nergy Levels Including Spin-Orbit Perturbatio Vc ... - ACS Publications

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3401

nergy Levels Including Spin-Orbit Perturbatio rama R. Perumareddi Depnrtment of Chemistry, Florida Atlantic University, Boca Raton, Florida $3@9

(Received June 7 , 1973)

Starting from strong-field octahedral representation, the symmetry adapted quadrate wave functions are derived in both coordinate and spiri space for d2 and d8 electronic configurations. The corresponding energy matrices are constructed as parametrically dependent on ligand field, electron correlation, and spin-orbit interaction perturbations in the coupling scheme in which the spin-orbit perturbation is applied last. The energy diagrams displaying the splittings of the cubic energy levels by the additive axial part of the quadrate ligand field and further by the spin-orbit perturbations are obtained by considering octahedral d* configuration as an example. The importance of full configuration interaction on the quadrate splittiuga of the cubic energy levels is emphasized. Applications of our energy equations to optical spectral and magnetic studies on quadrately distorted or substituted octahedral, quadrately distorted tetrahedral, and five-coordinate square pyramidal systems, of d2 and dg electronic configurations, are pointed out.

.

Introductio

The few attempts that have been made so far in interpreting, the electronic spectra of quadrate (tetragonal) nicliel(I[) complexes made use of energy levels with no spin-orbit perturbation, and either with complete or partial neglect of configuration interaction. 1 Treatment of octahedral nickel(I1) complexes with full configuration interaction and spin-orbit perturbation m the past2 gave rise to a better understanding of their electronic s truelures Inclusion of configuration interaction is very impor.lant in that it may alter the energies considerably and in some instances even affect the assignments. Spin-orbit interaction, though it acts as a minor perturbation in 1,Be case of the early members of the 3d transition series elements, can be dominant for the later rnerribei*sof the series. It is certainly necessary to include spin-orbit perturbation for meaningful studies on 5ystems iiivolviiig elements of 4d and 5d transition series. The purposle ol‘ this investigation is to derive the energy levels of d2 and d8 electronic configurations immersed in quadra t)e ligand fields including spin-orbit perturbation with complete configuratioii interaction a,nd interpret the electronic spectra of appropriate compounds, ‘Ths present report is concerned with the development of the underlying theory. Applications of the Iheory to opticai spectra and other experimental data will be described in future papers. The appropriate compound$, of d2 and d8 electronic configurations that can be studicd by the theory developed here are the tetragonally distorteld or substituted octahedral and tetrahedral complexes and the five-coordinate square pyramidal sys1,ems. The tetragonally substituted octahedral complexes include the mono- and the trans disubstituted octahedral system. The cis disubstituted octahedral cornplcxes can also be treated by the same theory if it i 4 assumed that the ligand fields of trans ligand pair-, can be

11. Theory Octahedral Orientation. In the systems for which the theory is developed here, the additive axial ligand field acts as a minor perturbation relative t o the major cubic ligand field so that these systems can be treated as slightly distorted cubic field complexes. 1n other words, we shall use the two electron cubic wave functions as the basis set which will then be decomposed properly to correspond to the symmetry adapted functions of quadrate symmetry. The decomposition of the cubic functions relative to the representations of quadrate symmetry is given in Table J. Coupling Scheme. We utilize the coupling scheme in which the spin-orbit perturbation is applied last. If Vc, V,, .Z2>%e2/rt3, and Z & ( r J c * s’, denote, respectively, the perturbations due to the cubic ligand field, axial ligand field, electron correlation, and spin-orbit interaction, the sequence4 in which these perturbations are applied is as follows Vc >

C e 2 / r L j> V , > 3>‘1

1.

Wave Functions. The procedure of fabricating symmetry adapted two-electron cubic wave functions in coordinate space is well known.b Once these are ob(1) (a) See for a review of these, N. S. Hush and R. J. M. Nobbs, “Progress in Inorganic Chemistry,” Vol. 10, F. A. Cotton, Ed., Interscience Publishers, New York, 1968. See also (b) 6 . W. R.eimann, J . Phys. Chem., 74, 561 (1970); (c) R. L. Chia.ng and R. S.Drago, Inorg. Chem., 10,453 (1971), and references therein. (2) A. D. Liehr and C. J. Ballhausen, Ann, Ph,ys., 6,134 (1959) (3) (a) R. Krishnamurthy, ‘W. B. Schaap, and S. R.Perumareddi, Inorg. Chem., 6, 1338 (1967); (b) J. R. Perumareddi, Coord. Chem. Ecv., 4,73 (1969). (4) For a review and importance of other coupling schemes possible for low-symmetry ligand fields, read (a) J. R. Perumareddi, J. Phys. Chem., 71, 3144 (1967); (b) J. R. Perumareddi and A . D. Liehr, Abstracts, Symposium on Molecular Structure and Spectroscopy, Columbus, Ohio, June 14-19, 1965; (c) J,. R. Perumareddi, t o be submitted for publication. (5) See, for instance, C. J. Ballhausen, “Introduction t o Ligand Field Theory,” McGraw-Hill, New York, N. Y., 1962. ~

The Journal of Physical Chemietry, Vo1. 76, KO.9S7197%

JAYARAMA R. PERUMAREDDI

3402 I

-

~

I

Table 1: Decomposition of Cubic Representations Relative to Quadrate Symmetry

Table I1 : Symmetry Transformation Properties of Quadrate Oriented Cubic Orbital and Spin Functions Function

t,ained, they are decomposed to quadrate functions according to the Table I. The next step is to make these functj o m symmetry adapted including spin space. It should be noted that the spin-singlet functions are Bymmetry a,dapted both in coordinate and spin space to begin with W e need only make the spin-triplet functions spin oriented. This is done by first studying the symmetry transformation properties of spin functions. If a ,and 6 represent the one-electron spin states, their transforamlion properties can be obtained by the substitution of angular values into the spinor array of Goldsteins after the angles are determined by identifying the transformation matrix of the symmetry operatiori with the Eulerian matrix.6 The two-electron spin functions prooerly transforming as the cubic and quadrate represent ations can then be constructed by examining the transformation properties of the twoelectron spin “product” functions. The spin functions so constructed and their symmetry transformation properties aire Tiven in Table I1 which also includes the transformation properties of orbital representations. The inclusion of spin space is then achieved by appropria tely combining these spin functions with the orbital funcitions res’ulting in quadrate functions which are again made to transform in exactly the same way as the symmetry transformations of quadrate representations shown in Table 11. The two-electron wave functions which are symmetry adapted both in coordinate and spin spaceTderived by this procedure are given in Tablc 111. The corresponding eight-electron wave functions can be easily obtained if needed from those listed by considering the two electrons as holes. Energy Matrices. The various perturbations can be Thus, the ligand field gives parametrized (1s rise to a cubic Dq and axial Ds, Dt parameters, the electron correlation yields the A , B , C parameters, and the spin- orbit perturbation results in a one-electron spin-orbit coupling constant parameter The secular determinants obtained as a function of these parametcrs with the use of the wave functions of Table III are given in Tables IVa to e. The cubic ligand field ( 3 q ) ifi completely diagonalized, as it should be, in these energy matrices. The electron correlation ( A , B , C ) and the axial ligand field (Ds, Dt) are diagonal, respectively, in the cubic quantum number { X J C1 and cluadrate quantum number { X j Q

r.

1.

The Journal of Physical Chemistry., Val. 76,No. 23, 1972

-Representation-----Symmetry Cubic Quadrate Cr(z) C&)

- t1a

operation--. Ca(z’)

tlb

tlb

$1,

-t 1 o

tl,

The spin-orbit interaction (p), of course, is diagonal only in the double group quadrate quantum number {TjQ]. It should be noted that in the limit of zero spin-orbit and axial ligand field perturbations, these energy matrices become identical with the cubic d2 energy matrices of Tanabe and Sugano8 within phases. The correctness of our quadrate energy matrices had been checked by directly calculating all the matrix elements from the corresponding energy matriccs of the other coupling schemes possible and the unitary transformation matrices connecting these coupling schemes with the coupling scheme used in this rciport. The quadrate energy matrices of d8configuration are obtained simpIy by changing the sign of Dt, Ds, and p. The constant additive energy term in the diagonal elements, 27A - 426: 21C, only shifts the d8energies but does not affect the energy differences. With the appropriate sign for S, the energy equations listed here are applicable to quadrately distorted or substituted octahedral d8 and quadrately distorted tetrahedral d8

+

(6) H. Goldstein, “Classical Mechanics,” Addison-Wesley, Gambridge, Mass., 1951. ( 7 ) The symmetry adapted wave functions with and without spin space, respectively, are given the Bethe’s r.L (z = 1 to 5 ) and Mulliken’s X , (X,’ = AI, Az, E, TI, Tz and X,Q = Ai, A2, Bi, Bz, and E) classifications. It should be noted that although the energy levels in the diagrams have superscripts Q and C t o denote, respectively, the quadrate symmetry designation and cubic parentage, these superscripts have been omitted in the main text for convenience. Similarly quadrate levels have been written variously with and without a g subscript in this report. Strictly speaking, if the complex is noncentrosymmetric (e.g., C4*), the quadrate levels should not have the g subscript. The g subscript should be retained for the quadrate representations of a centrosymmetric complex (e.g., Dad. (8) (a) Y. Tanabe and S. Sugano, J. Phys. Soe. Jap., 9, 7 5 3 (1954); (b) see also S. Sugano, Y. Tanabe, and H. Xamimura, “‘Multiplets of Transition-Metal Ions in Crystals,” Academic Press, h’ew York, N. Y.. 1970.

The Journal of Physical Chemistry, Vol. 7 6 , N o . 25, 1972

JAYARAMA R. PERUMAREDDI

3404

systems and b j changing the sign of Dq, they will be applicable t o quadrately distorted or substituted octahedral dlLand quadrately distorted tetrahedral d2 systems. Similarly the d2 energy matrices and the d8 energy matrices obtained by changing the sign of {, Dp, Dt, and Ds can 8180 be utilized in studying the electronic structures of the corresponding five-coordinate square pyramidal systems orice the Dt and Ds are considered as symmetry parameters.s Energy Dictgnzrns. The splitting of cubic levels by the additive rtxjd ligand field perturbation of quadrate symmetry in the limit of zero spin-orbit interactionlois The Journal of€'hy::ieal Chemistry, Val. 76, No. $8,1972

shown in Figure 1 by considering octahedral ds configuration as an example. A B value of 800 cm-l is chosen in this diagram to represent a series of octahedral (9) (a) K. G. Caulton, I n o r g . Chem., 7,392 (1968); (b) see also J. R. Perumareddi, J . Phys. Chem., 71,3155 (1967). (10) I n the limit of zero spin-orbit interaction, with the exception of *E,, 'Egwhich are 3 X 3 matrices and 'AI, which is a 4 X 4 matrix, all other levels are either single (~BI,,~Bz,, or a t the most 2 X 2 matrices ( ~ A z~~ B , I'Bz,) ~ , . The solutions of 2 X 2 secular determinants yield the energy equations: 3Aze = (-31)s - Ds - 3Dt 1/zB) '/z[(lODq - BDs lODt 9B)z 144B2]"2, 'Big = (2Dp DS - '/zDt '/zB 2C) rt 1/2[(20D4 - 208 15Dt B )* 48B211'2, and 1Btg = (- 3Dq Ds - '/zDt 1/zB 2 C ) 3~ '/z[(lODp - 209 15Dt - B)2 48B211'2.

*

+

+

+

+

+

+

+

+

+

+

+

+

+

-?-

f

-"

Q

0

0

0

+

-t

+

4-+

The Journal

of

Phvsical Chemistry, VQZ.7 6 , S o . 25, 1972

JAYARAMA R. PERUMAREDDI.

3406 __________

~

~

~~

~~~

Table IVb : Quddrate Energy Matrices of d* Electronic Configuration,

3Eg[*Tl,(W)l

-8Dq A

- Ds - 3Dt

+

ra Representation

$r

+ 2s1

-6B

- 5B + -I 1 2

7 2Dp - -Dt 4

+A -

-4(4Ds 4

+ 5Dt) +

i a

- -- I 4

2Dq

A

~

-

_

.

.

_

I

-

_

- 4Ds + 2Dt + + 4B + 2C

_

Table IVc : Quadrate Energy Matrices of d2 Electronic Configuration, r3 Representation

0

2Dq A

+ 7Dt + - 8B

ss

- d 4 5

7 2Dq - 4 -Dt A-8B1 4s

+

+

-G(4Ds 4 Dt)

id2

--s2

0

fi - -r 4

+ + + +

2Dq 20.3 3 -Dt A 4

4B

-ir

0

4

idzf

idG -I 2

+ 4.1r + + + +

-8Dq ~ D -s 8Dt A B 2C

- 2.\/SB 12Dq A

+ 7Dt +

+ 2C

Ni(I1) complexes and the theoretical ratio of 4 for C to B is used. Employing fixed values of Dt and ~ , * the DsJDt ratio, the energy levels are plotted as a function of Dq which is varied from 0 t o 4000 cm-1. At The Journal of Physical Chemistry, Vol. 76,No. 23, 19'72

l

-

Dq = 0, the levcls correspond to those of cylindrical