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Ligand field analysis of the 3dN ions at orthorhombic or higher symmetry sites
Compufers Chem. Vol. 16, No. 3, pp. 207-216. 1992 Printed in Great Britain. All rights reserved
0097~8485/92
ss.cie + 0.00
Copyright Q 1992 Psrgamon Press Ltd
LIGAND FIELD ANALYSIS OF THE 3dN IONS AT ORTHORHOMBIC OR HIGHER SYMMETRY SITES Y. Y.
YEUNG’*
and C. RUDOWICZ~
‘Department of Applied Physics, Hong Kong Polytechnic, Hunghom, Hong Kong and ‘Department of Applied Science, City Polytechnic of Hong Kong, Kowloon, Hong Kong (Received
30 December
1991; in revised form 28 February
1992)
Ahatrae-A computer package has been developed to calculate the energy levels and the ligand field states for a full Hamiltonian including the electrostatic terms, Trees correction, the spin-orbit interaction and the ligand field interactions within the 3dN configuration (N = 2 to 8). The program can deal with various cubic and non-cubic site symmetries of the 3d ion, namely the tetragonal, trigonal and orthorhombic ones. The package may be very useful in analyzing the EPR spectra of 3dN ions and in correlating the ligand field parameters with spin Hamiltonian parameters derived from the orbital singlet states for some F-term (S = 1, 3/2) and S-term (S = 5/2) ions as well as the 3d” and 3d6 ions with high-spin S = 2. Numerous examples of these ions in various crystals exist in the literature. Other spin states arising from strong ligand fields can also be dealt with.
1. INTRODUCTION
Optical spectroscopy is a powerful tool in studying the electronic structure of transition-metal ions in a variety of materials (Lever, 1984; Sugano et al., 1970). A knowledge of the energy levels and ligand field parameters, or “crystal field” (CF) parameters as they are often called, enables a better understanding of the physical interactions, as well as predicting spectroscopic and magnetic properties required for specific technological applications. This. demands a means for correlating the optical spectroscopy data with electron paramagnetic resonance (EPR) data as well as Miissbauer spectroscopy data. In the transition-metal series, the class of 3d2 to 3d* ions, e.g. Tiz+, V2+, C?, Mr?+, Fe’+, Mn’+, F&‘, Co3*, Ni4+ and Cu3+, plays an important role in the physics and chemistry of materials (for references see e.g. Lever, 1984). Our recent literature survey which focused on 3d4 and 3d6 ions has revealed numerous cases of FeZ+ (3d6) ions with high-spin S = 2 at the orthorhombic symmetry as well as the tetragonal symmetry sites (Rudowicz, 1991). Several pertinent cases of the 3d4 (Cr2+, Mn’+, Fe4+) and 3d6 (Co”) ions have also be identified. For the Cr3+ (3d3) ion at noncubic sites in MgO, the splittings of the ground state 4A, and the first excited state *E have recently been studied (see e.g. Yeung, 1990; Choy & Yeung, 1990) using the CF parameters derived from data on the strain-induced splittings of Cr’+:Al,O, (Yeung & Newman, 1986). The splitting of the ground state %& for 3d5 (Fe3+ and Mn2+) ions with S = 5/2, which is of great theoretical interest (for references see e.g. Rudowicz, 1987a), has also been studied by us * Author
for correspondence. 207
(Yeung, 1988) using full diagonalization of the noncubic CF Hamiltonian. Independently, CF analysis within the 5D approximation (Abragam & Bleaney, 1986) for tetragonal and orthorhombic symmetry is currently being carried out. Applications to specific compounds dealt with so far include: Fez+ and Fe4+ ions in a high T, superconductor YBa,(Fe,Cu, _,),O, _6 (Rudowicz & Yu, 1991) and CrZ+ ion in the mixed system Rb*Mn,Cr, _XCl4 (Rudowicz & Zhou, 1992). These studies provide the energy levels and CF mixing coefficients, as well as the values of the spin-orbit and spin-spin coupling constants to be used as input data for the subsequent microscopic spin Hamiltonian analysis (for references see e.g. Rudowicz, 1989). CF analysis within the 5D approximation also provides background for more accurate analysis of optical and spin Hamiltonian data using the computer program for CF analysis within the whole 3d4 and 3d6 configuration presented here. In order to analyze the optical spectroscopy data using the crystal-field approach, an efficient computer program for diagonalization of a complete Hamiltonian (including electrostatic terms, Trees correction, spin-orbit interaction and crystal-field) within the 3dN configuration (N = 2 to 8) is indispensable. At present no such program is available in the public domain for general usage. Hence, individual researchers have either to spend a lot of time in developing their own programs or to rely on the TanabSugano ligand-field diagrams (see Kijnig & Kremer, 1977 for detailed tabulation) for partial information pertinent to axial symmetry only. In order to rectify the situation we have recently undertaken systematic development of a set of computer programs for 3dN configuration (N = 2 to 8) and for symmetry as low as orthorhombic.
Y. Y. YNNG and C.
208
The present version of the CF analysis computer programs is suitable for orthorhombic symmetry (point groups: Dz. CIv, DZh), and due to the ascent in symmetry method, for tetragonal symmetry (point groups: D,, C,, D,,, D,,) as well as cubic symmetry (point groups: T, Th, 0, Tdr 0,). Trigonal symmetry (point groups: DX, Ck, D3,) and hexagonal symmetry (point groups: C,, C,, , %, D,, G, h,, Dsh) have also been considered separately. Extensions to monoclinic and triclinic as well as trigonal and tetragonal symmetry cases involving “imaginary” CF terms are in progress. Note that in the latter cases (point groups: CJ, C,i and C,, C,, S,, respectively), the complex CF parameter can be made real by an appropriate rotation around the z-axis. Other low-spin states, e.g. S = 1, 0, for 3d4 and 3d6 ions and S = 3/2, l/2 for 3d5 ions, arising at strong ligand fields, as well as the low spin to high spin transition can also be studied using the package.
RUDOWICZ
matrix elements for the operators (4) are evaluated as follows: (i)
(@[X,1+‘)
in equations (2) to
= c { dN&SIIJ’ll dNa’L’S’>Fk k X ALL
where the two-particle
* 6,.
* hfs,
9 * Rm
tensor operators
4Ckfj )
fk = 1 P(i)
i
and their reduced matrix elements (llf11) are evaluated according to the formula given in the Appendix. The Slater integrals F* (k = 0, 2 and 4) are related to the Racah parameters A, B and C as follows (Gerloch & Slade, 1973, p. 50): A=F,-49F, B = F2 - 5Fa C = 35F,
2. THEORY For the transition metal 3dN ions, all the electronic shells except the 3d shell are spherically symmetric and so will not cause splittings of the energy levels. Hence, the Hamiltonian for a 3dN ion in crystal can be written as (for details see e.g. Ballhausen, 1962; Schlafer & Gliemann, 1969; Gerloch & Slade, 1973; Hfifner, 1978):
The free-ion Hamiltonian &?n (excluding the kinetic energy of electrons and their Coulomb attraction with the nucleus) consists of electrostatic repulsion amongst these 3d electrons:
2&f,i<,rij the spin-orbit
= i l@ik. i-1
10
(ii)
{tiIJP”,IJI’)=[ x
x(-l)S+L-“S-M
(dNaLS 1)Y” 1)d”a ‘L’S’)
x [(21 + 1)(6 + 1)1]“2c (- 1)’ P 1 S’ L 1 S
X
(
-MS
4
-M
>(
-q
L’ M’>
@I
(3)
(iii)
and the Trees correction &‘,, describing the twobody orbit-orbit polarization interaction (see e.g. Gerloch & Slade, 1973, p. 54). The crystal (ligand) field Hamiltonian in equation (1) in Wybourne’s notation (1965) is given by
(Jll.WT,I$‘>
= aL:(L + 1) x &,. &X6,.
where Bkqare the crystal field parameters and C$) are the renorrnalized spherical tensor operators. Since the ligand field is in the weak to intermediate range for most of the 3d transition metal ions in crystals, so the basis of states in the LS coupling scheme can be taken as: (5)
where GLis an extra quantum number (seniority) and the orbital quantum number I = d = 2. Hence the
6, MS
where a is the Trees parameter.
(4)
IQ> = (d%SM,LU)
M;
where (a * -) denotes the 3 -j symbols whereas the reduced matrix elements (11V”ll) are calculated explicitly according to the formula given in the Appendix. The spin-orbit coupling constant < is treated as an adjustable free-ion parameter for the 3d shell.
interaction: z,,
and F, = p, F2 = F2{49, F4 = F4/441. Usually we set A = 0 since it will cause no energy level splittings apart from shifting the centroid of the whole spectrum.
($lcy$)
=(-1y-M
L -M
k q
L’ M’>
x &.w&Mj where /3 1 dNaLS and = {dIICkll d> where
, with d=2.
Ligand field analysis of 3dN ions The reduced matrix elements for the unit operator Uk are calculated in the Appendix.
tensor
3. THE PROGRAM Our ligand field analysis package which is specifically developed for PC users consists of the following items: fi) Program name HefIdN.bas-source file can be edited or executed in QuickBasic version 4.5 or QBasic in MS DOS 5.0 in conjunction with the supplied library AtomLib. HcfLdN.EXE&standalone file executable in any IBM 286PC (preferably equipped with a 287 coprocessor) or compatible. (ii) Libraries AtomLll.Liba library developed by Y. Y. Yeung for general matrix manipulations and for calculating the atomic coefficients such as the 3j, 6j and 9j symbols. (iii) Supplementary jites Atomlib.INC-the included file, for declaration of subroutines/functions in Atomlib.Lib. DnRMVlLval, DnRMU4.va1, DnRMU2.va1, DnRMf2,val and DnRMF4,vrl-the reduced matrix elements of the operators Vu, Vz, U4, f2 and f4, respectively. DnRM . Lab-the aSL labels for the basis states defined by Nielson & Koster (1964). It should be kept in mind that since the electronic configuragion 3dN, N < 5, is conjugate to the configuration 3d lo-“, the matrix elements are identical in both cases (apart from a sign difference arising from the electron-hole equivalence). Hence, in giving supplementary file names for the reduced matrix elements, we should set n = N and n = 10 -N for N < 5 and N 2 5, respectively. Our package itself automatically inserts the appropriate signs to those matrix elements. Within our program, the matrix of the full Hamiltonian, equation (l), is partitioned into n submatrices according to the number n of the n-fold symmetry in a given (n, y, z) axes system when the spin-orbit interaction is ignored. However, if the spin-orbit interaction is included, then such a partition is not feasible as all LM quantum numbers will he mixed up. The flow-chart of our program is given in Fig. 1and the associated subprograms are defined as follows: (a) Calt Sub BuildCkL&-to build the reduced matrix elements for the operator Cc&)in aLS basis from the given reduced matrix elements of U”) in &S basis. CkFact-to
calculate
(!IICLII1) =(-
the quantities 1)‘(21+ 1) :, (
“D ; >
209
CetLSRM--to get reduced matrix elements for U2, U’ or Y” in the aLS basis from a given data file. HamiltonianAo construct the partitioned Hamiltonian .%?’in the aSMJ,M basis and to find its eigenvectors and eigenvalues using an external procedure in the library Atomlib.lib. InitStateLS--to initialize the aSL term labels. InitStateLM-to construct the c&44, LA4 basis. LSValue-to assign the numerical values to L and S from a given set of aLS labels. SortEstore-to sort all eigenvalues in increasing order. (‘b) GoSub Monltorlneto select what system to work on. Working-to work in a system, including construction, finding solution and storing up. ConstruetLSRM--to construct all the necessary reduced matrix elements in the c&L basis, including -)epcF, JEP,and Se,. ElectrostaticMat-to calculate the matrix elements in the aLS basis. for X_ + Xr_ CetParameters-to input parameters for a given system. PrintEnergy-to print out all energies. Prlntvector-to print out the energies and eigenvectors of a given number of low-lying states in each partitioned Hamiltonian with coefficients greater than a prescribed value. StoreNZvector-to store up a given number of eigenvectors in a file for each partitioned Hamiltonian with coefficients greater than a prescribed value. StoreEnergy--to store up all sorted energies in a file. Note that the ground state energy is set to zero. 4. DATA INPUT AND PARAMETERS CONVERSION The format of the input data file for the free ion parameters, viz. A, B, C, a and 4, and the CF parameters Bkr is given in Table 1. For illustration, the values of the parameters (in cm-‘) describing the tetragonal CF levels of Cr3+ in MgO (Yeung, 1990) without the spin-orbit interaction (i.e. c = 0) are presented in Table 2. Note that we may alternatively input the Slater integrals F’ instead of the Racah parameters A, B and C and the program makes the conversion itself. Ideally, the numerical values of the input parameters can either be. obtained from parameter fitting to the observed optical spectra or from ab initio calculations. However, the linewidth of the spectral lines for the 3d ions is usually quite broad, rendering an insufficient number of well-identified energy levels to fit all the necessary parameters. On the other hand, the ab initio calculations still cannot yield reliable CF parameter values especially for low symmetry cases. Therefore we may adopt the idea of transferability of
Y, Y. YEUNG and C. RUWWICZ
210
variable
types
Inifialization
:
InitStateLS,
LSVafue, GetParameters:, ConstructLSRM GkFac t , GetLSRM, BuildCkLS, ElectrostaticMat:
InitStateIM,
k--7l
Hamiltonian,
(Printvector:],
StoreNZvector:
I
StoreEnergy: Pr intEnergy
:
Monitoring:
Fig. 1.
Flow chart of the @and iield analysis package.
Table 1. Formal of the inputpaametcrs file “Title” “Parameters dexsipGm’* No. of the free iw pmmetem [default = 51 Values of A, B, c, 1. a No. of the n-fold symmetry around the z-axis if c = 0; otherwise I No. of the CF parameters k, q* B,, values
~ilcnamqt for the i.e. DnRMW.val, DnRMFO.vd.
matrixekmcnts of u’, U’, V”,f* andP DnRMUZ.val. DnRMVi Lval, DIIRMFZ.V&
reduced
parameters to estimate the parameter values of a given 3d ion doped in a new host crystal, based on its values in a similar but well-studied host crystal. This strategy is applied in a slightly different way for the free-ion parameters (i.e. A, 8, C, c and IX)and for the CF parameters. Since the former parameters are less sensitive to the crystalline environment, they can lx taken directly from the literature (e.g. Abragam & Bleaney, 1986). The fatter parameters strongly depend on &and type as well as on site symmetry. Fortunately, their number can be reduced by using
Ligand field analysis of 3dN ions Table 2. An example of the input parameters file called CRMGO.PAR. All the free-ion and CF parameters are in nn-’ “Cr3+ at C4v site in MgO without spin-orbit interaction.” “Parameters from Fairbank L Klauminrncr 1973 Phys.Rev.B7, 5 0, 570, 3165, 0, 70 : 2 :
0 04
211
arising from the p or dN configuration, then we have
respectively;
500.”
&‘c.=c
4550 20645 37065
4 -4 20645 “RME\D3RMU4,VAL” “RME\D3RMUZ.VAL” “RME\D3RMVI I.VAL” “RME\D3RMFZ,VAL” “RME\DJRMF4.VAL”
the superposition model (Newman, 1971; Rudowicz, 1987b) as a further parameterization as follows
where (I?,, 0,, 4j) are the polar coordinates of the ith ligand. The “intrinsic” parameter Bk represents the strength of the crystal field contribution from a given ligand type and the site symmetry effect is described by the “coordination factors” Kkq. & can be fitted either to a limited number of optical spectral levels or to the EPR spectra on the straininduced ground state splittings (see e.g. Yeung & Newman, 1986). The transferability of & for Cr3+ in oxygen ligand coordination has been found to be highly satisfactory (Yeung, 1988, 1990) in the sense that the calculated energy levels bear about 10% uncertainties and the eigenvector coefficients are fairly insensitive to the input parameters. Hence we may subsequently employ, with good confidence, the known values of & to predict the unknown values of the Bb parameters according to the superposition model, or alternative crystal field models such as the angular overlap model (Gerloch & Slade, 1973).
BIOl(J) k.q
or
HcF=xBXO@,), k.q
(7)
where the CF parameters B’: = AX{rk)Or, now incorporate the relevant radial (rk> and multiplicative factors 6, which depend on k only (Wyboume, 1965; Morrison & Leavitt, 1982). For orthorhombic symmetry the Sk, parameters in equation (4) can be chosen to be real. Then we have in equations (4), (6) and (7): k = 2, q = 0, 2, k = 4, q = 0, 2, 4 for the transition metal (3dN) ions. The relationships between Re(B@) in equation (4) and Az(rk> in equation (6) have been given by Wybourne (1965), Morrison & Leavitt (1982) and Hiifner {1978). Note the corrections to Wyboume’s (1965) relations provided by Eremin et al. (1970) and Kassman (1970). For easy reference the conversion factors are listed in Table 3. The ascent in symmetry method can be used to obtain, from the orthorhombic results, the corresponding ones for the tetragonal sites with symmetry given by the point groups: D4, Cl”, DX , and D4,,. By increasing symmetry from orthorhombic to tetragonal by setting B,, = Bd2-0, each of the orthorhombic cases reduces to an appropriate tetragonal case. Various types of conventional notations for CF parameters can be encountered for the transition metal (3d3 ions. In the present work, we choose the Wybourne’s Bkpparameters as they transform covariantly with the rotation matrix, rendering them much easier for transformation from one coordinate system to another. Usually, the ligand field is separated into a cubic component with parameters BzblC plus a noncubic one with parameters E& so that Bkq= Bzbic + BL are defined as follows: (a) In a rrigonal axis system, equation (4) becomes
4. I. Conversion relations for crystal jeld parameters There are two major types of operators used in the area of optical spectroscopy, viz. the spherical tensor operators [STO] (Wyboume, 1965; Morrison & Leavitt, 1982) and the (extended) Stevens operators [ESO] (see e.g. Newman, 1971; Rudowicz, 1985). The confusion concerning the properties of the latter operators has recently been clarified thus enabling extension of these operators to the negative (q < 0) components (Rudowicz, 1985, 1987). The CF Hamiltonian in terms of ST0 has been given in equation (4), whereas its equivalent form in terms of ES0 can be written as
(6) I
However, the Stevens operators have been used only for &+cF within a given J- or L-multiplet
@) In a terragonal axis system, equation (4) can be written as
We shall consider the two main kinds of noncubic axes systems mentioned above, viz. trigonal and tetragonal, for the F-term ions (i.e. 3dZ, 3d3, 3d’ and 3d*) and the D-term ions (i.e. 3d’, 3d4, 3d6 and 3d9). The relationships between Wyboume’s (1965) CF
Table 3. Conversion factors Re(Bk,)/Al(rX> for the crystal field parameters in Wyboume’s E*, and Stevens’ A: notations k/a
0
2
3
4
2 4
2
Jri3 2J@
-2/J%
4&G
8
Y. Y. YEUNGand C. Ruwwlcz
212
Table4. Relationships ktween the Wyboume’s(1965) cubic B;” and noncubic B;, crystal field paramctemand the convenrianalligandfieldparametersdefinedfor the F-tern and D-termions in the literature. IODq = A is the splitting of the 3dN ground tern due to the cubic ligand field D-term ions
F-term iohs
Trigonal axis system BP= -_14Dq
p”c 43 -- p*ic 1.-l = +2fi Pryce d; Runciman’s (1958) u & u’ B;, I v - bv’/.,/Z
Dq Ballhausen’s (1962) Ds & DI Bh= -7Da
EL= -21 Dr
B;, = $(v + 3v’l,,f2) T&ago4
axis system
B:* -21 Dq Byp=By!$= Grifith’s (1962) 6 & p B&=6-a B;= --;(?p +a)
k3mDq Bdlhawen’s (1962) Ds & Df Bi,= -7Ds B&= -21 Dt orrhorhombicDC & Dq B&z= B;,,,=
parameters and the various ligand field parameters introduced by Pryce & Runciman (1958), Ballhausen (1962) and Griffith (1961) for the F-term and D-term
*Cr3+ 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
at
C4v site 0.0000 14516.1725 16450.3298 21266.7009 21928.5482 29396.8140 30988.6819 32419.2891 34727.4453 36990.5090 38287.0118 41112.9414 4839841017 49625.7463 52466.1650 64112.4352 70184.2802
in ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
MgO without 26) 2 13) 5 28) 29) 1; 16) 14 4) 17 30) 20 18) 23 44) 26 45) 29 46) 32 33) 35 22) 38 48) 41 11) 44 36) 47 12) 50
Dq
of the apparent higher rhombicity expressed by the experimental parameter B: or BIL(Rudowicz, 1991).
ions are given in Table 4. Relations with other less
common ligand field parameters have been discussed by Ktinig & Kremer (1977). It has been shown (Rudowicz & Bramley, 1985) that for an orthorhombic Xc, the ratio of @/BP = I and & /Bm= K can always be confined, by a pro r transformation, to the range (0, 1) and (0, l/ Je 6), respectively. This intrinsic property of the orthorhombic &‘cr has an important physical comequence which has not been fully recognized in the CF literature as yet. It leads to an upper limit on the maximum rhombicity, i.e. on the asymmetry parameter 1 (K), for a given ion/crystal case irrespective
7Jsizw - 21@
5. OUTPUT AND RESULTS An output file of a given name will store the sequential numbers as well as quantum numbers of all KSM, LM states used, a selected set of eigenvalues
and eigenvectors for all or some of the low-lying states and a complete set of energies sorted in ascending order. The intermediate results can be directed either to the screen or to a printer for monitoring purposes. The resulting energies without spin-orbit interaction for our example, i.e. the tetragonal system Cr’+:MgO, are given in Fig. 2. The numbers in brackets are the original sequential numbers for the
spin-orbit 14248.4146 14516.1725 17081.4374 21343.5845 21928.5482 30681.1895 31108.2676 32728.8558 34727.4453 36990.5090 38287 a0118 47712.3741 48398.1017 50424.2988 53507.0755 68506 a0326 72497.3988
interaction ( 27) 3 ( 38) 6 ( 14) 9 ( 15) 12 ( 41) 15 ( 17) 18 ( 5) 21 ( 6) 24 ( 19) 27 ( 20) 30 ( 21) 33 ( 34) 36 ( 47) 39 ( 10) 42 ( 49) 45 ( 50) 48 ( 37)
in crmgo.par” 14331.5756 14741.5702 17081.4374 21343.5845 24946.5297 30681.1895 32419.2891 32850.5677 35953.8000 37795.5165 40597.0693 47994.3050 49625.7463 50425.0650 53507.0755 68506.0326
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
Fig. 2. Calculated energies in an-’ for Cr’+ at a tetragonal site in MgO without spin-rbit interaction.
1) 2) 39) 40) 3) 42) 43) 31) 7) 8) 32) 9) 23) 35) 24) 25)
213
Ligand field analysis of 3dN ions
“Cr3+
at
C4v
4 12 1 4P 4 2Dl 7 2G 10 2H 12 1 -0.100075 0.300483 ii.495454 0.363706 3 0.719840 4 -0.101886 -0.042328 5 0.562266 -0.471915 6 -0.290370 -0.492535 7 -0.694140 8 -0.409866 -0.760460 9 -0.029536 0.006583 10 0.535514 0.254733 11 -0.519153 -0.258207 12 0.786698 -0.316921 13 1 4P 4 2P 7 2F 10 2G 13 2H 12 13 0.487888 -0.256908
site
1.5 0.5 0.5 0.5
4 10 3 11 1 5 12
0 0 -4 -4
2 5 a 11
4F 2D2 2G 2H
spin-orbit
1.5 0.5 0.5 0.5
0 0 0 0
-13958.58953880421 0.421988 0.460192 5 -0.300483 12 -13548.59485818768 0.429871 -0.233067 6 0.314917 12 -3343.635388705245 0.694140 2 1106.648915496702 0.748701 0.461239 7
3 11 4 10 1 3 11 3 11 4 10 3 11 4 10 1.5 0.5 0.5 0.5 0.5
4 9
in MgO without
2818.102502052525 0.386718 0.181470 6 0.254086 12 4438.69071384086 0.027682 0.645806 5 0.492535 12 7663.634935948367 0.719840 2 9505.351418168408 -0.500391 6 -0.035478 -0.02oola 12 19704.13993272921 0.003054 6 0.405239 -0.579068 12 22134.13367304788 0.591274 5 -0.238636 -0.254733 12 24175.99987334538 0.813848 6 0.013541 0.023395 12 41894.11510452662 0.226873 -0.105698 5 0.316921 12 1 1 -3 1 5
2 5 8 11
4F 2Dl 2F 2H
1.5 0.5 0.5 0.5
-3 1 1 -3
-13773.99258536732 0.105351 5 -0.058724 0.187813 -0.536350 10
Fig. 3. Partial printout of the eigenvcctors
interaction
3 6 9 12
2P 2F 26 2H
in crmgo . par M
0.5 0.5 0.5 0.5
0 0 4 4
7 7 -0.491406
8
0.421988
9
7 7 -0.429871
9
0.314917
10
0.461239
9
0.042328
10
7 7 -0.386718
9
0.254086
10
7 7
0.109115
a
0.027682
9
0.035478
9
-0.020018
10
-0.405239
9
-0.579068
10
0.346334
8
-0.238636
9
7 7 -0.013541
9
0.023395
10
7 7 -0.257125
8
0.226873
9
-0.094114 -0.416296
8 13
2 6 8
2 7 7 7 7
77
3 6 9 12
4F 2D2 2G 2H
1.5 0.5 0.5 0.5
10 6 -0.217547 11 -0.359146
for Cr3+ at tetragonal interaction.
1 1 -3 1
7 12
site in MgO without
spin-t-bit
Y. Y. YEWNG and C. Ruwwtcz
214
eigenvectors. A partial printout of the eigenvectors is given in Fig. 3 and the format of the printout is defined as follows: Title line annexed with the filename for input pornmeters.
n (fold of symmetry axis).
No. of uSM,ILM states in the Hamilionian s&matrix = Ns.
N, sets of (state sequential No., aSL label, M,, M). No. of low-lying eigenstates stored = N.~ N, sets of eigenstates {eigenstate sequential No., elgenvalue, No. of eigenuector components = NV, NVsets of (eigenvecfar coejkient, state sequential No.)]. RESULTS FOR ANOTHER TONIAN SUBMATRIX. a.
“Cr3+ 1
4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 103 106 109 112 115 118
PARTITIONED
at C4v site in MnQ 0.0000
0.1450 14298.8625 14458.0997 14742.9698 16446.3262 17078.1811 17106.2920 17141.6475 21168.5657 21351.5865 21897.9488 21983.8602 22029.8284 24971.5134 29406.5924 30745.1260 30991.4628 32372.5213 32442.2775 32858.0161 34706.4995 35986.6199 35995.1489 36979.0938 37009.0319 37838.0467 38260.5491 40611.1821 41128.2561 48008.9676 48387.5938 49637.7029 49654.5894 50442.9093 52473.3393 53527.5551 64127.6906 68611.4600 70207.2622
(
i,
( 4) ( 7) ( 10) ( 13) ( 16) ( 19) ( 22) ( 25) ( 28) ( 31) ( 34) ( 37) ( 40) ( 43) ( 46) ( 49) ( 52) ( 55) ( 58) ( 61) ( 64) ( 67) ( 70) ( 73) ( 76) ( 79) ( 82) ( 85) ( 88) ( 91) ( 94) ( 97) (100) (103) (106) (109) (112) (115) (118)
HAMIL-
To include the spin-orbit interaction, we put 1; = 240 cm-’ and replaced the quantity n = 4 by 1 in Table 2. Then the calculated values of the ground state ‘A, splitting and the first excited state *E splittings are found to he 0.145 cm-’ and 87.1 cm-‘, respectively, matching fairly well with the experimental values of 0.164 and 93.5 cm-’ (for references see Yeung, 1990). It has been demonstrated by Yeung (1990) that the conventional perturbation calculation would give a very significant error for the 2E splitting in comparison with the complete matrix diagonalization approach. In fact, our whole set of calculated energies (Fig. 4)
in crmgoso
2 5 8 11 14 17 20 23 26 29 32 35 38
41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98 101 104 107 110 113 116 119
.uar” b.0000 14211.7265 14298.8625 14535.1735 14742.9698 16449.5172 17078.1811 17107.7445 17141.6475 21350.4800 21351.5865 21981.7571 21983.8602 24960.0686 24971.5134 30611.0773 30745.1260 31121.0513 32372.5213 32749.1740 32858.0161 34748.8478 35986.6199 36949.4934 36979.0938 37061.6063 37838.0467 38376.7124 40611.1821 47726.2946 48008.9676 48436.6284 49637.7029 50438.7875 50442.9093 53525.0825 53527.5551 68436.0107 68611.4600 72521.0614
Fig. 4. Calculated energies in cm-’ for Cr’+ at a tetragonal I = 24ocm-‘.
( 2) ( 5) ( 8) ( 11) ( 14) ( 17) ( 20) ( 23) ( 26) ( 29) ( 32) ( 35) ( 38) ( 41) ( 44) ( 47) ( 50) ( 53) ( 56) ( 59) ( 62) ( 65) ( 68) ( 71) ( 74) ( 77) ( 80) ( 83) ( 86) ( 89) ( 92) ( 95) ( 98) (101) (104) (107) (110) (113) (116) (119)
3 6 9 12 15 18 21. 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120
0.1450 14211.7265 14458.0997 14535.1735 16446.3262 16449.5172 17106.2920 17107.7445 21168.5657 21350.4800 21897.9488 21981.7571 22029.8284 24960.0686 29406.5924 30611.0773 30991.4628 31121.0513 32442.2775 32749.1740 34706.4995 34748.8478 35995.1489 36949.4934 37009.0319 37061.6063 38260.5491 38376.7124 41128.2561 47726.2946 48387.5938 48436.6284 49654.5894 50438.7875 52473.3393 53525.0825 64127.6906 68436.0107 70207.2622 72521.0614
site in MgO with spin-orbit
( 3) ( 6) ( 9) ( 12) ( 15) ( 18) ( 21) ( 24) ( 27) ( 30) ( 33) ( 36) ( 39) ( 42) ( 45) ( 48) ( 51) ( 54) ( 57) ( 60) ( 63) ( 66) ( 69) ( 72) ( 75) ( 78) ( 81) ( 84) ( 87) ( 90) ( 93) ( 96) ( 99) (102) (105) (108) (111) (114) (117) (120)
interaction
Ligand field analysis is identical to that given by Fairbank & Klauminzer (1973), Table VII), thus verifying the validity of our package for a 36’ ion.
6. CONCLUSIONS The program calculates the energy levels and CF states in the aSMsLM basis of states for any 3dN (N = 2 to 8) ion, at various symmetry sites including the orthorhombic, tetragonal and trigonal ones. This program has been tested and found to be valid for the CT)+ ion at a tetragonal site in MgO. It should facilitate identification of the energy level schemes for each particular ion/symmetry-type case occurring in crystals. Especially useful applications are envisaged for the energy level schemes with an orbital singlet ground state, e.g. the high-spin (S = 2) 3d6 and 3d4 ion complexes, since high frequency and highmagnetic field EPR studies have recently become feasible (for references see e.g. Rudowicz, 1989). Another important point is that the full ligand (crystal) field analysis provides a check of the validity of the 5D approximation used in the previous papers. The program enables consideration of the role of states arising from the higher-lying *‘+‘L multiplets which may be of comparable energies with the ‘D term states due to strong CF. Preliminary calculations indicate this is the case for Fe2+ ion at some orthorhombic sites in YBa,(Fe,Cu, _X)3O7_6. Furthermore, the calculated crystal field eigenvectors in the present c&&LM basis of states (instead of the alternative uSLJM, basis), are particularly useful for the subsequent calculation of the spectroscopic splitting g-factor under magnetic field as well as for the analysis of EPR data (see e.g. Yeung & Newman, 1986; Yeung, 1988), with or without strain, on the ground state splittings of orbital singlets arising from the S-term or F-term ions in some crystals.
Program availability-A copy of the package which includes the computer program, the state labels, the relevant reduced matrix elements and the user’s document is obtainable from Dr Y. Y. Yeung by sending a 5;” 1.2M high density diskette and an address label to the Department of Applied Physics, Hong Kong Polytechnic, Hunghom, Hong Kong. Acknowledgemenrs-Financial support from University and Polytechnic Grants Committee and the City Polytechnic of Hong Kong as well as Hong Kong Polytechnic Research Subcommittee is gratefully acknowledged.
REFERENCES Abragam A. & Bleaney B. (1986) Electron Paramagnetic Resonance of Transition Ions. Dover, New York. Ballhausen C. J. (1962) Introducrion 10 Ligand Field Theory. McGraw-Hill, New York. Choy T. L. & Yeung Y. Y. (1990) Phys. Status Solidi B 161, K107.
of 3dN ions
215
Condon E. U. & Odabasi H. (1980) Alomic Swucture. Cambridge University Press, London. Eremin M. V., Kurkin 1. N., Mar’yakhina 0. I. & Shekun L. Ya. (1970) Sovie1 Phys. Solid State 11, 1697. Fairbank W. M. & Klauminzer G. K. (1973) Whys. Rev. B 7, 500. Gerloch M. and Slade R. C. (1973) LigandField Parameters. Cambridge University Press, London. Griffith J. S. (1961) Theory of Transirion Metal Ions. Cambridge University Press, London. Hiifner S. (1978) Optical Spectra of Transparent Rare Earth Compounds. Academic Press, New York. Kassman A. J. (1970) J. Chem. Phys. 53, 4118. KBnig E. and Kremer S. (1977) Ligand Field Energy Diagrams. Plenum Press, New York. Lever A. B. P. (1984) Inorganic Spectroscopy, 2nd edn. Elsevier, Amsterdam. Morrison C. A. & Leavitt R. P. (1982) Handbook on Ihe Physics and Chemistry of Rare Earths (Edited by Gschneider K. A. Jr & Erying L.), Vol. 5, p. 461. Amsterdam, North-Holland. Newman D. J. (1971) Adv. Phys. 20, 197. Nielson C. W. & Koster G. F. (1964) Spectroscopic Coeflcients for p”, d” and f” Configurations. MIT Press, Massachusetts. hyce M. H. L. & Runciman W. A. (1958) discuss. Faraday sot. 26, 34. Rudowicz C. (1985) J. Phys. C: Solid State Phys. 18, 1415; Erratum. Ibidem C18, 3837. Rudowicz C. (1987a) Magn. Res. Rev. 13, I. Rudowicz C. (1987b) J. Phys. C 20, 6033. Rudowicz C. (1989) Physica B 155, 336. Rudowicz C. (1991) Mol. Phys. 74, 1159. Rudowicz C. % Bramley R. (1985) J. Chem. Phys. 83, 5192. Rudowicz C. & Yu W-L. (199t) Supercond. Sri. Technoi. 4, 535. Rudowicz C. & Zhou Y. Y. (1992) J. Mag. Mug. Mat. In press. Schlafer H. L. and Gliemann G. (1969) Basic Principles of L&and Field Theory. Wiley-Interscience, New York. Sugano S., Tanabe Y. & Kamimura H. (1970) MuhQlers of Transition-Metal Ions in Ctrystals. Academic Press, New York. Wybourne B. G. (1965) Spectroscopic Properties of Rareearth. Wiley, New York. Yeung Y. Y. & Newman D. J. (1986) Phys. Rev. B 34,2258. Yeung Y. Y. (1988) J. Phys. C: Solid State Phys. 21, 2453. Yeung Y. Y. (1990) J. Phys. C: Condens. Matter 2, 2461.
APPENDIX The reduced matrix elements for the orbital tensor operators U” s 5 &i) i= I and the double tensor operator N
If”‘)
s
N
C u’*(i) = C s,u’(i),
can be evaluated from the coefficients of fractional parentage (iN, aSL{ll”-‘a’S’L’) tabulated by Nielson & Koster (1964). From Condon and Odabasi (1980, p. 273, Eq. (E)), we have
; ;,,} x ,_E‘ (--I) L+L.+,+k.{;, x (I”, aSLIIIN-‘a”S”L”)(lN,
a’S’L’{IIN-‘a”S”L”)
216
Y. Y. YEUNGand C. RUMIWICZ
and
(PaLSII Y(“)IIPa’L’S’)
=:(lllCq1)*
= N[(2L + 1)(2S + 1)]“‘[(2L’+ 1)(2S’+ I)]‘” L+L”+I*x+s+s+3,*
X ~ti~‘_(-l,
x {i X (IN,
7
:,,}I;,
1
~,,}(rN,~SL{l~N--I~.ISYLI)
K’S’L’{I/N-‘K”S”L”).
From Condon and Odabasi (19g0, p. 275), we obtain also
1 X (iNK,iSI(
-h’6,‘1
&mzyw+L”
(/k(~~NK”~“S”)(~NK”~“S*~~~k~~fNK’~~)
which reduces to (N(N - 1)/2) 6,. for k. = 0. All the spectroscopic coefficients used above have been calculated numerically with double precision and stored in data files to be used as input data for our program.
0097~8485/92
ss.cie + 0.00
Copyright Q 1992 Psrgamon Press Ltd
LIGAND FIELD ANALYSIS OF THE 3dN IONS AT ORTHORHOMBIC OR HIGHER SYMMETRY SITES Y. Y.
YEUNG’*
and C. RUDOWICZ~
‘Department of Applied Physics, Hong Kong Polytechnic, Hunghom, Hong Kong and ‘Department of Applied Science, City Polytechnic of Hong Kong, Kowloon, Hong Kong (Received
30 December
1991; in revised form 28 February
1992)
Ahatrae-A computer package has been developed to calculate the energy levels and the ligand field states for a full Hamiltonian including the electrostatic terms, Trees correction, the spin-orbit interaction and the ligand field interactions within the 3dN configuration (N = 2 to 8). The program can deal with various cubic and non-cubic site symmetries of the 3d ion, namely the tetragonal, trigonal and orthorhombic ones. The package may be very useful in analyzing the EPR spectra of 3dN ions and in correlating the ligand field parameters with spin Hamiltonian parameters derived from the orbital singlet states for some F-term (S = 1, 3/2) and S-term (S = 5/2) ions as well as the 3d” and 3d6 ions with high-spin S = 2. Numerous examples of these ions in various crystals exist in the literature. Other spin states arising from strong ligand fields can also be dealt with.
1. INTRODUCTION
Optical spectroscopy is a powerful tool in studying the electronic structure of transition-metal ions in a variety of materials (Lever, 1984; Sugano et al., 1970). A knowledge of the energy levels and ligand field parameters, or “crystal field” (CF) parameters as they are often called, enables a better understanding of the physical interactions, as well as predicting spectroscopic and magnetic properties required for specific technological applications. This. demands a means for correlating the optical spectroscopy data with electron paramagnetic resonance (EPR) data as well as Miissbauer spectroscopy data. In the transition-metal series, the class of 3d2 to 3d* ions, e.g. Tiz+, V2+, C?, Mr?+, Fe’+, Mn’+, F&‘, Co3*, Ni4+ and Cu3+, plays an important role in the physics and chemistry of materials (for references see e.g. Lever, 1984). Our recent literature survey which focused on 3d4 and 3d6 ions has revealed numerous cases of FeZ+ (3d6) ions with high-spin S = 2 at the orthorhombic symmetry as well as the tetragonal symmetry sites (Rudowicz, 1991). Several pertinent cases of the 3d4 (Cr2+, Mn’+, Fe4+) and 3d6 (Co”) ions have also be identified. For the Cr3+ (3d3) ion at noncubic sites in MgO, the splittings of the ground state 4A, and the first excited state *E have recently been studied (see e.g. Yeung, 1990; Choy & Yeung, 1990) using the CF parameters derived from data on the strain-induced splittings of Cr’+:Al,O, (Yeung & Newman, 1986). The splitting of the ground state %& for 3d5 (Fe3+ and Mn2+) ions with S = 5/2, which is of great theoretical interest (for references see e.g. Rudowicz, 1987a), has also been studied by us * Author
for correspondence. 207
(Yeung, 1988) using full diagonalization of the noncubic CF Hamiltonian. Independently, CF analysis within the 5D approximation (Abragam & Bleaney, 1986) for tetragonal and orthorhombic symmetry is currently being carried out. Applications to specific compounds dealt with so far include: Fez+ and Fe4+ ions in a high T, superconductor YBa,(Fe,Cu, _,),O, _6 (Rudowicz & Yu, 1991) and CrZ+ ion in the mixed system Rb*Mn,Cr, _XCl4 (Rudowicz & Zhou, 1992). These studies provide the energy levels and CF mixing coefficients, as well as the values of the spin-orbit and spin-spin coupling constants to be used as input data for the subsequent microscopic spin Hamiltonian analysis (for references see e.g. Rudowicz, 1989). CF analysis within the 5D approximation also provides background for more accurate analysis of optical and spin Hamiltonian data using the computer program for CF analysis within the whole 3d4 and 3d6 configuration presented here. In order to analyze the optical spectroscopy data using the crystal-field approach, an efficient computer program for diagonalization of a complete Hamiltonian (including electrostatic terms, Trees correction, spin-orbit interaction and crystal-field) within the 3dN configuration (N = 2 to 8) is indispensable. At present no such program is available in the public domain for general usage. Hence, individual researchers have either to spend a lot of time in developing their own programs or to rely on the TanabSugano ligand-field diagrams (see Kijnig & Kremer, 1977 for detailed tabulation) for partial information pertinent to axial symmetry only. In order to rectify the situation we have recently undertaken systematic development of a set of computer programs for 3dN configuration (N = 2 to 8) and for symmetry as low as orthorhombic.
Y. Y. YNNG and C.
208
The present version of the CF analysis computer programs is suitable for orthorhombic symmetry (point groups: Dz. CIv, DZh), and due to the ascent in symmetry method, for tetragonal symmetry (point groups: D,, C,, D,,, D,,) as well as cubic symmetry (point groups: T, Th, 0, Tdr 0,). Trigonal symmetry (point groups: DX, Ck, D3,) and hexagonal symmetry (point groups: C,, C,, , %, D,, G, h,, Dsh) have also been considered separately. Extensions to monoclinic and triclinic as well as trigonal and tetragonal symmetry cases involving “imaginary” CF terms are in progress. Note that in the latter cases (point groups: CJ, C,i and C,, C,, S,, respectively), the complex CF parameter can be made real by an appropriate rotation around the z-axis. Other low-spin states, e.g. S = 1, 0, for 3d4 and 3d6 ions and S = 3/2, l/2 for 3d5 ions, arising at strong ligand fields, as well as the low spin to high spin transition can also be studied using the package.
RUDOWICZ
matrix elements for the operators (4) are evaluated as follows: (i)
(@[X,1+‘)
in equations (2) to
= c { dN&SIIJ’ll dNa’L’S’>Fk k X ALL
where the two-particle
* 6,.
* hfs,
9 * Rm
tensor operators
4Ckfj )
fk = 1 P(i)
i
and their reduced matrix elements (llf11) are evaluated according to the formula given in the Appendix. The Slater integrals F* (k = 0, 2 and 4) are related to the Racah parameters A, B and C as follows (Gerloch & Slade, 1973, p. 50): A=F,-49F, B = F2 - 5Fa C = 35F,
2. THEORY For the transition metal 3dN ions, all the electronic shells except the 3d shell are spherically symmetric and so will not cause splittings of the energy levels. Hence, the Hamiltonian for a 3dN ion in crystal can be written as (for details see e.g. Ballhausen, 1962; Schlafer & Gliemann, 1969; Gerloch & Slade, 1973; Hfifner, 1978):
The free-ion Hamiltonian &?n (excluding the kinetic energy of electrons and their Coulomb attraction with the nucleus) consists of electrostatic repulsion amongst these 3d electrons:
2&f,i<,rij the spin-orbit
= i l@ik. i-1
10
(ii)
{tiIJP”,IJI’)=[ x
x(-l)S+L-“S-M
(dNaLS 1)Y” 1)d”a ‘L’S’)
x [(21 + 1)(6 + 1)1]“2c (- 1)’ P 1 S’ L 1 S
X
(
-MS
4
-M
>(
-q
L’ M’>
@I
(3)
(iii)
and the Trees correction &‘,, describing the twobody orbit-orbit polarization interaction (see e.g. Gerloch & Slade, 1973, p. 54). The crystal (ligand) field Hamiltonian in equation (1) in Wybourne’s notation (1965) is given by
(Jll.WT,I$‘>
= aL:(L + 1) x &,. &X6,.
where Bkqare the crystal field parameters and C$) are the renorrnalized spherical tensor operators. Since the ligand field is in the weak to intermediate range for most of the 3d transition metal ions in crystals, so the basis of states in the LS coupling scheme can be taken as: (5)
where GLis an extra quantum number (seniority) and the orbital quantum number I = d = 2. Hence the
6, MS
where a is the Trees parameter.
(4)
IQ> = (d%SM,LU)
M;
where (a * -) denotes the 3 -j symbols whereas the reduced matrix elements (11V”ll) are calculated explicitly according to the formula given in the Appendix. The spin-orbit coupling constant < is treated as an adjustable free-ion parameter for the 3d shell.
interaction: z,,
and F, = p, F2 = F2{49, F4 = F4/441. Usually we set A = 0 since it will cause no energy level splittings apart from shifting the centroid of the whole spectrum.
($lcy$)
=(-1y-M
L -M
k q
L’ M’>
x &.w&Mj where /3 1 dNaLS and
, with d=2.
Ligand field analysis of 3dN ions The reduced matrix elements for the unit operator Uk are calculated in the Appendix.
tensor
3. THE PROGRAM Our ligand field analysis package which is specifically developed for PC users consists of the following items: fi) Program name HefIdN.bas-source file can be edited or executed in QuickBasic version 4.5 or QBasic in MS DOS 5.0 in conjunction with the supplied library AtomLib. HcfLdN.EXE&standalone file executable in any IBM 286PC (preferably equipped with a 287 coprocessor) or compatible. (ii) Libraries AtomLll.Liba library developed by Y. Y. Yeung for general matrix manipulations and for calculating the atomic coefficients such as the 3j, 6j and 9j symbols. (iii) Supplementary jites Atomlib.INC-the included file, for declaration of subroutines/functions in Atomlib.Lib. DnRMVlLval, DnRMU4.va1, DnRMU2.va1, DnRMf2,val and DnRMF4,vrl-the reduced matrix elements of the operators Vu, Vz, U4, f2 and f4, respectively. DnRM . Lab-the aSL labels for the basis states defined by Nielson & Koster (1964). It should be kept in mind that since the electronic configuragion 3dN, N < 5, is conjugate to the configuration 3d lo-“, the matrix elements are identical in both cases (apart from a sign difference arising from the electron-hole equivalence). Hence, in giving supplementary file names for the reduced matrix elements, we should set n = N and n = 10 -N for N < 5 and N 2 5, respectively. Our package itself automatically inserts the appropriate signs to those matrix elements. Within our program, the matrix of the full Hamiltonian, equation (l), is partitioned into n submatrices according to the number n of the n-fold symmetry in a given (n, y, z) axes system when the spin-orbit interaction is ignored. However, if the spin-orbit interaction is included, then such a partition is not feasible as all LM quantum numbers will he mixed up. The flow-chart of our program is given in Fig. 1and the associated subprograms are defined as follows: (a) Calt Sub BuildCkL&-to build the reduced matrix elements for the operator Cc&)in aLS basis from the given reduced matrix elements of U”) in &S basis. CkFact-to
calculate
(!IICLII1) =(-
the quantities 1)‘(21+ 1) :, (
“D ; >
209
CetLSRM--to get reduced matrix elements for U2, U’ or Y” in the aLS basis from a given data file. HamiltonianAo construct the partitioned Hamiltonian .%?’in the aSMJ,M basis and to find its eigenvectors and eigenvalues using an external procedure in the library Atomlib.lib. InitStateLS--to initialize the aSL term labels. InitStateLM-to construct the c&44, LA4 basis. LSValue-to assign the numerical values to L and S from a given set of aLS labels. SortEstore-to sort all eigenvalues in increasing order. (‘b) GoSub Monltorlneto select what system to work on. Working-to work in a system, including construction, finding solution and storing up. ConstruetLSRM--to construct all the necessary reduced matrix elements in the c&L basis, including -)epcF, JEP,and Se,. ElectrostaticMat-to calculate the matrix elements in the aLS basis. for X_ + Xr_ CetParameters-to input parameters for a given system. PrintEnergy-to print out all energies. Prlntvector-to print out the energies and eigenvectors of a given number of low-lying states in each partitioned Hamiltonian with coefficients greater than a prescribed value. StoreNZvector-to store up a given number of eigenvectors in a file for each partitioned Hamiltonian with coefficients greater than a prescribed value. StoreEnergy--to store up all sorted energies in a file. Note that the ground state energy is set to zero. 4. DATA INPUT AND PARAMETERS CONVERSION The format of the input data file for the free ion parameters, viz. A, B, C, a and 4, and the CF parameters Bkr is given in Table 1. For illustration, the values of the parameters (in cm-‘) describing the tetragonal CF levels of Cr3+ in MgO (Yeung, 1990) without the spin-orbit interaction (i.e. c = 0) are presented in Table 2. Note that we may alternatively input the Slater integrals F’ instead of the Racah parameters A, B and C and the program makes the conversion itself. Ideally, the numerical values of the input parameters can either be. obtained from parameter fitting to the observed optical spectra or from ab initio calculations. However, the linewidth of the spectral lines for the 3d ions is usually quite broad, rendering an insufficient number of well-identified energy levels to fit all the necessary parameters. On the other hand, the ab initio calculations still cannot yield reliable CF parameter values especially for low symmetry cases. Therefore we may adopt the idea of transferability of
Y, Y. YEUNG and C. RUWWICZ
210
variable
types
Inifialization
:
InitStateLS,
LSVafue, GetParameters:, ConstructLSRM GkFac t , GetLSRM, BuildCkLS, ElectrostaticMat:
InitStateIM,
k--7l
Hamiltonian,
(Printvector:],
StoreNZvector:
I
StoreEnergy: Pr intEnergy
:
Monitoring:
Fig. 1.
Flow chart of the @and iield analysis package.
Table 1. Formal of the inputpaametcrs file “Title” “Parameters dexsipGm’* No. of the free iw pmmetem [default = 51 Values of A, B, c, 1. a No. of the n-fold symmetry around the z-axis if c = 0; otherwise I No. of the CF parameters k, q* B,, values
~ilcnamqt for the i.e. DnRMW.val, DnRMFO.vd.
matrixekmcnts of u’, U’, V”,f* andP DnRMUZ.val. DnRMVi Lval, DIIRMFZ.V&
reduced
parameters to estimate the parameter values of a given 3d ion doped in a new host crystal, based on its values in a similar but well-studied host crystal. This strategy is applied in a slightly different way for the free-ion parameters (i.e. A, 8, C, c and IX)and for the CF parameters. Since the former parameters are less sensitive to the crystalline environment, they can lx taken directly from the literature (e.g. Abragam & Bleaney, 1986). The fatter parameters strongly depend on &and type as well as on site symmetry. Fortunately, their number can be reduced by using
Ligand field analysis of 3dN ions Table 2. An example of the input parameters file called CRMGO.PAR. All the free-ion and CF parameters are in nn-’ “Cr3+ at C4v site in MgO without spin-orbit interaction.” “Parameters from Fairbank L Klauminrncr 1973 Phys.Rev.B7, 5 0, 570, 3165, 0, 70 : 2 :
0 04
211
arising from the p or dN configuration, then we have
respectively;
500.”
&‘c.=c
4550 20645 37065
4 -4 20645 “RME\D3RMU4,VAL” “RME\D3RMUZ.VAL” “RME\D3RMVI I.VAL” “RME\D3RMFZ,VAL” “RME\DJRMF4.VAL”
the superposition model (Newman, 1971; Rudowicz, 1987b) as a further parameterization as follows
where (I?,, 0,, 4j) are the polar coordinates of the ith ligand. The “intrinsic” parameter Bk represents the strength of the crystal field contribution from a given ligand type and the site symmetry effect is described by the “coordination factors” Kkq. & can be fitted either to a limited number of optical spectral levels or to the EPR spectra on the straininduced ground state splittings (see e.g. Yeung & Newman, 1986). The transferability of & for Cr3+ in oxygen ligand coordination has been found to be highly satisfactory (Yeung, 1988, 1990) in the sense that the calculated energy levels bear about 10% uncertainties and the eigenvector coefficients are fairly insensitive to the input parameters. Hence we may subsequently employ, with good confidence, the known values of & to predict the unknown values of the Bb parameters according to the superposition model, or alternative crystal field models such as the angular overlap model (Gerloch & Slade, 1973).
BIOl(J) k.q
or
HcF=xBXO@,), k.q
(7)
where the CF parameters B’: = AX{rk)Or, now incorporate the relevant radial (rk> and multiplicative factors 6, which depend on k only (Wyboume, 1965; Morrison & Leavitt, 1982). For orthorhombic symmetry the Sk, parameters in equation (4) can be chosen to be real. Then we have in equations (4), (6) and (7): k = 2, q = 0, 2, k = 4, q = 0, 2, 4 for the transition metal (3dN) ions. The relationships between Re(B@) in equation (4) and Az(rk> in equation (6) have been given by Wybourne (1965), Morrison & Leavitt (1982) and Hiifner {1978). Note the corrections to Wyboume’s (1965) relations provided by Eremin et al. (1970) and Kassman (1970). For easy reference the conversion factors are listed in Table 3. The ascent in symmetry method can be used to obtain, from the orthorhombic results, the corresponding ones for the tetragonal sites with symmetry given by the point groups: D4, Cl”, DX , and D4,,. By increasing symmetry from orthorhombic to tetragonal by setting B,, = Bd2-0, each of the orthorhombic cases reduces to an appropriate tetragonal case. Various types of conventional notations for CF parameters can be encountered for the transition metal (3d3 ions. In the present work, we choose the Wybourne’s Bkpparameters as they transform covariantly with the rotation matrix, rendering them much easier for transformation from one coordinate system to another. Usually, the ligand field is separated into a cubic component with parameters BzblC plus a noncubic one with parameters E& so that Bkq= Bzbic + BL are defined as follows: (a) In a rrigonal axis system, equation (4) becomes
4. I. Conversion relations for crystal jeld parameters There are two major types of operators used in the area of optical spectroscopy, viz. the spherical tensor operators [STO] (Wyboume, 1965; Morrison & Leavitt, 1982) and the (extended) Stevens operators [ESO] (see e.g. Newman, 1971; Rudowicz, 1985). The confusion concerning the properties of the latter operators has recently been clarified thus enabling extension of these operators to the negative (q < 0) components (Rudowicz, 1985, 1987). The CF Hamiltonian in terms of ST0 has been given in equation (4), whereas its equivalent form in terms of ES0 can be written as
(6) I
However, the Stevens operators have been used only for &+cF within a given J- or L-multiplet
@) In a terragonal axis system, equation (4) can be written as
We shall consider the two main kinds of noncubic axes systems mentioned above, viz. trigonal and tetragonal, for the F-term ions (i.e. 3dZ, 3d3, 3d’ and 3d*) and the D-term ions (i.e. 3d’, 3d4, 3d6 and 3d9). The relationships between Wyboume’s (1965) CF
Table 3. Conversion factors Re(Bk,)/Al(rX> for the crystal field parameters in Wyboume’s E*, and Stevens’ A:
0
2
3
4
2 4
2
Jri3 2J@
-2/J%
4&G
8
Y. Y. YEUNGand C. Ruwwlcz
212
Table4. Relationships ktween the Wyboume’s(1965) cubic B;” and noncubic B;, crystal field paramctemand the convenrianalligandfieldparametersdefinedfor the F-tern and D-termions in the literature. IODq = A is the splitting of the 3dN ground tern due to the cubic ligand field D-term ions
F-term iohs
Trigonal axis system BP= -_14Dq
p”c 43 -- p*ic 1.-l = +2fi Pryce d; Runciman’s (1958) u & u’ B;, I v - bv’/.,/Z
Dq Ballhausen’s (1962) Ds & DI Bh= -7Da
EL= -21 Dr
B;, = $(v + 3v’l,,f2) T&ago4
axis system
B:* -21 Dq Byp=By!$= Grifith’s (1962) 6 & p B&=6-a B;= --;(?p +a)
k3mDq Bdlhawen’s (1962) Ds & Df Bi,= -7Ds B&= -21 Dt orrhorhombicDC & Dq B&z= B;,,,=
parameters and the various ligand field parameters introduced by Pryce & Runciman (1958), Ballhausen (1962) and Griffith (1961) for the F-term and D-term
*Cr3+ 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
at
C4v site 0.0000 14516.1725 16450.3298 21266.7009 21928.5482 29396.8140 30988.6819 32419.2891 34727.4453 36990.5090 38287.0118 41112.9414 4839841017 49625.7463 52466.1650 64112.4352 70184.2802
in ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
MgO without 26) 2 13) 5 28) 29) 1; 16) 14 4) 17 30) 20 18) 23 44) 26 45) 29 46) 32 33) 35 22) 38 48) 41 11) 44 36) 47 12) 50
Dq
of the apparent higher rhombicity expressed by the experimental parameter B: or BIL(Rudowicz, 1991).
ions are given in Table 4. Relations with other less
common ligand field parameters have been discussed by Ktinig & Kremer (1977). It has been shown (Rudowicz & Bramley, 1985) that for an orthorhombic Xc, the ratio of @/BP = I and & /Bm= K can always be confined, by a pro r transformation, to the range (0, 1) and (0, l/ Je 6), respectively. This intrinsic property of the orthorhombic &‘cr has an important physical comequence which has not been fully recognized in the CF literature as yet. It leads to an upper limit on the maximum rhombicity, i.e. on the asymmetry parameter 1 (K), for a given ion/crystal case irrespective
7Jsizw - 21@
5. OUTPUT AND RESULTS An output file of a given name will store the sequential numbers as well as quantum numbers of all KSM, LM states used, a selected set of eigenvalues
and eigenvectors for all or some of the low-lying states and a complete set of energies sorted in ascending order. The intermediate results can be directed either to the screen or to a printer for monitoring purposes. The resulting energies without spin-orbit interaction for our example, i.e. the tetragonal system Cr’+:MgO, are given in Fig. 2. The numbers in brackets are the original sequential numbers for the
spin-orbit 14248.4146 14516.1725 17081.4374 21343.5845 21928.5482 30681.1895 31108.2676 32728.8558 34727.4453 36990.5090 38287 a0118 47712.3741 48398.1017 50424.2988 53507.0755 68506 a0326 72497.3988
interaction ( 27) 3 ( 38) 6 ( 14) 9 ( 15) 12 ( 41) 15 ( 17) 18 ( 5) 21 ( 6) 24 ( 19) 27 ( 20) 30 ( 21) 33 ( 34) 36 ( 47) 39 ( 10) 42 ( 49) 45 ( 50) 48 ( 37)
in crmgo.par” 14331.5756 14741.5702 17081.4374 21343.5845 24946.5297 30681.1895 32419.2891 32850.5677 35953.8000 37795.5165 40597.0693 47994.3050 49625.7463 50425.0650 53507.0755 68506.0326
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
Fig. 2. Calculated energies in an-’ for Cr’+ at a tetragonal site in MgO without spin-rbit interaction.
1) 2) 39) 40) 3) 42) 43) 31) 7) 8) 32) 9) 23) 35) 24) 25)
213
Ligand field analysis of 3dN ions
“Cr3+
at
C4v
4 12 1 4P 4 2Dl 7 2G 10 2H 12 1 -0.100075 0.300483 ii.495454 0.363706 3 0.719840 4 -0.101886 -0.042328 5 0.562266 -0.471915 6 -0.290370 -0.492535 7 -0.694140 8 -0.409866 -0.760460 9 -0.029536 0.006583 10 0.535514 0.254733 11 -0.519153 -0.258207 12 0.786698 -0.316921 13 1 4P 4 2P 7 2F 10 2G 13 2H 12 13 0.487888 -0.256908
site
1.5 0.5 0.5 0.5
4 10 3 11 1 5 12
0 0 -4 -4
2 5 a 11
4F 2D2 2G 2H
spin-orbit
1.5 0.5 0.5 0.5
0 0 0 0
-13958.58953880421 0.421988 0.460192 5 -0.300483 12 -13548.59485818768 0.429871 -0.233067 6 0.314917 12 -3343.635388705245 0.694140 2 1106.648915496702 0.748701 0.461239 7
3 11 4 10 1 3 11 3 11 4 10 3 11 4 10 1.5 0.5 0.5 0.5 0.5
4 9
in MgO without
2818.102502052525 0.386718 0.181470 6 0.254086 12 4438.69071384086 0.027682 0.645806 5 0.492535 12 7663.634935948367 0.719840 2 9505.351418168408 -0.500391 6 -0.035478 -0.02oola 12 19704.13993272921 0.003054 6 0.405239 -0.579068 12 22134.13367304788 0.591274 5 -0.238636 -0.254733 12 24175.99987334538 0.813848 6 0.013541 0.023395 12 41894.11510452662 0.226873 -0.105698 5 0.316921 12 1 1 -3 1 5
2 5 8 11
4F 2Dl 2F 2H
1.5 0.5 0.5 0.5
-3 1 1 -3
-13773.99258536732 0.105351 5 -0.058724 0.187813 -0.536350 10
Fig. 3. Partial printout of the eigenvcctors
interaction
3 6 9 12
2P 2F 26 2H
in crmgo . par M
0.5 0.5 0.5 0.5
0 0 4 4
7 7 -0.491406
8
0.421988
9
7 7 -0.429871
9
0.314917
10
0.461239
9
0.042328
10
7 7 -0.386718
9
0.254086
10
7 7
0.109115
a
0.027682
9
0.035478
9
-0.020018
10
-0.405239
9
-0.579068
10
0.346334
8
-0.238636
9
7 7 -0.013541
9
0.023395
10
7 7 -0.257125
8
0.226873
9
-0.094114 -0.416296
8 13
2 6 8
2 7 7 7 7
77
3 6 9 12
4F 2D2 2G 2H
1.5 0.5 0.5 0.5
10 6 -0.217547 11 -0.359146
for Cr3+ at tetragonal interaction.
1 1 -3 1
7 12
site in MgO without
spin-t-bit
Y. Y. YEWNG and C. Ruwwtcz
214
eigenvectors. A partial printout of the eigenvectors is given in Fig. 3 and the format of the printout is defined as follows: Title line annexed with the filename for input pornmeters.
n (fold of symmetry axis).
No. of uSM,ILM states in the Hamilionian s&matrix = Ns.
N, sets of (state sequential No., aSL label, M,, M). No. of low-lying eigenstates stored = N.~ N, sets of eigenstates {eigenstate sequential No., elgenvalue, No. of eigenuector components = NV, NVsets of (eigenvecfar coejkient, state sequential No.)]. RESULTS FOR ANOTHER TONIAN SUBMATRIX. a.
“Cr3+ 1
4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 103 106 109 112 115 118
PARTITIONED
at C4v site in MnQ 0.0000
0.1450 14298.8625 14458.0997 14742.9698 16446.3262 17078.1811 17106.2920 17141.6475 21168.5657 21351.5865 21897.9488 21983.8602 22029.8284 24971.5134 29406.5924 30745.1260 30991.4628 32372.5213 32442.2775 32858.0161 34706.4995 35986.6199 35995.1489 36979.0938 37009.0319 37838.0467 38260.5491 40611.1821 41128.2561 48008.9676 48387.5938 49637.7029 49654.5894 50442.9093 52473.3393 53527.5551 64127.6906 68611.4600 70207.2622
(
i,
( 4) ( 7) ( 10) ( 13) ( 16) ( 19) ( 22) ( 25) ( 28) ( 31) ( 34) ( 37) ( 40) ( 43) ( 46) ( 49) ( 52) ( 55) ( 58) ( 61) ( 64) ( 67) ( 70) ( 73) ( 76) ( 79) ( 82) ( 85) ( 88) ( 91) ( 94) ( 97) (100) (103) (106) (109) (112) (115) (118)
HAMIL-
To include the spin-orbit interaction, we put 1; = 240 cm-’ and replaced the quantity n = 4 by 1 in Table 2. Then the calculated values of the ground state ‘A, splitting and the first excited state *E splittings are found to he 0.145 cm-’ and 87.1 cm-‘, respectively, matching fairly well with the experimental values of 0.164 and 93.5 cm-’ (for references see Yeung, 1990). It has been demonstrated by Yeung (1990) that the conventional perturbation calculation would give a very significant error for the 2E splitting in comparison with the complete matrix diagonalization approach. In fact, our whole set of calculated energies (Fig. 4)
in crmgoso
2 5 8 11 14 17 20 23 26 29 32 35 38
41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98 101 104 107 110 113 116 119
.uar” b.0000 14211.7265 14298.8625 14535.1735 14742.9698 16449.5172 17078.1811 17107.7445 17141.6475 21350.4800 21351.5865 21981.7571 21983.8602 24960.0686 24971.5134 30611.0773 30745.1260 31121.0513 32372.5213 32749.1740 32858.0161 34748.8478 35986.6199 36949.4934 36979.0938 37061.6063 37838.0467 38376.7124 40611.1821 47726.2946 48008.9676 48436.6284 49637.7029 50438.7875 50442.9093 53525.0825 53527.5551 68436.0107 68611.4600 72521.0614
Fig. 4. Calculated energies in cm-’ for Cr’+ at a tetragonal I = 24ocm-‘.
( 2) ( 5) ( 8) ( 11) ( 14) ( 17) ( 20) ( 23) ( 26) ( 29) ( 32) ( 35) ( 38) ( 41) ( 44) ( 47) ( 50) ( 53) ( 56) ( 59) ( 62) ( 65) ( 68) ( 71) ( 74) ( 77) ( 80) ( 83) ( 86) ( 89) ( 92) ( 95) ( 98) (101) (104) (107) (110) (113) (116) (119)
3 6 9 12 15 18 21. 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120
0.1450 14211.7265 14458.0997 14535.1735 16446.3262 16449.5172 17106.2920 17107.7445 21168.5657 21350.4800 21897.9488 21981.7571 22029.8284 24960.0686 29406.5924 30611.0773 30991.4628 31121.0513 32442.2775 32749.1740 34706.4995 34748.8478 35995.1489 36949.4934 37009.0319 37061.6063 38260.5491 38376.7124 41128.2561 47726.2946 48387.5938 48436.6284 49654.5894 50438.7875 52473.3393 53525.0825 64127.6906 68436.0107 70207.2622 72521.0614
site in MgO with spin-orbit
( 3) ( 6) ( 9) ( 12) ( 15) ( 18) ( 21) ( 24) ( 27) ( 30) ( 33) ( 36) ( 39) ( 42) ( 45) ( 48) ( 51) ( 54) ( 57) ( 60) ( 63) ( 66) ( 69) ( 72) ( 75) ( 78) ( 81) ( 84) ( 87) ( 90) ( 93) ( 96) ( 99) (102) (105) (108) (111) (114) (117) (120)
interaction
Ligand field analysis is identical to that given by Fairbank & Klauminzer (1973), Table VII), thus verifying the validity of our package for a 36’ ion.
6. CONCLUSIONS The program calculates the energy levels and CF states in the aSMsLM basis of states for any 3dN (N = 2 to 8) ion, at various symmetry sites including the orthorhombic, tetragonal and trigonal ones. This program has been tested and found to be valid for the CT)+ ion at a tetragonal site in MgO. It should facilitate identification of the energy level schemes for each particular ion/symmetry-type case occurring in crystals. Especially useful applications are envisaged for the energy level schemes with an orbital singlet ground state, e.g. the high-spin (S = 2) 3d6 and 3d4 ion complexes, since high frequency and highmagnetic field EPR studies have recently become feasible (for references see e.g. Rudowicz, 1989). Another important point is that the full ligand (crystal) field analysis provides a check of the validity of the 5D approximation used in the previous papers. The program enables consideration of the role of states arising from the higher-lying *‘+‘L multiplets which may be of comparable energies with the ‘D term states due to strong CF. Preliminary calculations indicate this is the case for Fe2+ ion at some orthorhombic sites in YBa,(Fe,Cu, _X)3O7_6. Furthermore, the calculated crystal field eigenvectors in the present c&&LM basis of states (instead of the alternative uSLJM, basis), are particularly useful for the subsequent calculation of the spectroscopic splitting g-factor under magnetic field as well as for the analysis of EPR data (see e.g. Yeung & Newman, 1986; Yeung, 1988), with or without strain, on the ground state splittings of orbital singlets arising from the S-term or F-term ions in some crystals.
Program availability-A copy of the package which includes the computer program, the state labels, the relevant reduced matrix elements and the user’s document is obtainable from Dr Y. Y. Yeung by sending a 5;” 1.2M high density diskette and an address label to the Department of Applied Physics, Hong Kong Polytechnic, Hunghom, Hong Kong. Acknowledgemenrs-Financial support from University and Polytechnic Grants Committee and the City Polytechnic of Hong Kong as well as Hong Kong Polytechnic Research Subcommittee is gratefully acknowledged.
REFERENCES Abragam A. & Bleaney B. (1986) Electron Paramagnetic Resonance of Transition Ions. Dover, New York. Ballhausen C. J. (1962) Introducrion 10 Ligand Field Theory. McGraw-Hill, New York. Choy T. L. & Yeung Y. Y. (1990) Phys. Status Solidi B 161, K107.
of 3dN ions
215
Condon E. U. & Odabasi H. (1980) Alomic Swucture. Cambridge University Press, London. Eremin M. V., Kurkin 1. N., Mar’yakhina 0. I. & Shekun L. Ya. (1970) Sovie1 Phys. Solid State 11, 1697. Fairbank W. M. & Klauminzer G. K. (1973) Whys. Rev. B 7, 500. Gerloch M. and Slade R. C. (1973) LigandField Parameters. Cambridge University Press, London. Griffith J. S. (1961) Theory of Transirion Metal Ions. Cambridge University Press, London. Hiifner S. (1978) Optical Spectra of Transparent Rare Earth Compounds. Academic Press, New York. Kassman A. J. (1970) J. Chem. Phys. 53, 4118. KBnig E. and Kremer S. (1977) Ligand Field Energy Diagrams. Plenum Press, New York. Lever A. B. P. (1984) Inorganic Spectroscopy, 2nd edn. Elsevier, Amsterdam. Morrison C. A. & Leavitt R. P. (1982) Handbook on Ihe Physics and Chemistry of Rare Earths (Edited by Gschneider K. A. Jr & Erying L.), Vol. 5, p. 461. Amsterdam, North-Holland. Newman D. J. (1971) Adv. Phys. 20, 197. Nielson C. W. & Koster G. F. (1964) Spectroscopic Coeflcients for p”, d” and f” Configurations. MIT Press, Massachusetts. hyce M. H. L. & Runciman W. A. (1958) discuss. Faraday sot. 26, 34. Rudowicz C. (1985) J. Phys. C: Solid State Phys. 18, 1415; Erratum. Ibidem C18, 3837. Rudowicz C. (1987a) Magn. Res. Rev. 13, I. Rudowicz C. (1987b) J. Phys. C 20, 6033. Rudowicz C. (1989) Physica B 155, 336. Rudowicz C. (1991) Mol. Phys. 74, 1159. Rudowicz C. % Bramley R. (1985) J. Chem. Phys. 83, 5192. Rudowicz C. & Yu W-L. (199t) Supercond. Sri. Technoi. 4, 535. Rudowicz C. & Zhou Y. Y. (1992) J. Mag. Mug. Mat. In press. Schlafer H. L. and Gliemann G. (1969) Basic Principles of L&and Field Theory. Wiley-Interscience, New York. Sugano S., Tanabe Y. & Kamimura H. (1970) MuhQlers of Transition-Metal Ions in Ctrystals. Academic Press, New York. Wybourne B. G. (1965) Spectroscopic Properties of Rareearth. Wiley, New York. Yeung Y. Y. & Newman D. J. (1986) Phys. Rev. B 34,2258. Yeung Y. Y. (1988) J. Phys. C: Solid State Phys. 21, 2453. Yeung Y. Y. (1990) J. Phys. C: Condens. Matter 2, 2461.
APPENDIX The reduced matrix elements for the orbital tensor operators U” s 5 &i) i= I and the double tensor operator N
If”‘)
s
N
C u’*(i) = C s,u’(i),
can be evaluated from the coefficients of fractional parentage (iN, aSL{ll”-‘a’S’L’) tabulated by Nielson & Koster (1964). From Condon and Odabasi (1980, p. 273, Eq. (E)), we have
; ;,,} x ,_E‘ (--I) L+L.+,+k.{;, x (I”, aSLIIIN-‘a”S”L”)(lN,
a’S’L’{IIN-‘a”S”L”)
216
Y. Y. YEUNGand C. RUMIWICZ
and
(PaLSII Y(“)IIPa’L’S’)
=:(lllCq1)*
= N[(2L + 1)(2S + 1)]“‘[(2L’+ 1)(2S’+ I)]‘” L+L”+I*x+s+s+3,*
X ~ti~‘_(-l,
x {i X (IN,
7
:,,}I;,
1
~,,}(rN,~SL{l~N--I~.ISYLI)
K’S’L’{I/N-‘K”S”L”).
From Condon and Odabasi (19g0, p. 275), we obtain also
1 X (iNK,iSI(
-h’6,‘1
&mzyw+L”
(/k(~~NK”~“S”)(~NK”~“S*~~~k~~fNK’~~)
which reduces to (N(N - 1)/2) 6,. for k. = 0. All the spectroscopic coefficients used above have been calculated numerically with double precision and stored in data files to be used as input data for our program.