Research: Science and Education
Group Theory and Crystal Field Theory: A Simple and Rigorous Derivation of the Spectroscopic Terms Generated by the t2g2 Electronic Configuration in a Strong Octahedral Field Simone Morpurgo Dipartimento di Chimica, Università degli Studi di Roma “La Sapienza”, P.le A. Moro 5, 00185 Roma, Italia;
[email protected]
The crystal field theory, in spite of age and relative simplicity, is a powerful tool to elucidate many spectroscopic, magnetic, and thermodynamic properties of transition-metal complexes and solid-state compounds. It finds application in the field of inorganic and materials chemistry, catalysis, metallorganic, and bioinorganic chemistry and a vast literature is available on the topic. General inorganic chemistry textbooks (1–3) provide a descriptive approach to the theory, which is suitable for first- or second-year undergraduates. More detailed descriptive textbooks (4–7) are suitable for upper-level undergraduates. Some pedagogical descriptions of the subject are also available in the literature (8–11). The quantum chemical details of the theory are treated in some textbooks (12–16), which are appropriate for advanced undergraduate and graduate students. Despite the long teaching tradition, some mathematical features of the theory are still difficult for students or graduates who wish to achieve a more in-depth knowledge of the matter. This is due either to oversimplification of the pictorial descriptions or to excessive complexity of the quantum-mechanical presentations, which are not always easy to read or to study. Notwithstanding the problem of spin–orbit coupling, the crystal field theory is based on two main approximations, namely the weak-field and the strong-field approximations, depending on the relative weight of interelectronic repulsion and crystal field as energy contributions. Symmetry and spin multiplicity of the spectroscopic terms of a given transition-metal complex are independent of the field strength and depend only (i) on the dn electron configuration of the transition metal involved and (ii) on the geometrical properties (point group) of the complex itself. The relative energy of the terms depends, instead, on the field strength. For any given dn electron configuration, the Orgel (17) and the Tanabe–Sugano (18) diagrams, reproduced in many textbooks, relate the relative energy of the terms with the field strength. Although there is a one-to-one correspondence between weak- and strong-field terms, the terms and the related wavefunctions are derived in a different way, depending on the approximation adopted. Weak-field terms are derived from the free-ion terms by a conceptually simple procedure. A relatively difficult point, instead, is the definition of strong-field terms. In this case, a simple group-theory procedure based on the direct product is generally shown by textbooks (7, 13–16) for the d2 electronic configuration. For more than two d electrons, such a procedure fails because it produces a higher number of terms than there should be (see ref 7, section 5.5). Suitable corrections, such as the “Hyde method” (19, 20) and the “spin-factor method” (21, 22), were successfully proposed for a schematic definition of the free-ion terms but can also
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be applied to crystal field theory. However, the definition of energy terms does not say anything about the real procedure to obtain the related wavefunctions and about the wavefunction itself, which should be known in order to perform energy calculations. The aim of this article is to show (i) how to apply in a simple but rigorous way the principles of symmetry and group theory to the zero-order wavefunctions associated with the strong-field t2g2 configuration and (ii) how to derive their symmetry-adapted linear combinations (SALC) associated with the generated energy terms. The approach is widely adopted as an application of group theory but deserves a more careful pedagogic development. A full comprehension of the topic will also allow the student to better understand the use of schematic methods for the identification of strong-field terms. It is assumed that the reader knows the background of group theory, crystal field theory, and the perturbation method at least to the first-order energy correction. Discussion The application of an octahedral crystal field to the five monoelectronic d functions splits them into two subsets, having t2g (dxy, dxz, dyz) and eg (dz2, dx2−y2) symmetry, respectively. The related polyelectronic configurations are obtained by assigning the electrons to each subset in all possible ways, that is, for a d2 complex, t2g2, t2geg, and eg2, in order of increasing energy. The zero-order polyelectronic functions (microstates) associated with each configuration are built by distributing the two electrons across the orbitals, in all possible ways, taking into account the electron spin, the Pauli exclusion principle, and the indistinguishability of the electrons. The procedure is shown in refs 1 and 2 for the d2 configuration and, in more detail, in ref 23 for the p2 configuration of a free atom. Placing two electrons across three p or three t2g orbitals is a conceptually equivalent procedure. The 15 functions shown in Table 1 were obtained for the t2g2 configuration. Each one is a Slater determinant defined, as usual, by the elements of its main diagonal. The functions are grouped according to their MS eigenvalue (15, 23) (i.e., the projection of the total spin along a reference axis, usually taken as the z axis). After defining the polyelectronic functions, the symmetry properties of the monoelectronic d functions contained in each Slater determinant should be established. This is not a difficult task, since it is known that the dxy, dxz, and dyz orbitals have the same transformation properties as the binary products of the Cartesian coordinates. As explained in many textbooks (12–16), in an octahedral environment (Oh)
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Research: Science and Education Table 1. Polyelectronic Functions (Slater Determinants) Associated with the t2g2 Electronic Configuration in a Strong Octahedral Field Monoelectronic Functions dxy
dxz
↑
↑
↑ ↑ ↑
↓
↓
↑
↑
dyz 1 ↑
1
↑
1 0 0
↓
↓
0
↑
0
↑
↓
0
↓
↑
0
↑ ↓
0 ↑ ↓
↓
MS
0
↓
F1 = |xy(1)α(1), xz(2)α(2)| F2 = |xy(1)α(1), yz(2)α(2)| F3 = |xz(1)α(1), yz(2)α(2)| F4 = |xy(1)α(1), xz(2)β(2)|
C3F1 = C3|xy(1)α(1), xz(2)α(2)| = | yz(1)α(1), xy(2)α(2)| = ᎑F2 C3F2 = C3|xy(1)α(1), yz(2)α(2)| = | yz(1)α(1), xz(2)α(2)| = ᎑F3 C3F3 = C3|xz(1)α(1), yz(2)α(2)| = | xy(1)α(1), xz(2)α(2)| = ᎑ F1
F7 = |xy(1)β(1), yz(2)α(2)| F8 = |xz(1)α(1), yz(2)β(2)| F9 = |xz(1)β(1), yz(2)α(2)|
C2´F1 = C2´|xy(1)α(1), xz(2)α(2)| = | xy(1)α(1), ᎑yz(2)α(2)| = ᎑F2 C2´F2 = C2´|xy(1)α(1), yz(2)α(2)| = | xy(1)α(1), ᎑xz(2)α(2)| = ᎑ F1 C2´F3 = C2´|xz(1)α(1), yz(2)α(2)| = |᎑yz(1)α(1), ᎑xz(2)α(2)| = ᎑F3
F10 = |xy(1)α(1), xy(2)β(2)| F11 = |xz(1)α(1), xz(2)β(2)| F12 = |yz(1)α(1), yz(2)β(2)|
NOTE: The table was compiled according to the following three rules: (i) If one column (i.e., a monoelectronic orbital) of the transformed determinant appears with the minus sign, the determinant changes its sign with respect to a given reference determinant reported in Table 1. (ii) If the symmetry operation interchanges the columns, the sign of the determinant changes. (iii) If the above points are combined in any way an even number of times, the sign of the determinant does not change. If the number of combinations is odd, the sign of the determinant changes.
0 ᎑1
↓
᎑1
F13 = |xy(1)β(1), xz(2)β(2)| F14 = |xy(1)β(1), yz(2)β(2)|
↓
᎑1
F15 = |xz(1)β(1), yz(2)β(2)|
the operations of the pure rotational subgroup O can be employed to this purpose, owing to the intrinsic symmetry of the d orbitals with respect to the inversion operation. Such operations, grouped in classes, are respectively E , 6C 4 , 3C 2 (= C 42 ), 8C 3 , 6C 2 ′ To define the above symmetry operations, we refer to the following system of Cartesian axes, taking into account that (i) the C4 and C2 (= C42) axes coincide with the z axis, (ii) the C3 axis comes through the origin and forms 45⬚ angles with the three coordinate axes, (iii) the C2ⴕ axis lies on the xy plane and forms 45⬚ angles with the x and y axes, and (iv) all rotations are conventionally assumed to be anti-clockwise or, alternatively, they correspond to a clockwise rotation of the coordinate system.
The effect of the rotations on each single Cartesian coordinate is C4
x → y y → − x z → z x → y
C 3 y → z z → x
152
C2F1 = C2|xy(1)α(1), xz(2)α(2)| = | xy(1)α(1), ᎑xz(2)α(2)| = ᎑ ᎑F1 C2F2 = C2|xy(1)α(1), yz(2)α(2)| = | xy(1)α(1), ᎑yz(2)α(2)| = ᎑F2 C2F3 = C2|xz(1)α(1), yz(2)α(2)| = |᎑xz(1)α(1), ᎑yz(2)α(2)| = ᎑F3
F5 = |xy(1)β(1), xz(2)α(2)| F6 = |xy(1)α(1), yz(2)β(2)|
↑ ↓ ↓
↓
Slater Determinant (main diagonal)
Table 2. Effect of the Symmetry Operations of the O Group on the F1–F3 Slater Determinants C4F1 = C4|xy(1)α(1), xz(2)α(2)| = |᎑xy(1)α(1), yz(2)α(2)| = ᎑F2 C4F2 = C4|xy(1)α(1), yz(2)α(2)| = |᎑xy(1)α(1), ᎑xz(2)α(2)| = ᎑ F1 C4F3 = C4|xz(1)α(1), yz(2)α(2)| = | yz(1)α(1), ᎑xz(2)α(2)| = ᎑F3
x → − x C 2 y → − y z → z x → y ′ C 2 y → x z → − z
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from which it is easily deduced how the binary products of the coordinates transform xy → − xy xz → yz
C4
xy → xy C2
yz → − xz xy → yz xz → xy
C3
yz → xz
xz → − xz yz → − yz xy → xy
C 2 ′ xz → − yz y z → − xz
The Slater determinants F1, F2, and F3, corresponding to the MS = 1 spin eigenvalue (both electrons have α spin), can be considered first to establish the effect of the symmetry operations on the polyelectronic functions. By applying the symmetry operations, they are transformed into each other as reported in Table 2. Taking into account that each function gives a ±1 contribution to the character of the transformation matrix only when it is transformed into itself, with the + or the − sign, the following characters are obtained χ(E)
χ(C4)
χ(C2)
χ(C3)
χ(C2´)
3
1
᎑1
0
᎑1
By comparison with the character table (16) of the O group, it can be verified that they correspond to the T1 irreducible representation. This implies that F1, F2, and F3 are basis for T1 in O, which corresponds to T1g in Oh. The F13, F14, and F15 determinants are equivalent to F1, F2, and F3 in the spatial part of the wavefunction and differ from the former ones in the spin part (both electrons have β spin). It can be easily
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deduced that they have exactly the same symmetry properties as F1, F2, and F3; that is, they are also basis for T1. Until now, we have identified six basis functions for the T1 irreducible representation, three of which with MS = 1 (F1, F2, F3) and three with MS = ᎑1 (F13, F14, F15). It is evident that such functions belong to a 3T1 (3T1g in Oh) term. The remaining three functions of 3T1 are to be searched for within the Slater determinants with MS = 0, most probably as a linear combination of them. An examination of the Slater determinants from F4 to F12 in Table 1 shows that three of them, F10, F11, and F12, have both electrons in the same orbital, with paired spin. It is obvious that any symmetry operation will transform them into each other and that they will never be transformed into a F4–F9 determinant, where the electrons are placed in different orbitals. Therefore, it can be foreseen that the two function sets, F 4–F 9 and F 10–F 12, will be basis for distinct representations of the O group. The application of the symmetry operations to F4–F9 is reported in Table 3. As usual, taking into account that each function gives a ±1 contribution to the character of the transformation matrix only when is transformed into itself, with the + or the − sign, the following characters are obtained
Table 3. Effect of the Symmetry Operations of the O Group on the F4–F9 Slater Determinants C4F4 = C4|xy(1)α(1), xz(2)β(2)| = |᎑xy(1)α(1), yz(2)β(2)| = ᎑F6 C4F5 = C4|xy(1)β(1), xz(2)α(2)| = |᎑xy(1)β(1), yz(2)α(2)| = ᎑F7 C4F6 = C4|xy(1)α(1), yz(2)β(2)| = |᎑xy(1)α(1), ᎑xz(2)β(2)| = F4 C4F7 = C4|xy(1)β(1), yz(2)α(2)| = |᎑xy(1)β(1), ᎑xz(2)α(2)| = F5 C4F8 = C4|xz(1)α(1), yz(2)β(2)| = | yz(1)α(1), ᎑xz(2)β(2)| = F9 C4F9 = C4|xz(1)β(1), yz(2)α(2)| = | yz(1)β(1), ᎑xz(2)α(2)| = F8 C2F4 = C2|xy(1)α(1), xz(2)β(2)| = | xy(1)α(1), ᎑xz(2)β(2)| = ᎑F4 C2F5 = C2|xy(1)β(1), xz(2)α(2)| = | xy(1)β(1), ᎑xz(2)α(2)| = ᎑F5 C2F6 = C2|xy(1)α(1), yz(2)β(2)| = | xy(1)α(1), ᎑yz(2)β(2)| = ᎑F6 C2F7 = C2|xy(1)β(1), yz(2)α(2)| = | xy(1)β(1), ᎑yz(2)α(2)| = ᎑F7 C2F8 = C2|xz(1)α(1), yz(2)β(2)| = |᎑xz(1)α(1), ᎑yz(2)β(2)| = F8 C2F9 = C2|xz(1)β(1), yz(2)α(2)| = |᎑xz(1)β(1), ᎑yz(2)α(2)| = F9 C3F4 = C3|xy(1)α(1), xz(2)β(2)| = | yz(1)α(1), xy(2)β(2)| = ᎑F7 C3F5 = C3|xy(1)β(1), xz(2)α(2)| = | yz(1)β(1), xy(2)α(2)| = ᎑F6 C3F6 = C3|xy(1)α(1), yz(2)β(2)| = | yz(1)α(1), xz(2)β(2)| = ᎑F9 C3F7 = C3|xy(1)β(1), yz(2)α(2)| = | yz(1)β(1), xz(2)α(2)| = ᎑F8 C3F8 = C3|xz(1)α(1), yz(2)β(2)| = | xy(1)α(1), xz(2)β(2)| = F4 C3F9 = C3|xz(1)β(1), yz(2)α(2)| = | xy(1)β(1), xz(2)α(2)| = F5
χ(E)
χ(C4)
χ(C2)
χ(C3)
χ(C2´)
6
0
᎑2
0
0
C2´F4 = C2´|xy(1)α(1), xz(2)β(2)| = | xy(1)α(1), ᎑yz(2)β(2)| = ᎑F6 C2´F5 = C2´|xy(1)β(1), xz(2)α(2)| = | xy(1)β(1), ᎑yz(2)α(2)| = ᎑F7
By means of the common reduction procedure (16), it turns out that such a representation is the sum of the T1 and T2 irreducible representations. As a consequence, the F4–F9 functions will give rise to (i) three linear combinations with T1 symmetry, which belong to the 3T1 term (with F1–F3 and F13–F15) and (ii) three T2 combinations, which necessarily belong to a 1T2 term. In principle, the above linear combinations should be obtained by the projection-operator technique (16), which, for multi-dimensional irreducible representations, requires the knowledge of all transformation matrices associated with the group operations. Such a procedure is conceptually simple but very tedious in practice. In the present case, however, it is possible to derive the correct functions in a simple way: by imposing that the corresponding linear combinations are eigenfunctions of Sˆ2 (23). Obviously, the functions of the 3T1 term will have S(S + 1) = 2 and 2S + 1 = 3, whereas those of the 1T2 term will have S(S + 1) = 0 and 2S + 1 = 1. In brief, the polyelectronic Sˆ2 operator can be expressed as follows (23)
ˆ2
S
= Sˆx + Sˆ y 2
2
+ Sˆ z
∑ ˆs x
i
∑ ˆs y
+
i
2
+
i
∑ ˆs z
i
i
where ˆs x i , sˆ y i , and sˆ z i are the components of the ith monoelectronic spin operator. Once the square of each sum is developed, the terms can be grouped in the following way
Sˆ 2 =
∑ (ˆs xi 2
+ ˆs yi 2 + ˆs z i 2
i
+
∑ ˆs zi ˆsz j
i ≠ j
+
C2´F8 = C2´|xz(1)α(1), yz(2)β(2)| = |᎑yz(1)α(1), ᎑xz(2)β(2)| = ᎑F9 C2´F9 = C2´|xz(1)β(1), yz(2)α(2)| = |᎑yz(1)β(1), ᎑xz(2)α(2)| = ᎑F8 NOTE: The table was compiled according to the rules listed in the footnote of Table 2.
The first term is the sum of the one-electron ˆs 2 operators, whereas the third term can be rewritten as in the equation below ˆs 2 + sˆ ˆs Sˆ 2 =
∑ i
)
∑ (ˆs xi ˆs x j
+ ˆs y i ˆs y j
i ≠ j
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)
∑
i
+
∑ (sˆ x
i ≠ j
2
i
C2´F7 = C2´|xy(1)β(1), yz(2)α(2)| = | xy(1)β(1), ᎑xz(2)α(2)| = ᎑F5
2
2
=
C2´F6 = C2´|xy(1)α(1), yz(2)β(2)| = | xy(1)α(1), ᎑xz(2)β(2)| = ᎑F4
zi z j
i ≠ j i
+ isˆ yi
) (sˆx
j
− isˆ y j
)
The terms (sˆxi + isˆyi) and (sˆxj + isˆyj) are, respectively, the oneelectron spin-up and spin-down operators; that is, when applied to a monoelectronic function, they have the effect to rise and to lower, respectively, the z component of the spin by one unit (in terms of ប). In other words, the above operators transform the α one-electron spin function into the β one, and vice versa. In general, a Slater determinant, such as any of the F1–F15 functions we are considering in the present example, is not an eigenfunction of the polyelectronic Sˆ2 operator. Reference 23 shows in detail how the Sˆ2 operator applies to any Slater determinant. For the sake of brevity, the procedure is omitted and only the final expression (eq 9-34
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ˆ 2 Operator Table 4. Application of the S to the Symmetric and to the Antisymmetric Linear Combinations of the F4–F9 Slater Determinants Operator Application
S
2S+1
ˆ 2(F + F ) = 2(F + F ) = 1(1 + 1)(F + F ) S 4 5 4 5 4 5
1
3
ˆ 2(F + F ) = 2(F + F ) = 1(1 + 1)(F + F ) S 6 7 6 7 6 7 Sˆ 2(F8 + F9) = 2(F8 + F9) = 1(1 + 1)(F8 + F9) ˆ 2(F − F ) = 0 = 0(0 + 1)(F − F ) S 4 5 4 5 Sˆ 2(F6 − F7) = 0 = 0(0 + 1)(F6 − F7) ˆ 2(F − F ) = 0 = 0(0 + 1)(F − F ) S 8
9
8
9
1
3
1
3
0
1
0
1
0
1
Table 5. Effect of the Symmetry Operations of the O Group on the F10–F12 Slater Determinants C4F10 = C4|xy(1)α(1), xy(2)β(2)| = |᎑xy(1)α(1), ᎑xy(2)β(2)| = F10 C4F11 = C4|xz(1)α(1), xz(2)β(2)| = | yz(1)α(1), yz(2)β(2)| = F12 C4F12 = C4|yz(1)α(1), yz(2)β(2)| = |᎑xz(1)α(1), ᎑xz(2)β(2)| = F11 C2F10 = C2|xy(1)α(1), xy(2)β(2)| = | xy(1)α(1), xy(2)β(2)| = F10 C2F11 = C2|xz(1)α(1), xz(2)β(2)| = |᎑xz(1)α(1), ᎑xz(2)β(2)| = F12 C2F12 = C2|yz(1)α(1), yz(2)β(2)| = |᎑yz(1)α(1), ᎑yz(2)β(2)| = F11 C3F10 = C3|xy(1)α(1), xy(2)β(2)| = | yz(1)α(1), yz(2)β(2)| = F12 C3F11 = C3|xz(1)α(1), xz(2)β(2)| = | xy(1)α(1), xy(2)β(2)| = F10 C3F12 = C3|yz(1)α(1), yz(2)β(2)| = | xz(1)α(1), xz(2)β(2)| = F11 C2´F10 = C2´|xy(1)α(1), xy(2)β(2)| = | xy(1)α(1), xy(2)β(2)| = F10 C2´F11 = C 2´|xz(1)α(1), xz2)β(2)| = |᎑yz(1)α(1), ᎑yz(2)β(2)| = F12 C2´F12 = C2´|yz(1)α(1), yz(2)β(2)| = |᎑xz(1)α(1), ᎑xz(2)β(2)| = F11 NOTE: The table was compiled according to the rules listed in the footnote of Table 2.
in ref 23) is reported below Sˆ 2 F
{(n , l , m , s ), (n , l , m , s ), ..., 1
1
1
1
2
2
2
2
..., (ni , l i , mi , si ) , ..., (n j , l j , m j , s j ) , ...,
=
∑ si ( si i
F
+ 1) +
∑ si s j
}
2 h ×
i ≠ j
{(n , l , m , s ), (n , l , m , s ), ..., 1
1
1
1
2
2
2
2
..., (ni , l i , mi , si ) , ..., (n j , l j , m j , s j ) , ...,
+
3 + si 2
∑
i ≠ j
F
1 − si 2
3 − sj 2
1 + sj 2
{(n , l , m , s ), (n , l , m , s ), ..., 1
1
1
1
2
2
2
}
1/ 2
2 h ×
2
..., (ni , l i , mi , si + 1) , ..., (n j , l j , m j , s j −1) , ...,
}
F{(n1, l1, m1, s1), (n2, l2, m2, s2), …, (ni, li, mi, si ), …, (nj, lj, mj, sj ), …} is a Slater determinant expressed by its main diagonal; (ni, li, mi, si ) and (nj, lj, mj, sj ) are the monoelectronic functions, respectively, associated with the ith and the jth elec-
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trons; n, l, m are the well-known quantum numbers; and si and sj are the corresponding eigenvalues of the monoelectronic ˆs z operator, that is, the projection of the electron spin along the z axis. In summary, (i) when the first two terms of Sˆ2 (i.e., Σisˆi2 + Σi≠jˆs ziˆs zj ) are applied to a Slater determinant, the determinant itself is obtained, multiplied by the [Σi|si|(|si| + 1) + Σi≠jsisj]ប2 coefficient and (ii) the application of Σi≠j(sˆxi + isˆyi)(sˆxj − isˆyj ) generates a sum of determinants, each one having a couple of monoelectronic functions (defined by the i and j indices) with the si and sj eigenvalues respectively augmented (si + 1) and diminished (sj − 1) by one unit with respect to the original determinant. The [(3/2 + si )(1/2 − si )]1兾2 coefficient is associated with the monoelectronic function with augmented si eigenvalue, whereas the [(3/2 − sj )(1/2 + sj )]1兾2 one is associated with the function with diminished sj eigenvalue. It should be noted that, when an electron has α spin, its si = 1/2 eigenvalue cannot be augmented because, for such a si value, the [(3/2 + si)(1/2 − si)]1兾2 coefficient vanishes. For the same reason, the sj = ᎑1/2 eigenvalue cannot be diminished. As an example, let us apply Sˆ2 to F4 and F5 (Table 1). By applying Sˆ2 to F4, (i) F4 is re-obtained and (ii) a new determinant is generated where the spin of the electron 1 is diminished by one unit (α → β) and the spin of the electron 2 is correspondingly augmented (β → α). The latter determinant is nothing but F5. The application of Sˆ2 to F5 yields the same result. In synthesis, the application of Sˆ2 to F4–F9 gives (ប2 is omitted) 2 Sˆ F 5 = F 4 + F 5 2 Sˆ F 7 = F 6 + F 7 2 Sˆ F = F + F
ˆS2F = F + F 4 4 5 ˆS2F = F + F 6 6 7 ˆ2F = F + F S 8 8 9
9
8
9
It can be observed that the F4–F9 determinants are not eigenfunction of Sˆ2, but give, in couples, the same result. As a consequence, the symmetric and the antisymmetric linear combinations of the functions of each couple turn out to be eigenfunctions of Sˆ2, as in Table 4. The symmetric linear combinations (S = 1) are the appropriate eigenfunctions for a triplet term, whereas the antisymmetric ones (S = 0) are associated with a singlet term. Therefore, it should be checked whether the symmetric functions have really T1 symmetry and the antisymmetric ones have T2 symmetry. Taking into consideration the results of Table 3, the transformation properties of the symmetric combinations are C4(F4 + F5) = ᎑(F6 + F7)
C2(F4 + F5) = ᎑(F4 + F5)
C4(F6 + F7) = (F4 + F5)
C2(F6 + F7) = ᎑(F6 + F7)
C4(F8 + F9) = (F9 + F8)
C2(F8 + F9) = (F8 + F9)
C3(F4 + F5) = ᎑(F6 + F7)
C2´(F4 + F5) = ᎑(F6 + F7)
C3(F6 + F7) = ᎑(F8 + F9)
C2´(F6 + F7) = ᎑(F4 + F5)
C3(F8 + F9) = (F4 + F5)
C2´(F8 + F9) = ᎑(F8 + F9)
with the following characters of the corresponding transformation matrices χ(E)
χ(C4)
χ(C2)
χ(C3)
χ(C2´)
3
1
᎑1
0
᎑1
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Since these are the very characters of the T1 irreducible representation in O, it can be deduced that the symmetric linear combinations are the correct eigenfunctions of the 3T1 term (3T1g in Oh). In the same way, the transformation properties of the antisymmetric combinations are C4(F4 − F5) = ᎑(F6 − F7)
C2(F4 − F5) = ᎑(F4 − F5)
C4(F6 − F7) = (F4 − F5)
C2(F6 − F7) = ᎑(F6 − F7)
C4(F8 − F9) = ᎑(F8 − F9)
C2(F8 − F9) = (F8 − F9)
C3(F4 − F5) = (F6 − F7)
C2´(F4 − F5) = ᎑(F6 − F7)
C3(F6 − F7) = (F8 − F9)
C2´(F6 − F7) = ᎑(F4 − F5)
C3(F8 − F9) = (F4 − F5)
C2´(F8 − F9) = (F8 − F9)
from which the following characters are obtained χ(E)
χ(C4)
χ(C2)
χ(C3)
χ(C2´)
3
᎑1
᎑1
0
1
They correspond to the T2 irreducible representation in O, so that the antisymmetric linear combinations are the eigenfunctions associated with the 1T2 term (1T2g in Oh). Finally, the transformation properties of the F10, F11, and F12 functions are reported in Table 5. The characters of the transformation matrices are χ(E)
χ(C4)
χ(C2)
χ(C3)
χ(C2´)
3
1
3
0
1
The corresponding representation is easily decomposed into A1 + E in O (A1g + Eg in Oh ). In this case, also, the symmetry-adapted linear combinations have to be formed by means of the projection operator method. The procedure, for the sake of brevity, is omitted. It should be noted that the A1g and Eg terms must be singlet terms, since they are only composed by functions with MS = 0. To this purpose, it can be checked that the F10, F11, and F12 determinants are eigenfunctions of Sˆ2 with the S(S + 1) = 0 eigenvalue. In summary, it was shown that, in case the strong-field approximation is valid, the spectroscopic terms generated by the t2g2 configuration in an octahedral complex are 1
A1g +
1
Eg +
3
T1g +
1
T2g
Once the eigenfunctions associated with each term are known, the first-order perturbative problem can be solved to establish the relative energy of the terms. The appropriate matrix elements of the secular determinant arising from the perturbation treatment at the first order (23) should be calculated. From a practical point of view, only one ψ function can be selected within each term (all functions of a given term are degenerate) to compute the 兰ψ*Vˆeeψdτ integral, where Vˆee is the operator corresponding to the interelectronic repulsion (12–16, 23).
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Conclusions The procedure presented here is rigorous from the point of view of the identification of strong-field spectroscopic terms by means of group-theory principles. A more elaborate treatment should obviously be expected for more complex configurations. It should be warned once again that (i) the symmetry-adapted linear combinations (SALC) have, more formally, to be obtained by means of the projection operator method and (ii) the eigenfunctions of Sˆ2 for the 3T1g and 1T2g terms were derived in an intuitive way but a more systematic procedure (23) should be adopted for more complex cases. Schematic representations of strong crystal field terms for d1–d9 electronic configurations are available, for instance, in refs 4–7 and 13. Literature Cited 1. Kettle, S. F. A. Physical Inorganic Chemistry; Spectrum Academic Publishers: Oxford, 1996. 2. Shriver, D. F.; Atkins, P. W.; Langford, C. H. Inorganic Chemistry; Oxford University Press: New York, 1994. 3. Cotton, F. A.; Wilkinson, G. W. Advanced Inorganic Chemistry; John Wiley and Sons: New York, 1962. 4. Orgel, L. E. An Introduction to Transition-Metal Chemistry; John Wiley and Sons: New York, 1960. 5. Lewis, J.; Wilkins, R. G. Modern Coordination Chemistry; Interscience Publishers Inc.: New York, 1960. 6. Dunn, T. M; McClure, D. S.; Pearson, R. G. Crystal Field Theory; Harper and Row: New York, 1965. 7. Kettle, S. F. A. Coordination Compounds; Appleton-CenturyCrofts: New York, 1969. 8. Sutton, L. E. J. Chem. Educ. 1960, 37, 498. 9. Carlin, R. L. J. Chem. Educ. 1963, 40, 135. 10. Cotton, F. A. J. Chem. Educ. 1964, 41, 466. 11. Johnson, R. C. J. Chem. Educ. 1965, 42, 147. 12. Ballhausen, C. J. Ligand Field Theory; McGraw-Hill: New York, 1962. 13. Figgis, B. N. Introduction to Ligand Fields; John Wiley and Sons, London, 1966. 14. Lever, A. B. P. Inorganic Electronic Spectroscopy; Elsevier: Amsterdam, 1968. 15. Schlaefer, H. L.; Gliemann, G. Basic Principles of Ligand Field Theory; John Wiley and Sons: London, 1969. 16. Cotton, F. A. Chemical Applications of Group Theory; John Wiley and Sons: New York, 1990. 17. Orgel, L. E. J. Chem. Phys. 1955, 23, 1004. 18. (a) Tanabe, Y.; Sugano, S. J. Phys. Soc. Jpn. 1954, 9, 766. (b) Tanabe, Y.; Sugano, S. J. Phys. Soc. Jpn. 1954, 9, 753. 19. Hyde, K. E. J. Chem. Educ. 1975, 52, 87. 20. Vicente, J. J. Chem. Educ. 1983, 60, 560. 21. McDaniel, D. H. J. Chem. Educ. 1977, 54, 147. 22. Guofan, L; Ellzey, M. L., Jr. J. Chem. Educ. 1987, 64, 771. 23. Eyring, H.; Walter, J.; Kimball, G. E. Quantum Chemistry; John Wiley and Sons: New York, 1944.
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