Group Theory Applied to Chemistry Arnout Jozef Ceulemans

Arnout Jozef CeulemansTheoretical Chemistry and Computational ModellingGroup Theory Applied to Chemistry201310.1007/978-94-007-6863-5© Springer Science+Business Media Dordrecht 2013

Appendix A Character Tables

Contents

A.1

Finite Point Groups 192

C ₁ and the Binary Groups C _s , C _i , C ₂ 192

The Cyclic Groups C _n ( n =3,4,5,6,7,8) 192

The Dihedral Groups D _n ( n =2,3,4,5,6) 194

The Conical Groups C _nv ( n =2,3,4,5,6) 195

The C _nh Groups ( n =2,3,4,5,6) 196

The Rotation–Reflection Groups S _{2 n} ( n =2,3,4) 197

The Prismatic Groups D _nh ( n =2,3,4,5,6,8) 198

The Antiprismatic Groups D _nd ( n =2,3,4,5,6) 199

The Tetrahedral and Cubic Groups 201

The Icosahedral Groups 202

A.2

Infinite Groups 203

Cylindrical Symmetry 203

Spherical Symmetry 204

Character tables were introduced to chemistry through the pioneering work of Robert Mulliken [ 1 ]. The book on “Chemical Applications of Group Theory” by F. Albert Cotton has been instrumental in disseminating their use in chemistry [ 2 ]. Atkins, Child, and Phillips [ 3 ] produced a handy pamphlet of the point group character tables. ¹

A.1 Finite Point Groups

C ₁ and the Binary Groups C _s , C _i , C ₂

The Cyclic Groups C _n ( n =3,4,5,6,7,8)

The Dihedral Groups D _n ( n =2,3,4,5,6)

The Conical Groups C _nv ( n =2,3,4,5,6)

The C _nh Groups ( n =2,3,4,5,6)

The Rotation–Reflection Groups S _{2 n} ( n =2,3,4)

The Prismatic Groups D _nh ( n =2,3,4,5,6,8)

The Antiprismatic Groups D _nd ( n =2,3,4,5,6)

The Tetrahedral and Cubic Groups

The Icosahedral Groups

A.2 Infinite Groups

Cylindrical Symmetry

Spherical Symmetry

Appendix B Symmetry Breaking by Uniform Linear Electric and Magnetic Fields

Contents

B.1

Spherical Groups 205

B.2

Binary and Cylindrical Groups 205

B.1 Spherical Groups

B.2 Binary and Cylindrical Groups

The ∥ notation refers to a field oriented along the principal cylindrical axis; in the ⊥ direction several symmetry breakings are possible: C ₂ symmetry implies that the field coincides with the $\hat{C}_{2}$ axis; a magnetic field perpendicular to a symmetry plane or an electric field in a symmetry plane will conserve at least C _s symmetry.

Appendix C Subduction and Induction

Contents

C.1

Subduction G ↓ H 207

C.2

Induction: H ↑ G 211

C.1 Subduction G ↓ H

C.2 Induction: H ↑ G

Ascent in symmetry tables have been provided by Boyle [ 4 ]. Fowler and Quinn have listed the irreps that are induced by σ -, π -, and δ -type orbitals on molecular sites [ 5 ]. These tables are reproduced below. They are useful for the construction of cluster orbitals. $\varGamma_{\it reg}$ always denotes the regular representation. Γ _σ corresponds to the positional representation. The mechanical representation is the sum Γ _σ + Γ _π .

Appendix D Canonical-Basis Relationships

The importance of canonical-basis relationships was demonstrated by Griffith in his monumental work on the theory of transition-metal ions [ 6 ]. The icosahedral basis sets were defined by Boyle and Parker [ 7 ].

(See Fig. D.1 .) Transformation to trigonal basis set:

$\begin{aligned} |E_{\theta}\rangle =& d_{z^2} = \frac{1}{\sqrt{3}} (-d_{x'^2-y'^2}- \sqrt{2}d_{y'z'} ) \\ |E_{\epsilon}\rangle =& d_{x^2-y^2} = \frac{1}{\sqrt{3}} (d_{x'y'}+ \sqrt{2} d_{x'z'} ) \\ |T_{1a}\rangle =& \frac{1}{\sqrt{3}} \bigl( |T_{1x}\rangle + |T_{1y}\rangle+|T_{1z}\rangle \bigr) = p_{z'} \\ |T_{1\theta}\rangle =& \frac{1}{\sqrt{2}} \bigl( |T_{1x}\rangle - |T_{1y}\rangle \bigr) = p_{x'} \\ |T_{1\epsilon}\rangle =& \frac{1}{\sqrt{6}} \bigl( |T_{1x}\rangle + |T_{1y}\rangle-2|T_{1z}\rangle \bigr) = p_{y'} \\ |T_{2a}\rangle =& \frac{1}{\sqrt{3}} \bigl( |T_{2\xi}\rangle + |T_{2\eta}\rangle+|T_{2\zeta}\rangle \bigr) = d_{z'^2} \\ |T_{2\theta}\rangle =& \frac{1}{\sqrt{6}} \bigl( |T_{2\xi}\rangle + |T_{2\eta}\rangle-2|T_{2\zeta}\rangle \bigr) = \frac{1}{\sqrt{3}} ( \sqrt{2}d_{x'^2-y'^2}- d_{y'z'} ) \\ |T_{2\epsilon}\rangle =& \frac{1}{\sqrt{2}} \bigl( |T_{2\eta}\rangle - |T_{2\xi}\rangle \bigr) = \frac{1}{\sqrt{3}} (- \sqrt{2}d_{x'y'}+ d_{x'z'} ) \end{aligned}$

Fig. D.1

Octahedron with x , y , z coordinates in D ₄ and D ₃ setting

Fig. D.2

Icosahedron with x , y , z coordinates in D ₂ setting

It is important to note that in the Boyle and Parker basis the | Hθ 〉 and | Hϵ 〉 components do not denote components that transform like the functions $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ , but refer to linear combinations of these:

$\begin{aligned} |H\theta\rangle =& \sqrt{\frac{3}{8}} d_{z^2} +\sqrt{ \frac{5}{8}}d_{x^2-y^2} \\ |H\epsilon\rangle =& -\sqrt{\frac{5}{8}} d_{z^2} +\sqrt{ \frac{3}{8}}d_{x^2-y^2} \end{aligned}$

Griffith has presented the subduction of spherical | JM 〉 states to point-group canonical bases for the case of the octahedral group. Similar tables for subduction to the icosahedral canonical basis have been published by Qiu and Ceulemans [ 8 ]. Extensive tables of bases in terms of spherical harmonics for several branching schemes are also provided by Butler [ 9 ].

Appendix E Direct-Product Tables

Extensive direct-product tables are provided by Herzberg [ 10 ]. Antisymmetrized and symmetrized parts of direct squares are indicated by braces and brackets, respectively.

Appendix F Coupling Coefficients

Coupling coefficients are denoted as 3 Γ symbols: 〈 Γ _a γ _a Γ _b γ _b | Γγ 〉. Their symmetry properties were given in Sect. 6.3 . Octahedral coefficients have been listed by Griffith. Icosahedral coefficients are taken from the work of Fowler and Ceulemans [ 11 ].

In the icosahedral tables, ϕ denotes the golden number $(1+\sqrt{5} )/2$ , and α =3 ϕ −1, β =3 ϕ ⁻¹ +1.

Appendix G Spinor Representations

Contents

G.1

Character Tables 235

G.2

Subduction 237

G.3

Canonical-Basis Relationships 237

G.4

Direct-Product Tables 240

G.5

Coupling Coefficients 241

Extensive character tables for double groups were provided by Herzberg. The ℵ symbol in the present table corresponds to the Bethe rotation through an angle of 2 π . Spin-orbit coupling coefficients for the icosahedral double group have been listed by Fowler and Ceulemans [ 12 ]. The notation ρ ₁ , ρ ₂ for conjugate components follows Griffith. The single-valued irreps in Appendix A also represent the double groups. The rotation through 2 π leaves these irreps invariant. Their characters under $\hat{R}$ and $\aleph \hat{R}$ are thus the same.

G.1 Character Tables

G.2 Subduction

G.3 Canonical-Basis Relationships

The components of the fourfold-degenerate G _3/2 irrep in O ^∗ and I ^∗ are labeled as κ , λ , μ , and ν . For a quartet spin, these labels correspond to M _S =+3/2,+1/2,−1/2, and −3/2, respectively.

G.4 Direct-Product Tables

G.5 Coupling Coefficients

Canonical complex T ₁ and T ₂ basis functions:

$\begin{aligned} T_1: \ \ & |1\rangle = \frac{1}{\sqrt{2}} \bigl[ -|T_{1x} \rangle - i |T_{1y}\rangle \bigr] \\ \ \ & |{0}\rangle =|T_{1z}\rangle \\ \ \ & |{-}1\rangle = \frac{1}{\sqrt{2}} \bigl[ |T_{1x}\rangle - i |T_{1y}\rangle \bigr] \\ T_2: \ \ & |1\rangle = \frac{1}{\sqrt{2}} \bigl[ -|T_{2x} \rangle - i |T_{2y}\rangle \bigr] \\ \ \ & |{0}\rangle = |T_{2z}\rangle \\ \ \ & |{-}1\rangle = \frac{1}{\sqrt{2}} \bigl[ |T_{2x}\rangle - i |T_{2y}\rangle \bigr] \end{aligned}$

Solutions to Problems

1.1

The diagram for the product $\hat{C}_{2}^{z} \;\hat{\imath}$ is the same as in Fig. 1.1 , except for the intermediate point P ₂ , which should be denoted by a circle instead of a cross, since it is now below the gray disc. However, the end point P ₃ remains the same, irrespective of the order of the operators. This implies that their commutator vanishes.

1.2

Represent the rotation of the coordinates by the rotational matrix $\mathbb{D}$ as given by

$\begin{aligned} &\left ( {\begin{array}{c} {{\mathrm{x}_2}} \\ {{\mathrm{y}_2}} \\ \end{array} } \right ) = \mathbb{D} \left ( {\begin{array}{c} \mathrm{x}_1 \\ \mathrm{y}_1 \\ \end{array} } \right ) =\left ( {\begin{array}{c@{\quad}c} {\cos \alpha } & { - \sin \alpha } \\ {\sin \alpha } & { \cos \alpha } \\ \end{array} } \right ) \left ( {\begin{array}{c} \mathrm{x}_1 \\ \mathrm{y}_1 \\ \end{array} } \right ) \\ &\left ( {\begin{array}{c@{\quad}c} {{\mathrm{x}_2}} & {{\mathrm{y}_2}} \end{array} } \right ) = \left ( {\begin{array}{c@{\quad}c} {{\mathrm{x}_1}} & {{\mathrm{y}_1}} \end{array} } \right ) \mathbb{D}^T =\left ( {\begin{array}{c@{\quad}c} {{\mathrm{x}_1}} & {{\mathrm{y}_1}} \end{array} } \right ) \left ( {\begin{array}{c@{\quad}c} {\cos \alpha } & { \sin \alpha } \\ {-\sin \alpha } & { \cos \alpha } \\ \end{array} } \right ) \end{aligned}$

Express the sum $\mathrm{x}_{2}^{2} +\mathrm{y}^{2}_{2}$ as the scalar product of the coordinate row with the coordinate column and verify that this scalar product remains invariant under the matrix transformation.

1.3

In general, the radius does not change if $\mathbb{D}$ is orthogonal, i.e., if

$\mathbb{D}^T\times\mathbb{D} =\mathbb{I}$

1.4

Apply the general rule that a displacement of the function corresponds to an opposite coordinate displacement. As a result of the transformation, the function acquires an additional phase factor:

$\mathcal{T}_a \; e^{ikx} =e^{ik(x-a)} = e^{-ika}e^{ikx}$

1.5

The action of a rotation about the z -axis can be expressed by a differential operator as

$\hat{O}(\alpha) = \cos{\alpha} \biggl( x\frac{\partial}{\partial x} +y\frac{\partial}{\partial y} \biggr) + \sin{\alpha} \biggl( y\frac{\partial}{\partial x} -x\frac{\partial}{\partial y} \biggr)$

The unit element corresponds to α =0, and hence,

$\hat{E} =\hat{O}(0) = \biggl( x\frac{\partial}{\partial x} +y\frac{\partial}{\partial y} \biggr)$

The angular momentum operator is given by

$\begin{aligned} \mathcal{L}_z =& xp_y - yp_x \\ =& \frac{\hbar}{i} \biggl(x\frac{\partial}{\partial y} -y\frac{\partial}{\partial x} \biggr) \\ =& - \frac{\hbar}{i}\lim_{\alpha \rightarrow 0} \frac{\hat{O}(\alpha) -\hat{E}}{\alpha} \end{aligned}$

The angular momentum operator thus is proportional to an infinitesimal rotation in the neighborhood of the unit element.

2.1

The condition that $\mathbb{C}$ be unitary gives rise to six equations:

$\begin{aligned} 1 =& |a|^2 + |b|^2 \\ 1 =&|a|^2 +|c|^2 \\ 1 =&|b|^2 +|d|^2 \\ 1 =& |c|^2 + |d|^2 \\ 0 =&|ac|e^{i(\alpha-\gamma)} +|bd| e^{i(\beta-\delta)} \\ 0 =&|ab|e^{i(\alpha-\beta)} +|cd| e^{i(\gamma-\delta)} \end{aligned}$

From these equations it is clear that | a |=| d | and | b |=| c |. The phase relationships may be reduced to

$e^{i(\beta + \gamma)} = -e^{i(\alpha + \delta)}$

With the help of these results the four matrix entries can be rewritten as

$\begin{aligned} |a|e^{i\alpha} =& |a|e^{i(\alpha+\delta)/2} e^{i(\alpha - \delta)/2} \\ |d|e^{i\delta} =& |a|e^{i(\alpha+\delta)/2} e^{-i(\alpha - \delta)/2} \\ |b|e^{i\beta} =& |b|e^{i(\alpha+\delta)/2} e^{i [\beta -\frac{\alpha + \delta}{2} ]} \\ |c|e^{i\gamma} =& -|b|e^{i(\alpha+\delta)/2} e^{i [-\beta +\frac{\alpha + \delta}{2} ]} \end{aligned}$

The general U (2) matrix may thus be rewritten as

$\mathbb{U} = e^{i(\alpha+\delta)/2}\left ( {\begin{array}{c@{\quad}c} |a| e^{i(\alpha - \delta)/2} & |b|e^{i [\beta -\frac{\alpha + \delta}{2} ]} \\ -|b| e^{i [-\beta +\frac{\alpha + \delta}{2} ]} & |a|e^{-i(\alpha - \delta)/2} \\ \end{array} } \right )$

with | a | ² +| b | ² =1. Note that a general phase factor has been taken out. The remaining matrix has determinant +1 and is called a special unitary matrix (see further in Chap. 7 ).

2.2

The relevant integrals are given by

$\begin{aligned} \int_0^{2\pi}e^{-ik\phi}e^{ik\phi}d \phi = [\phi]_0^{2\pi} = 2\pi& \\ \int_0^{2\pi}e^{\pm 2ik\phi}d\phi = \frac{1}{\pm 2ik}\bigl[e^{\pm 2ik\phi}\bigr]_0^{2\pi} = 0& \end{aligned}$

The normalized cyclic waves are thus given by

$|\pm k\rangle = \frac{1}{\sqrt{2\pi}} e^{\pm i k\phi}$

and these waves are orthogonal: 〈− k | k 〉=0.

2.3

The combination of transposition and complex conjugation is called the adjoint operation, indicated by a dagger. A Hermitian matrix is thus self-adjoint. An eigenfunction of this matrix, operating in a function space, may be expressed as a linear combination

$|\psi_m\rangle = \sum_{k}c_{k} |f_k\rangle$

We may arrange the expansion coefficients as a column vector c . This is called the eigenvector. Its adjoint, c ^† , is then the complex-conjugate row vector. The corresponding eigenvalue is denoted as E _m . Now start by writing the eigenvalue equation and multiply left and right with the adjoint eigenvector:

$\begin{aligned} \mathbb{H} \mathbf{c} =& E_m \mathbf{c} \\ \mathbf{c}^\dagger \mathbb{H} \mathbf{c} =& E_m \mathbf{c}^\dagger \mathbf{c} \end{aligned}$

Now take the adjoint and use the self-adjoint property of $\mathbb{H}$ :

$\begin{aligned} \mathbf{c}^\dagger \mathbb{H}^{\dagger} \mathbf{c} =& \bar{E}_m\mathbf{c}^\dagger \mathbf{c} \\ \mathbf{c}^\dagger \mathbb{H} \mathbf{c} =& \bar{E}_m \mathbf{c}^\dagger \mathbf{c} \end{aligned}$

A comparison of both results shows that the eigenvalue must be equal to its complex conjugate and hence be real. If $\mathbb{H}$ is skew-symmetric, a similar argument shows that the eigenvalue must be imaginary.

3.1

The table is a valid multiplication table of a group that is isomorphic to D ₂ . The element C is the unit element. There are six ways to assign the three twofold axes to the letters A , B , D .

3.2

Any nonlinear triatomic molecule with three different atoms has only C _s symmetry, e.g., a water molecule with one hydrogen replaced by deuterium. C ₂ symmetry requires a nonplanar tetra-atomic molecule, such as H ₂ O ₂ . In the free state the dihedral angle of this molecule is almost a right angle (see the figure). To realize C _i symmetry, one needs at least six atoms. Since three atoms are always coplanar, the smallest molecule with no symmetry at all has at least four atoms.

3.3

There are only three regular tesselations of the plane: triangles, squares, and hexagons.

3.4

The rotation generates points that are lying on a circle, perpendicular to the rotation. If the rotational angle is not a rational fraction of a full angle, every time the rotation is repeated, a new point will be generated. To obtain an integer order, the additional requirement is to be added that the original point is retrieved after one full turn.

3.5

Consider a subgroup H ⊂ G such that | G |/| H |=2. Then the coset expansion of G will be limited to only two cosets:

$G= H + \hat{g}H$

Here $\hat{g}$ is a coset generator outside H . The subgroup is normal if the right and left cosets coincide, Since there is only one coset outside H , it is required that

$\hat{g} H = H \hat{g}$

Suppose that this equation does not hold. Then this can only mean that there are elements in H such that

$\hat{h}_x \hat{g} = \hat{h}_y$

But then the coset generator must be an element of H , which contradicts the staring assumption.

3.6

Soccer ball: I _h . Tennis ball: D _{2 d} . Basketball: D _{2 h} . Trefoil knot: D ₃ .

3.7

The figure (from Wikipedia) shows the helix function for n =1. One full turn is realized for t / a =2 π ≈6.283. This is a right-handed helix.

The enantiomeric function reads:

$\begin{aligned} x(t) =& a\cos\biggl(\frac{nt}{a}\biggr) \\ y(t) =&a \sin\biggl(-\frac{nt}{a}\biggr) \\ z(t) = & t \end{aligned}$

Note that a uniform sign change of t would leave the right-handed helix unchanged. For the discrete helix, the screw symmetry consists of a translation in the z -direction over a distance 2 πa / m in combination with a rotation around the z -axis over an angle 2 πn / m . If m is irrational, the helix will not be periodic, and the screw symmetry is lost.

4.1

The site symmetry of a cube is T _h . The cube is an invariant of its site group and transforms as a _g in T _h . The set of five cubes thus spans the induced representation: aT _h ↑ I _h . Applying the Frobenius theorem to the subduction (see Sect. C.1 ), one obtains

$aT_h\uparrow I_h = A_g + G_g$

(1)

4.2

The irreps can be obtained from the induction table in Sect. C.2 , as Γ _π C _{3 v} ↑ T _d :

$\varGamma_{\pi} C_{3v} \uparrow T_d = E + T_1 + T_2$

(2)

The SALCs shown span the tetrahedral E irrep, the one on the left is the E _θ component, and the one on the right is the E _ϵ component. Note that they transform into each other by rotating all π -orbitals over 90 ^∘ in the same sense [ 13 ].

4.3

The 24 carbon atoms of coronene form three orbits: two orbits of six atoms, corresponding to the internal hexagon and to the six atoms on the outer ring that have bonds to the inner ring, and one orbit of the twelve remaining atoms. The elements of the 6-orbit occupy sites of $C'_{2v}$ symmetry, based on $\hat{C}_{2}', \hat{\sigma}_{h}, \hat{\sigma}_{v}$ in D _{6 h} . The p _z orbitals on these sites transform as b ₁ , and hence the induced irreps are as in the case of benzene:

$b_1 C_{2v}\uparrow D_{6h} = B_{2g} + A_{2u} + E_{1g} + E_{2u}$

(3)

The remaining 12-orbit connects carbon atoms with only C _s site symmetry, the p _z orbitals on these sites transforming as a ″. The induced irreps read:

$a'' C_{s}\uparrow D_{6h} = B_{1g} + B_{2g} + A_{1u} + A_{2u} + 2E_{1g} + 2E_{2u}$

(4)

The A _{1 u} and B _{1 g} irreps only appear in the 12-orbit, so we can infer that the molecular orbitals with this symmetry will entirely be localized on the 12-orbit. The SALCs can easily be constructed, as they should be antisymmetric with respect to the $\hat{\sigma}_{v}$ planes in order not to hybridize with the SALCs based on the 6-orbits.

4.4

The tangential π -orbitals transform as Γ _π in the C _{5 v} site group of I _h . According to Sect. C.2 , one has:

$\varGamma_\pi \; C_{5v}\uparrow I_h = T_{1g}+T_{1u}+G_g+G_u+H_g+H_u$

4.5

When the projector that generated the component is characterized as $\hat{P}^{\varGamma_{i}}_{kl}$ , the other components may be found by varying the k index.

4.6

Act with an operator $\hat{S}$ on the projector and carry out the substitution $\hat{R}=\hat{S}^{-1}\hat{T}$ :

$\begin{aligned} \hat{S} \hat{P}^{\varGamma_0}_{11} =& \hat{S} \frac{1}{|G|} \sum_R \hat{R} \\ =& \frac{1}{|G|}\sum_R \hat{S}\hat{R} = \frac{1}{|G|} \sum_T \hat{T} = \hat{P}^{\varGamma_0}_{11} \end{aligned}$

4.7

Applying the inverse transformation to the SALCs of the hydrogens in ammonia yields

$\left ( {\begin{array}{c@{\quad}c@{\quad}c} |sp^2_{ {A}}\rangle & |sp^2_{ {B}}\rangle & |sp^2_C\rangle \end{array} } \right ) = \left ( {\begin{array}{c@{\quad}c@{\quad}c} |2s\rangle & |2p_x\rangle & |2p_y\rangle \end{array} } \right ) \left ( {\begin{array}{c@{\quad}c@{\quad}c} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{2}{\sqrt{6}} & -\frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{6}}\\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \end{array}} \right )$

4.8

This mode transforms as E _y . It can be written as a linear combination of a radial and a tangent mode:

$Q = \frac{-1}{\sqrt{2}}Q^{\mathrm{rad}}_y + \frac{1}{\sqrt{2}}Q^{\mathrm{tan}}_{y}$

with

$\begin{aligned} Q^{\mathrm{rad}}_y =& \frac{1}{\sqrt{2}} ( \Delta R_{{B}} - \Delta R_{{C}} ) \\ Q_{y}^{\mathrm{tan}} =& \frac{1}{\sqrt{6}} R ( 2 \Delta \phi_{{A}} - \Delta \phi_{{B}} - \Delta \phi_{{C}} ) \end{aligned}$

This mode preserves the center of mass and is a genuine normal mode.

4.9

Since all irreps are one-dimensional, the characters can only consist of a phase factor:

$\mathbb{D}(C_5) = e^{i\lambda} \mathbb{I}$

(5)

The fifth power of the generator will yield the unit element, and hence,

$e^{5i \lambda} =1$

(6)

This is the Euler equation. Its solutions are the characters in the table of C ₅ , as given in Appendix A .

4.10

The product of inversion with a $\hat{C}_{2}$ axis must yield a reflection plane, perpendicular to this axis. As an example, a product of type $\hat{\imath} \cdot \hat{C}_{2}'$ must yield a reflection plane of $\hat{\sigma}_{d}$ type, as this is perpendicular to the primed twofold axis. For the one-dimensional irreps of D _{6 h} , one thus should have

$\chi(\imath) \chi \bigl(C_2'\bigr) = \chi( \sigma_d)$

(7)

This is indeed verified to be the case.

4.11

The $a_{2}''$ distortion is antisymmetric with respect to $3\hat{C}_{2}, \hat{\sigma}_{h}$ , and $2\hat{S}_{3}$ . As a result, when the mode is launched, all these symmetry elements will be destroyed, and the symmetry reduces to the subgroup C _{3 v} . In general, the result of a distortion will always be the maximal subgroup for which the distortion is totally symmetric [ 14 ].

4.12

The group of this fullerene is D _{6 d} . The 24 atoms separate into two orbits: a 12-orbit containing the top and bottom hexagons and another 12-orbit containing the crown of the 12 atoms, numbered from 7 to 18. In both cases the site group is only C _s , and hence both orbits will span the same irreps:

$a' C_s\uparrow D_{6d} = A_1 + B_2 + E_1 + E_2 + E_3 +E_4 + E_5$

Quite remarkably, the Hückel spectrum for this fullerene has a nonbonding level of E ₄ symmetry.

5.1

Let r _i and r _j denote the position vectors of electrons i and j . The electron repulsion operator contains the distance between both electrons as | r _i − r _j |. The matrix $\mathbb{D}(R)$ expresses the transformation of the Cartesian coordinates under a rotation. This matrix will also rotate the coordinate differences :

$\hat{R} \left ( {\begin{array}{c} {{\mathrm{x}_i}-\mathrm{x}_j} \\ {{\mathrm{y}_i}-\mathrm{y}_j} \\ {{\mathrm{z}_i}-\mathrm{z}_j} \\ \end{array} } \right ) = \mathbb{D}(R) \left ( {\begin{array}{c} {{\mathrm{x}_i}-\mathrm{x}_j} \\ {{\mathrm{y}_i}-\mathrm{y}_j} \\ {{\mathrm{z}_i}-\mathrm{z}_j} \\ \end{array} } \right )$

(8)

Exactly as in the derivation for Problem 1.2, the square of the distance between the two electrons is then found to be invariant under any orthogonal transformation of the coordinates.

5.2

For the G irrep, it is noted from Sect. C.1 that a tetrahedral splitting field will branch G into A + T . It thus acts as a splitting field to isolate the unique Ga component. Symmetry adaptation to $\hat{C}_{2}^{z}$ will yield two totally symmetric components, one of which will be the Ga already obtained; the remaining one is then Gz . The corresponding Gx and Gy may then be found by cyclic permutation under the $\hat{C}_{3}^{xyz}$ axis.

For the H irrep, one may make use of the $\hat{C}_{3}^{xyz}$ axis again. It resolves H into A ₁ +2 E . This unique A ₁ component will be the sum Hξ + Hη + Hζ . We can project the Hζ component out of this sum by using the $\hat{C}_{2}^{z}$ axis. Although the H level subduces three totally symmetric irreps in C ₂ , there will be no contamination with Hθ and Hϵ since these were already removed in the first step by projecting out the trigonal A ₁ .

5.3

The total number of nuclear permutations and permutation-inversions for CH ₃ BF ₂ is 24. This is the product of six permutations of the protons, two permutations of the fluorine nuclei, and the binary group of the spatial inversion. However, as the fluxionality of this molecule is limited to free rotations of the methyl group, the operations should be limited to those permutations or permutation-inversions that lead to structures that can be rotated back to the original frame or to a rotamer of this frame . Only half of the operations will comply with this requirement. As an example, the odd permutations of the protons are not allowed since the resulting structure cannot be turned into the original one by outer rotations or by rotations of the methyl group around the C-B bond. The results are given in [ 15 ]. The corresponding symmetry group is isomorphic with D _3
h .

5.4

Ferrocene is a molecule with two identical coaxial rotors. Its nuclear permutation-inversion group consists of 100 elements. It has a halving rotational subgroup of 50 proper permutations: for each of the cyclo-pentadienyl rings, there are 5 cyclic permutation operations, yielding a total of 5 ² =25 operations, and this number must be doubled to account for the permutation of the upper and lower rings. In addition, there is a coset of improper permutation-inversions containing the other 50 elements. This coset also contains two kinds of elements. In the table we summarize the structure of the group. The carbon atoms are numbered 1,…,5 in the upper ring and 6,…,10 in the lower ring.

6.1

The ( t _{1 u} ) ² configuration gives rise to 15 states. The direct product decomposes as follows (see Appendix D ):

$T_{1u}\times T_{1u} = [A_{1g}+E_g + T_{2g}] + \{T_{1g}\}$

The symmetrized part will give rise to six singlet functions, while there are nine triplet substates, forming a ³ T _{1 g} multiplet. Since the 3-electron Ψ state is a quartet, the singlet states cannot contribute, and we need to couple the triplet to a ² T _1
u state, resulting from a ( t _1
u ) ¹ configuration. The orbital part of the triplet is obtained from the T ₁ × T ₁ = T ₁ coupling table in Appendix F :

$\begin{aligned} |T_{1g}x\rangle =& \frac{1}{\sqrt{2}} \bigl[ - y(1)z(2) + z(1)y(2) \bigr] \\ |T_{1g}y\rangle =& \frac{1}{\sqrt{2}} \bigl[ x(1)z(2) - z(1)x(2) \bigr] \\ |T_{1g}z\rangle =& \frac{1}{\sqrt{2}} \bigl[ - x(1)y(2) + y(1)x(2) \bigr] \end{aligned}$

The coupling with the third electron can yield A _{1 u} , E _u , T _{1 u} , and T _{2 u} states. Our results is based on the A _{1 u} product. This yields

$\begin{aligned} A_{1u} =& \frac{1}{\sqrt{3}} \bigl[|T_{1g}x\rangle | x(3) \rangle +|T_{1g}y\rangle |y(3)\rangle +|T_{1g}z\rangle |z(3) \rangle \bigr] \\ =& -\frac{1}{\sqrt{6}}\left \vert {\begin{array}{c@{\quad}c@{\quad}c} x(1) & y(1) & z(1) \\ x(2) & y(2) & z(2) \\ x(3) & y(3) & z(3) \\ \end{array} } \right \vert \end{aligned}$

This should be multiplied by the product of the three α -spins, α ₁ α ₂ α ₃ , to obtain the ⁴ A _1
u ground state of the ( t _{1 u} ) ³ configuration.

6.2

The JT problem is determined by the symmetrized direct product of T _1
u . As we have seen in the previous problem, this product contains A _{1 g} + E _g + T _{2 g} . Since A _{1 g} modes do not break the symmetry, the JT problem is of type T ₁ ×( e + t ₂ ). In the linear problem only two force elements are required. The distortion matrix is thus as follows:

$\mathcal{H}' = \frac{F_E}{\sqrt{6}} \left ( {\begin{array}{c@{\quad}c@{\quad}c} Q_\theta & 0 & 0 \\ 0 & Q_{\theta} & 0 \\ 0 & 0 & -2Q_\theta \\ \end{array} } \right ) +\frac{F_T}{\sqrt{2}} \left ( {\begin{array}{c@{\quad}c@{\quad}c} 0 & -Q_\zeta & -Q_\eta \\ -Q_\zeta & 0 & -Q_\xi \\ -Q_\eta & 0_\xi& 0 \\ \end{array} } \right )$

6.3

The magnetic dipole operator transforms as T _{1 g} , while the direct square of e _g irreps yields A _{1 g} + A _{2 g} + E _g . Since the operator irrep is not contained in the product space, the selection rules will not allow a dipole matrix element between e _g orbitals.

6.4

We first draw a simple diagram representing the R -conformation. The point group is C ₂ . The twofold-axis is oriented along the y -direction, and the centers of the two chromophores are placed on the positive and negative x -axes. The dipole moments are then oriented as

$\begin{aligned} \boldsymbol{\mu}_1 =& \mu \biggl( 0,\cos{\frac{\alpha}{2}},-\sin {\frac{\alpha}{2}} \biggr) \\ \boldsymbol{\mu}_2 =& \mu \biggl( 0,\cos{\frac{\alpha}{2}},\sin { \frac{\alpha}{2}} \biggr) \end{aligned}$

The exciton states on both chromophores are interchanged by the twofold axis and can be recombined to yield a symmetric and an antisymmetric combination, denoted as A and B , respectively. One has:

$\begin{aligned} |\varPsi_A\rangle =& \frac{1}{\sqrt{2}} \bigl( |\varPsi_1 \rangle + |\varPsi_2\rangle \bigr) \\ |\varPsi_B\rangle =& \frac{1}{\sqrt{2}} \bigl( |\varPsi_1 \rangle - |\varPsi_2\rangle \bigr) \end{aligned}$

The corresponding transition dipoles are oriented along the positive y - and negative z -direction, respectively:

$\begin{aligned} \boldsymbol{\mu}_A =& \sqrt{2} \mu \biggl( 0,\cos{ \frac{\alpha}{2}},0 \biggr) \\ \boldsymbol{\mu}_B =& \sqrt{2} \mu \biggl( 0,0,-\sin{ \frac{\alpha}{2}} \biggr) \end{aligned}$

The dipole-dipole interaction is given by

$V_{12} = \frac{1}{4 \pi \epsilon_0} \frac{\cos \alpha}{R_{12}^3}$

(9)

For α < π /2, the dipole orientation is repulsive. As a result, the in-phase coupled exciton state | Ψ _A 〉 will be at higher energy than the out-of-phase | Ψ _B 〉 state. Finally, we also calculate the magnetic transition dipoles, using the expressions from Sect. 6.8 :

$\begin{aligned} \mathbf{m}_A =& \frac{i \pi \nu}{\sqrt{2}} (\mathbf{r}_1 \times \boldsymbol{\mu}_1 + \mathbf{r}_2\times \boldsymbol{\mu}_2 ) = \frac{i \pi \nu \mu}{\sqrt{2}} R_{12} \sin \frac{\alpha}{2} (0,1,0) \\ \mathbf{m}_B =& \frac{i \pi \nu}{\sqrt{2}} (\mathbf{r}_1 \times \boldsymbol{\mu}_1 - \mathbf{r}_2\times \boldsymbol{\mu}_2 ) = \frac{i \pi \nu \mu}{\sqrt{2}} R_{12} \cos \frac{\alpha}{2} (0,0,1) \end{aligned}$

These results are now combined in the Rosenfeld equation to yield the rotatory strength of both exciton states:

$\begin{aligned} \mathcal{R}_A =& \frac{\pi \nu \mu^2}{2} R_{12} \sin \alpha \\ \mathcal{R}_B =& -\frac{\pi \nu \mu^2}{2} R_{12} \sin \alpha \end{aligned}$

This result predicts a normal CD sign, with a lower negative branch (B-state) and an upper positive branch (A-state) [ 16 ]. This is a typical right-handed helix, corresponding to a rotation of the dipoles in the right-handed sense when going from chromophore 1 to chromophore 2 along the inter-chromophore axis. In the S -conformation the sign of α will change, and the CD spectrum will be inverted.

6.5

The direct square of the e -irrep in D _{2 d} yields four coupled states:

$e\times e = A_1 + A_2 + B_1 + B_2$

(10)

The corresponding coupling coefficients are given in the table below. This table is almost the same as the table for D ₄ in Appendix F , but note that B ₁ and B ₂ are interchanged. Such details are important, and therefore we draw again a simple picture of the molecule in a Cartesian system. Both in D ₄ and in D _{2 d} , the B ₁ and B ₂ irreps are distinguished by their symmetry with respect to the $\hat{C}_{2}'$ axes.

In the orientation of twisted ethylene, as indicated in the figure below, the directions of these axes are along the bisectors of x and y . In contrast, in the standard orientation for D ₄ they are along the x and y axes, while the bisector directions coincide with the $\hat{C}_{2}''$ axes, and hence the interchange between B ₁ and B ₂ .

Note that the two-electron states are symmetrized, except the A ₂ combination. The symmetrized states will combine with singlet spin states, while the A ₂ state will be a triplet. One thus has:

$\begin{aligned} {}^1A_1 =& \frac{1}{\sqrt{2}} \bigl( x(1)x(2) + y(1)y(2) \bigr) \frac{1}{\sqrt{2}} \bigl( \alpha(1)\beta(2) -\beta(1)\alpha(2) \bigr) \\ =& \frac{1}{\sqrt{2}} \bigl( \big|(x\alpha) (x\beta)\big| + \big|(y\alpha) (y\beta)\big| \bigr) \\ {}^1B_1 =& \frac{1}{\sqrt{2}} \bigl( \big|(x\alpha) (y \beta)\big| + \big|(y\alpha) (x\beta)\big| \bigr) \\ {}^1B_2 =& \frac{1}{\sqrt{2}} \bigl(-\big|(x\alpha) (x \beta)\big| + \big|(y\alpha) (y\beta)\big| \bigr) \\ {}^3A_2 =& \big|(x\alpha) ( y\alpha)\big| \end{aligned}$

The ¹ A ₁ and ¹ B ₂ states are the zwitterionic states , while the ¹ B ₁ and ³ A ₂ states are called the diradical states. It is clear from the expressions that in both cases the two radical carbon sites are neutral. The zwitterionic states are easily polarizable though.

6.6

The carbon atoms form two orbits. The p _z orbital on the central atom is in the center of the symmetry group and transforms as $a_{2}''$ . The three methylene orbitals are in C _{2 v} sites, transforming as the b ₂ irrep of the site group, i.e., they are antisymmetric with respect to $\hat{\sigma}_{h}$ and symmetric with respect to $\hat{\sigma}_{v}$ . The induced representation is

$b_2C_{2v}\uparrow D_{3h} = a_2'' + e''$

(11)

The SALCs are entirely similar to the hydrogen SALCs in the case of ammonia; this implies, for instance, that the component labeled x is symmetric under the vertical symmetry plane through atom A. It will be antisymmetric for the twofold-axis going through atom A since the relevant orbital is of p _z type:

$\begin{aligned} |\varPsi_a\rangle =& \frac{1}{\sqrt{3}} \bigl(|p_A\rangle + |p_B\rangle +|p_C\rangle \bigr) \\ |\varPsi_x\rangle =& \frac{1}{\sqrt{6}} \bigl(2|p_A\rangle - |p_B\rangle -|p_C\rangle \bigr) \\ |\varPsi_y\rangle =& \frac{1}{\sqrt{2}} \bigl(|p_B\rangle -|p_C\rangle \bigr) \end{aligned}$

The $a_{2}''$ orbitals interact to yield bonding and antibonding combinations at $E=\alpha \pm \sqrt{3} \beta$ . Since the graph is bipartite, the remaining e ″ orbitals are necessarily nonbonding and will be occupied by two electrons. The direct square of this irrep yields symmetrized $A_{1}'$ and E ′ states and an antisymmetrized $A_{2}'$ state. The expressions for these states are obtained from the coupling coefficients for D ₃ in Appendix F :

$\begin{aligned} {}^1A_1' =& \frac{1}{\sqrt{2}} \bigl( x(1)x(2) + y(1)y(2) \bigr) \frac{1}{\sqrt{2}} \bigl( \alpha(1)\beta(2) -\beta(1) \alpha(2) \bigr) \\ =& \frac{1}{\sqrt{2}} \bigl(\big|(x\alpha) ( x\beta)\big| + \big|(y\alpha) ( y\beta)\big| \bigr) \\ {}^1E_x' =&\frac{1}{\sqrt{2}} \bigl(\big|(x \alpha) ( y\beta)\big| +\big |( y\alpha) (x\beta)\big| \bigr) \\ {}^1E_y' =&\frac{1}{\sqrt{2}} \bigl(-\big|(x \alpha) ( x\beta)\big| + \big|(y\alpha) ( y\beta)\big| \bigr) \\ {}^3A_2 =&\big|(x\alpha) ( y\alpha)\big| \end{aligned}$

Note that the distinction between zwitterionic and diradical states does not hold in this case. Formally, TMM can be described as a valence isomer between three configurations in which one of the peripheral atoms has a double bond to the central atom and the other two sites carry an unpaired electron.

7.1

In a cube the d -shell also splits in e _g + t _{2 g} , but the ordering is reversed. Explicit calculation of the potential shows that the splitting is reduced by a factor 8/9:

$\Delta_{\mathrm{cube}} = -\frac{8}{9} \Delta_{\mathrm{octahhedron}}$

7.2

Perform the matrix multiplication and verify that the product matrix is of Cayley–Klein form. The multiplication is not commutative:

$\left ( {\begin{array}{c@{\quad}c} a_1 & b_1 \\ -\bar{b}_1 & \bar{a}_1 \\ \end{array} } \right ) \times \left ( {\begin{array}{c@{\quad}c} a_2 & b_2 \\ -\bar{b}_2 & \bar{a}_2 \\ \end{array} } \right ) = \left ( {\begin{array}{c@{\quad}c} a_1a_2-b_1\bar{b}_2 & a_1b_2+\bar{a}_2b_1 \\ -\bar{a}_1\bar{b}_2-{a}_2\bar{b}_1 & \bar{a}_1\bar{a}_2- \bar{b}_1b_2 \\ \end{array} } \right )$

(12)

7.3

The double group $D_{3}^{*}$ contains 12 elements. In Table 7.5 we have listed the six representation matrices for the elements on the positive hemisphere. The $\hat{C}_{2}^{A}$ axis is along the x -direction, $\hat{C}_{2}^{B}$ is at −60 ^∘ and $\hat{C}_{2}^{C}$ is at +60 ^∘ . The derivation of the multiplication table and the underlying class structure (see Table 7.6 ) is based on a straightforward matrix multiplication.

7.4

The action of the spin operators on the components of a spin-triplet can be found by acting on the coupled states, as summarized in Table 7.2 . As an example, where we have added the electron labels 1 and 2 for clarity:

$\begin{aligned} S_x |+1\rangle =& S_x\bigl [ |\alpha_1\rangle | \alpha_2\rangle \bigr] =\bigl[S_x |\alpha_1\rangle\bigr] | \alpha_2\rangle + |\alpha_1\rangle \bigl[S_x| \alpha_2\rangle\bigr] \\ =& \frac{\hbar}{2} \bigl[ |\beta_1\rangle |\alpha_2 \rangle + |\alpha_1\rangle |\beta_2\rangle \big] = \frac{\hbar}{\sqrt{2}}|0\rangle \\ S_y |-1\rangle =& -\frac{i\hbar}{\sqrt{2}} |0\rangle \end{aligned}$

These results can be generalized as follows:

$\begin{aligned} {S}_z |M_S\rangle =& \hbar M_S |M_S\rangle \\ ( {S}_x \pm i {S}_y )|M_S\rangle =& \hbar \bigl[ (S\mp M_S) (S\pm M_s + 1) \bigr]^{\frac{1}{2}} |M_s \pm 1\rangle \end{aligned}$

The action of the spin Hamiltonian in the fictitious spin basis gives then rise to the following Hamiltonian matrix (in units of μ _B ):

$\mathcal{H}_{Ze}$	\|0〉	\|+1〉	\|−1〉
〈0\|	0	$g_{\perp} \frac{1}{\sqrt{2}}(B_{x}+iB_{y})$	$g_{\perp} \frac{1}{\sqrt{2}}(B_{x}-iB_{y})$
〈+1\|	$g_{\perp} \frac{1}{\sqrt{2}}(B_{x}-iB_{y})$	g _\|\| B _z	0
〈−1\|	$g_{\perp} \frac{1}{\sqrt{2}}(B_{x}+iB_{y})$	0	− g _\|\| B _z

We can now identify these expressions with the actual matrix elements in the basis of the three D ₃ components, keeping in mind the relationship between the complex and real triplet basis, as given in Eq. ( 7.39 ). One obtains:

$\begin{aligned} \langle 0 |\mathcal{H}_{Ze} |+1\rangle =& -\frac{1}{\sqrt{2}}\langle A_1|\mathcal{H}| E_x + i E_y\rangle = \frac{1}{\sqrt{2}} \bigl[ -a +d +i (-b-c) \bigr] \\ \langle 0 |\mathcal{H}_{Ze} |-1\rangle =& \frac{1}{\sqrt{2}}\langle A_1|\mathcal{H}| E_x - i E_y\rangle = \frac{1}{\sqrt{2}} \bigl[ a +d +i (b-c) \bigr] \\ \langle \pm 1|\mathcal{H}_{Ze}|\pm 1\rangle =& \frac{1}{2} \bigl[ \langle x|\mathcal{H}|x\rangle +\langle y|\mathcal{H}|y\rangle \pm i \bigl( \langle x|\mathcal{H}|y\rangle -\langle y|\mathcal{H}|x\rangle \bigr)\bigr] = \pm f \end{aligned}$

From these equations the parameters may be identified as follows:

$\begin{aligned} a =& 0 \\ b =& -g_{\perp}B_y \\ c =& 0 \\ d =& g_{\perp}B_x \\ e =& 0 \\ f =& g_{||} B_z \end{aligned}$

The Zeeman Hamiltonian does not include the zero-field splitting between the A ₁ and E states. This can be rendered by a second-order spin operator, which transforms as the octahedral E _g θ quadrupole component:

$\mathcal{H}_{ZF}= \frac{D}{3\hbar^2} \bigl( 2\tilde{S}_z^2 -\tilde{S}_x^2-\tilde{S}_y^2 \bigr) = \frac{D}{\hbar^2}\biggl(\tilde{S}_z^2 - \frac{1}{3}\tilde{S}^2\biggr)$

One then obtains

$D= 3 \Delta$

7.5

The action of the components of the fictitious spin operator on the Γ ₈ basis is dictated by the general expressions for the action of the spin operators on the $S=\frac{3}{2}$ basis functions. It is verified that the spin-Hamiltonian that generates the J _p part of the matrix precisely corresponds to

$\mathcal{H}_p = J_p \mathbf{B} \cdot \tilde{ \mathbf{S}}$

The fictitious spin operator indeed transforms as a T ₁ operator and has the tensorial rank of a p -orbital. However, as we have shown, the full Hamiltonian also includes a J _f part, which involves an f -like operator. To mimic this part by a spin Hamiltonian, one thus will need a symmetrized triple product of the fictitious spin, which will embody an f -tensor, transforming in the octahedral symmetry as the T ₁ irrep. These f -functions can be found in Table 7.1 and are of type z (5 z ² −3 r ² ). But beware! To find the corresponding spin operator, it is not sufficient simply to substitute the Cartesian variables by the corresponding spinor components, i.e., z by $\tilde{S}_{z}$ , etc.; indeed, while products of x , y , and z are commutative, the products of the corresponding operators are not. Hence, when constructing the octupolar product of the spin components, products of noncommuting operators must be fully symmetrized. For the $f_{z^{3}}$ function, this is the case for the functions 3 zx ² and 3 xy ² , which are parts of 3 zr ³ . As an example, the operator analogue of 3 zx ² reads

$3zx^2 \rightarrow \tilde{S}_z \tilde{S_x} \tilde{S_x}+ \tilde{S_x}\tilde{S}_z \tilde{S_x}+ \tilde{S_x}\tilde{S_x} \tilde{S}_z$

One then has for the operator equivalent of 3 z ( x ² + y ² ):

$\begin{aligned} &\tilde{S}_z \tilde{S_x}\tilde{S_x}+ \tilde{S_x}\tilde{S}_z \tilde{S_x}+ \tilde{S_x}\tilde{S_x}\tilde{S}_z + \tilde{S}_z \tilde{S_y}\tilde{S_y}+ \tilde{S_y}\tilde{S}_z \tilde{S_y}+ \tilde{S_y}\tilde{S_y}\tilde{S}_z \\ &\quad = 3\tilde{S}_z \bigl(\tilde{S}_x^2 + \tilde{S}_y^2\bigr) +i\hbar (\tilde{S}_x \tilde{S}_y - \tilde{S}_y\tilde{S}_x ) =3 \tilde{S}_z \bigl(\tilde{S}_x^2 + \tilde{S}_y^2\bigr) -\hbar^2 \tilde{S}_z \end{aligned}$

where we have used the commutation relation for the spin-operators:

${S}_x{S}_y - {S}_y{S}_x =i\hbar {S}_z$

The octupolar spin operator will then be of type

$\begin{aligned} \mathcal{H}_f =& \frac{\mu_B}{\hbar^3} g_f B_z \biggl( \tilde{S}_z^3 - \frac{3}{5} \tilde{S}_z \tilde{S}^2 + \frac{1}{5}\hbar^2\tilde{S}_z \biggr)+ B_x \biggl( \tilde{S}_x^3 - \frac{3}{5} \tilde{S}_x \tilde{S}^2 + \frac{1}{5}\hbar^2 \tilde{S}_x \biggr) \\ &{}+B_y \biggl( \tilde{S}_y^3 - \frac{3}{5} \tilde{S}_y \tilde{S}^2 + \frac{1}{5}\hbar^2\tilde{S}_y \biggr) \end{aligned}$

In order to identify the parameter correspondence, let us work out the action of this operator on the quartet functions. As an example for a magnetic field along the z -direction, the matrix is diagonal, and its elements (in units of μ _B ) are given by

$\begin{aligned} \biggl\langle \pm \frac{3}{2}\bigg |\mathcal{H}_f\bigg| \pm \frac{3}{2}\biggr\rangle =& \pm g_f B_z \frac{3}{2} \biggl(\frac{9}{4} -\frac{45}{20} + \frac{1}{5} \biggr) = \pm \frac{3}{10} g_f B_z \\ \biggl\langle \pm \frac{1}{2} \bigg|\mathcal{H}_f\bigg| \pm \frac{1}{2}\biggr\rangle =& \mp \frac{9}{10} g_fB_z \end{aligned}$

By comparing these elements to the results in Table 7.8 we can identify the parameter correspondence as

$J_f = -\frac{3}{10}g_f$

(13)

References

Mulliken, R.S., Ramsay, D.A., Hinze, J. (eds.): Selected Papers. University of Chicago Press, Chicago (1975)

Cotton, F.A.: Chemical Applications of Group Theory. Wiley, New York (1963)

Atkins, P.W., Child, M.S., Phillips, C.S.G.: Tables for Group Theory. Oxford University Press, Oxford (1970)

Boyle, L.L.: The method of ascent in symmetry. I. Theory and tables. Acta Cryst. A 28 , 172 (1972)

Fowler, P.W., Quinn, C.M.: Theor. Chim. Acta 70 , 333 (1986)

Griffith, J.S.: The Theory of Transition-Metal Ions. Cambridge University Press, Cambridge (1961)

Boyle, L.L., Parker, Y.M.: Symmetry coordinates and vibration frequencies for an icosahedral cage. Mol. Phys. 39 , 95 (1980)

Qiu, Q.C., Ceulemans, A.: Icosahedral symmetry adaptation of | JM 〉 bases. Mol. Phys. 100 , 255 (2002)

Butler, P.H.: Point Group Symmetry Applications, Methods and Tables. Plenum Press, New York (1981)

10.

Herzberg, G.: Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules. Van Nostrand, Princeton (1966)

11.

Fowler, P.W., Ceulemans, A.: Symmetry relations in the property surfaces of icosahedral molecules. Mol. Phys. 54 , 767 (1985)

12.

Fowler, P.W., Ceulemans, A.: Spin-orbit coupling coefficients for icosahedral molecules. Theor. Chim. Acta 86 , 315 (1993)

13.

Stone, A.J.: A new approach to bonding in transition-metal clusters. Theory. Mol. Phys. 41 , 1339 (1980)

14.

Ceulemans, A., Vanquickenborne, L.G.: The epikernel principle. Structure and Bonding 71 , 125–159 (1989)

15.

Longuet-Higgins, H.C.: The symmetry group of non-rigid molecules. Mol. Phys. 6 , 445 (1963)

16.

Kobayashi, N., Higashi, R., Titeca, B.C., Lamote, F., Ceulemans, A.: Substituent-induced circular dichroism in phthalocyanines. J. Am. Chem. Soc. 121 , 12018 (1999)

Index

Symbols

10 Dq

Δ-enantiomer

Γ ₈ quartet

π -modes

σ -mode

Abel

Abelian

Abragam

Absorption spectra

acac ⁻

Aldridge

Allene

Alternant

Altmann

Ammonia

Dynamic symmetry

Permutation symmetry

Point group

SALC

Angular momentum

Annulene

Antiprisms

Antisymmetrization

Archimedean solids

Archimedene

Aromaticity

Associativity

Atkins

Atomic population

Automorphism group

Azimuthal coordinate

Balabanov

Barut

Barycentre rule

Basis set, canonical

Bending modes

Benfey

Benzene

Bernoulli

Berry

Berry phase

Bersuker

Bethe

Beyens

Biel

Bilinear interaction

Binary elements

Bipy

Bisphenoid

Bleaney

Boggs

Bohr magneton

Bond order

Born-Oppenheimer condition

Boundary condition

Boundary operation

Boyle

Bra function

Bracket

Branching rule

Braun

Bruns

Buckminsterfullerene

Bunker

Butler

C ₆₀

Calabrese

Cauchy theorem

Cayley

graph

theorem

Cayley–Klein parameters

Centrosymmetry

Ceulemans

Character

string

table

theorem

Charge-transfer (CT) transitions

Chatterjee

Chemical bonding

Chemical shift

Chibotaru

Child

Chirality

Chromophore

Circular dichroism

Circular polarization

Clar

Class

Clebsch-Gordan (CG) coefficient

Closed shell

Closure

Clusters

Commutator

Compernolle

Condon

Condon–Shortley convention

Cone

Conical intersection

Conjugation

class

complex

Conrotatory

Contact term

Continuity condition

Coordinate system

Cartesian

D2 setting

tetragonal

trigonal

Coordination compounds

Coset

Cotton

Coulomb interaction

Coulson

Coulson–Rushbrooke theorem

Coupling channel

Coupling coefficients

exchange symmetry

Crystal-field potential

Crystallography

Cube

Cuboctahedron

Curie principle

Curl

Cvetković

Cyclobutene

Cyclohexadiene

Cylinder

Day

Day and Sanders model

Degeneracy

Deltahedron

Determinant

unimodular

Diatomic

Dihedral

Dipole

induced

moment

Dirac

Dirac notation

Direct square

Dish

Dish Archimedene

Disrotatory

Dissymmetry

Distortion modes

Dodecahedrane

Dodecahedron

Domcke

Donor–acceptor interactions

Doob

Double group

Dual

Dynamic symmetry

Edge representation

Edmonds

Eigenfunctions

Eigenvalues

Electric

see Stark effect

crystal field

dipole

field (E)

symmetry breaking

Electron diffraction

Electron precise

Enantiomers

Equivalent electrons

Euclid

Euler equation

Euler theorem

Excited state

Exciton

Face representation

Fagan

Faraday effect

Ferrocene

Fibre bundle

Flint

Fowler

Franck-Condon principle

Fries

Frobenius

Frontier orbitals

Fullerenes

Function space

Gauge

Genealogical tree

GFP protein

Gilmore

Graph

automorphism

bipartite

Great Orthogonality Theorem (GOT)

Griffith

Group

see Lie groups

see point groups

Abelian

alternating

cyclic

definition

double

generator

halving

orthogonal

permutation

symmetric

unitary

Haake

Halevi

Half-integral momentum

Hamiltonian

Heath

Helicity

Hemisphere

Hermitian

Hessian

Hexadecapole

Hilbert space

Hilton

Hoffmann

HOMO

Homomorphism

Hückel theory

Hydrocarbon

Icosahedron

Indistinguishability

Induction

Integral

hopping

overlap

resonance

Intensity

Intra-ligand (IL) transitions

Inversion

Irrep

Isolobal analogy

Isomorphism

Isotope shift

Jahn

Jahn–Teller effect

Judd

Katzir

Kinetic energy

Klein

see Cayley

four-group

Kobayashi

Köppel

Kramers’ degeneracy

Kronecker delta

Kroto

Lagrange theorem

Lagrangian

Lanthanides

Le Bel

Leapfrog

Lie groups

SO(3), O(3)

SU(2), U(2)

Ligand orbitals

Ligator

Lijnen

Linear dichroism

Linearly polarized

Lipscomb

London

London approximation

Longuet-Higgins

Lulek

LUMO

Magnetic

see Faraday

see London

dipole

field (B)

flux

symmetry breaking

Mallion

Malone

Manifold

Manolopoulos

Martins

Mass-weighted coordinates

Matrix

adjacency

circulant

complex conjugate

diagonalization

element

orthogonal

trace

transposed

unitary

Matsuda

M(CO) ₃ fragments

Melvin

Methane

Mexican hat potential

Mingos

Miura

Molecular-symmetry group

Monopole

Mulliken

Mulliken symbols

Multiplication table

Mys

Neumann principle

Nordén

Normal modes

O’Brien

OCAMS

Octahedron

Octupole

Odabaşi

Ojha

O’Leary

Omnicapping

Opechowski

Opechowski theorem

Operator

action on a function

action on a point

action on an operator

anti-linear

congruence

idempotent

inverse

inversion

ladder

linear

projection

proper and improper

rotation

rotation-reflection

spin

Orbitals

molecular

Order of a group

Organo–transition–metal complexes

Orgel

Orthonormality

Orthorhombic

Paquette

Parity

permutational

space

Parker

Partial derivative

Partitioning

Pauli exclusion principle

Pauling

Pekker

Permutation group

Perturbation theory

Phase convention

Phillips

Platonic solids

Point groups

C ₂ , C _i , C _s

C _{3 v}

C _n , C _nh , C _nv

D ₂

D _{2 h}

D _{6 h}

D _n , D _nd , D _nh

I , I _h

O , O _h

S _{2 n}

T , T _d , T _h

Polarization function

Polyhedrane

Polyhedron

Potential energy

Prisms

Product

antisymmetrized

direct group

direct representation

multiplicity

scalar

symmetrized

Pseudo Jahn–Teller effect (PJT)

Pseudo-doublet

Pseudo-scalar representation

Pythagorean tradition

Qiu

Quantum chemistry

Quartet spin state

Quinn

Rank

Reciprocity theorem

Reduced matrix element

Representation

determinantal

faithful

ground

irreducible

mechanical

positional

pseudoscalar

Reversal

space

time

Rhombohedral

Right-thumb rule

Ring closure

Rodger

Rodrigues

Rosenfeld equation

Rotation

bodily

matrix

operator

optical

pole

Rotatory strength

Rouvray

Ru(bipy) ₃

Ruthenocene

Sachs

SALC

Salem

Samuel

Sanders

Satten

Schäffer

Schrödinger

stationary equation

time-dependent equation

Schulman

Schur

Secular equation

see Zeeman

Selection rule

Shapere

Shimanouchi

Similarity transformation

Singleton

Site symmetry

Smalley

sp ³ -hybridization

Spherical harmonics

Spin, spinor

Spin-orbit coupling

Splitting scheme

Stabilizer

Standard fibre

Stanger

Stark effect

Stereo-isomers

Stone

Stretching modes

Subduction

Subelement

Subgroup

Subporphyrin

Subrepresentation

Sum rule

Symmetry

see operators

see point groups

active definition

breaking

dynamic

hidden

spherical

Takeuchi

Tangential modes

Teller

Tetragonal compression

Tetragonal elongation

Tetrahedron

Tight-binding model

Time reversal

Time-even, time-odd

Topology

Trace

Transfer term

Transition dipole

Translation

Triangular condition

Triphenylmethyl

Trischelate complex

Trivalent polyhedron

Troullier

Truncation

Twisted cylinder

Two-well potential

Uni-axial

Unit element

Uranium

Vanquickenborne

Van’t Hoff

Vector

axial

polar

potential

row versus column

Vertex representation

Vibrational modes

Vibronic interaction

Vollhardt

Walçerz

Wales

Walsh diagram

Wigner

Wigner–Eckart theorem

Wilczek

Woodward–Hoffmann rule

Wunderlich

Wylie

Yarkony

Yersin

Yu-De

Yun-Guang

Zeeman interaction

Footnotes

In the tables the columns on the right list representative coordinate functions that transform according to the corresponding irrep. The symbols R _x , R _y , R _z stand for rotations about the Cartesian directions.

Appendix A Character Tables

A.1 Finite Point Groups

C 1 and the Binary Groups C s , C i , C 2

The Cyclic Groups C n ( n =3,4,5,6,7,8)

The Dihedral Groups D n ( n =2,3,4,5,6)

The Conical Groups C nv ( n =2,3,4,5,6)

The C nh Groups ( n =2,3,4,5,6)

The Rotation–Reflection Groups S 2 n ( n =2,3,4)

The Prismatic Groups D nh ( n =2,3,4,5,6,8)

The Antiprismatic Groups D nd ( n =2,3,4,5,6)

The Tetrahedral and Cubic Groups

The Icosahedral Groups

A.2 Infinite Groups

Cylindrical Symmetry

Spherical Symmetry

Appendix B Symmetry Breaking by Uniform Linear Electric and Magnetic Fields

B.1 Spherical Groups

B.2 Binary and Cylindrical Groups

Appendix C Subduction and Induction

C.1 Subduction G ↓ H

C.2 Induction: H ↑ G

Appendix D Canonical-Basis Relationships

Appendix E Direct-Product Tables

Appendix F Coupling Coefficients

Appendix G Spinor Representations

G.1 Character Tables

G.2 Subduction

G.3 Canonical-Basis Relationships

G.4 Direct-Product Tables

G.5 Coupling Coefficients

Solutions to Problems

Group Theory Applied to Chemistry

C ₁ and the Binary Groups C _s , C _i , C ₂

The Cyclic Groups C _n ( n =3,4,5,6,7,8)

The Dihedral Groups D _n ( n =2,3,4,5,6)

The Conical Groups C _nv ( n =2,3,4,5,6)

The C _nh Groups ( n =2,3,4,5,6)

The Rotation–Reflection Groups S _{2 n} ( n =2,3,4)

The Prismatic Groups D _nh ( n =2,3,4,5,6,8)

The Antiprismatic Groups D _nd ( n =2,3,4,5,6)