J. Phys. Chem. 1993,97, 99961ooO1
9996
Relativistically Parametrized Extended Hiickel Calculation of Net Charges on Atoms in YBazCu30.1 Anusuya Datta and C. M. Srivastava Department of Physics, Indian Institute of Technology, Powai, Bombay-400076, India
Sambhu N. Datta' Department of Chemistry, Indian Institute of Technology, Powai, Bombay-400076, India Received: May IO, 1993.
The electronic structure of YBazCusOt has been investigated by the method of relativistically parametrized extended Hiickel (REX)calculations. Calculations have been carried out in steps, for different clusters of atoms, so as to obtain a proper picture of the charge distribution in the solid state. We have calculated the total energies of different electronic configurations for each cluster. The bulk has been simulated as a vast collection of appropriate point charges for each electronic configuration of the cluster. Interaction of the cluster with its counterion environment has been treated in the point charge interaction model. The ground-state electronic configuration has then been determined for each cluster embedded in solid. These calculations show that on average the hole has a higher probability to reside on the O(4) atoms and a lower probability to be found in the Cu( I)-O( 1) chains. These results are in good agreement with experimental observations.
1. Introduction One of the most important problems in the field of hightransition-temperature superconductivity is to understand the precise mechanism responsible for electrical transport in the normal state of oxidic superconductors.' Various theoretical models have been proposed, and all of these attempt to explain the anomalous properties of the superconducting perovskites in their normal as well as superconducting states. It is important to know about the net charges on different atoms of the high-T, cuprates, for with this knowledge one can identify the mechanism responsible for normal-state transport in these systems. Sulaiman et a1.2 carried out an ab-initio unrestricted HartreeFock calculation on the [ C U J O I ~ I -cluster, I~ where the environmental effect was generated by employing the Watson sphere model for the basis set for oxygen atoms and by augmenting the Hartree-Fock potential experienced by the electrons in thecluster with the potential from the rest of the lattice while treating the lattice ions as point charges. They found that the hole resides on O(4). They also found a net charge of +1.49 on Cu(1). However, X-ray absorption and photoemission spectroscopies3v4 point out that in YBa2Cu307 both Cu( 1) and Cu(2) are in the +2oxidationstate. Thisclearlyindicates the need for a calculation on bigger clusters which can be better representatives of the material. Among other theoretical investigations, MCSCF calculations on various Cu-0 clusters have been carried out by Kamimura and Eta? who have shown that in the hole-doped Cu06 cluster near the doping the ground state changes from 'AI, to ~BI, concentrationof the onset of superconductivity. Whangbo et al.6 have demonstrated the important point that the structural characteristicsof Y B ~ ~ C that U ~depend O ~ upon ~ the oxygen atom vacancies in the Cu( 1) atom plane are reproduced by employing the empirical atom-atom potentials derived from binary oxides. Saalfrank et al.7 have carried out an ab-initio band structure investigation of the cluster [Cu04l4 with finite point charge environment for La2CuO4. They have found that, upon doping, holes of predominantly oxygen character are formed within the CuOl planes of 2- 1-4. With this background we have investigated the electronic structure of YBazCusO7 (henceforth abbreviated as YBCO) by *Abstract published in Aduunce ACS Absrrucfs, September 1, 1993.
0022-3654/93/2097-9996$04.00/0
the relativistically parametrized extended Hiickel (REX) method developed by Lohr and Pyykkii8 The REX program, available from Quantum Chemistry Program Exchange? has been used. Although the method is approximate, computations are faster and we could study clusters which could not be easily investigated by more sophisticatedmethods. The extended Hiickel technique is known to provide useful insight into the electronic structure of large molecules.1° These calculations have been carried out in steps, for an increasingly large size of the cluster, so that a proper picture of the charge distribution in the solid state may emerge. However, one still has to account for the environment of a cluster embedded in a crystal. The effect of the environment of a small cluster in a (conducting) crystal can be visualized in two ways: (i) There will be extensive conjugation,that is, overlap and interaction of orbitals of different clusters, in thecrystal, but the resulting orbitals will still be standing waves of usual quantum chemistry. (ii) Further, crystal orbitals are to be formed as Bloch sums of cluster orbitals with well-defined characteristics of propagation. In fact the second point-the departure from the usual quantum chemical approach-is inherent in all band structure calculations.'lJ1J2 In ionic solids, the valence and conduction bands are more or less developed from the valence and outervalence orbitals of the repeatingunit. In conducting crystals, however, orbitals on atoms lying on a chain or a plane of conduction are substantially mixed to induce a facile delocalizationof electrons. In order to obtain the correct charge distribution in a conducting crystal from an extended Hiickel calculation on a small representative cluster, the crystal environment should be simulated with care (via point charges, Ewald sums, crystal potentials, etc.). This paper is organized as follows: (1) In Section 2, we discuss the calculations [systems, Figure la-g, and basis sets employed]. From each calculation one may obtain a set of net atomic charges, but the distribution of the net charges depends on the scheme of orbital population. (2) The extended Hiickel method is normally used for organic and inorganic molecules which are isolated or solvated, or which form insulating crystals. In such cases the calculation on a carefullychosen single cluster (molecule or complex) would more or less hold for the crystal. All the lowest energy orbitals are occupied-so all the ligand orbitals are fully occupied-in the ground-state electronic configuration.1° In Section 3 we discuss 0 1993 American Chemical Society
Net Charges on Atoms in YBa2Cu307 the net charges calculated with the usual population distribution for clusters a-f. These indicate the existence of Cu+3 which is entirely expected. In a small electrondeficient cluster containing transition metal ions, the hole would reside in the metal ion orbitals (giving rise to Cu+’) and not on the ligands, as the metal ion orbitals are of higher energy. These results would have held for the solid state had the crystal been an insulator; since YBCO forms a conducting crystal, these are purely fictitious. Indeed, a host of band structure calculations has revealed that the oxygen atomic orbitals participate in the formation of energy bands crossing the Fermi surface in Cu3Ol2 and similar clusters.12 This effectively calls for a rearrangement of cluster orbital populations in order to account for the situation in the solid state. The net charges calculated with the usual scheme of orbital population do not even apply to isolated clusters, as in the isolated case the equilibrium geometry will be highly different. (3) In Section 4 we carry out two more steps of calculation to determine the electronic configuration of lowest energy. The first step involves the addition of a correction term to the total energy calculated by the REX method, as the REX parametrization is inadequate for the calculationof the ground-stateenergies of electron-deficient clusters. The second step is based on the realization that the strand model of the infinite complex bulk solid is much too small to be physically meaningful. Embedding to model the bulk is essential and almost universally employed in treating small clusters. This will be especially required for very highly charged clusters, for without an embedding to neutralizethe charge there will beenormous surface chargeeffects. We have modeled the bulk as a vast collection of point charges appropriate for each electronic configuration of each cluster. We then consider electrostatic interaction of the net charges on the atoms of each cluster with the charges of counterions forming the environment and determine the minimum-energy electronic configuration of each embedded cluster. The electronic population in the resulting ground state yields net charges in agreement with experimental results. (4) We study in Section 5 a prototype, case g, where the crystal environment is approximately reproduced in the neighborhood of all metal atoms except yttrium. In this case the distribution of net charges calculated with the usual scheme of orbital population is seen to be somewhat comparable to those obtained for the ground-state configurations in the earlier cases (a-f). Nevertheless, the ground-state configuration is found to be different from the usual configuration. There are 0 ( 1 ) and O(4) holes in the ground state. ( 5 ) Conclusions derived from this investigation are discussed in the last section.
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9991
(a1
(g)
Figure 1. Clusters examined for the calculation of net atomic charges: (a) YBazCusO7; (b) YBasCusOs; (e) YBazCu08; (d) [CUSO~O]-~~; (e) [ C u n 0 i ~ l - ~( f~) ;[CUsOzil-? (B) [YBaCusOz~l-~~.
2. Calculation
To date, almost all theoretical investigations (on YBCO) have been on very smalF5g7or large13Cu-0 clusters, without including the other atoms of YBCO in the calculation. The presence of other atoms has been considered only through the formation of the environment, the crystal potential, etc. The large CuO clusters studied by Gopinath et aL1fhave been expanded in the direction of conduction. Here our attempt has been to take as many types of atoms of the unit species in our calculations as possible. So most of the strands chosen are along the direction of the unit species and perpendicular to the direction of conduction. The inherent logic is that if all the necessary theoretical considerations are included in the treatment, a charge distribution typical of the conducting crystal should result even if the calculation is carried out for strands perpendicular to the axes of conduction. The clusters studied are depicted in Figure la-g. In part a we have taken one formula unit with 13 atoms. In part b we have added one BaO to YBazCu@7such that thecluster exhibitscrystal periodicity in the c-direction. In part c instead of BaO we have
added CuO to make the configuration symmetric. In part d we have removed Ba and Y to fully incorporate the planar atomic arrangement around Cu(2) in the unit cell to obtain [Cu3010]-13, In part e we have incorporated this symmetry around Cu( 1) to obtain the cluster [ C ~ 3 0 ~ 2 ] - 1In~ .part f we have extended the cluster to two edges of the unit cell. The result is the cluster [ C ~ a 0 2 1 ] - ~In~part . g half of the unit cell framework with cluster configuration YBaCusO22 containing 32 atoms and a net charge of -21 has been considered. In all these cases we have adopted the atomic coordinates given in ref 14 for a unit cell of YBa2CU307. The REX program (QCPE Program No. 387)9 allows for different choices of the valence basis set. In each calculation the basis set consists of (i) 2s and 2p orbitals of the oxygen atom, (ii) 3d, 4s, and 4p orbitals of the copper atom, (iii) 4d, 5s. and Sp orbitals of the yttrium atom, and (iv) Sd, 6s, and 6p orbitals of the barium atom. All the computations were performed with default parameters (Table I).
Datta et al.
9998 The Journal of Physical Chemistry, Vol. 97, No. 39, I993 TABLE I: Parameters Used in CalculationP Hll in eV S P atom i= i = I l 2 , 312 0
a
-34.08013 (-34.02433)
cu
-6.66069 (-6.48982)
Y
-5.47904 (-5.33739)
Ba
4.44029 (4.28658)
exponent d
i = 312, s / ~
-16.78449, -16.74528 (-16.76756) -3.39483, -3.37400 (-3.35920) -3.55702, -3.48872 (-3.51 523) -3.105 15, -2.99 144 (-3.06268)
P
S
-13.27600, -12.90638 (-1 3.36702) -6.33053, -6.27372 (-6.79876) -3.625 15, -3.58159 (4.14307)
i= '12
i = l l z , )I2
2.19396 (2.1 9159)
2.01953, 2.01669 (2.01763) 0.98871, 0.9795 3 (0.97373) 1.09664, 1.07666 ( 1.07903) 1.09714, 1.05909 (1.07132)
1.37972 (1.35089) 1.30614 (1.27906) 1.27867 (1.23649)
d
i = 3/2, s l ~
3.53220, 3.49287 (3.53125) 1.79906, 1.78481 (1.84788) 1.87935, 1.84159 ( 1.96032)
These are the default values used in the ITEREX-87 pr~gram.~ Values in parentheses are for extended Htickel calculations.
C
TABLE IW Net Charges on Atoms in Isolated Clusters of cbosea Geometries, calculated w w i tbe usual Population Scheme (above) and for the Ground-State Configuration (below).
0 -1.96 +2.77 -1.90 +2.10 -1.94 -1.93 +1.44 +2.83 +1.41 -1.94 -1.94 +2.10 -1.90 +2.77 -1.96
Y Cu(2) O(2) 2.83 1.88 -1.93 1.92 -1.93 (282-+281431) 2.83 1.86 -1.91 1.92 -1.91 YBaKurOR 2.83 1.44 -1.93 . (322j01 2.83 1.42 -1.86 32050151'52' (322501 2.83 1.91 -1.88 32l50051l52l) [cu3olol-" 2.85 -1.96 (3225321.95 -1.88 32'53I54l55I) fcu3olzl-~' 1.93 -1.98 (48ii481631) 1.93 -1.98 [CuSO211-'8 1.91 -1.98
TABLE II: Net Charges on Atoms in Isolated Strands EHT'
cluster charge
REXb
REX
a
a
a
0
0
-1
-1.99 +2.05 -1.92 -1.92 +1.88 +2.81 +1.91 -1.92 -1.92 +2.09 -1.88 +2.75 -1.95
-1.99 +2.06 -1.93 -1.93 +1.92 +2.83 +1.88 -1.93 -1.93 +2.10 -1.90 +2.78 -1.96
-1.99 +2.06 -1.94 -1.94 +1.42 +2.83 +1.42 -1.94 -1.94 +2.10 -1.90 +2.77 -1.96
41(2)
42(1)
REX
b 0
O(1)
ci(i)
.,
O(4)
Ba
O(3) O(2)
Cu(2) Y Cu(2)
O(2) O(3)
Ba
O(4)
Cu(1) O(1) o(4)
Ba HOMOC
41(2)
-1.99 +2.05 -1.93 -1.93 +1.90 +2.82 +1.86 -1.93 -1.93 +2.11 -1.91 +2.73 -1.96 -1.93 +2.02 45(2)
REX
50(1)
Orbital Energy in eV HOMO LUMO
-13.122 -13.116
-12.731 -12.726
-12.726 -12.628
-12.731 -12.726
-12.726 -12.628
EHT stands for the nonrelativistic extended Htickel calculation. REX is its relativistic version. The HOMO population is shown in a
parentheses.
TABLE IIk HOMO and LUMO Energies for Isolated Clusters of Chosen Geometries' HOMO LUMO cluster no. energy no. energy YBa~Cu307 41 -12.73 42 -12.72 YBaICG08 50 -12.72 51, 52 -12.63 53 -12.56 54, 55 -12.47 [cu,olol-'3 61,62 -12.47 [CY0211-28 110 -12.53 [ Y B ~ C U ~ O ~ J -123 ~~ -12.57 a All orbital energies are in eV. [~@121-"
63 111, 112 124
-12.22 -12.41 -12.49
3. Usual Population Scheme Results of our calculations are given in Tables I1 and 111. For configurationa we have performed both the nonrelativistic(EHT) and relativistic (REX) extended Hackel calculations. There is not muchdifferencebetween EHTand REX chargedistributions, TheREXcalculation showsa slightly enhanced chargeseparation. However,the REXcalculation givesbetter orbitalenergies (Table 11) due to two reasons: (i) relativity causes a contraction of s and p orbitals and a self-consistent expansion of d orbitals of the atoms as shown by the orbital exponents in Table I, and (ii) the effects of the spin-orbit interaction are included in the orbital exponents and the Hn matrix elements chosen for the REX
cluster YBszCu30,
-
-
(622 62°11311141)
1.91 -1.93
[YBaCul022]-21 2.87 2.26 -1.98 (four) -1.88 (two) (8S 2.85 2.03 -1.98 (four) 850124'1 371) -1.87
-
(two)
O(3) -1.93 -1.93 -1.91 -1.92 -1.94 -1.91
Ba 2.10 2.06 2.10 2.06 2.10 2.10
-1.91
2.10 -1.73
-1.97 -1.98
-1.97 -1.65
O(4) Cu(1) O(1) -1.90 2.78 -1.96 -1.99 -1.50 1.86 -1.52 -1.99 -1.90 2.77 -1.96 -1.50 1.85 -1.54
-1.99 -1.93 -1.98 -1.71 -1.99 -1.92 (four) -1.92 (two) -1.94 -1.62 (four) -1.92 (two) -1.98 2.67 -1.95 (four) -1.87 (two) -1.98 1.98 -1.70 (four) -1.88 (two)
1.83 -1.76 0.97 1.88 2.79 -1.97 1.90 -1.73 2.78 -1.97 (two) -1.91 (one) 1.90 -1.97 (two) -1.95
(one)
1.81 -1.98
1.73 -1.89 (four) -1.61 (two)
The population revision leading to the ground-stateconfiguration is shown in parentheses. calculation (Table I). We have therefore performed only the REX calculation for configurations b-g. Net charges on atoms in isolated strands are shown in Table 11. HOMO and LUMO energies of the clusters are given in Table 111. Net charges on atoms in clusters a and c-f are shwon in Table IV. The YBazCu307 strand (Figure la) has net charges on respective atoms as Per their normal valencies (Table 11). The only significantdeviationis exhibited by Cu( 1),which has aquired an additional 3/4 positive charge. Both the highest occupied Orbital and the lowestunmupied molecular orbital (LUMO) are built up from the 3d orbitals of the two cU(2) atoms. The (orbital 41) is a bonding orbital and the LUMO (orbital 42) is an antibonding orbital. SO the assignment of an overall negative charge to the strand does not remove the hole on Cu(l), but it reduces the positive charge on each Cu(2) atom by about 0.5. In order to investigate the effect of periodicity, we have attached oneextra BaO unit to the Cu( 1) end of the strand (Figure Ib). However, we have found almost
Net Charges on Atoms in YBa2Cu307 the same charge distribution, and this case will not be discussed further. If BaO is attached, the periodicity is exhibited but the strand becomes very unsymmetric. To preserve the symmetry of one edge of the unit cell, we have attached one Cu( 1)0( 1) unit to the BaO end (Figure IC). The resultant, cluster YBa2Cu40s,has 1.5 holes shared by the two Cu( 1) atoms and one extra electron shared by two Cu(2) atoms, and the charge distribution is much like that in the [YBazCu307]- cluster (Table 11). The HOMO (orbital 50), which is half-filled, is mainly an antibonding linear combination of the 3d orbitals of the two Cu(2) atoms. The LUMOs (orbitals 5 1and 52) aredoubly degenerate. Themain contribution to each LUMO comes from the 3d orbitals of the Cu(1) atoms. Addition of an overall negativecharge to the strand decreases the positive charge on each Cu(2) atom from nearly 1.5 to about 1 without affecting the charge on the Cu(1) atom. From the previous calculations we obseme that the HOMO(s) and the LUMO(s) are always built from the 3d orbitals of copper atoms and oxygen atomicorbitals form the lowest lying molecular orbitals of each strand. Barium and yttrium are always found to be more or less at their normal valencies with closed-shell configurations. So, instead of considering more strands along the edge of the unit cell, we have investigated three bigger clusters. Results found for these clusters are discussed in the following. First we have taken the cluster [Cu3010]-~3along one edge of the unit cell (Figure Id). The charge distributions are as follows: all oxygen atoms have net charges as per their normal valencies whereas Cu(1) has a net positive charge of about 1 and each Cu(2) has a net charge of +2.85 (Table 111). The nondegenerate HOMO (orbital 53) is a mixture of Cu(1) 4s and 3d orbitals. The doubly degenerate LUMOs (orbitals 54 and 5 5 ) are mainly nonbonding 3d orbitals of Cu(2) atoms. This clusters has been extended by including two 0(1) atoms along the Cu(l)-O(l) chain. In the resulting cluster [Cu3012]-17 (Figure le) all atoms except Cu( 1) have net charges as per their normal valencies and Cu(1) has a net charge of +2.79 (Table 111). Each of the doubly degenerate HOMOS (orbitals 61 and 62) is mainly a nonbonding orbital-a linear combination of the 3d orbitals of a particular Cu(2) atom. The LUMO (orbital 63) is mainly a nonbonding 3d orbital of Cu(1). Instead of extending the cluster further along the edge of the unit cell, we have now taken up the cluster [C~,jO21]-~~ (Figure If) distributed about a face of the unit cell. The charge distribution is very similar to that in the previous case (Table 111). All these examples show that the hole is associated with copper. Most of these indicate the hole to be on Cu( l), but the calculation on [Cu3010]-'3 points out the hole to be on Cu(2). In summary, the location of the hole depends on the size, shape,and composition of the cluster under investigation.
The Journal of Physical Chemistry, Vol. 97,NO. 39, 1993 9999 the largest normal negative charge, but this situation rarely arises for the ground state of an electron-deficient cluster. We correct the calculated total energy by writing
4. Most Stable Coflgurntion
where the sum is over all the positive ions belonging to the cluster under study and &&) and ZR&) are the experimental and REX ionization potentials for the pth ion exceeding the largest normal charge. The quantity APp is the excess population of positive charge on the pth ion and is obtained from the REX calculation. Equation 1 has been used in our calculation of the total energies of different electronic configurations. Environment. In the second step of the calculation of total energy we have considered the cluster to be embedded in the bulk of the solid material YBCO. The material has been simulated as an aggregate of 48 X 48 X 16 unit species with the cluster placed in the center. The surroundings of the cluster have been represented by a collection of counterions placed at appropriate atomic sites. Thus the geometry chosen for all the atoms in the aggregate has been kept unaltered while replacing the ions surrounding the central cluster by point charges. For an ion type common to both the cluster and its environment, the point charge (counterion) equals the net atomic charge calculated for the particular electronic configuration of the cluster. Thus for different electronic configurations of the same cluster we have selected different environments. For instance, the net atomic charges given in Table IV equal the counterion charges for common ions for two configurationsof the embedded cluster, (i) the configuration the same as the ground configuration of the isolated cluster and (ii) the ground configurationof the embedded cluster. Tomaintain the electrical neutrality of each unit species, we have sometimes modified the charges representing ions of the additional types belonging to the environment from their normal charges. No alteration is needed for cluster a. For cluster c the environment of a has been retained. For d the point charge standing for the O( 1) ion has been adjusted. For the other two Cu-0 clusters e and f, the environment contains charges representing copper and oxide ions as well as +3 charges representing yttrium ions and charges slightly greater than +2 representing barium ions. The counterion environment thus produced has been assumed to interact electrostatically with the net atomic charges in the cluster. The average interaction energy per cluster, Eht,equals half the energy of electrostatic (Coulombic) interaction of the particular cluster with its environment, so that no interaction is counted twice. The other half clusterenvironment interaction energy modifies the energy of the environment. The size of the material has been chosen in such a way that the interaction energy Eint converges through four significant digits. The total energy of the cluster in the solid is given by
Correction of REX Total Energy. The REX parameters have been chosen in such a way that good orbital energies are obtained in the case of nonconducting ionic crystals. The REX total energy (ERM) is expressed as the sum of the energies calculated by multiplying orbital energies with corresponding orbital occupancies. Thus in most cases the usual population scheme yields the ground-state configuration. But a problem arises when the total energy of an electron-deficient system is calculated. The REX energy parameters are largely inadequate to account for the energy change that occurs when the valence of an ion exceeds its largest normal ionicvalency. For instance, the third ionization potential of copper (that corresponds to the process Cu+2+ C U + ~ ) is known to be 3545 kJ mol-l or 36.74( 146) eV whereas the REX scheme yields a value of only 12.906 38 eV. Therefore, in the REX method the total energy is underestimated whenever the valency of copper or that of barium exceeds +2. A similar underestimation happens when thechargeof a negative ion exceeds
General Trends. A direct consequence of using eq 1 is that the transfer of a single electron from the calculated highest energy (antibonding) oxygen orbitals to the LUMOs yields the lowest energy electronic configurations. The consequenceof considering the electrostatic interaction of a cluster with its environment is that the transfer from the highest antibonding orbital formed from the atomic orbitals of 0(1) and O(4) [and not from the orbitals of O(2) and 0(3)] gives the most stable electronic configuration. When two orbitals, one of O( 1) and the other of O(4). are concerned, the transfer from O(4) is more favored, as 0(1) is closer to the metal atoms. Ground-state configurationsof the clusters studied are shown in Table V. Net charges on atoms in the ground state of each cluster are shown in Table IV against the net charges calculated from the usual scheme of population distribution. Total energies
Datta et al.
loo00 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 TABLE V: Evolution of the Ground-State Confimation Using Eq 1' cluster
Y BazCu307 Y BazCw08 [cu3olol-'~
[~~Oiz1-" [Cu60211-=
higheat energy antibonding ortital on nei&oring a t o d no. energy atoms
28 32 32
-16.42 -16.42 -16.67
0(1)-0(4) 0(1)-0(4) 0(2)-0(4)
48 62
-16.30 -16.69
0(1)-0(4) O(4)
-16.32
0(1)-0(4)
[ Y B ~ C U ~ O ~ ] -85 ~~
lowest energy, unoccupied, nonbonding orbital of metal atomsc no. energy atoms 43 51,52 53 54,s 63 113d 114' 124 137
-12.63 -12.63 -12.56 -12.47 -12.22 -12.30 -12.15 -12.49 -2.693
Cu(1) Cu(1) Cu( 1) CU(2) Cu(1) Cu(1) CU( 1) Cu(2) Ba
evolution of ground-state configuration
-- ... -.- ...
282...412- 281...412431 322...501 32°...501511521,321...500511521 32?..532 321 531541551 482...611621 48I...6l162l63l 622...1102 620...11311141 852...1232
850
12411371
a All orbital energies are in eV. Orbital occupied in the usual population scheme. e Orbital 53 of [ C U ~ O ~ O is] an - ~ occupied ~ orbital. Bonding combination of 3d orbitals of Cu(1) atoms. *Antibonding combination of 3d orbitals of Cu(1) atoms.
TABLE VI: Calculationof TOMEnergy for clusters Embedded in the Bulk of YBCO
config- location urationse of hole Cu(1) YBalCU39 u 0(1), O(4) gg Cu(1) Y BalCN0: u 0(1),0(4) gg CU(2) [cu3olol-'~ ug o(4) g Cu(1) [~~Oid-~' u O(l), O(4) gg Cu(1) ~CU60211-~ u o(4) 88 [YBaCueO&zl ub Ba, Cu(2) 88 0(1), O(4) cluster
energy in au average embedded isolated interaction cluster cluster ( E ) (Ei,,,) ( E + Ebt) -57.333 -55.249 -2.084 -57.554 -55.789 -1.765 -67.499 -65.095 -2.403 -68.339 -66.308 -2.031 -72.988 -17.1 l(3) -90.101 -74.319 -14.92(9) -89.249 -86.163 -20.12(5) -106.29 -86.708 -20.28( 1) -106.99 -1 53.767 -47.16(7) -200.93 -1 54.810 -5 1.24(9) -206.06 >-168.567 -169.314 -26.59(9) -195.91
0 Configuration according to the usual population scheme is denoted by u. The ground-state configuration found by using cq 1 is indicated by g. The second g indicates the ground-state configuration evolved after considering the interaction of the cluster with its environment! The lower limit of E has been calculated from the adopted lower limit of the third ionization potential of Ba.
calculated for the ground-state configuration and the usual one are compared in Table VI. These have been calculated by applying eq 1 only for the excess charge on each copper atom. The excess charge on a barium atom has been overlooked, as it is at most about 0.1 and the molecular orbitals formed from barium atomic orbitals lie at least 8.8 eV above the HOMO. The removal of the excess charge from the barium ion becomes energetically favorable only when the charge is large enough. This situation has been encountered for cluster g and is discussed in Section 5. The evolution of the ground-state configuration is discussed for each cluster individually in the following. Individual Clusters. For the YBa2Cu307 strand, the HOMO (orbital 41) energy is -12.73 eV. Orbital 43 (energy -12.63 eV) is a nonbonding d orbital on Cu( l), and orbital 28 (energy -1 6.42 eV) is the highest antibonding ligand molecular orbital spread over 0(1) and the O(4) next to Cu(1). The most stable configuration results from the reallocation of one electron from orbital 28 to orbital 43. In this configuration the hole is almost equally shared by 0(1) and the O(4) atoms (Table IV). In the case of the symmetric cluster YBa2Cu408, the highest antibonding molecular orbital spread over the oxygen atoms is orbital 32 that is formed from O( 1) and O(4) atomic orbitals. If two electronsmigrate from orbital 32 to the two LUMOs which are nonbonding 3d orbitals of Cu( 1) atoms, we get 1/2 hole on each O(1) or O(4) atom and l / z extra electron on each Cu(2) atom. The resulting configuration320 49250151l521 has a lower E and a small Ert. The configuration 321...492500511521has the lowest E as well as a larger interaction energy Er, (which is negative) and correspondsto the ground state. As a result there is only one hole and it is almost equally distributed over the O( 1) and O(4) atoms.
...
In the case of [Cu3010]-~3,the HOMO (orbital 53) is a nonbonding orbital on Cu( 1). The ground-state configurationis 321...531541551,and the hole has more or less migrated to the two O(4) atoms. The transition from orbital 53 is essential for decreasing E. Compared to the migration from orbital 40, the migration from orbital 32 leads to a lower Ert,but the interaction with the environment stabilizes the usual configurationto a great extent-so much so that the possibility of finding a hole on Cu(2) is indicated. The result is presumably less reliable, since the environment of Cu( 1)within thecluster is far from being adequate. There are two O( 1) atoms in cluster [ C U ~ O ~ Z The ] - ~ oxygen ~. molecular orbitals are all mixed up, the 0(1) orbitals and the O(4) orbitals contributing together. The ground-state configuration hasbeenfoundtobe481...631.This bringstheholealmost equally to 0(1) and O(4) atoms. However, the O( 1)-0(4) orbital separation has been achieved with the [Cy021]-28cluster. Orbital 1lOis theHOMO. Orbitals 113 and 114 are combinationsof the two nonbonding Cu(1) 3d orbitals for Cu3O12 fragments, and both are very close to the HOMO in energy. Since 0 ( 1 ) is closer to metal atoms than 0(4), antibonding molecular orbitals formed from O( 1) atomic orbitals (such as orbital 84) will be relatively more stabilized in the crystal environment. In the point charge interaction model the decrease of the negative charge on O(4) rather than 0(1) leads to a greater stability. Hence the ground-stateconfiguration results when two electrons relocate to orbitals 113 and 114 from orbital 62, an antibonding orbital formed from the O(4) atomic orbitals, and not from orbital 84. Consequently the holes migrate to the O(4) atoms. In short, the calculation of the energetic ground state almost always gives holes of oxygen character, that is, a charge distribution compatible with the results of band structure calculations7J1J2 on similar clusters.
5. A Larger Test Case In the calculations described above the yttrium atom has always been found to have a net positive charge of about 3 and a closedshell configuration. Yttrium atomic orbitals always contribute to the higher energy unoccupied orbitals. Therefore, we have considered thecluster [YBaCu80~]-21(Figure lg), whichisabout half of the unit cell frame with atomicenvironmentsapproximately reproduced for all metal atoms except yttrium. It is clearly seen from Table IV that the charge distribution calculated from the usual population scheme is in some respectssimilar to the groundstate charge distributions of the earlier cases (8-0. The hole is not confined to a particular atom. On average, about 1/3 of the hole is situated on each Cu(2)-0 unit and the rest of the hole resides in the Ba-0(4) unit. So the hole has much greater probability to reside in the Ba-0(4) network. However, the calculated result shows 0.67 hole on Ba and 0.05 hole on O(4). This distribution is an artifact of surrounding the Ba atom with four O(4) atoms and arises due to the inadequacy of REX parametrizationas discussed earlier in section 4. A strongsurface
Net Charges on Atoms in YBa2Cu307 charge effect has developed, for on the basis of the calculated net atomic charges we find that the unit species would have a net charge of +0.914. The result is therefore far from being representativeof the bulk solid, and an environmentalcalculation has not been performed. The third ionization potential of barium should be at least 0.872667 au (minus the Dirac-Fock 5p3p orbital energy of a neutral barium atomls). In the absence of reliable information on the third ionizationenergy of barium, this value has been used by us to calculate the lower limit of E. Using eq 1 we find that the reallocation of two electrons from orbital 85 [0(1) and O(4) orbitals] to orbitals 124 [Cu(2) orbital] and 137 [Ba orbital] leads to the lowest E (Tables V and VI). Only a minor adjustment of the environmentalcharges is required now. For environmental calculations the net charge (4.352) has been compensated by increasing the charge on the second barium atom in the unit species. The configuration 850...12321241...1371 is found as the ground-state configuration even after the cluster-nvironment interaction has been taken into account (Table IV). We find about 0.3 hole on each of the four O(4) atoms in the Ba-0(4) frameworkand about 0.8 hole distributed over the six O( 1) atoms in the Cu(l)-O(l) plane in the embedded cluster (Table IV). 6. Conclusions The main interest of this work is to show that essential physics is in fact recaptured with a very simple calculation. There is a simple remedy to the unrealistic treatment, within the extended Hackel framework, of very highly charged species. It is also interesting to note that finding the minimum total energy of the embedded system may require electron assignment to orbitals which make a configuration far from the ground configuration of the isolated fragment. In the isolated clusters of chosen geometries, a-f, the hole resides perferably on one of the copper atoms. The location depends on the size, shape, and composition of the cluster. We have corrected the calculation of the REX total energy and considered the environmental effects by simulating the bulk as a large collection of point charges. We have found that theclusters in the solid state have the hole mainly in the Ba-0 plane, more specifically, on the O(4) atoms, and to a lesser extent on O( 1) atoms belonging to the Cu-0 chains. The calculation on the largecluster [YBaC~*022]-~' with each metal atom experiencing an approximate crystal environment also shows the hole to be mainly in the Ba-O(4) plane. Our result is in general agreement with the experimental observationof Bianconietal.3 and Fujimori4 that the hole in YBa2Cu3074 systems (partly) arises as a hole on the apical O(4) atoms. Recently Mazin et a1.16have presented theoretical arguments and pointed out experimentalevidence for the existence of two close-lying, nearly parallel, energy bands in
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 loo01 YBa2CuoO7, one of which crosses the Fermi level. The electrons in these bands are located in the Ba-O planes. From a set of extended Hiickel calculationson Cu-0 clusters, Gopinath et al.I3 have found that, with an assumed core charge of +2 on Cu, the calculated HOMO energy as a function of oxygen stoichiometry shows an excellent correlation with the variation of Tc. Our treatment traces the reason for the need of fixing the copper atoms with core charges and suggests a systematic way to deal with the problem. Our calculations also indicate that relativistic effects play a less vital role in determining the distribution of charge carriers. The carrier distribution and stability depend mainly on the crystal structure.
Acknowledgment. The authors are grateful to the reviewers for many valuable suggestions and comments. References and Notes (1) Anderson, P. W.; Zou, 2.Phys. Rev. Lett. 1988,150, 132. Varma, C. M.; Littlewood, P. E.; Schmitt-Rink, S. Phys. Rev. Lett. 1989,63, 1996. Varma, C. M. Int. J. Mod.Phys. 1989,133,2083. Anderson,P. W.; Schrieffer, J. R. Phys. Today 1991, 44, (6), 54. (2) Sulaiman, S. E.; Sahoo, N.; Markert, S.; Stein, J.; Das, T. P.; Nagamine, K. Bull. Mater. Sei. 1991, 14, 149. (3) Bianconi, A. International Conferenceon Superconductivity-ICSC, Bangalore, India, January 10-14, 1990, Joshi, S. K., Rao, C. N. R., Subramanyam,S. V., Us.; World Scientific: Singapore, 1990; pp 448-469. (4) Fujimori, A. In MechamsinsofHigh TemperatureSuprrconductivity; Kamimura, H., Oshiyama, A., Eds.;Springer-Verlag: Berlin, 1988; p 176. (5) Kamimura, H.; Eto, M. J. Phys. Soc. Jpn. 1990,59,3053. Eto, M.; Kamimura, H. J . Phys. Sa.Jpn. 1991,150.23 11. (6) Whangbo, M.-H.; Evain, M.; Beno, M. A,; Geiser, U.;Williams, J. M. Inorg. Chem. 1988, 27,467. (7) Saalfrank,P.; Ladik, J.; Wood, R. F.;Abdel-Raouf, M. A,; Liegener, C.-M. Physica 1992, C196, 340. (8) Lohr, L. L., Jr.; Pykk6,P. Chem. Phys. Lett. 1979, 62, 333. (9) Lohr, L. L., Jr.;Hotokka, M.; PykkrB, P. REX. QCPE 1980,12,387. Rbsch, N. QATREX. QCPE 1983,3,468. Pykk8, P. ITEREX-85. Report HUKI 1-86; Department of Chemistry, University of Helsinki, 1986. ITEREX-87 was modified from ITEREX-85 by PykkrB, P., and Larsson, S. (10) Hoffman, R. J. Pure Appl. Chem. 1986, 58, 481. Hoffman, R. J. Solids and Surfaces. A Chemist's View of Bonding in Extended Structures; VCH Press: New York, 1988. Trindle, C. In Topics in Chemistry; Dstta, S. N., Ed.; Omega Scientific: New Delhi, 1992; Vol. 2. (11) Hughbanks,T.;Hoffman,R. J. J. Am. Chem.Soc. 1983,105,1150. (12) Massidda, S.; Yu, J.; Freeman, A. J.; Koelling, D.D.Phys. Lett. 1987, A122, 1987. Yu, J.; Massidda, S.; Freeman, A. J.; Koelling, D.D. Phys. Lett. 1987, A122,203. Hybcnten, M. S.; Matheiss, L. F. Phys. Rev. Lett. 1988, 60, 1661. Massidda, S.; Yu, J.; Freeman, A. J. Physica 1988, C152. 251. Freeman. A. J.: Yu. J.: Massidda. S.In Mechanisms ofHiph Temperature Superconductivity; Kamimura, H., Oshiyama, A.; Ed;.; Springer-Verlag: Berlin, 1989; p 99. (13) Gopinath,C. S.; Ninnala, R.; Subramanian,S. Physica 1991, C177, .)1
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