Charge fluctuations in high-tc superconducting cuprates: A reformulation of the “symmetry dilemma” of the mean-field approximation

Chemical Physics 135 (1989) 27-36 North-Holland, Amsterdam

CHARGE FLUCTUATIONS IN HIGH-I’, SUPERCONDUCTING CUPRATES: A REFORMULATION OF THE “SYMMETRY DILEMMA” OF THE MEAN-FIELD APPROXIMATION Michael C. BOHM Institut ftir Physikalische

Chemie, Physikalische

Chemie III, Technische Hochschule

Darmstadt. D-6100 Darmstadt,

FRG

Gerd BUBECK and Andrzej M. OLES ’ Max-Planck-Institutflr

Festkijrperforschung,

D-7000 Stuttgart 80, FRG

Received 15 February 1989

We investigate the strength and symmetry-dependence of the mean-square deviation of the electronic charge in a CulzO~~cluster which has been used as a model for the CuOz plane of the recently discovered high-T, superconducting copper oxides. The charge fluctuations are derived in the mean-field ( 1hscF)) and the correlated ground state ( 1ty,,)), respectively. The interatomic correlations in the cuprate cluster are described by modifying 1q&) obtained in a semi-empirical SCF formalism via the “local approach”. A comparison of the mean-field data and the correlated ones leads to a physically transparent alternative formulation of the “symmetry dilemma” of the restricted Hartree-Fock approximation. The mean-field errors in the simulation of the charge fluctuations are largest in the limit of highly symmetric wavefunctions. Electronic correlations lead here to a sizeable charge localization with the largest effect in the Cu 3d,~_,~ orbitals. Our results show that the high-Tc superconducting oxides belong to the class of strongly correlated systems with fluctuating valences. They are furthermore characterized by a strong coupling of local phonons (breathing mode) to the redistribution of the hole density and to changes in electron correlations.

1. Introduction The so-called “symmetry dilemma” of the restricted Hat-tree-Fock (HF) wavefunction has been formulated many years ago by Liiwdin who has shown that the energetically best HF wavefunction does not necessarily transform according to constants of motion and particularly to the symmetry group of the corresponding Hamiltonian [ I]. This leads to the situation that a large part of the correlation energy depends apparently on the symmetry constraints of the electronic wavefunction. I.e. there exists a clear relation between the strength of electronic correlations in a fermion system and the tendency to minimize the mean-field energy due to symmetry-breaking single-particle excitations. A quantitative approach to estimate the stability of the symmetry adapted HF wavefunction has been derived by Thou’ Permanent address: Institute of Physics, Jagellonian University, PL-30-059 Kralcbw, Poland.

less in the early sixties [ 2,3 1. The corresponding HF (in)stability conditions are given by characteristic eigenvalue problems. In recent years these instability conditions have been employed by several authors to emphasize the importance of many-body effects in electronic systems [ 4-81. But investigations based on an analysis of total energies or roots of an instability problem have however a common shortcoming. A transparent demonstration of the physical limitations of the mean-field approximation following from the improper description of the mean-square deviations of the electronic charge around the mean-values ffi is not possible by means of the aforementioned theoretical schemes. This shortcoming may be avoided by a physically improved treatment of the charge fluctuations in the correlated ground state. In the present contribution we suggest an alternative approach to visualize the connection between the magnitude of electronic correlations and the spatial symmetry of the corresponding wavefunction. For this purpose we analyze correlation-strength param-

0301-0104/89/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

28

MC. BBhm et al. / Charge.fluctuations in high-T; superconducting cuprates

eters di and Ziprespectively, in a system with stronger correlated valence electrons as a function of a spatial displacement coordinate. The di and ,Zi measure the difference in the charge fluctuations in the correlated ground state 1v/~) and the one which follows from the SCF approximation I kF). In previous work [ 911 ] we have used such electron-correlation parameters to classify the many-particle character of chemical bonding. On the basis of the material collected in refs. [ 9, lo] we have shown that the different bonds can be divided into characteristic subclasses according to their deviations from the independent-particle limit. In the following we will demonstrate that the magnitude of the correlation-strength parameters is reduced with an increasing perturbation of. the symmetry in (rye) . Then we adopt the electron-correlation parameter di defined for a given orbital i to calculate the probabilities P, (n ) of finding n = 0, 1, 2 electrons in this orbital. By means of the P,(n), e.g., it is possible to analyze the formation of magnetic local moments in the considered region i due to correlations. In this way one can relate the properties of the correlated ground state 1v/~) to experimental observables. The method of the “local approach” (IA) [ 12141 has been used to derive the electron-correlation parameters di and Xi. For the SCF part we have adopted a simple model-Hamiltonian of the INDO (intermediate neglect of differential overlap) type [ 151. Previously we have shown [ 10,13,14,16,17] that numerical data of sufficient accuracy can be derived by combining the LA with the SCF operator of ref. [15]. To demonstrate the numerical connection between the importance of electronic correlations and the spatial symmetry of the wavefunction we have studied a CulzOt$- cluster (see fig. 1) as a function of spatial distortions in the central part. The corresponding cluster unit should be a reliable model for the electronic states of a CuOZ plane, a common structural unit of high-T, copperoxide superconduo tors. The charge on the cluster has been fixed to simulate the electronic structure in “pure” systems of formal stoichiometries LqCuO, or YBazCu30a, having excess holes in the CuOZ planes; Detailed experimental investigations have shown that electronic correlations are sizeable in these materials and that their

Fig. 1. Schematic display of the CulzO~~- cluster. The transition-metal atoms are labeled by full circles, the oxygen sites by empty ones. We have schematized an antibonding combination of Cu 3d,2_,2 and 0 2p,/2p,. The variation of the CuO bondlengths is indicated by arrows. The correlation approach has been restricted to the shaded area in the center of the model-cluster.

high T,-values cannot be explained in terms of the conventional pairing theory restricted to lattice degrees of freedom [ 18 1. The importance of electronic correlations in the high-T, superconductors has been deduced from the antiferromagnetic long-range order in LaZCu04 [ 19 ] and YBa2Cu306 [ 201, from the satellite structure in the corresponding photoelectron spectra [ 2 l-231 and the nonmetallic behaviour of, e.g., IqCuO, [ 241. Previous investigations of finite and infinite models of high-T, copper oxides have shown that these cuprates are characterized by nearly half-filled antibonding states which result from Cu 3d,2_,2 and 0 2pJ2p, orbitals in the CuOZ planes [ 25 1. Thus, they can be modelled via two-dimensional networks of Cu and 0 atoms. A typical arrangement is shown in fig. 1. By using such structural models, the electronic correlations in these systems have been studied in previous work [ 11,26-281. In agreement with experimental observations, the latter contributions have shown that the many-particle interactions are particularly strong in the Cu 3d,2_+ orbitals. In fig. 1 we have symbolized the considered displacement in the cluster geometry of Cu,,Of?. In this way we simulate a breathing mode due to the oxygen atoms transforming according to e, symmetry (in D4,,). For our model calculations we have as-

MC. Biihm et al. / Chargejluctuations

sumed a CuO equilibrium bondlength of 189.4 pm as found in the X, y plane of tetragonal LaZCu04 [ 29 1. The transition-metal atom in this phase is located in the center of a distorted octahedron. The CuO separations in the direction of the Cartesian z axis amount to 242.8 pm and are large in comparison to the CuO distances in the x, y plane. In the subsequent numerical simulations we have studied small changes in the CuO equilibrium bondlength in the x, y plane up to 2.5%. The organization of the present contribution is as follows. In the next section we summarize the theoretical background of the LA and the determination of the correlation-strength parameters di and & In the present work we have employed a definition for the electron-correlation parameters, which differs partially from our previous approaches [9-l 1] ; see below. The suggested many-particle indices have however the advantage that also stronger correlations can be studied with reliable accuracy. In section 3 we discuss the numerical results for the Cu,,O]?cluster. We have generated wavefunctions 1yo) of reduced symmetry via the geometric deformations schematized in fg 1 and not via “instabilities” of the symmetry-adapted wavefiiction in a lattice of higher symmetry. But our general findings (i.e. relation between the correlation strength and symmetry of the wavefimction) are independent of the microscopic mechanism leading to a wavefunction of reduced symmetry; for some related discussions, see refs. [ 30,3 11. Final remarks and conclusions are given in section 4.

2. Thearetlcal basis and numerical framework In several previous contributions [9-l 1,13, 14,16,17 ] we have combined the LA with the all-valence Hamiltonian of ref. [ 15 1. The theoretical background of the LA to treat the correlations has been documented in detail in these investigations. Therefore only the outline of this method is given in the present work. The adaptation of a semi-empirical allvalence operator restricts the determination of the electronic correlations to the interatomic contributions [ 13,14,16,17 1. The latter have been calculated for the half-tilled antibonding Cu 3dX2_+ and 0 2pJ 2p, states. But this limitation in the active space is

29

in high-T, superconducting cuprates

without significance for the scope of the present work which is confined to an analysis of the correlation strength in the valence shell. The correlated ground state 1yo) in the LA is expressed as follows [ 12 ] IVo>=exP( - 5

tlijOij)l@scF)

3

(1)

where the ?,Qare variational parameters found by minimization of the energy. It results from I &--) by its modification due to local projection operators 0, which allow for a reduction of charge fluctuations in localized bonds or orbitals (i=j) and between localized regions (i #j) due to virtual two-particle excitations. The corresponding projection operators are given as (2)

ah2’=

C nbn,,, Q,d

i#j.

(3)

They are used to define the operators 0, as their irreducible parts which go beyond the one-particle operators as obtained from the a,, i.e. Oij=Oij-[Oij]SCF*

For (nit)

(4)

instance, [nit nil ] scF= ni, ( t~i,) + ( n, ) nji ( tZ, ) , where the symbol ( ... ) stands for

(&cFI...]@sSCF). In eqs. (2) and (3) we have employed density operators ni= C,,n, associated to the electron densities in the considered Cu 3dX2_y2and 0 2p,/2p, orbitals which contribute to the half-filled antibonding band in the Cu02 plane. The corresponding local creation and destruction operators are abbreviated by bz and bio, respectively. The nti are tht~given by nin=b&b,. In section 1 we have already mentioned that charge

fluctuations are not properly described in the inde pendent-electron approximation. It is thus convenient to define an electron-correlation strength parameter di by the difference between the mean-square deviation of the electronic charge in I vo) and I q&,--) , (5)

A4.C. Bdhm et al. /Charge fluctuations in high-T, superconducting cuprates

30

In eq. ( 5 ) we have adopted a hypothetical SCF state )q& ) where the one-particle densities are the same as those as in the correlated ground state, (hGIniI&F)=
*

(6)

The changes in the density distribution which arise due to electron correlations are described by operators 0::) = ,2:oni,,. The definition of a reference state I q&z ) has the conceptual advantage that identical particle numbers, i.e. densities, are taken into account in the evaluation of the mean-square deviation of the electronic charge in the correlated ground state I vo) and the mean-field one, (q&). Previously we have shown that electronic redistributions are of particular importance in heteropolar bonds with stronger correlated electrons [ 14 1. In refs. [ 9,101 we have adopted a linearized approximation for the correlation-strength parameter A, which is valid for not too strongly correlated electrons. For stronger correlations one has to use a variational formula which results from expanding the wavefunction I tyo) up to first order in the variational parameters vi, both in the numerator and denominator of the mean-square deviation of the electronic charge in I ly,). This leads to the expression for A,, Ai=Ci

[

(

(An:)= -I

x 1+ m.k,?mn~kl

>I

.

(7)

(An;), stands for the charge fluctuations in the hypothetical SCF state I e ) whose density has been derived by making use of eq. (6). Our choice of the reference state IeF ) gives a particularly simple form of the numerator and allows to compare charge fluctuations in states of the same density. In our recent analysis of electronic correlations in high-Z’=derivatives the identity (6) has not been taken into ao count [ 11,26 1. The corresponding Ai and Ci elements should be accepted as an upper limit. The numerator of eq. (7) is given by

(8)

This expression contains only elements of the type nitnil. Physically they express the suppression of charge fluctuations in the correlated ground state. The correlation-strength parameter Ai can take values between Ai= 0, which indicates the strict validity of the independent-particle picture, and Ai=A? in the limit of completely correlated electrons. The upper boundary AY is determined by the expectation value of the number operator Ai. One finds Af”““=A,/(2-IT,;) =(2-ri,)/fi,

forO
,

(9)

The two expressions in eq. (9) are related by the particle-hole symmetry of the electron correlations around the occupation of tiz= 1. The many-particle indices A, have obviously the conceptual disadvantage that they depend on the actual densities fi, in the considered localized bonds or orbitals. An “invariant” correlation-strength parameter Zi is feasible via the definition c =A,/Ay

,

(10)

where we have eliminated the fi, dependence of the Ai. The two boundaries of the invariant many-parti-

cle index are Zi = 0 and 2YF = 1. The former value indicates again the validity of an independent-electron description and ,Zy = 1 stands for the maximum possible correlations. By means of the electron-correlation parameter A, and the mean-values of the density H-,it is straightforward to calculate the probabilities Pj( n ) of finding n=O, 1, 2 electrons in the ith spatial region. For the P, (n ) we have the standard relations ;

P,(n)=1 ,

(11)

“=I

Eq. ( 11) corresponds just to the normalization condition, while eqs. ( 12) and ( 13) define the average number and number square of the valence-electron density in the ith orbital. The correlation parameter Aj allows for an explicit calculation OfPi( 1) and Pi( 2 ) ,

MC. Biihm et al. / Charge fluctuations in high-T, superconducting cuprates

Pj(l)=Ai(l--ii,/2)(1+di) P,(2)=(~i/4)[~ji(2-ffi)di]

9

(14) *

(15)

Of course, the expressions ( 14) and ( 15 ) give also the probabilities of single and double occupancy in the ith region as realized in the mean-field approximation if di = 0 is used. The increasing Pi( 1) describe obviously the formation of magnetic local moments in the correlated ground state ) yo) . The latter equations are the prerequisite to relate a theoretical manyparticle index to measurable expectation values. The enhancement of the local moment L f as a result of electronic relations is defined by a similar expression toeq. (5), (16)

where S, denotes the spin operator for an electron in the ith region. At the end of this section we give a short descrip tion of the computational framework for the described below cluster calculations. The original SCF formalism [ 151 has been designed in terms of screened interaction potentials. For the calculation of the electronic correlations we have to replace the dressed two-electron part by a bare interaction. The Cu 3d on-site repulsion used in the subsequent model calculations amounts to 27.8 eV corresponding to 85% of the ab initio value. This setup should simulate the screening resulting from Rydberg transitions and core-valence excitations which are beyond the active space of our simple all-valence Hamiltonian. To restore the charge-neutrality of the copperoxide cluster we have adopted a plain point-charge approximation. This allows for the simulation of the surrounding La atoms in, e.g., La2Cu04, one prototype in the class of the high-T, superconducting oxides. For previous descriptions of point-charge models in combination with the SCF Hamiltonian of ref. [ 15 1, see refs. [ 32,331. To attenuate surface effects in the calculations of the Cu lZOly- cluster we have limited the many-body part to the shaded interval symbolized in fig. 1. In this domain we have taken into account all operators of the O&’) type for the half-filed antibonding states built by Cu 3d,2_,2 and 0 2p,/2p,. As interorbital operators O,C:)we have considered all coupling ele-

31

ments between adjacent bonds. Taking into account the local nature of the electronic correlations, we believe that the adopted setup renders possible a realistic simulation of interatomic correlation effects in high-T, copper oxides in the solid state.

3. Results and discussion We have restricted the subsequent discussion to the strongly correlated Cu 3dXt_,z electrons in the halffilled antibonding Cu 3d,2_,2-0 2pX/2p, band. The weaker correlation effects in the oxygen sublattice are not important for the scope of the present work. The computational results for Cur2018- are summarized in table 1. In detail we give the correlation-strength parameters Bi and Zi ( i = 3d,2 _,,2) for the two copper atoms in the center of the adopted cluster together with the one-particle density tii in the corresponding orbital as a function of the displacement coordinate ArcUo (in per cent ). Additionally we summarize the probabilities Pi(n) of double, single and empty occupation in the Cu 3d,2_,2 orbital, corresponding to Cu configurations of 3d”, 3d9 and 3d*, respectively. The P,( n ) have been calculated for the correlated ground state 1tyo) and the mean-field wavefunction 1q&--). A schematic display of the Pi(n) of CU 3dXa_yzas a function of the displacement coordinate ArcUois shown in fig. 2. At the bottom of table 1 we have collected electron-correlation parameters di,,, and Z;,a, (i= 3dX2_Y2)averaged over the two central Cu sites of Cu120]y- together with the corresponding mean value of the density fii,av.The latter data indicate immediately that the net correlation strength in the Cu 3dX2_,2orbitals is reduced with increasing perturbation of the mirror-symmetry in the copper oxide cluster. For the symmetric arrangement an electron-correlation parameter Z, of 0.68 is found, indicating the sizeable correlations in the Cu 3d sublattice. This result is in good agreement with the data obtained by using a tight-binding model Hamiltonian and speo troscopic values of the interaction parameters [ 28 1. A mutual variation of 1%in the CuO bondlength leads already to a Ci reduction from 0.68 to 0.635. For mutual CuO displacements of 2% or 2.5% further reduced Ci,,v values of 0.59 and 0.5 5 are observed. It is seen that the averaged density iii,aVin Cu 3d,2_,2 is

32

M.C. Bijhm et al. / Chargefluctuations in high-T, superconducting cuprates

Table 1 Electron-correlation parameters A,, Zl and the electron densities fi, for the Cu 3d,2_,* orbitals in Cu,,O~$‘- as well as the probabilities P, ( n ) as a function of the spatial deformations. The two Cu atoms are indicated in fig. 1. On the bottom we have given the mean values of A,, A, and Z, when averaged over the two copper sites

‘2

1.38 0.29 0.68

3d8 3d9 3d”

P,(O) P,(l) P,(2)

0.04 0.55 0.4 1

0.05 0.58 0.37

0.05 0.60 0.35

0.07 0.61 0.32

0.08 0.62 0.30

3d8 3d9 3d”’

P,(O) Pi(l) Pt(2)

0.10 0.43 0.47

0.11 0.45 0.44

0.13 0.46 0.41

0.13 0.47 0.40

0.15 0.48 0.37

Cu(2)

fi, A, z

1.38 0.29 0.68

1.43 0.27 0.69

1.49 0.24 0.70

1.54 0.21 0.71

1.60 0.18 0.71

3d8 3d9 3d”

Pi(O) P,(l) P,(2)

0.04 0.55 0.41

0.03 0.52 0.45

0.02 0.47 0.51

0.01 0.43 0.56

0.01 0.38 0.61

3d8 3d9 3d”

P,(O) P,(l) P,(2)

0.10 0.43 0.47

0.08 0.40 0.52

0.07 0.37 0.56

0.06 0.35 0.59

0.04 0.32 0.64

%a”

1.38 0.29 0.68

1.38 0.290 0.655

1.39 0.275 0.635

1.40 0.260 0.615

1.41 0.245 0.590

Cu(1)

A,

4

AIS” z I,B”

not substantially changed due to the deformation in the CulzO~?- lattice, i.e. a geometry dependent variation of the metal-l&and charge transfer is not observed. However, the individual densities on both atoms Cu ( 1) and Cu ( 2 ) change quite considerably. Table 1 is clear evidence for the intimate connection between the importance of electronic correlations and the spatial symmetry of the corresponding wavefunction. The correlations are largest in the case of the symmetric arrangement and thus also for the symmetry-adapted wavefunction. This result has been corroborated in a larger number of model calculations of different systems. We feel strongly that the present data offer a physically intelligible access to visualize the “symmetry dilemma” of the restricted mean-field wavefunction. In addition to this formal aspect of electronic-structure calculations our analysis allows to quantify changes in expectation values which are the result of electronic correlations.

1.33 0.31 0.62

1.29 0.31 0.57

1.26 0.31 0.52

1.22 0.31 0.48

But at next let us explain the theoretical source leading to the Zi,,y reduction with increasing perturbation in the symmetry of the electronic wavefunction. Table 1 shows that the net reduction Of Ci,,, follows from the dominating modification of the charge fluctuations in the compressed CuO., unit. The respective reduction of Z, is not compensated by an enhancement of the electron-correlation parameter at the second Cu atom. In fact, almost a saturation in the latter C, value is found in the considered ArcUo interval. The computational results given in table 1 and fig. 2 show the sizeable influence of the electronic correlations in the Cu 3d,2_,2 orbital on the respective probability distribution P,( n ) . One important consequence of the many-particle interactions in CuIZO!y- is the suppression of the 3dS [email protected]. For the symmetric structure of the cluster we observe P,(0)= 0.04 in 1%). This is in agreement

33

M.C. Biihm et al. /Charge fluctuations in high-T, superconductingcuprates

I P,kd

1%

2% 3% -Arr,cI%l

2%

1% -

3% ArcUcWI

1%

2% -

3% Arr,c[%l

Fig. 2. Variation of the probabilities P,( n) of double, single and empty occupation in the Cu 3dxz_-yzorbitals as a function of the displacement coordinate btio. The curves associated to the correlated ground state 1y,,) are symbolized by full lines, the mean-field values of the P,(n) are labeled by broken curves. On the left-hand side we have displayed the P,(n) data for the elongated Cu04 unit and in the middle the curves for the compressed copper oxygen fragment. On the extreme right the superposition of both diagrams is given. For the evaluation of the curves nine datapoints have been adopted. The considered Arc.ovaluesamount toO.O, 0.125,0.25,0.50,0.75, 1.0, 1.5, 2.0 and 2.5%, respectively.

with experimental measurements indicating that the amount of 3ds in, e.g., LazCu04 does not exceed 5% in the electronic ground state and agrees perfectly with previous results of the tight-binding model [ 28 1. The experimental estimate is based on ESR, XPS and XANES (X-ray absorption near-edge structure) data [21-23,34,35]. The 3d8 suppression in the copperoxide cluster is accompanied by a reduction of the 3d” configuration. In 1q&) a difference of 0.04 between Pi( 1) andP,(2)withPi(2)>Pi(l) isobservedatAr,,o=O. In 1yo) the latter sequence is changed; Pi( 1) exceeds here P, (2 ) by 0.14 indicating the formation of magnetic local moments due to correlations. The nonintegral one-particle density in 3d+-,z prevents the complete localization of the electrons also in the presence of stronger correlations. Valence fluctuations between 3d9 and 3d” are still possible. This indicates that the high-T, copper oxides belong to the class of correlated valence fluctuating systems, as already pointed out in several previous studies [ 27,36,37]. In the following we analyze the Arc,, dependent variations in the mean-square deviations of the electronic charge confmed to 3d,2_,2 of Cu,,O$ . For

this purpose we adopt the diagram on the extreme right in fg 2. Here we have given the averaged probabilities Pi(m) of 3dx2_Y2as observed in Leo) and I&--) . The maximum charge localization in 1vo) is predicted for the symmetric structure. Increasing values of ArcUo lead to a reduction of Pi( 1) in the correlated ground state. The fluctuations between 3d9 and 3d’O are enlarged correspondingly. A comparison of the averaged Pi( II) curves obtained with I yo) and ]hcF) shows in a simple way the aforementioned “symmetry dilemma” of the restricted HF wavefunction. The overestimation of the charge fluctuations in the mean-field description is largest for the symmetric wavefunction ( Arcllo = 0 ) . The meansquare deviations in the electronic charge are then attenuated with increasing perturbations of the cluster symmetry. Fig. 2 indicates that mainly the relative weights of 3d9 and 3d” are influenced by the value of Artio. The magnitude of P, ( 0 ) is only a slowly increasing function of ArcUo. To summarize, in the mean-field description of Cu,,O]?- unphysically large charge fluctuations are predicted for the symmetric wavefunction. In the correlated ground state the fluctuations between 3d9 and 3d’Oare strongly reduced due to correlations. This

34

MC. Bdhm et al. / Charge fluctuations in high-T, superconducting cuprates

reduction is largest for the symmetric cluster arrangement, so these fluctuations appear to be enlarged with increasing violations in the symmetry of the charge distribution. Especially the strong mean-field fluctuations encountered at Arcuo = 0 can be considered as a reference to possible HF instabilities of the symmetry-adapted SCF wavefunction. By means of a symmetry-breaking single-particle excitation an electronic configuration can be formed where the fluctuations in the mean-field picture are reduced. Inspection of the two remaining plots in fig. 2 shows that the symmetry-dependent HF errors are largest for the probability functions P, ( 1) and P,(0)in the elongated Cu04 unit. The gradients of the Pi(n) curves associated to ) yo) and ]&cF) are roughly comparable in the compressed Cu04 fragment. The aforementioned splitting between the P,(n)curves associated to 1yo) and l&c.) is here Arc,, independent. Up to now we have used the Cui20t?- system prevailingly as a general model to present an alternative formulation for the “symmetry dilemma” of the restricted HF wavefunction. In detail we have demonstrated that the mean-field ability to reproduce the charge fluctuations around the corresponding meanvalue depends on symmetry-constraints in I hF). At the end of this section we show that some important electronic-structure properties of the high-T, superconducting cuprates can be reproduced with reliable accuracy by the presently adopted Hamiltonian. The calculated one-particle density ?ii= 1.38 in the Cu 3dX2_+ orbital is close to populations derived from photoemission measurements. For La2Cu04 3dX2_,2 densities between 1.27 and 1.40 have been deduced from the corresponding spectra [ 24,26 1. We have already mentioned that electronic correlations lead to the formation of magnetic local moments in the Cu sublattice of the high-T, cuprates. If we define a magnetic moment which can be reached in a given state as m = 2Sefl, where S,, is the effective spin responsible for the local moment in Iyo), we find with the Zj values of 0.68 (symmetric structure) or 0.48 (0.71) attheatomsCu(1) andCu(2)withthe 2% displacement in the CuO bondlength m = 0.78 PB or 0.76 (0.63) &$, respectively, in the NCel state. Taking into account the presence of quantum fluctuations which reduce the moment from the value in the NCel state, these theoretical estimates are in suf-

ficient agreement with experimental data of (O.SO?O.lO) ,& in LazCuO, [ 19,381 or 0.64 pa to 0.66 pugin YBazCu30b [ 20,391, respectively.

4. Conclusions In previous contributions we have quantified the many-particle nature of chemical bonding by comparing the mean-square deviations of the electronic charge in the mean-field approximation and the correlated ground state. This work allowed for the formulation of some general order principles in molecules and solids beyond the frequently aspired collection of one-electron expectation values. Electronic-structure investigations in quantum chemistry are conventionally restricted to such quantities. This limitation is still conserved in computational studies where also the electronic correlations are taken into account. In the present work we have used the meansquare deviations of the electronic charge to derive an alternative formulation of the “symmetry dilemma” of the restricted Hartree-Fock approximation. We believe, that this proceeding is of some physical transparence. The symmetry constraints encountered in the independent-particle approximation are emphasized by comparing the strength of valence fluctuations in I yo) and I &--) , respectively, as a function of the symmetry of the electronic wavefunction. As a model system we have adopted a CulzO~!cluster which should allow for a reliable description of the high-T, superconducting copper oxides. Detailed experimental investigations have shown that electronic correlations are sizeable in these materials and should be taken into account in models intended to explain their superconducting properties. We have confirmed this point of view and demonstrated that the stronger correlations in Cu 3d,2_,2 are a sensitive function of the actual symmetry in the corresponding wavefunction. The fluctuations between the Cu 3d9 and 3d” configurations depend on the actual symmetry/geometry in the planar Cu02 network. Their proper description is only possible via the correlated ground state wavefunction. The mean-field errors are enhanced with increasing symmetry of the cluster. The strong charge fluctuations (mixed valences) in

MC. Biihm et al. /Charge fluctuations in high-T, superconductingcuprates

the limit of sizeable correlations are the result of an electrondensity & > 1 which prevents the entire suppression in the mean-square deviations of the valence electrons. The present computational results support the suggestion based on previous investigations of high-Tc cuprates that the motion of the oxygen atoms in the x, y plane may play an important role in the actual mechanism microscopic of superconductivity [ 19,40,4 11. Our calculations have shown a remarkable coupling between electronic and lattice degrees of freedom. The actual value of the mean-square deviation of the electronic charge in the Cu 3d,z_+ orbitals depends critically on the local symmetry of the copper-oxygen units. In addition, we have shown that the magnitude of the magnetic local moment is coupled to vibrational modes of the oxygen atoms. While the magnetic moments on the Cu atoms surrounded by the short oxygen bonds are almost unchanged, the ones on the Cu atoms surrounded by the long bonds are suppressed due to the increasing density in the Cu 3dX2_+ orbitals.

Acknowledgement This work has been supported by the Deutsche Forschungsgemeinschaft due to a Heisenberg grant and the Fonds der Chemischen Industrie (MCB), by the Max-Planck-Gesellschaft (GB and AM0 ) , as well as by the Polish Research Project CPBP 01.09 (AMO).

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