J. Phys. Chem. 1995,99, 6493-6501
6493
Garnet Crystal Structures. An ab Initio Perturbed Ion Study A. Beltran, J. Andres,* J. A. Igualada, and J. Carda Department of Experimental Sciences, Universitat Jaume I, P. 0. Box 242, 12080 Castellb, Spain Received: December 22, 1994@
An ab initio perturbed ion (aiPI) study using X-ray diffraction data has been carried out for pure and doped calcium aluminum silicate garnet, Ca3A12Si3012(grossularite, GROS), and yttrium aluminum garnet, Y3A15012(YAG), crystal structures. Different clusters containing from 55 to 139 ions have been built up, using large Slater type orbitals (STOs) to represent each atomic center. Basis sets and geometry optimizations have been performed with the aim of determining the relative stability, cell parameters, force constants, and vibrational frequencies of radial displacements associated with the local relaxation for pure and doped structures. Numerical results are compared with experimental data, and the geometrical cell parameters of different structures obtained by computer simulation are found to be similar to the experimental results. This comparison validates the aiPI methodology used in the theoretical characterization of the local properties of complex ionic systems. For GROS, the substitution of Cr3+ for A13+ at the octahedral site is energetically favorable while the substitution of CF+ for Si4+ at the tetrahedral site is unstable. For YAG, the substitution of Cr3+ for A13+ at octahedral or tetrahedral sites is energetically unfavorable. The differences between the ionic radii reported by Shannon and Prewitt for the species concerned in the doping process are capable of explaining the relaxation of crystal lattice parameters for tetrahedral sites. However, the relaxation in octahedral sites is lower than the differences in ionic radii. The doping process produces a decrease of force constant (k) values associated with the breathing fundamental vibrational mode for YAG garnet while an opposite effect appears in GROS. The k associated with the radial displacement in octahedral substitution in grossularite is especially high. The bulk modulus of the pure structures has also been theoretically calculated, GROS being less compressible than YAG.
1. Introduction Garnets are of considerable interest in chemistry due to their complex crystal structure and high thermochemical stability.' Many hundreds of synthetic compounds and minerals have this structure, and they present a wide range of applicability; they can be used as laser2 and luminescent source^,^ they present ferromagnetic4 and antiferromagnetic5properties, and they are slightly birefringent6 and moreover can be spontaneously polarized in the presence of an electromagnetic field.7 Garnets also constitute a particularly attractive structural family of singlecrystal oxides which have been identified as useful materials in the development of new high-temperature ceramic reinforcement fibers due to their thermochemical stability and creep resistance.8 Among these, calcium aluminum silicate garnet, Ca3A12Si3012 (grossularite, GROS), and yttrium aluminum garnet, Y3A15012 (YAG), are very interesting materials in the development of ceramic pigments by means of the introduction of chromophore cations through the formation of solid solution~.~ The physical and chemical properties which garnets exhibit are intimately related to the particular characteristics that their crystal structures present. Features such as site preference following selective doping, lattice stability, and crystal response under local distortion around impurity centers are far from being theoretically established. In this work, we systematically analyze a number of properties associated with garnet crystals through a theoretical model based on the ab initio perturbed ion (aiPI) method.'O This approach is conceptually conceived within a cluster-in-the-lattice scheme which involves the rigorous quantum mechanical solution of large clusters embedded in a quantum crystal lattice. @
Abstract published in Advance ACS Abstracts, April 1 , 1995.
The relative stability of the impurity center Cr3+ in GROS and YAG garnets, as well as Ci'+ and GROS, is calculated in order to determine favorable substitutional sites dictated by the energetic balance between pure and doped geometrically optimized structures. The response of the crystal environment under substitutional effects is analyzed by simulating the host lattice with large clusters. The optimized geometries are tested against systematic relaxation of various shells of ions placed around the impurity center, and local distortion effects are investigated via analysis of the breathing vibrational modes around the optimized substitutional centers. The paper is organized as follows: in section 2, structural and unit cell parameters for the garnet structure are given together with a brief account of the experimental synthetic procedure followed by our group. In section 3, the method and model are presented. In section 4, numerical results are reported and compared with experimental data. A short summary of conclusions is given in section 5.
2. Structural and Synthetic Details of Garnets Garnets have the general formula {A3)[B2](C3)012,where {
1, [ 1, and ( ) denote dodecahedral, octahedral, and tetrahedral
coordination, respectively.' I The structure has the cubic space group Za3d, and all cation positions are fixed by symmetry. The anion occupies the general position and thus has three degrees of positional freedom. This structure can be visualized in Figure 1 as a lattice formed by BO6 octahedra and C 0 4 tetrahedra. Each BO6 octahedra is connected to another six octahedra through comer-sharing tetrahedra. Each C04 tetrahedra, in turn, shares comers with four octahedra, so the structure is 02t3 = (B03)2(C02)3 = B2C3012. The largest ions (A) occupy eightcoordination sites in the lattice interstices.'* Garnets have usually been manufactured by means of the
0022-365419512099-6493$09.00/0 0 1995 American Chemical Society
6494 J. Phys. Chem., Vol. 99, No. 17, 1995
Beltrh et al.
TABLE 1: Different Cluster Models for Garnets, GROS and YAG, Representing Dodecahedral, Octahedral, and Tetrahedral Sites garnet dodecahedral site octahedral site tetrahedral site
to obtain structural data. Powcal and Lsqc20 programs were used for this purpose.
3. Method and Model
e AP+ AP @ si4+ AI" Figure 1. Garnet structure. Dodecahedral,octahedral, and tetrahedral coordinations have been depicted about an oxygen ion."
heat 709C
stirring
reflux 24h.
Inorganic soluble salts t Water stirring
stirring +-heat
70tC
F; Catalyst HCI 3M I
stirring
70W
1-1 VIA COLLOIDAL GEL
VIA POLYMERIC GEL
Figure 2. Scheme for the synthesis of garnets by the sol-gel method. ceramic method, but this technique generally requires intensive mechanical mixing and high temperature^.'^,'^ One technique widely used to yield multicomponent crystals is the sol-gelI5 process. There have been reports on the synthesis of garnet phases by the sol-gel method,I6-l8 starting with a component with gel formation ability (colloidal silica) or tetraethoxysilane (TEOS) and inorganic salts of the other ions. In both colloidal and polymeric gel networks the necessary thermal processing needs lower temperaturesthan the conventionalceramic method. In Figure 2, the processes we have already used for the preparation of colloidal and polyemric gels are set out schematiand YAG have been synthesized using this ~ a l 1 y . l GROS ~ method, and then X-ray analyses have been carried out in order
3.1. aiPI Method. The basic idea of the aiPI methodology, based on an improved formulation of the theory of electronic separability proposed by McWeenyz' and Huzinaga?2 consists of replacing inactive core electrons by an effective potential in order to characterize the crystalline environment of clusters embedded in ionic lattices as an approximationin the calculation of the electronic structures of pure and doped crystals. The aiPI method is summarized in the appendix. The aiPI method provides an adequate quantum mechanical treatment of the atom-in-the-lattice structure and offers a computationally tractable approach to the analysis of the electronic structure in solids. Properties such as bonding, stability, and local defects can be studied in terms of clusters that properly simulate the crystal environment and the way it reacts to the presence of impurity ions. This method has been successfully applied to the calculation of the electronic structures of different systems like halides,23 hydride^?^ binary oxides,25 fluoroperovskites,26 and more complex systems such as the zircon structure.27 In this sense, the present work can be considered as a further application of this methodology by considering a more complex and large crystal system. This is again a computationally challenging case, since a large unit cell is involved and a large number of degrees of freedom are to be optimized. The bulk geometry of the different crystal structures was first optimized by varying the positional parameters and cell length and calculating the effective energy (the definition of this variable is given in the appendix) until a minimum was found. Geometry optimization was achieved by means of the Powell subroutine.28 The local optimized distances were found by minimizing the effective energy of the cluster as a function of radial displacement of the neighbor ions from their experimental positions. 3.2. Model. The different structures and the impurity centers (C$+ and @+) have been simulated by a large cluster. The electronic structure is computed via optimization of local geometries for the pure and doped structures around the impurity centers. Selective doping effects in the crystal structure are probed by studying lattice stability under substitution and local distortion around the replaced sites. The impurity ions occupying cationic vacancies in the crystal are simulated assuming that the substitution occurs in a single center, which is the reference origin in the construction of the clusters. The clusters have been built successivelyadding shells of ions that are symmetrically distributed around the origin. In this work, we have self-consistently computed the electronic structure of various clusters including from 55 to 139 atoms (14 to 30 shells of neighbors). Previous calculations have been carried out in order to test the convergence of the method, showing that an increase in cluster size does not affect the results obtained. The composition of the different cluster models is summarized in Table 1. 3.3. Basis Sets. Large Slater type orbital (STO) basis sets are used for every ion: (5s4p) for A13+ and Si4+, (5s5p) for
J. Phys. Chem., Vol. 99, No. 17, 1995 6495
Garnet Crystal Structures
TABLE 2: Crystal Lattice Parameters for GROS and YAG garnet a(A) X L' Z origin GROS 11.845 0.0381 0.0449 0.6514 ref 31 11.813 0.0410 0.0366 0.6519 this work 0.0512 0.6500 ref31 YAG 12.000 0.0306 this work 11.733 0.0297 0.0471 0.6507 TABLE 3: Calculated and Experimental Bond Lengths for Pure and Dowd GROS neighbor optimized experimental site ions distance (A) distance (A) Ca2+ 8 x 02- 2.140 and 2.442 2.319 and 2.491 AI3+ (pure structure) 6 x 02- 1.965 1.924 Cr3+(doped structure) 6 x 02- 1.981 Si4+(pure structure) 4 x 02- 1.538 1.645 Cr"+ (doped structure) 4 x 02- 1.697 TABLE 4: Bond Lengths for Pure and Chromium-Doped YAG at Octahedral and Tetrahedral A13+ Sites neighbor optimized experimental distance (A) distance (A) site ions Y3+ 8 x 02- 2.360 and 2.558 2.303 and 2.432 octahedral A13+(pure) 6 x 02- 1.753 1.937 octahedral Cr3+ (doped) 6 x 02- 1.678 tetrahedral A13+(pure) 4 x 02- 1.707 1.761 tetrahedral Cr3+(doped) 4 x 02- 1.784
02-,(7s5p) for Ca2+,(8s6p5d) for C$+ and C#+, and (10s9p5d) for Y3+.29 The wave functions and effective potentials, used to represent the lattice environment on each cluster, have been obtained from a previous aiPI calculation on the pure, infinite cluster. The optimization of these basis sets was done in order to minimize the total energy while maintaining SCF stability. The descriptions of orbitals of s, p, and d symmetry for all ions involved are listed in Tables 10-16. The Madelung potential, responsible for the largest part of the interaction energies, has been analytically integrated. The quantum mechanical contributions to the interaction energies have been considered for a large number of neighbor shells until a convergence of hartree is achieved in the effective energy of the clusters. The effective energy includes correlation estimated by means of the unrelaxed Coulomb-Hartree-Fock formalism.30 The term unrelaxed is used because the aiPI wave functions are not affected by this correction.
4. Results and Discussion 4.1. Structures. Optimization of GROS and YAG crystal structures yields the lattice parameters reported in Table 2, which come close to the experimental data obtained with the help of precise structure refinements3, In Tables 3 and 4 we summarize the calculated distances for the pure and doped structures. The experimental distances are obtained from the lattice parameters in Table 2, considering the Za3d space group. The optimized bond distances for both garnets are in close agreement with experimental data. In the case the dodecahedral sites (Ca2+and Y3+ in GROS and YAG, respectively), larger discrepancies with respect to experiment are found. For GROS, the optimization process shows that the first shell of four oxygen ions comes closer to the central cation while the distance of the second shell of four oxygens remains constant. This result agrees reasonably well with the structural studies performed by Novak and G i b b ~ , where ~ * they conclude that the distance A-0(1), 2.140 A, is about 0.3 8, shorter than that of A-0(2), 2.442 A, in dodecahedral sites for all the silicate garnets. In the case of the optimization process for YAG, the distance of the first shell is keep nearly constant while in the second
TABLE 5: Lattice Energy for Experimental and Optimized Structures garnet lattice energy (kJ/mol) GROS, experimental -20 375.85 -20 543.88 GROS, optimized -15 228.16 YAG, experimental -15 293.54 YAG, optimized shell it increases slightly. The difference between optimized and experimental distances is very small, although a larger difference exists for the octahedral site. The possibility of obtaining solid solutions, although necessarily under extreme conditions, of uvarovite (Ca3Cr2Si3012)GROS by isovalent substitution of Cr3+ for A13+ was already mentioned by Roy and Roy.33 On the other hand, detailed investigations of garnet crystals doped with Cr3+ have established that the replacement is isovalent in the octahedral position.34 Our group was able to synthesize solid solutions up to 30% molar of GROS with polymeric gels at 1260 C/3 days. The decrease in cell length with increasing degree of substitution, determined with the programs Powcal and Lsqc, agrees with the substitution of Cr3+ for A13+ in the octahedral site.35 The substitution of C#+ for Si4+ in GROS produces an optimized Cfl+-02- distance of 1.697 8,, which is in excellent agreement with the value found experimentally by our group in uvarovite-GROS solid solutions.36 The effect of doping tetrahedral Cr3+ into the YAG structure is that the four oxygen ions move away from their original positions, altering the lattice ion to oxygen bond lengths from 1.707 A (A13+-02-) to 1.784 8, (Cr3+-02-). This is in agreement with the difference between the ionic radii in A13+ and Cr3+ and verifies the study by Novak and G i b b ~ where , ~ ~ they conclude that garnets can be rationalized in terms of a hard sphere model in spite of the diverse nature of the substituent cations. This also agrees with a previous study carried out by our group on solid solution in garnets by applying only a purely electrostatic model3' At the octahedral position an opposite effect occurs; the doping process decreases these distances. A good correlation between lattice relaxation in the doping process and the difference between ionic radii exists for tetrahedral sites. However, the differences between ionic radii of the species concerned in the doping process at octahedral sites do not explain the relaxation of crystal lattice parameters for the GROS and YAG structures. These results are in agreement with the structural study carried out by Hawthorne3' on about fifty refined experimental garnet structures, concluding that the mean bond length forecast from the sum of the cation and anion radii is in fairly close agreement for tetrahedral coordination but the scatter increases significantly with increasing coordination number. 4.2. Energetics. In Table 5, lattice energy (Elatt)values, as it is defined in the Appendix, are reported for the experimental and optimized structures. The Elattvalue and its decrease when switching from experimental to optimized geometry are higher in GROS than in YAG. The defect reactions involved in the substitution processes are as follows: Cr3+(g)
+ n(Ca3A1,Si301,)(s) c~,~AI,-,c~s~,,o,,(s)
Cr4+(g)
+ x(Ca,Al,Si,O,,)(s)
-
+ ~ 1 ~ + ( (1) g)
6496 J. Phys. Chem., Vol. 99, No. 17, 1995 Cr3+(g)
Beltrh et al.
+ x(Y3A1,A1301,)(s) y3fi1,-,crAl3,O,,(s)
+ AI3+@) (3)
In order to analyze the lattice stability on each substituted center of the doping process, we can consider eqs 1-4 in the following general form: (X"'-cluster):GARNET
+ Ym+(Ym+ -cluster) :GARNET
+ X"+ ( 5 )
where X"+ refers to A13+ or Si4+, Ym+ is Cr3+ or Cfl+, and GARNET corresponds to GROS or YAG. The doping process is govemed by the intemal energy, obtained in our quantum mechanical calculation as
+ &right) = E,,,,,(Y"+:cluster:lattice) + Eo(X"+)
E(1eft) = Et0,,,(Xn+:cluster:lattice) Eo(Ym+)
(6) (7)
where E(1eft) and ,?(right) denote the left- and right-hand sides of eq 5, respectively, and EO is the energy of the free ion. The difference between the two former energies yields the substitutional energy, which in our case measures the stability of Ym+ in the lattice. Although the total energy of the crystal is in general difficult to calculate, it can be estimated in a simplified way by assuming that the system is divided into the impurity center, the cluster surrounding the defect, and the remainder of the crystal. Accordingly the total energy for the pure and doped structures can be written as
E,,,,,(X"+:cluster:lattice) = E,,(X"+:cluster:X)
+ E,,,,
(8)
The stabilization energy, AE, would thus be given as
TABLE 6: Defect Reaction Energies garnet site doping ion GROS A13+ (octahedral) Cr3+ Si4+(tetrahedral) Cfl+ YAG A13+ (octahedral) Cr3+ A13+ (tetrahedral) Cr3+
-65.64 1864.11 262.55 735.14
TABLE 7: Bulk Modulus for GROS and YAG Garnets garnet bulk modulus (Pa) 1.209 x 1O'O GROS 1.074 x 1O1O YAG TABLE 8: Force Constants (k) and Vibrational ~ Pure ) and Doped GROS Frequencies ( v ~for site Ca2+ AI3+ (pure structure) Cr3+(doped structure) Si4+(pure structure) Cf'+ (doped structure)
k (N m-I)
valg
1385.66 3339.60 3830.03 2210.83 2343.17
(cm-9 429 769 823 766 789
TABLE 9: Force Constants (k) and Vibrational Frequencies (vslg)for Pure and Chromium-Doped YAG at Octahedral and Tetrahedral A13+ Sites site
k (N m-I)
vale(cm-9
octahedral A13+(pure) octahedral A13+(doped) tetrahedral A13+ (pure) tetrahedral AI3+ (doped)
1292.25 2584.49 1175.48 1486.86 1206.62
414 676 456 628 566
Y3+
TABLE 10: Orbital Energies (au), Orbital Exponents, and 'Sa Expansion Coefficients for Calcium: ls22s23s22p63p6 @( 1s) expansion 442s) expansion 4(3s) expansion (s) orbital coefficients coefficients coefficients n, L exponents (-149.147 141)b (-16.603 315)b (-2.022 641)b IS 19.856 100 0.945 523 46 -0.284 294 80 0.096 123 48 I S 32.439 700 0.020 163 80 -0.001 063 97 0.001 297 32 2s 7.799 210 0.004 751 81 1.049 641 46 -0.409 709 58 2s 17.360 700 0.042 805 01 -0.140 329 51 0.057 118 32 3s 4.183 230 0.001 507 27 -0.000 514 40 0.608 780 89 3s 2.731 330 -0.000 479 75 0.001 825 60 0.630 946 04 3s 6.358 940 -0.002 977 86 0.088 454 66 -0.210 167 20 ~
~~
4(2p) expansion n, 1
Eefdenotes the effective energy of the cluster in the lattice, and Erest,the energy associated with the rest of the crystal. In the above equations it is assumed that Erest, which labels a quantity other than the effective energy of the cluster, remains unchanged upon substitution; i.e., Erest= Erest,.This is due to the fact that the effective energy of the cluster contains all the components of the total energy in which the wave functions of the substituted center and the ions in the cluster directly participate. This approximation should be more adequate for isovalent impurities as the cluster size grows. The defect reaction energies for the substitution of Cr3+ for A13+ in GROS and both sites in YAG, as well as for the substitution of C#+ for Si4+ in GROS, are listed in Table 6. AE values correspond to the optimized structures shown in section 4.1. These results conclude that Cr3+ substitution for A13+ in the octahedral site is energetically favorable in GROS (AE = -65.64 kJ/mol), and as mentioned in section 2, this agrees with our experimental results. The same substitution for the octahedral A13+ site in YAG is slightly unfavorable. Substitutions of Cr3+ for A13+ at the tetrahedral site in YAG are energetically
AE (Hlmol)
2P 2p 3P 3P 3P
(p) orbital exponents 9.1 14 330 15.863400 3.568 610 2.271 630 7.379 930
coefficients (-13.411 685)b 0.677 120 89 0.044 196 48 0.011 130 84 -0.002 550 33 0.333 242 35
~
~~~
4 4 3 ~ expansion ) coefficients (-1.113 298)b -0.226 924 74 -0.013 483 83 0.593 856 74 0.5 16 378 43 -0.137 029 04
This notation is taken from Clementi and RoettLz9 Orbital energies in au. unfavorable, while the substitution of C#+ for Si4+ in GROS is clearly unfavorable, requiring very intense processing conditions in order to occur. We could therefore conclude that Cr3+ can substitute A13+only in octahedral sites and that Cfi+ cannot substitute Si4+ in GROS. This fact can be explained with the ionic radii listed by Shannon and P r e ~ i t t . ~The * difference between octahedral C$+ and A13+is 0.08 A, while the difference between C#+ and Si4+ is 0.15 A.
4.3. Bulk Modulus. Isotropic volume changes were used to determine the bulk modulus, B, which can be calculated by changing the cell edge and finding the resulting lattice energy. The bulk modulus is then calculated using the following expression:
J. Phys. Chem., Vol. 99, No. 17,1995 6497
Garnet Crystal Structures
B=V-
d2Elatt 6 V2
where V and El,,, are the unit cell volume and lattice energy, respectively. A fit to a fifth-degree polynomial of lattice energy versus volume has been performed, and the values obtained are presented in Table 7. We have not found experimental values in the literature, but from these results we can conclude that GROS must be less compressible than YAG. 4.4. Force Constants and Vibrational Frequencies. In this section we simulate lattice relaxation around a center and use this to evaluate the energy and force constant (k) associated with the symmetric radial (breathing) fundamental vibrational mode. Vibrational frequencies of breathing normal modes for the pure and doped structures have been determined approximately by calculating the cluster effective energy Eefas a function of the small alg symmetry-adapted displacement, 6, of the f i s t coordination sphere of oxygen ions around the origin from their equilibrium positions (optimized distance in Tables 3 and 4). These displacementshave been treated as approximate normal modes, and the frequency Y has been calculated from the relation
v=L$ 2n
with IC=(-)
m
a24f
aQ2
0
where the symmetry-adapted coordinates Q are defined as
Q, = 6, Ig
Q, = 6, Ig
Q, = 6, Ig
18
k
4
TABLE 11: Orbital Energies (au), Orbital Exponents, and Expansion Coefficients for Aluminum: ls22s22p6'S 4(ls) expansion qj(2s) expansion (s) orbital coefficients coefficients n, 1 exponents (-58.400 497)" (-4.759 466)" 1s 12.819 300 0.904 849 93 -0.233 485 06 1s 19.468 800 0.044 542 73 -0.012 674 51 0.005 509 53 0.400 300 30 2s 5.508 510 2s 11.650 600 0.062 196 69 -0.121 433 49 2s 3.799 200 -0.001 054 53 0.709 151 65 4(2p) expansion (p) orbital
n, 1
2P 2P 2P 2P a
TABLE 12: Orbital Energies (au), Orbital Exponents, and Expansion Coefficients for Silicon: ls22s22p6IS 4( 1s) expansion qj(2s) expansion (s) orbital coefficients coefficients n, 1 exponents (-69.034 733)" (-6.279 255)a 1s 13.817 700 0.913 441 79 -0.242 778 11 1s 21.221 700 0.039 605 00 -0.011 118 18 2s 6,006 480 0.006 071 53 0.375 072 85 2s 12.530 600 -0.125 027 87 0.057 494 60 -0.001 540 85 2s 4.284 800 0.736 053 37
for the tetrahedral site
for the octahedral site
2P 2P 2P 2P
for the dodecahedral site IP a
and m is the reduced mass. The calculated force constants and the vibrational frequencies of the breathing mode, alg, for the pure and doped GROS and YAG structures at the octahedral, tetrahedral, and twelvefold sites are listed in Tables 8 and 9. Force constants are higher for GROS than YAG. These values increase in the following order: dodecahedral (1292- 1386 N m-l), tetrahedral (14872211 N m-l), and octahedral sites (2584-3340 N m-l). The vibrational frequencies are 414-429 cm-I for dodecahedral, 628-766 cm-I for tetrahedral, and 676-769 cm-I for octahedral sites. The doping process produces a decrease of k values for the YAG structure while an opposite effect appears in GROS.
5. Conclusions A theoretical study with the aiPI method of pure and doped (Cr3+ and C8+) garnet structures (Ca3A12Si3012, GROS, and Y3Al5Ol2, YAG) in conjunction with X-ray diffraction data has been carried out to obtain a systematic characterization and analysis of the geometry, energy, bulk modulus, force constants, and vibrational frequencies associated with these pure and doped crystals. The results of the present work can be summarized as follows: (i) The calculated lattice parameters and internal atomic coordinates agree well with experiments. (ii) The substitution of Cr3+ for A13+ in GROS is energetically favorable. (iii) Substitutions of Cr3+ for A13+ in YAG are unfavorable at both sites. Substitution is less unfavorable at the octahedral site. (iv) The substitution of Cfi+ for Si4+ in GROS is energetically unfavorable. (v) The structures of the cubic silicate garnets can
coefficients (-3.082 294)" 0.561 558 31 0.263 097 53 0.223 072 84 0.008 032 62
Orbital energies in au.
n,1
&
exponents 3.923 300 6.835 530 2.938 790 14.244 100
(p) orbital exponents 4.445 630 7.617 870 3.435 410 15.879 700
4(2p) expansion coefficients (-4.406 817)a 0.557 112 14 0.246 298 96 0.239 596 77 0.006 326 15
Orbital energies in au.
be rationalized in terms of a hard sphere model in spite of the diverse nature of the substituent cations, although some deviation appears in octahedral sites. (vi) Bulk modulus is higher for GROS. This means that GROS is less compressible than YAG. (vii) Force constants are higher for GROS. The doping process produces a decrease in the force constants of YAG, while the opposite effect appears in GROS.
Acknowledgment. Financial support from research funds of Fundaci6 Bancaixa and Universitat Jaume I is gratefully acknowledged. We are most indebted to V. Luaiia and A. Martin-Pendds for helpful discussions and to Centre de Processament de Dades at Universitat Jaume I for providing us with multiple computing facilities. Appendiv The particular features of the aiPI method have been developed by Pueyo et al.,'O and it is suitable for studying crystal defects in terms of clusters whose size and characteristics can properly model the effects induced by the presence of impurity ions. The most important characteristics of this method are briefly explained below: The aiPI method is an approximated cluster-lattice model based on the general theory of electronic separability. This method stresses the cluster-lattice interaction by reducing the cluster to an active group (A) coupled to the lattice through a rigorous quantum mechanical cluster-lattice interaction. The system wave function is written as an antisymmetrized product
6498 J. Phys. Chem., Vol. 99, No. 17,1995
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TABLE 13: Orbital Energies (au), Orbital Exponents, and Expansion Coefficients for Oxygen: ls22s22p6lS @(ls)expansion 4(2s) expansion 4(2p) expansion (s) orbital coefficients coefficients (p) orbital coefficients n, 1 exponents (-20.130 592)" (-0.993 419)" n, 1 exponents (-0.266 009)" 1s 13.425 500 0.038 376 77 -0.005 051 30 2P 7.820 440 0.028 975 17 1s 7.614 670 0.937 078 33 -0.203 353 07 2P 3.429 750 0.182 820 36 2s 6.314 910 0.035 320 98 -0.096 904 56 2P 1.743990 0.789 91 1 47 2s 3.212 690 0.003 323 19 0.419 114 55 2P 0.863 930 0.112 334 27 2s 1.750 760 0.000 361 34 0.704 002 66 2P 0.408 310 -0.095 16406 Orbital energies in au. TABLE 14: Orbital Energies (au), Orbital Exponents, and Expansion Coefficients for Yttrium: ls22s23s24s22p63p64p63d10 'S @( 1s) expansion 4(2s) expansion 4(3s) expansion 4(4s) expansion (s) orbital coefficients coefficients coefficients coefficients n, 1 exponents (-616.606 330)" (-85.673 877)" (-14.615 947)" (- 1.999 509)" 0.322 474 29 0.472 262 12 1s 32.835 100 -0.174 519 01 -0.086 121 19 1s 40.944 700 0.685 379 00 -0.048 772 89 0.005 567 65 0.017 525 99 -0.009 663 24 2s 27.458 000 -0.004 748 18 0.075 130 75 -0.080 334 79 2s 16.066600 -0.002 210 77 - 1.224 536 56 0.648 229 02 0.210 456 91 3s 14.296200 0.002 740 74 0.072 094 52 0.1 19 870 93 0.125 507 63 3s 9.109 370 -0.002 090 60 -0.056 964 81 -0.818 703 03 -0.493 520 20 3s 6.371 810 0.001 087 60 0.024 065 69 -0.443 798 61 -0.105 443 90 4s 3.845 460 -0.000 375 75 -0.007 430 87 0.021 084 48 0.878 568 02 0.000 278 04 0.005 339 36 -0.017 157 09 0.270 556 95 4s 2.579 020 -0.000 098 90 -0.001 836 52 0.006 690 68 -0.057 423 89 4s 1.771 920 ~~
4(2p) expansion coefficients (-78.028 299)" 0.815 211 46 0.111 308 34 0.086 742 23 0.022 166 66 -0.007 446 79 0.002 162 56 -0.000 482 54 0.000 213 09 -0.000 069 99
(p) orbital
exponents 17.036 600 26.043 800 15.510000 9.494 030 6.572 750 5.385 070 3.156 030 2.029 660 1.427 330
n, 1
2P 2P 3P 3P 3P 4P 4P 4P 4P
a
n, 1
(d) orbital exponents
3d 3d 3d 3d 3d
5.306 500 3.362 400 7.949 630 10.354 300 17.114 200
4(3p) expansion coefficients (-11.711 360)" 0.378 888 82 0.345 521 77 0.089 827 41 -0.534 074 64 -0.621 868 07 -0.000 474 96 -0.006 385 06 0.002 430 10 -0.000 352 46
4(4p) expansion coefficients (-1.111 973)" 0.148 280 05 0.005 101 95 0.004 479 25 -0.1 12 275 83 -0.438 793 43 0.315 049 76 0.771 498 15 0.219 631 01 -0.097 844 91
4(3d) expansion coefficients (-6.454 946)" 0.407 522 65 0.004 834 67 0.337 997 11 0.278 625 04 0.055 434 27
Orbital energies in au.
of the local wave function describing each group. Additional orthogonality requirements among groups allow us to express the total energy as the sum of intragroup or net energies plus intergroup or interaction e n e r g i e ~ . ~ ~ , ~ ~ Given a set of frozen groups (S f A), the minimization of the effective energy will produce the best wave function for the active group. This restricted variational procedure can be succesively applied to every different group, in order to determine the best system wave function compatible with the initial assumption of separability, as well as fully consistent group wave functions for the system. The contributions of the A group to the total energy can be collected in the effective energy:
'2
=:'et
+
C Ep,S =
e e t +E p t
('41)
SPA)
which, by minimization, gives the best @ A for a set of a given frozen groups. The total energy of the system is not the sum of the group effective energies. However, we can define the additive energy of the A group as
For an A3B2C3012 ionic crystal, such as the garnet structure, cations (A, B, C) and anions (0)are stabilized by ion-lattice interaction energy, and the crystal energy per molecule is Ecryst
= 3% ' dd = 3E:et
+ 2Efdd
'
3cdd
+ l2gdd
+ 2E:e, + 3ce+t12e', +
The lattice energy (Elatt)in the aiPI method is given by
E,,,, = Ecvst- (3Et
+ 2E: + 3 6 + 12%)
(A4)
where the subscript 0 stands for free-ion values. When the frozen groups are described by single Slater determinants (e.g. closed shells in a Hartree-Fock formalism), the effective energy can be expressed as the expected value of an effective Hamiltonian:
J. Phys. Chem., Vol. 99, No. 17, 1995 6499
Garnet Crystal Structures
TABLE 15: Orbital Energies (au), Orbital Exponents, and Expansion Coefficients for Chromium: ls22s23s22p63p63d3 4F (s) orbitai 4( 1s) expansion 4(2s) expansion 4(3s) expansion n, 1 exponents coefficients (-220.205 257)" coefficients (-26.033 944)" coefficients (-2.965 065)y 1s 24.202 500 -0.927 370 55 -0.276 763 65 0.112 920 74 -0.004 227 91 1s 35.466 500 -0.022 708 51 -0.007 667 47 -0.158 582 86 0.045 353 86 2s 21.575 400 -0.061 667 69 -0.331 368 62 2s 10.055 300 -0.001 476 67 -0.943 738 97 -0.290 385 01 -0.000 148 75 9.620 320 0.189 517 91 3s 0.518 685 64 0.OOO 738 10 3s 5.654 070 0.027 058 55 0.176 681 78 -0.012 850 70 -0.OOO 727 86 3s 4.080 700 0.000 311 64 0.537 943 03 3.329 770 0.007 587 41 3s (p) orbital exponents 10.788 100 16.807 300 9.459 240 5.877 480 3.489 390 1.769 970
n, 1
2P 2P 3P 3P 3P 3P
3d 3d 3d 3d 3d
4(3p) expansion coefficients (-1.661 031)" -0.231 402 06 -0.034 243 59 -0.181 121 92 0.327 801 30 0.771 896 12 0.078 483 24 4(3d) expansion coefficients (0.032 468)" 0.198 859 07 0.023 154 73 0.297 254 30 0.377 398 84 0.305 729 04
(d) orbital exponents 3.317 650 10.497 700 5.298 670 2.312 230 1.282 570
n, 1
a
4(2p) expansion coefficients (-21.959 532)" 0.674 898 84 0.087 641 10 0.271 762 28 0.019 539 04 -0.000 620 39 0.OOO 363 08
Orbital energies in au. The coupling coefficients have been taken from Malli and Oli~e.4~
where n refers to nuclei and e refers to electrons. is The active group effective Hamiltonian
where g runs over all occupied orbitals with orbital S energies cg. Interactions between the active group and the frozen groups nuclei, as well as between the active group electrons and the frozen groups nuclei, in eq A5 can be described as
(gf)
VP
where htdi) is the effective Hamiltonian of the ith electron, i and j run over the NA electrons of the active group, and a and /3 refer to the nuclei: VA
+ C[l$(i)+ ?(i)]
htdi) = [T(i)- ~ Z a r i a - ' ] a= 1
(A7)
StA
In eq A7, qdi) represents the potential energy of the ith electron of the active group in the field due to the S frozen group. This effective potential contains nuclear attraction, Coulombic electronic repulsion, and exange attraction contributions. Only the first two components contribute to the energy of the A nuclei in the field due to S : vs
+ c(i)+ C(i)= eoc +
e&i) = -CZprip-l p= 1
(e,)
(cc)
(A8)
If S has a closed shell structure, these terms turn into
4
where represents the effective potential of the S frozen group in the location of nucleus a in the active group. Please note that only the nuclear and Coulombic terms of the effective potential contribute to energy in eq A l l , since there are no interchange terms between electrons and nuclei. On the other hand, this energy remains constant if the nuclear geometries of the active and frozen groups are kept fixed, but it changes when one or more groups undergo vibrational motion. The electrostatic potential, S , can be conveniently divided and nonclassic terms: into classic
where g considers the occupied orbitals of S. and $ are the Coulomb and interchange operators, respectively. In eq A7, the projector p ( i ) tries to enforce the orthogonality constraints between the active group and the S frozen group. For a closed shell frozen group this operator takes the form
where v',Jr) represents the deviation arising from the finite extension of the ion's electronic density relative to a point-like charge. This operator can be adequately fit to BonifacicHuzinaga's expression?'
k= 1
The Madelung potential created by the surrounding ions
6500 J. Phys. Chem., Vol. 99, No. 17, 1995
Beltrh et al.
TABLE 16: Orbital Energies (au), Orbital Exponents, and Expansion Coemcients for Chromium: ls22s23s22p63p63d2 3F (s) orbital $(Is) expansion 4(2s) expansion 4(3s) expansion n, I exponents coefficients (-222.735 039)" coefficients (-28.631 847)" coefficients (-5.531 587)" 0.11000447 1s 23.751 800 -0.938 395 39 -0.281 700 91 0.001 025 03 1s -0.031 491 66 35.295 600 -0.009 292 84 0.054 877 72 -0.151 113 00 2s 2 1.472 900 -0.035 852 01 2s -0.372 267 35 -0.008 108 38 0.955 546 25 10.007 OOO 0.175 823 50 0.005 646 33 3s -0.243 963 67 9.573 950 0.258 861 53 5.855 610 0.026 618 20 -0.003 618 08 3s 0.003 543 80 0.446 262 21 -0.006 394 72 4.460 680 3s 0.001 749 70 3.670 590 -0.001 372 65 3s 0.488 863 70 n, 1
(p) orbital exponents
$(2p) expansion coefficients (-24.554 647)"
4(3p) expansion coefficients (-4.246 315)"
2P 2P 3P 3P 3P 3P
10.357 600 16.651 100 9.287 900 5.687 770 4.460 680 3.273 600
0.720 029 73 0.097 830 50 0.207 363 78 0.033 335 18 -0.012 607 46 0.002 245 62
-0.270 637 25 -0.035 246 70 -0.131 853 97 0.178 971 31 0.430 822 5 1 0.520 149 39 4(3d) expansion coefficients (-2.581 589)"
(d) orbital exponents 3.501 560 10.448 900 5.374 200 2.553 670 6.884 720
n,
3d 3d 3d 3d 3d
0.317 717 97 0.033 007 95 0.308 000 56 0.432 448 08 0.002 783 94
Orbital energies in au. The coupling coefficients have been taken from Malli and O l i ~ e . 4 ~
considered as point-like charges represents a large contribution to the interaction energy in ionic systems. The calculation of this term is difficult to overcome, being Macelung sums conditionally convergent: as the distance r = - RSI increases, the contribution of a single ion decrease as r-l, but the number of ions increases as 3. The angularly averaged Madelung potentials given by
('414) where S runs over the crystal i p s . To reach the above - RsI-' in terms of spherical expression, we can expand harmonics and perform the integration over the angles.I0 The exchange potential, however, is written as the nondiagonal spectral resolution:
where { lalp,S)} runs over the basis functions of S and 1 and m are the angular and azimuthal quantum numbers, respectively. The A(;lab,S)elements are obtained from the overlap (S) and exchange (K) one-center matrices according to
A = S-IKS-'
(A 16)
This spectral resolution is exact when a complete basis set is used. STO basis functions are used in this method because of their superior performance in all kinds of PI calculations.I0 The seminal idea of separability is to find electronic groups such that intergroup correlation energy plays a minor role and can be neglected. Intragroup correlation, however, contributes significantly to the cohesive properties, and it has to be computed using a size-consistent method that scales properly with the interionic distances. To estimate the magnitude of the
correlation energy, this method uses the Coulomb-HartreeFock (CHF) model proposed in 1965 by Clementi4*and recently reviewed by Chakravorty and Clementie30 In the CHF model, the J Coulomb-repulsionintegrals43are modified by introducing a spherical hole around each electron in which the other electrons cannot penetrate. The radius of this Coulomb hole for a particular integral depends on the overlap between the functions involved. Two parameters can scale the hole, and their values have been chosen to match the empirical correlation energy for He and Ne.30
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