1822
NOTES
Crystal Field Energy Levels PENTACOORDINATE bsh
for Various Symmetries
by Steven T. Spees, Jr.,l Jayarama R. Perumareddi12 and Arthur W. Adamson Department of Chemistry, University of Southern California, Los Angeles, California 90007 (Received November 20,1967)
The one-electron] d-orbital, crystal field energy levels for various symmetries have already been derived.3 (See also Appendix.) Except for the octahedral case which can be expressed in terms of only one radial parameter, all the others can be expressed in terms of two parameters
E. 1
b
+t
-6
P :/
and
(1)
where e and q are the electronic charge and the point charge of the ligand, respectively, R(r) is the radial function of the d orbital, and r and R are the electronic radius and the ligand distance from the central metal atom, respectively. For the octahedron, only the quartic parameter occurs and is usually expressed in terms of the well-known Dq parameter by the relation 1/6 ( p 4 ( r ) ) = Dq. We show in Figures 1, 2, 3, and 4 the variation of the crystal-field d-orbital energy levels as the ratio p = ( , a ( r ) ) / ( p k ( r ) ) is varied
-I
t.
-6
t
d
\
4I
\
1 0.5
I 1.0
I 1.5
I 2.0
1
2.5
1
3.0
I
3.5
6.0
P 2 /
Figure 1. Plot of the one-electron d-orbital energy levels us. the ratio of the radial parameters ( ~ z ( T ) )and ( p 4 ( ~ ) ) in a five-coordinated complex of Ctv symmetry (square pyramid). The Journal of Physical Chemistry
Figure 2. Plot of the one-electron d-orbital energy levels us. the ratio of the radial parameters ( p n ( r ) )and ( p 4 ( ~ ) ) in a five-coordinated complex of Dah symmetry (trigonal bipyramid).
from 0 to 4. Basolo and Pearson4 and subsequently Hush5 have used p = 2. There is not much experimental evidence upon which to base a reasonable value for the ratio, p . Although the theoretical evaluation by use of Slater orbitals and Hartree-Fock functions for the radial part of the functions seem to give a high value such as 2 or more,6the experimental evidence provided mainly bj7 Piper and his coworkers7in their studies on copper spectra points to a value of 0.9 or less. This ratio may not be a constant, but it may vary from one transition metal system to another, or, even for the same system, from one type of ligand to another. A comparison of the d-orbital energy levels is given in Table I for p equal to 1 and 2. The principal purpose of this note is to point out that in addition to all other approximations and pitfalls involved in discussions of CFSE’s, one should be aware (1) Department of Chemistry, Michigan State University, East Lansing, Mich. 48823. (2) Department of Chemistry, Florida Atlantic University, Boca Raton, Fla. 33432. (3) (a) C. J. Ballhausen, K g l . Danske Videnskab. Selskab. M a t - F y s . M e d d . , 29, No. 4 (1954); (b) C. J. Ballhausen and C. K. Jorgensen, ibid., 29, No. 14 (1955); see also ref 5 . (c) A generalized treatment of electrostatic potentials and d-orbital energy levels for various possible symmetries is given in “Geometry, Color, and Magnetism: The Ti(II1) and Cu(I1) Systems and Their Relatives,” by A. D. Liehr. These lecture notes are available upon request from Mellon Institute, Pittsburgh, Pa. 15213. (4) F. Basolo and R. G. Pearson, “Mechanisms of Inorganic Reactions,’’ 2nd ed, John Wiley and Sons, Inc., New York, N . Y., 1967, Chapters 2 and 3. ( 5 ) N. 6. Hush, Australian J . Chent., 15, 378 (1962). (6) (a) S. Koide and M. H. L. Pryce, P h i l . M a g . , 3, 607 (1958); (b) H. A. Weakliem, J . Chem. Phys., 36, 2117 (1962); (c) T. S. Piper and R. L. Carlin, ibid., 33, 1208 (1960); (d) J. P. Perumareddi, unpublished results. (7) (a) 8.G. Karipides and T. 9. Piper, Inorg. Chem., 1, 970 (1962) ; (b) W. E. Hatfield and T. S. Piper, ibid., 3, 841 (1964); (c) R. A. D. Rentworth and T. S.Piper, ibid., 4, 709 (1965).
1823
NOTES l
l
l
l
HEPTACOORDINATE
l
l
Table I : Comparison of the d-Orbital Energy Levels
05h
in Various Symmetries When p , the Ratio of the Radial Parameters, Is Given the Values 1 and 2
"t
i
-Energy
in units of D q - 7
(PzW) =
(PI(T))
-7
9.14 0.86 -0.86 -4.57
7.43 2.57 -2.57 -3.71
-I
7.07 -0.82 -2.72
6.21 0.04 -3.14
4.93 2.82 -5.28
5.79 1.96 -4.86
7.81 2.65 1.02 -3.09 -8.38
6.10 1.48 0.16 -1.92 -5.81
1
Or------
+[
-At
0
0.5
1 1.0
I 1.5
I 2.0
(pa(r)) =
2(P&))
1
I
I
I
25
3.0
3.5
40
P : (P2(rl)/(P4(rl)
Figure 3. Plot of the one-electron d-orbital energy levels vs. the ratio of the radial parameters ( p Z ( r ) ) and ( p 4 ( r ) ) in a seven coordihmted complex of D a h symmetry (pentagonal bipyramid).
Acknowledgment. J. R. P. wishes to acknowledge helpful discussions with Dr. Andrew D. Liehr. We thank Mr. G. B. Arnold and Mr. J. F. Benes of the Mellon Institute Research Drafting Department for their assistance in drawing the figures. Most of this work was completed while S. T. Spees was at the University of Minnesota and J. R. Perumareddi was at the Mellon Institute and was supported in part by the National Science Foundation.
Appendix Electrostatic Potentials and Crystal Field Matrix Elements (i) Pentacoordinate CdVGeometry (Square Pyramid). If one assumes a point charge, q, for ligands, the potential V when expanded in terms of spherical harmonics (retaining only those terms which give rise to nonzero matrix elements within the d manifold) takes the form
+
~y/41;yoo $y20{
-2p2(r)
+
P = /(P&!>
Figure 4. Plot of the one-electron d-orbital energy levels vs. the ratio of the radial parameters ( p Z ( r ) ) and ( p 4 ( r ) )
in a seven-coordinated complex of Czv symmetry (trapezoidal octahedron).
that a ratio of PZ(r)/p4(r) may be an additional variable. We do not believe that this has been explicitly discussed previously. T.his variation will be considered in a discussion of activation energies for the substitution reactions of inorganic complexes in a paper submitted by the authors for publication.
I-
I-
Volume 7.2, Number 6 Mau 1068
NOTES
1824 In the above, Y44cis the cosine combination of Yi4 and YdL4 harmonics. The p n ( y ) defined for the ligands in the xy plane is given by p n ( y ) = rn/Rn+l and corresporidingly p n ’ ( ~ ) = F/Rln+lfor the axial ligand, where r is the radius of the electronic charge, e, and R and R‘ are the ligand distances from the central metal atom. Use of the products of spherical harmonics8 gives the following crystal field matrix elements for the d orbitals (me shall use in the present and following cases the potential with p n ( r ) = p n ’ ( r ) )
v = eq[
~
~
G
Y
~
+ {;Yzo{ O
’
-:pz(rj
8 d $ p 4 ( ~ ) Y 4 ~ ](if p n ( y )
+
= pn’(r))
(6b)
where the primed p , ( ~ ) are for the axial ligands and the rionprimed are for the five ligands in the xy plane. The d-orbital energies are (when pn(r) = p n ’ ( r ) )
and all other elements are zero. The ( p n ( r ) j , which are regarded as parameters, are the average values of the nth power of the electronic radius, T , multiplied bj7 the factor (eq/R”+’). After substituting a proper relation between ( p 2 ( r ) ) and (pd(r)) and equating 1 / & 4 ( ~ ) ) to Dq, we find the correspondence with the octahedral case desired (ii) Pentacoordinate Osj1Geometry (Trigonal Bipyra-
and all others are zero. (iv) Heptacoordinate CzUGeometry (Trapezoidal Octahedro n)
mid)
2
where the p n ( r ) are for the ligands in the xy plane and ,on’(?) for the axial ligands. The one electron d-orbit a1 crystal-field energies are with the potential pn(?) = p n ’ ( ~ ) 1 ((x”lw(22))
(($2
- y2)
= +PZ(,.))
1 iq(.2 - y”) ((ZY) 1 VI (ICY)) ((yz)lT71(yx)))
)
=
=
(b)lVl(xx))
+ 25
gip4r))
1 25 - 7 ( p 2 ( r ) ) -I-
L(p2(r)) - -25 (p4(T)) 14
(if pn(r) = ,on’(?)) (8b)
where it is supposed that five of the ligands occupy the five corners of an octahedron at a distance R from the central metal atom (forming the square pyramid) and the remaining two ligands are at a distance R’ in the (x y)z plane or (x - y)x plane with an angle of 6, the tetrahedral angle. By assuming all ligands are at the same distance, R, we arrive at the following energies for the various d orbitals.
+
(5)
42
and all others are zero. (iii) HeptacooTdinate Djh GeometYy (Pentagonal Bipyramid) The Journal of Physical Chemistry
*Yd4c}] 18
( 8 ) It is very convenient to use Table A21 in J. S. Griffith, “The Theory of Transition Metal Ions,” Cambridge University Press, London, 1961.
NOTES
1825 30
-E
25
m
and all others zero. We notice in this case two nonvanishing nondiagonal matrix elements, which give rise to configuration interaction. The configuration interaction of the (yx) and (22) orbitals is just that needed to mix them into the properly symmewized combinations 1 / 4 3 [ (xx) f (yz) 1 which are written in the text for simplicity (z y)z. Then ((z
((x
+ Y>4II.‘i(X + !Ax)
- Y)ZlVl(X - Y)4
1
=
3
I
I
I
I
5
10
15
20
?(298)/(x
t y)
-
cal/”K
gm-atom
Figure 1. Entropy correlation for 298 and 1000’K.
102
standard partial molar entropy at one temperature, S o ( T 2 )to , that a t another temperature, TI), for a series of aqueous ions of similar charge and type (Le., simple ions, oxy anions, and acid oxy anions). The relationship between the partial molar entropies at two temperatures is linear and can be written in the form
- 189(P4(?))
-$Jz(Y))
0
22
- 189(P4(r))
= 7(PZW)
5
The configuration interaction of the (9)and (xy) orbitals is the usud type between two states of the same overall symmetry. So, by assuming equal mixing of the states, we solve the 2 X 2 secular determinant and denote the orbital with higher energy as (x2, zy)- and the one with lower energy (9,xy)+ (see also ref 5). It may be pointed out that neglecting the presence of Yooterms and putting pn’ (Y) = 0 in eq 2a, 4a, 6a, and 8a, which is equivalent to removing the corresponding ligands to infinity, there result the potentials, respectively, for a four-coordinated complex of square planar geometry (Ddh), a tricoordinate complex of trigonal planar geometry (D3h), a five-coordinated complex of pentagonal planar geometry (DSh),and a fivecoordinated complex of square pyramidal geometry (C4,).
S”(T2) = a(T2, Yl)
+ b(T2, Tds”(T1)
(1)
where the intercept and the slope depend on the two temperatures selected for the correlation and the ionic type. The relationship has been used to estimate partial molar entropies at elevated temperatures from room-temperature values. The absolute entropies of solid metals and compounds at room temperature and above are examined in this note and are found to obey a similar linear relationship.
Table I : Coefficients in the Entropy Correspondence Equation
Estimation of the High-Temperature Entropy of Solids by the Ehtropy Correspondence Principle’
by J. G. Eberhart and E. N. Gruetter
T,O
K
298 500 1000 1500
S a n d i a Laboratory, Albuquerque, N e w Mexico (Received October 6 , 1 9 6 7 )
871 16
In recent years, Cobble and C r i s ~ ~have - ~ developed an entropy correspondence principle, which relates the
a(T, 298), cal/deg g-atom
0.00
2.92 6.86 9.38
b ( T , 298)
1,000 1.038 1.105 1.141
(1) This work ‘was supported by the L’. S. Atomic Energy Commission. (2) C. M. Criss and J. W. Cobble, J . Am. Chem. Soc., 86,5385 (1964). (3) J. w. cobble, Science, 152,1479 (1966). (4) J. W. Cobble, Ann. Rev.Phys. Chem., 1 7 , (1966). ~ Volume 7St Number 6
M a y 1968
