Research: Science and Education
Splitting of One-Electron Levels in a Tetrahedral Environment Toyohiko J. Konno Institute for Materials Research, Tohoku University, Katahira, Aoba-Ku, Sendai, 980-8577, Japan;
[email protected] Background Group theory provides an appealing basis for finding the splitting of energy levels of hydrogen-like orbitals in a chemical environment (1, p 34; 2, p 253; 3, p 107). One of the most familiar examples is the splitting of d orbitals in an octahedral environment, Oh. To describe the splitting scheme, it is customary to start with the point group O, where all of the symmetry elements are simple rotation operations—that is, E, C3, C2 (= C42), C4, and C2. In this case, one can safely neglect the r and θ components of the d orbitals and deduce the splitting of the levels upon the application of the above symmetry operations simply by examining the changes in characters of the φ component: Φ(φ) = e imφ
e l iα l–1 iα
(2)
e
O
l–1 iα
→
p
(1)
That is, for the operation Cn on a (2l + 1)-fold set of functions, we need only evaluate the trace of the (2l + 1)-dimensional matrix (off-diagonal elements are all zero):
e
In the case of an Oh environment, we know that Oh is obtained by the combination of O and the inversion operation, i. Accordingly, in a number of introductory textbooks dealing with ligand field theory, the symbols for irreducible representation are derived by first using the character table for point group O and then attaching the subscript g or u to the appropriate Mulliken symbols, depending on the parity of the wave function. Hence it can be concluded, for instance, that p, d, f, and h orbitals are reduced under O and Oh environments as follows:
Oh
t1
→
t 1u
d
→
e + t2
→
e g + t 2g
f
→
a 2 +t 1 +t 2
→
a 2u +t 1u +t 2u
h
→
e +2t 1 +t 2
→
e u +2t 1u +t 2u
(5)
In textbooks, it is often suggested with no explanation that the same approach is applicable for other symmetries, including the tetrahedral one, Td (2, p 263). A moment of consideration reveals, however, that Td comprises not only proper rotation operations but also improper operations such as S4 and σd (Table 2).
e l iα Table 2. Character Table for Point Group Td
where
α = 2π n
(3)
The character χ(Cn) of the operation is further simplified by noting that the diagonal elements of eq 2 constitute simple geometric series. This leads to a concise formula often quoted in textbooks (1, p 45; 2, p 261; 3, p 137):
sin l + 1 α 2 = α sin 2
χ Cn
α≠0
(4)
Once the expression for the character of symmetry element Cn is attained, it is possible to explicitly evaluate the character for each n-fold rotation of the point group O and reduce it with the help of the character table O (Table 1). Table 1. Character Table for Point Group O O
E
A1
1
A2
8C3 3C2 6C4 6C2 1
1
1
1
1
1
1
1
1
E
2
1
2
0
0
T1
3
0
1
1
1
T2
3
0
1
1
1
674
x2 + y2 + z2 (2z2 – x2 – y2, x2 – y2) (Rx , Ry , Rz ); (x, y, z) (xy, xz, yz)
Td
E
8C3
3C2
6S4
A1
1
1
1
1
6σd 1
A2
1
1
1
1
1
E
2
1
2
0
0
T1
3
0
1
1
1
(Rx , Ry , Rz )
T2
3
0
1
1
1
(x, y, z)
x2 + y2 + z2 (2z2 – x2 – y2, x2 – y2) (xy, xz, yz)
On the other hand, S4 can be considered as a combination of C4 and σ, while σ can be regarded as a combination of C2 and inversion, i, the latter being a standard approach taken by crystallographers (4 ). Thus similarities between S4 and C4 and between σ and C2 do exist, suggesting that one may take advantage of the familiar expression (eq 4) once the transformation scheme of orbitals, or more exactly, the spherical harmonics Yl m(θ, φ), under S4 or σ are known. The purpose of this paper is therefore to describe the way of obtaining the characters of the orbitals with arbitrary l under S4 and σ operations, and thus to obtain irreducible representations under a Td environment. I will review some of the general transformation properties of the spherical harmonics and apply them to specific problems to obtain the general formula for the characters of orbitals with arbitrary l under S4 and σ. I will then reduce them to obtain irreducible representations of d, f, and other orbitals under a Td environment.
Journal of Chemical Education • Vol. 78 No. 5 May 2001 • JChemEd.chem.wisc.edu
Research: Science and Education
where m is understood to run from l to l. The character is then given by taking the trace of the matrix: θ r
χ Cn = α
φ
tr e i mα =
sin l + 1 α 2 1 sin α 2
(14)
This is the result given by eq 4. Note that in the special case of C2 (α = π), the spherical harmonics transform as C2
Yl m(θ, φ) → e iπmYl m(θ, φ) = (1)mYl m(θ, φ)
Figure 1. Simple rotation in the spherical coordinate system.
(15)
or equivalently, in the matrix form, Parity of Spherical Orbitals
C 2 = e iπ m = 1 m
(16)
Ignoring constants, the spherical harmonics are given by Yl m(θ, φ) Pl m(cos θ)e imφ
(6)
where θ and φ are the polar and azimuthal coordinates, respectively (Fig. 1) and Plm(x) is the associated Legendre function with l ≤ m ≤ l. The parity of the associated Legendre function is given by (5) Plm(x) = (1)l+mPlm(x)
i
With the mirror operation, σ, only the polar coordinate is changed: σ
σ
θ→θ+π
(i.e., cos θ → cos θ) σ
(7)
Now, under the inversion operation, the transformation of coordinates is given by θ→θ+π
Spherical Orbitals under Mirror Operation,
φ→φ
Therefore, in view of the parity of the associated Legendre function, eq 7, the spherical harmonics transform as
i
(i.e., cos θ → cos θ) i
φ→φ+π
σ
(8)
That is, the polar and azimuthal parts of the spherical coordinates transform as i
Plm(cos θ) → (1)l+mPlm(cos θ) i
e imφ → e im(φ+π) = (1)me imφ
(9)
i
Yl m(θ, φ) → (1)l+mYl m(θ, φ)
(18)
The combination of eqs 10 and 15 also automatically yields the above expression. This is because the mirror operation can be envisioned as the combination of i and C2. The transformation matrix for σ is thus given by σ = 1
l+ m
= 1
l
1
m
= 1 l C 2
(19)
Therefore the character, or the trace of the transformation matrix, of σ is related to that of C2 by the following equality:
Therefore, the whole spherical harmonics transform as Yl m(θ, φ) → (1)l+2mYl m(θ, φ) = (1)lYl m(θ, φ)
(17)
(10)
χ(σ) = (1)l χ(C2)
(20)
This expression defines the parity of the spherical harmonics. Spherical Orbitals under Improper Rotation, Sn
Spherical Orbitals under Rotation, Cn With a proper rotation operation, Cn, the transformation of coordinates is given by Cn
θ → θ Cn
Sn
(11)
φ → φ + α
Cn
(12)
The matrix form of this transformation scheme was given by eq 2. We will indicate matrices using braces; for example,
C n = e i mα
Yl m(θ, φ) → (1)l+me imαYl m(θ, φ)
(21)
Or equivalently, in the matrix form:
where α is the angle of rotation and is given by eq 3. Thus, the spherical harmonics transform as Yl m(θ, φ) → e im αYl m(θ, φ)
The transformation scheme of Ylm(θ, φ) under Sn, which is the combination of Cn and σ, can be obtained simply by combining eqs 12 and 18:
(13)
S n = 1
l+ m i mα e
(22)
Note that this expression differs from that for the proper rotation Cn (eq 13 matrix) only by the pre-exponential factor (1)l+m, which arises solely from the parity of the associated Legendre function, eq 7. Therefore our problem reduces to obtaining the trace of the eq 22 matrix. This straightforward procedure is given
JChemEd.chem.wisc.edu • Vol. 78 No. 5 May 2001 • Journal of Chemical Education
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Research: Science and Education
in the Appendix. The result is
One-Electron Energy Levels and Many-Electron States
cos l + 1 α 2 χ S n = tr 1 (23) 1 cos α 2 Comparison of the eq 23 with the expression for χ(Cn) leads to
Finally, we need to consider the difference between oneelectron levels and many-electron states expressed in a concise manner by term symbols. For example, 3F arising from a d 2 configuration is known to split into
l+ m i mα e =
χ S n = tan 1 α cot l + 1 α χ C n (24) 2 2 This expression is not particularly useful except in the case with n = 4 (α = π/2), when upon substitution it reduces to χ(S4) = (1)l χ(C4)
(25)
To summarize, the character of the mirror operation σ is the same as that of C2 when l is even, whereas it is the negative of the character of C2 when l is odd (eq 20); the same relationship exists between S4 and C4 (eq 25). The results obtained here are thus simple and intuitively acceptable: under these improper operations, an orbital with odd parity (l = odd) transforms into the minus of what is expected for the corresponding proper operation. One-Electron Energy Levels in a Td Environment The corollary of the above arguments is that orbitals with l = even in Td split into irreducible representations as in the O symmetry. For example, d → e + t2
F → 3A 2 + 3T 1 + 3T 2
(28)
in a Td environment (2, p 273). When we compare the splitting of many-electron terms with the splitting of oneelectron levels of the f orbital in Td (as given in eq 27), it is clear that it is not necessarily the same as the splitting of oneelectron energy levels with the same l value. In other words, we cannot use the splitting scheme of a wave function specified by l to obtain the corresponding splitting scheme of the manyelectron states specified by L having the same numerical value. To elucidate the differences in behavior between the f orbital and F state in Oh and Td environments, Table 4 was created to show the correlations of the bases in the orbital and in the state (based on the character table given in ref 6 ). Table 4. The Bases of f Orbital and F State Arising from a dn Configuration, in Oh and Td Environments f Orbital
Oh
Td
F State
Oh
Td
xyz
A2u
A1
d2 (e 2)
A2g
A2
(x 3, y 3, z 3)
T1u
T2
d2 (t 22)
T1g
T1
[x(z 2 – y 2), y (z 2 – x 2), z (x 2 – y 2)]
T2u
T1
d2 (t 2 e 2)
T2g
T2
(26)
Orbitals with l = odd in Td split differently from those in O. One has to resort to the character table for Td (Table 2), with the help of eqs 20 and 25 for χ(σ) and χ(S4), respectively, to obtain the irreducible representations. As an example, the characters of the f orbital in O and Td environments are given in Table 3. Table 3. Characters of f Orbital under O and Td Environments O
E
8C3
3C2
6C4
6C2
f orbital
7
1
1
1
1
Td
E
8C3
3C2
6S4
6σd
f orbital
7
1
1
1
1
With the conventional method (2, p 88), the representation in Table 3 can be reduced easily. The result, together with the results for p and h orbitals under a Td environment, is given below. p → t2 f → a1 + t1 + t2 (27) h → e + t1 + 2t 2 This splitting scheme should be compared with the schemes in an O environment, presented in eq 5. Observe that the effects of these cubic environments on the orbitals with l = odd are not identical. For example, in the f orbital, a2 in O is replaced by a1 in a Td environment (1, p 111).1 In a class where time is limited, the transformation scheme of orbitals under Td can thus be taught as described, with one caution: σ is given as the combination of C2 and i. 676
3
Each basis of the f orbital is represented by an irreducible representation with the ungerade character in an Oh environment, whereas the bases of the F states inherently possess the gerade character because these states are composed of d orbitals. Consequently, these irreducible representations correlate differently with their counterparts in a Td environment. Acknowledgment I thank students K. Irisawa and K. Ishikawa for bringing to my attention the difficulty of obtaining irreducible representations for the f orbital in a Td environment. Note 1. This is properly stated in the literature (see 1, p 111). Also, the character table (6 ) suggests that the f orbital transforms as given in eq 27 under Td . Unfortunately, the splitting scheme of the f orbital under Td is improperly designated as a2 + t1 + t2 on p 264 of ref 2.
Literature Cited 1. Ballhausen, C. J. Introduction to Ligand Field Theory; McGrawHill: New York, 1962. 2. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.; Wiley: New York, 1990. 3. Figgis, B. N. Introduction to Ligand Fields; Wiley: New York, 1966; Krieger: Malabar, FL, 1986. 4. Burns, G; Glazer, A. M. Space Groups for Solid State Scientists, 2nd ed.; Academic: Orlando, FL, 1990; p 12. 5. Arfken, G. Mathematical Methods for Physicists, 3rd ed.; Academic: Orlando, FL, 1985; p 668.
Journal of Chemical Education • Vol. 78 No. 5 May 2001 • JChemEd.chem.wisc.edu
Research: Science and Education 6. Harris, D. C.; Bertolucci, M. D. Symmetry and Spectroscopy; Oxford University Press: London, 1978; Dover: New York, 1989; p 473.
The matrix has the same form as that for proper rotation, except that α is replaced by π + α. Therefore we can use the same summation procedure as that used to obtain our familiar result eq 4:
Appendix The character of improper rotation, χ(Sn), is simply the trace of the transformation matrix: χ S n = tr 1
l+ m i mα e
(A1)
The summation is over all m and does not depend on l, except that l defines the limit of the summation.
χ S n = 1 l tr 1 me i mα = 1 l tr e i mπe i mα = 1 l tr e i m(π+α) (A2)
χ Sn
sin l + 1 π + α sin 1 π + lπ + l + 1 α 2 2 2 l = 1 = 1 = 1 1 1 sin π + α sin π + α 2 2 2 l
(A3) cos lπ + l + 1 α 2 = 1 1 l 1 cos α 2
1 cos l + 1 α cos l + 1 α 2 2 = 1 1 cos α cos α 2 2 l
l
JChemEd.chem.wisc.edu • Vol. 78 No. 5 May 2001 • Journal of Chemical Education
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