects the possible effects of the polarization of the orbitals on the tpd integrals and rests on a crude evaluation of the electrostatic effects on the denominators 共holes considered as centered on Cu sites, neglecting their oxygen character兲, but it seems to agree with the computed trends. 2. Magnetic coupling The same kind of analysis may be attempted for the JNN coupling constant. In the two-band model, the spin exchange between Cuaand Cubresults from an interaction through the bridge oxygen atoms. In a first step an electron is transferred between the oxygen atom and one of the Cu atoms. In Fig. 11 the open circle represents a bridge oxygen atom and closed circles are the copper atoms as in Fig. 10. The top pathway corresponds to an electron transfer between the oxygen and the Cuaatom. On the bottom pathway, the oxygen electron jumps in the opposite direction, toward the Cub atom. In absence of a hole in the neighborhood, both situations have the same energy: ⌬ECT1⫽⌬ECT2⫽⌬ECT . In the next step, an electron is transferred from the Cu atom (bin the top pathway, ain the bottom兲to the oxygen atom. Notice that the resulting determinants are the ionic forms 兩 aa ¯ 兩 and 兩 bb ¯ 兩 , which energies U1and U2correspond- ing to the Coulombic repulsion of two electrons placed in the same 3dorbital are equal in the undoped system, U1⫽U2 ⫽U. Following this perturbative development, the magnetic coupling is a fourth-order quantity: JNN 2⫽ ⫺2tpd 4 U共⌬ECT兲2.共21兲 Adding an external hole will not modify the electrostatic zero-order energy, but will affect the energies of all intermediate states, so that the coupling becomes JNN 2⫽ ⫺tpd 4 ⌬ECT1 2•1 U1 ⫹ ⫺tpd 4 ⌬ECT2 2•1 U2,共22兲 where the first term corresponds to the top pathway and the second term to the bottom one. Notice that while ⌬ECT1 ⫽⌬ECT⫹ ␦ ECT1and ⌬ECT2⫽⌬ECT⫹ ␦ ECT2have no reason to be related, U1⫽U⫹ ␦ Uand U2⫽U⫺ ␦ Uwhatever the outer charge distribution. The effective coupling is therefore CARMEN J. CALZADO AND JEAN-PAUL MALRIEU PHYSICAL REVIEW B 63 214520 214520-8 JNN 2⫽⫺tpd 4 冉 1 共⌬ECT⫹ ␦ ECT1兲2•1 U⫹ ␦ U ⫹1 共⌬ECT⫹ ␦ ECT2兲2•1 U⫺ ␦ U 冊 .共23兲 To the first order in ␦ U/Uand ␦ ECT /⌬ECT developments, one gets JNN 2⫽ ⫺2tpd 4 U共⌬ECT兲2 冉 1⫺ ␦ ECT1 ⌬ECT ⫺ ␦ ECT2 ⌬ECT ⫺ ␦ U U⫹ ␦ U U 冊 ⫹O共2兲.共24兲 The effect of the additional hole共s兲will go through their electrostatic interaction in the charge-transfer states. For the perpendicular situation 共Fig. 12兲 ␦ ECT1⫽0.187/Rand ␦ ECT2⫽⫺0.105/R. Hence, ␦ ECT1⫹ ␦ ECT2⫽0.082/Rand JNN is diminished by the presence of the adjacent hole in the perpendicular direction, as observed in the corresponding ab initio calculations (⌬JNN /JNN⫽⫺17%). Oppositely, in the linear situation the quantities are: ␦ ECT1⫽1/6Rand ␦ ECT2⫽⫺1/3R,so ␦ ECT1⫹ ␦ ECT2 ⫽⫺1/6Ris a negative quantity, hence the coupling constant should be increased. This is actually observed since ⌬JNN /JNN⫽⫹25%. The variation of JNN in the linear model is of the right sign and its amplitude is larger than for the perpendicular orientation. 3. Singlet-displacement operator The same kind of rationalization applies to the twoelectron/three-center operator hSD abc . In the one-band model as shown in Fig. 13, where only the Cu atoms have been included, the electron placed in site ais transferred to site b, followed by a second transfer toward c. This effect scales as tNN 2/U⬘, where U⬘⫽U⫺2Vab ⫹Vac ,Vij being the electrostatic interaction between the charges 共with respect to the basic undoped distribution兲. The denominator will be smaller for the linear configuration than for the perpendicular one since Vac is larger (1/ 冑 2Rinstead of 1/2R). This may explain the difference between the amplitudes of hSD abc found in the plaquette and in the linear cluster 共Table III兲. The more realistic two-band model introduces an alternative mechanism, since there exist two pathways to go from 兩 ab ¯ 兩 to 兩 bc ¯ 兩 , both involving electron-transfer processes between the Cu and oxygen atoms, 共cf. Fig. 14, where sites I and II correspond to oxygen atoms兲. The bottom pathway corresponds to the process appearing in the one-band model, which is not the case for the first one. Explicitly, the excitation energies are ⌬E1⫽Ed⫺Ep⫺Up⫽⌬ECT⫺Ud⫹2 R,共25兲 ⌬E2⫽2Ed⫺2Ep⫹Up⫺2Up⫺4 R⫹VI,II ⫽2⌬ECT⫺Ud⫹VI,II ,共26兲 ⌬E3⫽Ed⫺Ep⫺Up⫹Ud⫺3 R⫹Va,II⫽⌬ECT⫺1 R⫹Va,II , 共27兲 ⌬E1 ⬘⫽⌬E3,共28兲 ⌬E2 ⬘⫽Ud⫺2 R⫹Vac .共29兲 It is likely that ⌬E2 ⬘⬎⌬E2, i.e., that the bottom pathway has a larger contribution to the singlet displacement. The overall effect is hSD abc⫽tpd 4 ⌬E1 ⬘ 冉 1 ⌬E1•1 ⌬E2 ⫹1 ⌬E1 ⬘•1 ⌬E2 ⬘ 冊 ,共30兲 i.e., of the same order of magnitude as JNN . Regarding the orientation effect it is clear that the denominators are larger FIG. 8. Schematic representation of the singlet-displacement operator. The local singlet ab ¯ is displaced to bc ¯ , the hole moving from csite to asite. FIG. 9. The singlet-displacement operator acting over the diagonal of the plaquette. FIG. 10. Two-band model hole hopping mechanism, involving an intermediate charge-transfer state. Open circle corresponds to a bridge oxygen atom, and the closed circles to the copper atoms. TABLE IV. Hole-hole repulsions 共in eV兲. Values in parentheses take into account the polarizability of the outer environment PROPOSAL OF AN EXTENDED t-JHAMILTONIANFOR... PHYSICAL REVIEW B 63 214520 214520-9