3588
C. A. L. BECKER, D. W. MEEK,AND T. M. DUNN
tance-concentration curves (A os. X,) for this system.4~5 We interpret these concentrations to be the concentrations at which the solvent sheath of the electrolyte is completed. Two p-xylene molecules seem to be required to complete the solvation sheath of a pair of electrolyte ions. This suggests that probably only one
of the ions acquires a firmly bound solvation shell. This might be the smaller SCN- ion. More work is necessary to elucidate this interpretation. Acknowledgment. The author wishes to express his indebtedness to Dr. E. C. Evers for encouraging him to pursue this line of investigation.
Crystal Field-Spin Orbit Treatment in dl and d9 Trigonal Bipyramidal Complexes by Clifford A. L. Becker, Devon W. Meek, Department of Chemistry, The Ohio State University, Columbus, Ohio 43210
and T. M. Dunn Department of Chemistry, The University of Michigan, Ann Arbor, Michigan
48104
(Received September 11, 1967)
The energy matrices for combined crystal field-spin orbit perturbation of the d1 and d9 configurations in trigonal bipyramidal environment have been constructed. Basis functions are selected according to thg weak- and strong-crystalline-fieldapproximations,and the generalized D i ( a , 0, y) rotation matrices for angularmomenta eigenfunctions ( j = ”2, 5/2; I/%, 2) are constructed and used to determine their transformation properties. The crystal-field matrix elements are expressed in terms of the parameters Dq and Db. Secular determinants resulting from either approximation are equivalent and indicate perturbation of the d orbitals into five energy levels. Spacing between these energy levels depends on the relative magnitudes of crystallinefield parameters and spin-orbit coupling constants, so in their generalized forms, the energy matrices are applicable to any d1 or d9 configuration in trigonal bipyramidal environment.
Introduction Calculations of the electronic perturbations for the atomic d orbitals in the trigonal bipyramidal environment have usually dealt exclusively with Coulombic forces on the central metal ion arising from the effective point-charge or point-dipole natures of the neighboring ligands (that is, crystal-field theory). In this treatment of the perturbation energy levels within DBh symmetry, however, we wish to simultaneously include the perturbation arising from electromagnetic forces of coupled orbital and spin angular momenta (that is, spin-orbit interaction) and thereby achieve a “complete” calculation2 of the perturbation energies. This method of calculation is practical, if not actually necessary, for meaningful results from first-order perturbation theory for metal ions of the second and third transition series and probably also for the last members of the first row. I n these cations, of course, the crystal-field parameters are no longer sufficiently greater than the spin-orbit coupling constants that crystal-field effects overshadow those of spin-orbit interaction. For the purpose of calculation, Liehr’s convention The Journal of Physical Chemistry
of the weak-field and strong-crystalline-field approxim a t i o n ~has ~ been adopted. I n the weak-field approximation the Ij, m) wave functions for the free ion serve as basis functions which are then linearly combined to satisfy the symmetry requirements of the crystalline field. In the strong-field approximation, the basis functions are taken as the crystalline-field (orbital) eigenfunctions, and the symmetry (total) eigenfunctions are constructed after spin has been introduced. In both approximations the symmetry eigenfunctions are appropriate for the perturbation secular determinant of combined crystal field-spin orbital interaction
x’= vCF(r,e, ‘p) + E(Y)X
(1)
Before constructing the symmetry eigenfunctions, how(1) Calculations such as those given by, for example: (a) P. Day, Proc. Chem. SOC.,18 (1964); (b) M.Ciampolini, Inorg. Chem., 5, 35 (1966); (c) M.J. Norgett, J. H. M. Thornley, and L. M. Venanzi, J . Chem. SOC.,A , 540 (1967). (2) “Exact” or “complete” calculations within, of course, the natural limitations inherent to crystal-field and firstorder perturbation theories. (3) A. D. Liehr, J . Phys. Chem., 64, 43 (1960).
d1 AND de TRIGONAL BIPYRAMIDAL COMPLEXES
3589
ever, it is necessary and useful to consider the transformation matrices that will be used. Rotation Matrices. Many sets of Eulerian angles of rotation are used for orbital transformations, but perhaps the most familiar to chemists are the (cp, 0, 1c/) convention of Goldstein4 and the ( a , p, y) convention according to Rose.5 Both transformation schemes were used to investigate the specific rotations of DBhsymmetry, but the latter convention has been adopted for this work, since that is the scheme of Eulerian angles defined for the Dj(a, 0,y) transformation matrices according to Tinkham6r7and Roses5 I n the weak-field approximation the DaIZ(a,p, y) and D s ~ z ( 0, a , y) matrices may be applied directly in obtaining the transformation properties of the basis functions, since the j , m) functions are eigenfunctions of the total angular momentum. In the strong-field approximation, it is found most convenient to employ the D 2 ( a ,0,y) matrix for rotation of the orbital functions and the D1”(a, p, 7)matrix for rotation of the spin functions-this separability of the transformation operator is permitted since 1 and s remain commuting quantum numbers. The vectorial forms of the angular-momentum eigenfunctions must always be used with the Dj(a, p, y) matrices, and the Condon = and Shortley phase convention8 with regard to 2, mz = 1) is a p p r ~ p r i a t e . ~The explicit forms of W ( a , p, y), j = ”2, 2, and 5 / / 2 , are given in Appendix 1. Character I’ables. To calculate the symmetry properties for eigenfunctions of half-integral angular momenta, one may use the device of double groups. The double-group character tables pertinent to trigonal bipyramidal geometry are found in Appendix 2.1°
1
IZ
Weak-Field Approximation Basis Functions. I n the absence of the crystalline field, the Hamiltonian operator for a d’ configuration reduces to that for a free atom
x=
(-&2/2m)V2
+ U(r) + f ( r ) l . i
(2)
and the perturbation due to the spin-orbit interaction”
determines the form of the basis functions. Under spin-orbit coupling, 1 and s are no longer good quantum numbers, and the basis functions must now be eigen.-.functions of only the total angular momentum, j = s I. The transformation coefficients connecting the familiar { slmsmz]basis of the hydrogenic wave functions
+
{slm,mzj
=
1,
mz)Is, m,) =
~ ( n1, ,m,, mJ = R(nl>e(lmz>Nmz)+, mJ (4) with the { sljm) basis of the free atom assume the form of the Clebsch-Gordon or Wigner coefficients, available
in Table I3of Condon and Shortley’ss text and Table I11 of Wigner’s text12
or in the more abbreviated notation of Liehr3 and Griffith13 Ij, m) =
C/l,m d s , ms)(s,1, m,, m ~ l jm) ,
mzm
(6)
After the phase alteration needed to establish the conventional correlation of s, L, and J to J1, J z , and J , re~pectively,’~ the Ij, m) basis functions assume the orthonormal forms given in Table I, expressed in both the IZ, mz)ls, m,) and the trigonometric orbital notations. Symmetry Eigenfunctions. Transformations of the lj, m) weak-field basis functions under symmetry operations of the Dah point group, c&),C&), Cz(y) = uy (zx), CZ(X) = bh (xy), and ce/,(x)= f$(2),15,16 are listed in Table 11. I n this weak-field approximationj, not 1 or s, is a good quantum number, so we are working with ei(4) H. Goldstein, “Classical Mechanics,” Addison-Wesley, Cambridge, Mass., 1950, Chapter 4. (5) M. E. Rose, “Elementary Theory of Angular Momentum,” John Wiley and Sons, Inc., New York, N. Y., 1957, Chapter IV. (6) M. Tinkham, “Group Theory and Quantum Mechanics,” McGraw-Hill Book Co., Inc., New York, N. Y., 1964, Chapter 5. (7) Tinkham’s matrices for generalized angular-momentum eigenfunctions are defined for counterclockwise (positive) rotations of the functions with respect to a coordinate system fixed in space, the rotation by y being performed first, or equivalently, are defined for counterclockwise rotations of the functions about successively rotated Cartesian axes affixed to the functions themselves, the rotation by a being performed first: D i ( a , p , y) = Pz(a)Pg(p)Ps(y) = P,~~(y)Py~(p)P,(a). (Here P,(a) signifies the counterclockwise rotation through the angle a about the z axis. In general, the nomenclature used in this paper follows that of Tinkham quite closely.) (8) E. U. Condon and G. H. Shortley, “The Theory of Atomic Spectra,” Cambridge University Press, Ithaca, N. Y., 1964, Chapter 111. (9) Thus if it were desirable to represent the strong-field basis functions in terms of the trigonometric orbital functions, which it is not advantageous to do in Ds, symmetry, the D 2 ( a , p , y) matrix would operate on the vector { l/d$0, 0, 0, l/d%} to effect transformation of the dzz-,,2 orbital function, for example, and on the vector { 0, l/d$, 0, l/G,01 for the d,, orbital function. (10) As the double-group character tables are not uniquely specified by group theoretical conditions, the reducible representations generated by Dj may be ambiguously decomposable by the several character tables for the same double point group. (11) See ref 8, Chapter V. (12) E. P. Wigner, “Group Theory,” Academic Press Inc., New York, N. Y., 1959, Chapter 17. (13) J. S. Griffith, “The Theory of Transition-Metal Ions,” Cambridge University Press, Ithaca, N. Y., 1961, Chapter 2. (14) In this convention, which is Condon and Shortley’ss standard form, s is correlated with j~ and 1 with jz, necessitating a phase alteration from the initial form of the Wigner coefficients: $(s, 1, j, m) = ( - I)Z+*+$(l,s, j , m) = (-1)6/2-7212, mz)l1/%, m,)(jl/zm~mz~jm). (15) Mathematically, u,(zz) = CZ(V)~, uh(zy) = Cz(z)i,and SS(Z)= cd(z) Uh(Z1/) = Ca(z)Cz(z)i = C*/s(z)i or, in general, reflection through a plane of symmetry may be replaced by rotation about a twofold axis perpendicular to that plane of symmetry followed by reflection through an inversion center (whether or not i is an operation of the particular point group). Since both the d orbitals and the spin functions are gerade,S however, a group theoretical equivalency to the uv(zz) operation is obtained with CZ(V). (16) The Eulerian angle of rotation for C S / ~ (is Z )a = 5 r / 3 .
-
Volume 78, Number 10 October 1968
C. A. L. BECKER, D. W.
3590
MEEK, AND
T. nt. DUNN
Table I : The IJ,M ) Basis Functions
genfunctions of half-integral angular momentum ( j = 3//z and 5 / / 2 ) and consequently must use double-valued irreducible representations of the double groups t o describe symmetry properties of the crystalline-field eigenfunctions. Under the Dah' (double-group) character table, the D8,z reducible repre~entation'~-'~ decomposes into rS re and the D,,, representation into r7 rs re. Symmetry requirements following directly from the irreducible representations are sufficient to characterize the D S h symmetry eigenfunctionsZ0
+ +
+
The Journal of Physical Chemistry
(17) The DJ reducible representations are easily calculated by means of the character formula for rotation through an angle 01:W19 X J ( ~ ) = [sin (J l/z)a]/sin (01/2), valid for integral and half-integral values
+
of J. (18) M. Tinkham, "Group Theory and Quantum Mechanics,"
McGraw-Hill Book Co., Inc., New York, N. Y., 1964, Chapter 4. (19) B. G. Wybourne, "Spectroscopic Properties of Rare Earths," Interscience Publishers, New York, N. Y . ,1965, Chapter 6. (20) The symmetry eigenfunctions, T7, Fs, and re, are doubly degenerate wave functions and, therefore, must have two orthonormal components, rzaand ria. These symmetry requirements imposed upon the individual eigenfunction components are additional to the sample requirements of the Dah' character table and are, therefore, somewhat arbitrary and artificial, but they do give a means of identifying and separating the two eigenfunction components which will remain orthogonal throughout their perturbation matrices.
d1 AND d 9 TRIGONAL BIPYRAMIDAL COMPLEXES The simplest forms of the weak-field symmetry eigenfunctions, then., are the primitive Ij = 5/2, m) and Ij = 3/2, m) basis functions of the free atom as summarized in Table 111. Further linear combinations of the basis functions would render the energy computations more difficult and are not required by symmetry conditions.
3591 rically substituted as the customarily defined spin-orbit coupling c o n ~ t a n t , ~and p ~ the ~ + ~angular ~ portion of the spin-orbit operator has the known eigenvalues," ' / Z { j ( j 1) - "4 - Z(Z 1)]fi2,which are diagonal in j and m. The crystal-field perturbation matrix for a regular trigonal bipyramidal environment22is diagonal and completely specified by two parameters. From calculations using an ionic model, 2 3 matrix elements assume the forms
+
+
Ze2
Ze2
XZ = -
Symmetry eigenfunctions constructed for the D3' and C3y' double-group character tables are of the forms listed below, while th.e unmodified lj, m) basis functions completely satisfy the symmetry requirements of C3', but the perturbation secular determinants resulting for crystalline-field symmetry eigenfunctions under these four double-group character tables, Dsh', D3', C3y', and Cat, are equivalent and yield identical perturbation eigenvalues
Ze2 6a5
( R n J 2 ( r ) r 4 - dr r2
= -(r4)
( R , J 2 ( r ) r 2 . rdr 2
=
Ze
-(r2) 7a 3
(9) (10)
of which the first parameter is the familiar Dq from octahedral calculations, but the second has no exact equivalent in cubic symmetry and has been designated Db in this work. Within an ionic-model system, the electrostatic potential field of the ligands for both the tetragonally distorted octahedron (distortion along the fourfold x axis) V(r, 8, cp) = Ze2{(2/b) (4/a) [(3x2 r2)/b3] [(3x2 3y2 - 2r2)/a3] [(35x4 - 3 0 ~ 3r4)/4b5] [(35x2 35y2 - 30x2r2 - 30y2r2 6r4)/4a5]] and the trigonally distorted octahedron (distortion along the threefold (1, 1, 1) or x' axisz4) V(r,8, cp) = Z e 2 / b {6 { [18(cos2a)x2 9(sin2a ) ( x 2 y2) - 6r2]/2b2) { [840(cos4 a)z4 315(sin4 a)(x2 y2)2 2520(sin2 a)(cos2 a)x2(x2 y2)]/32b4] { [840(sin3a)(cos a)xx(x2 - 3y2) - 720(cos2 a)x2r2 360(sin2 a)(x2 y2)r2 72r4]/32b4]] contributes an average quadratic radical displacement, Le., ( r 2 ) . The respective parameters, Ds and Du, then assume upon integration the explicit formsz4Ds = 2Ze2/7[(l/a3) (l/b3)](r2)and Du = -3Ze2/7a3[3 cos2 a - l](r2) or on expansion of the azimuthal angle into the nondistorted value (i.e., a0 = 54'44') and the angle of disa, tortion (a)for the trigonal problem; i.e., a = a0 Du = 3Ze2/7a3[(sin2a) - 2 4 2 cos a sin a](r2). Both Ds and D a are seen to be parameters of their specific distortion then and must vanish identically in the regular geometry, i . e . , the octahedron. These parameters are also usually, though not necessarily, quite small with respect to Dq. The electrostatic potential for the regular trigonal bipyramidal geometry V ( r ,8, cp) = Ze2/a{5 [(9x2 9yz 12x2 - 10r2)/4a2] [(15x3 - 45xy2)/8a3] { [315(x2 y2)' 560~~ 120r2(r2- 3x2 - 3y2 - 4z2)]/64a4]] does indeed also contribute an average quadratic radial displacement
+
+
+
+
+
+
+ +
+
+
+
+ +
+ + +
+ +
+
+
+
+
+ +
+
+
+
+ +
(21) T.M.Dunn, D. S. McClure, and R. G. Pearson, "Some Aspects
of Crystal Field Theory," Harper and Row, New York, N. Y., 1965,
Chapter 3. (22) The regular trigonal bipyramidal environment is defined here as one in which the axial and equatorial bond lengths are equal. (23) For a detailed example of this method of crystal-field calculation, as applied to the octahedral geometry, see ref 21, Chapter 1. (24) T. S. Piper and R. L. Carlin, J. Chem. Phys., 33, 1208 (1960).
Volume 72,Number 10 October 1968
~
~
3592
C. A. L. BECKER,D. W. MEEK,AND T. M. DUNN
(the cubic term in the potential, however, vanishes under integration), but its matrix elements are not functions also of distorted bond lengths or distorted bond angles, so its crystal-field parameter in explicit form cannot be directly related to Ds or Da. For calculations within the ionic model, then, it seems most appropriate to assign the parameter of (r4) in terms of Dq, which follows directly, and to define a new parameter for the (r2) term, which is specific for the trigonal bipyramid and is not a function of a geometric distortion. If the crystalline-field Hamiltonian is constructed from the expansion of spherical harmonics according to symmetry requirements, 25 however, a different situation emerges. The trigonal bipyramidal potential must then assume the same form as that of the tetragonal and trigonal (axial) distortionsz6
+ BR4'(r)Y,O
VTB = AR2(r)Yz0
(*(O)lVl*(O))
or in the angular-momentum-operator notation
VTB =
3/2f2(r)(Z22
(XP(*l)lVl@{*l)}
- 2) '/2f4'(r)(35/12&4 -
155/12&2
+ 6)
and may, therefore, be characterized by the same p ammeters
Ds
=
J1R,z(r)12 3 / 2 j 2 (dT~ )= Du
D,
=
J/R,Z(T)~' 3 / 2 f 4 ' ( T ) dT
=
DT
the octahedral perturbation (Le., V o )and its parameter (Dq) vanishing completely in the trigonal bipyramidal geometry. The matrix elements now assume the familiar values3~24-27 (*(*2)IVIP(*2)) (!V(*l)IVl'P(*l))
=
=
2Da
-
DT
-Da
+ 407
(\.k(O)jVjXP(O)) = -2Da - GDT
which in the notation of this work indicates the equivalencies Db = -2Da and Dq = -"I25 0 7 . Now the trigonal bipyramidal crystal-field calculation becomes somewhat of a special case of the more generalized trigonal cal~ulation,~ but as different symmetry eigenfunctions have been constructed in the two examples much unscrambling is necessary in the perturbation matrices to obtain one from the other. (With these suggested substitutions, i.e., Dq = 0, Dr = -25/zaDq,and Da = - l/zDb, the l"4,bT perturbation matrix of ref 3 yields the The Journal of Physical Chemistry
rsweak-field matrix of this work within the off-diagonal phase factor, but the other weak-field and the strongfield representations of D3hsymmetry, however, require considerable unscrambling from those of D3d.) Within this perturbation-operator model, then, the crystal-field parameters needed for a trigonal bipyramid can be specified by those of the trigonal distortion of the octahedron, namely, Da and D T , but here it must be always borne in mind that any equivalency of Db to Da and of Dq(TB) to Dr is implicit, through their operator definitions, afid that no direct proportionality exists between thdse parameters in explicit form. Both methods of crystal-field calculation have, of course, their specific advantages and disadvantages. The authors' preference, however, is with the ionic model, so these are the calculations used throughout this work. The nonvanishing crystal-field matrix elements can now be expressed =
=
"/14Dq
+ Db
-25/7Dq
+ '/zDb
(11)
(*(*2)lVI@(*2)) = 25/280q- Db and from them the crystal-field perturbation elements of the Dah eigenfunctions are readily computed. The 10 X 10 perturbation secular determinant of the weakfield approximation readily blocks into the three doubly degenerate matrices now tabulated
Strong-Field Approximation
Basis Functions. The simplest eigenfunctions satisfying the diagonal trigonal bipyramidal crystal-field matrix are obviously the primitive atomic orbitals, po, p&1, and pfz, although linear combinations to the trigonometric orbitals are also satisfactory eigenfunctions. As little advantage can be gained with the trigonometric orbitals, the basis functions assumed for the strong-field approximation are those listed below, given in the notations of Liehr3 and Griffith,la respectively
(25) For illustration of this method of crystal-field calculations, see C. J. Ballhausen, "Introduction to Ligand Field Theory," McGraw-Hill Book Co., Inc., New York, N. Y.,1962, Chapters 4,5. (26) W.E.Moffitt and C. J. Ballhausen, Ann. Rev. Phys. Chem., 7, 107 (1956). (27) C. J. Ballhausen and W. E. Moffitt, J. Inorg. Nuclear Chem., 3 , 178 (1956).
d1 AND ds TRIGONAL BIPYRAMIDAL COMPLEXES
3593
al’{(m,) = 12, O)~’/Z, ma)
2Al’:
2E’: e,’{(m,)
=
12, 2)/’/2, m,)
ee’T(md = 12, -2)11/2, m J
2 E r ’ : e,”{(m,)
=
(13)
12, l)I1/2, m,)
eb”i-(m,) = 12, -~)I’/z, m,)
The transformation properties of the strong-field basis functions under the Dah symmetry operators are summarized in Table IV. It is now convenient to decompose the basis functions written in the Mulliken2*notation of the strong crystalline field into the BetheZ9notation for the doublegroup symmetry symmetry Eigenfunctions.
Cal’{(m,) = ms
rs;
Ce”t(m,) = ms
r8+ rg;
Ce’{(m,) =
r7
+
r9
(14)
As the same irreducible representations are required, the strong-field symmetry eigenfunctions must obey the same symmetry conditions as did the weak-field eigen-
Table IV : Transformation Properties of the Dah’ Strong-Field Basis Functions
Table
r4b”(’E) = 12, 1)11/2, functions and may be constructed in precisely the same manner. These D 3 h symmetry eigenfunctions are summarized in Table V. The strongfield symmetry eigenfunctions constructed under the D3’, C3v’,and C3’ double-group character tables become
-‘/2>
I’,’’(zE) = 1 / 4 2 , l)l’/~,‘/J
+ i / 4 2 , -1)l1/z,
-‘/z)
ra”(W = l/&P,
-w
-‘/2>
1)1’/2,
‘/2>
q 2 , -1)11/2,
(28) R. s. MuIliken, Phys. RW.,43, 279 (1932). (29) H.A. Bethe, Ann. Physik., 3, 133 (1929).
Volume 78,Number 10 October 1968
C. A. L. BECKER,D. W. MEEK,AND T. n4. DUNN
3594 C3' I?4(,d2A11 = 12, o)j'/z,
-'/2)
r4(P2)[2A11= 12, 0)/'/2, r3'('E) = 12, 2)1'/2,
-'/z)
ra"('E) = 12, -2)11/2, r4'(d
'/2)
'/2)
['El = 12, 2)1'/2,
'/2>
r 4 ' ( ~ 2 ) [ ~=E ]12, -2)1'/2, r3'('E) = 12, 1)1'/2,
'/2)
r3"(2E) = 12, -1)1'/2, r4"(p1)[~E] = 12,
-'/2)
-'/z)
-1)1'/2,
'/2)
r 4 ' ' ( d 2 ~= i12, 1)l%, -%>
(15)
and again give rise to perturbation secular determinants equivalent to those for Dah'. Energy Matrices. The eigenvalues for the 1 s operator of the spin-orbit interaction are given by eq 7a(3) of Condon and Shortleys for general wave functions in the {sZm,ml]basis
-
- 4
(ysZmsmz I L * S(y 'sl'm,'mz') =
+ 1/z6(m,', 1)d(1 - m + '/z)@ + m + (16)
626(ylm;y'l'm')( 6(m,, ms')mlm, ms f
l/df
The crystalline-field perturbation remains entirely diagonal and with the spin-orbit interaction gives rise to the blocked form of the perturbation secular determinant
The symmetry eigenfiinctions for the weak- and strong-field approximations are connected via the transformations 2E'
r7:
~ D ~ ~ J I + ~ I I 2A1' 2E" (18)
2E'
rg:
2EI' f
4%
and, therefore, the secular determinants must have the same eigenvalues, as can be illustrated upon solution of The Journal of Physical Chemistry
the energy matrices with appropriate parameter values. As they are calculated, the energy matrices are appropriate for a d' configurational metal ion subjected to a trigonal bipyramidal crystalline field. For the d9 electronic system the "hole formalism" dictates a simultaneous change of sign for the spin-orbit and crystal-field parameter^,^^ since the electronic d 9 configuration is equivalent to a d' positronic configuration. 927 The results of this crystal field-spin orbit perturbation calculation, and its contrast to the simple crystalfield calculations, can best be appreciated by means of an energy diagram. For the Cu(I1) species, a d9 system, reasonable approximated parametric v a l ~ e s ~ l - ~ ~ could be a Dq value of 1200 crn-l, the spin-orbit parameter assigned as 80% of its free-ion value (A 660 cm-l), and for Db, using a very crude a p p r o x i m a t i ~ n , ~ ~ Db = 2060 cm-l. The perturbation energy levels for these assigned parameter values are shown in Figure 1, for both the simple crystal field and the crystal fieldspin orbit calculations. A Distorted Trigonal Bipyramidal Environment. As a last consideration, the crystal field-spin orbit perturbation matrices for the distorted trigonal bipyramidal e n ~ i r o n m e n tare ~ ~ constructed. Since the symmetry remains D3h, the previously constructed eigenfunctions are still valid. Designating the ratio of equatorial-to-axial bond lengths as c, the nonvanishing elements of the crystal-field matrix assume the modified form of eq 19.
(30) A. D. Liehr and C. J. Ballhausen, Ann. Phys. (N. Y.), 6, 134 (1959). (31) The authors wished to apply these calculations to the known trigonal bipyramidal CuCla3- species3*133in a parametric manner but to date have not been able to obtain a satisfactory optical spectrum. Any crystal-field energy diagrams predicted from parameter values assigned on the basis of other inorganic complexes are a t best, of course, only speculative, as crystal-field parameters should be established experimentally. (32) M. Mori, Y. Saito, and T. Watanabe, Bull. Chem. SOC.Jap., 34, 295 (1961). (33) M. Mori, ibid., 34, 1249 (1961). (34) The approximation Db = 1.72Dq, or rather &-(square planar) = 3.44Dq, had its apparent origin in F. Basolo and R. G. Pearson, "Mechanisms of Inorganic Reactions," John Wiley and Sons, Inc., New York, N. Y., 1958, Chapter 2. (35) A distorted trigonal bipyramidal environment is considered here as a trigonal bipyramid of unequal axial and equatorial bond lengths, still rigorously Dah in symmetry but having a (equatorial)/ b (axial) = c # 1.
d1 AND d9 TRIGONAL BIPYRAMIDAL COMPLEXES P(O)IVl~(O))= 1:(27
3595 functions constructed under the Ds', Cay', or CS' groups.
+ 48c5)/14Dp + (4c3 - 3)Db
I v I*( =k1)) = - [(9 + 16cS)/7]Dq + ( 2 ~ '- 3/2)Db
Acknowledgments. The authors wish to acknowledge the National Science Foundation for support of this research and for a fellowship to C. A. L. B. during 1965-1967.
(\E ( f1)
(19)
(!l?(k2)lvp(+=2)) = [(9
+ 16c5)/28]Dq + (3 - 4~')Db
Appendix 1 The D'( a, p, 7)transformation matrices for angularmomenta eigenfunctions can be calculated through use
The energy matrices in the weak-field approximation now become
-
--/,A
r$
- 3c3)Db 14
9
-E
-"( 9 +141 6 ~Dq~ )- 4 6 -3)Db
X
(4~'~;
+ (2l - 28c5)DB - E -(9 +1416c5) D q + (12cSp ')Db
--/,A
-
+ 16c5
(+
>.P
4- 9 1416c6
4-
(16c3L
12)Db
=
0
(20)
-E
(9 +1416c6)Rq + (12c'5- ')Db - ("' iFb) D q + ("""53)Db - E
=o
~
while the strong-field energy matrices are calculated as
r7(2E'): IX
+ [(9 + 16c5)/28]0q + (3 - 4c')Db - El
+ 16c5
0
r~('E'')
W2A1')
9
=
- 4c3)Db - E
x '/2X
-
(9 +,16c5)Dq + (2c3 - "2)Db
-0
-E
Conclusion The method for "exact" electronic energy calculation according to Liehr3136and Liehr and B a l l h a u ~ e nhas ~~ been applied to the d' configuration in trigonal bipyramidal environment, and the D3(a, b, 7) rotation matrices for angular-momentum eigenfunctions have been illustrated as useful means for performing symmetry operations. The Dah secular determinants obtained as functions of three parameters yield identical eigenvalues in the weak- and strong-field approximations and indicate a complete perturbation of the d orbitals into five energy levels. The ordering of these first-order perturbation levels is obtained through substitution of appropriate parametric values, thus limiting application of these energy matrices only in regard to configuration and crystalline-field environment. Equivalent secular determinants with identical eigenvalues can also be obtained with symmetry eigen-
Figure 1. Solution of the Dah perturbation matrices with approximate values. For Cu(II), a d9 system, the assumed parameter values are Dq = 1200 cm-1, A = 0.8XO = 660 cm-1, and Db = 1 . 7 2 0 ~= 2060 cm-1. Volume 78,Number 10 October 1988
3596
C. A. L. BECKER, D. W. MEEK,AND T. M. DUNN
of the generating function for individual matrix elementsa
+
+
(-1N(j m > ! ( j - m > ! ( j m’)!(j - m’)! X k ! [ j m - k)!(j - m’ - k ) ! ( k m’ - m)! [cos (/3/2) 1-’ ( -sin (p/2) I”+ m’-
+
+
The matrix elements for D‘/’(a, p, y) and D1(a,p, y) are available from Tinkham,s so only the transformation matrices D’/*(a, p, y), D2(a, p, y), and D”/” (a,p, y)37will be listed below for future use.
(37) Since space limitations prohibit the usual horizontal format for these lengthy matrices, the respective elements for the 4 X 4, 6 X 6,and 6 X 6 matrices are indicated by the large bracea.
The Journal of Physical Chemistry
d1 AND d9 TRIGIONAL BIPYRAMIDAL COMPLEXES
3597
- [sin (P/2)]6e-5i(”-y)/2
-
4 5 cos (p/2) [sin ( p / 2 ) ]4e-i(3a-57)/2 -l/iO[cos (p/2) ]“sin ( ~ / 2 ) ] 3 e - ~ ( ~ - ~ ~ ’ / ~
d i b [cos ( p / 2 )13 [sin (p/2) 1 2 e ~ ( ~ + ~ ~ ) / ~
- 1/5[cos ( p / 2 )]4[sin ( p / 2 )]ei(3a+57)/2 [cos ( p / 2 )] ~ e ~ ~ ( ~ + y ) / ~
I
C. A. L. BECKER,D. W. MEEK,AND T. M. DUNN
3598
Table VI - ”” 3cz
E
R
1 1 2
Ai’ Az’
E’ Ai” Az” E“
2 -2 -2 -2
-
rl
r2{PZp1 rs r4{P8PI ” Where E
1 1 -1 1 1 -1 1
1 1
rl r8 rQ
Table VII“
2 Ca
=
1
Uh
+
3RCx
RUh
1 1
1 1 2
1 1
1 -1 0 1 -1
-1 -1 -1 2
0 0 0 0
-1
-2
+
2RCa
-1 -1
-2 0 0 0
2RSa
2S8
3U” t 3Ruv
1 1
1 1
1 -1
-1 -1 -1 1
-1 -1 -1 1
-1 1
43 -4 3
-4 3
0
0
43
0
0 0 0 0
obvious D”/”p6/a(0,T , 0) matrices were used for the C2(y) = ~ ( $ 2 )rotations. CS’
7
E
R
Ca
RCa
1 1 1 1 1 1
1 1 1 -1 -1 -1
1
1
€2
€2
1 ,p
,2*
,a*
€8
1
I
-1
--e
€2
--e2
-e*
,2*
-1 -e e*
CaP
RCas
1 €a* -e2
_$*
eTiI3.
&-I@, n, 0) o r ~ * / z ( O , T , T ) (butnotD’/z((n,n,o),whichlike D ’ / z ~ a / T, * (0) ~ , would introduce a - 1 factor). The
The Journal of Physical Chemietry
Appendix 2 Suitable double-group character tables may be constructed according to standard procedure^.^*^^^ Values for the Cav’character table were taken from Wybourne’s text,lg and the D3’ table (though not the Dgh‘ or C,’] used in this work is identical with that given by Koster, et uZ.,~* so only the D3h’ and C3’ character tables are listed in Tables VI and VII, respectively. (38) G . F. Koster, J. 0. Dimmock, R. G. Wheller, and H. Stats, “Properties of the Thirty-two Point Groups,” Part 2, Massachusetts Institute of Technology Press, Cambridge, Mass., 1963, p 66. The I’s and in the Da table of this reference have been used as the I’a(p1) and I’&z) representations, respectively, in the D8‘ of this work.
