6. Interactions

Operations and representations are merely theoretical constructs. What is actually observed are the interactions. In quantum mechanics, interactions are expressed as matrix elements of operators in a function space. When the operator is the unit operator, the matrix elements are just overlap integrals. These are the simplest form of interactions.

We start our analysis by examining symmetry selection rules for overlap integrals. Consider the overlap integral between the ith component of a function space which transforms according to the irrep Γ, and the kth component of another function space transforming as Γ′. The overlap integral, S _ki , is a scalar quantity and thus must be invariant under the action of linear symmetry operators acting on the functions. An integral being an infinite sum, the operator can be brought inside the bracket and then transform the bra and ket parts directly. By summing over all and dividing by the group order one obtains a form to which the GOT can be applied. We now rewrite this result in terms of elements of the overlap matrix : This simple derivation yields three important results: These results clearly illustrate the importance of the GOT. It not only provides a selection rule at the level of the irreps, but also at the level of the components. Of course, the latter selection rule will work only if we ensured that the symmetry adaptation of the basis set has been carried out at the component level, as was explained in Sect. 5.​3.

A further consequence is that SALCs on peripheral atom sites can quite often easily be derived from central symmetry-adapted orbitals. One simply has to make sure that the SALCs have the same nodal characteristics as the central functions, so as to guarantee maximal overlap. This is well illustrated in Fig. 4.​4.

Overlap integrals are scalar products of a bra and a ket function. A general matrix element is an integral of the outer product of a bra, an operator, and a ket, giving rise to a triad of irreps. The evaluation of such elements is based on the coupling of irreps. This concept refers to the formation of a product space. The simplest example is the formation of a two-electron wavefunction, obtained by multiplying two one-electron functions. This section will be devoted entirely to the formation of such product spaces.

Consider two sets of orbitals, transforming as the irreps Γ _a and Γ _b respectively, each occupied by one electron. A two-electron wavefunction with electron 1 in the γ _a component of the first set, and electron 2 in the γ _b component of the second set is written as a simple product function: |Γ _a γ _a (1)〉|Γ _b γ _b (2)〉. Clearly, since the one-electron function spaces are invariants of the group, their product space is invariant, too. Now the question is to determine the symmetry of this new space. The recipe to find this symmetry can safely be based on the character theorem: first determine the character string for the product basis, and then carry out the reduction according to the character theorem. Symmetry operators are all-electron operators affecting all particles together; hence, the effect of a symmetry operation on a ket product is to transform both kets simultaneously. The transformation of the product functions is thus expressed by a super matrix, each element of which is a product of two matrix elements for the individual orbital transformations. The trace of this super matrix is given by: This is a gratifying result. The character of a product space is simply the product of the characters of the factor spaces. Accordingly, the symmetry of the product space is identified as the direct product of the orbital irreps, and is denoted as Γ _a ×Γ _b . If both irreps are degenerate, the direct product will be reducible. Let c _Γ be the number of times that the irrep Γ occurs in the direct product: By straightforward application of the character theorem one obtains: Here we have, for the first time, a formula with a triad of irreps. This will form the basis for the symmetry evaluation of general matrix elements. The c _Γ coefficients are obtained by performing product manipulations on the character tables. As an example, Table 6.1 illustrates the reduction of the E _g ×T _2g product in O _h , as given in Eq. (6.9). Product tables are given in Appendix E.

Let us now proceed with the two-electron problem and address the next question, which is that, after having determined which symmetry species are present, we should like to know what the corresponding two-electron wavefunctions look like, i.e. we should like to construct the SALCs. This construction does not pose any new problems; the projection operators that were introduced in Sect. 4.​5 will do the job perfectly well. Some notation is important here. The product function will be written as: The combination coefficient is itself identified as a matrix element, by multiplying left and right with the one-electron bra functions and using orthonormality of the basis orbitals. This coefficient is known as a Clebsch–Gordan (CG) coupling coefficient and denoted by the 3Γ bracket 〈Γ _a γ _a Γ _b γ _b |Γγ〉. It indicates how the orbital irreps Γ _a and Γ _b have to be combined to yield a product ket that transforms as |Γγ〉. The CG-coefficients can be determined by using projection operators. The results are listed in Appendix F. It is often possible to obtain these results by a simpler procedure. We illustrate this for the components of the T _1g two-electron state, obtained in Eq. (6.9). The z-component of this state is the only component that is totally symmetric under the splitting field. It is clear that this symmetry can be obtained only by multiplying the |e _g ϵ〉 and |t _2g ζ〉 components, since these are both antisymmetric and thus will form a symmetric product. From here on we will adopt for the product functions the usual notation of small letters for the orbitals and capital letters for the coupled states. Hence: The coupling coefficient 〈E _g ϵT _2g ζ|T _1g z〉 is thus equal to 1. The x and y components may then immediately be obtained by applying the cyclic generator. As an example for the x-component: Thus: . The resulting coupling coefficients are shown in Table 6.2.

The CG-coefficients in the finite point groups stem from Wigner’s celebrated coupling coefficients for the spherical symmetry group [1]. Wigner proposed reformulating these coefficients in terms of more primitive 3j symbols, which contain, in a uniform way, the permutational properties of the spherical coupling coefficients. Several attempts have been made to define similar 3Γ symbols for the point group, but this requires the introduction of quite detailed phase conventions, which limits the efficiency of this formalism [2, 3]. We shall therefore not engage in a further factorization of the coupling coefficients, but, express the important symmetry properties of the couplings at the level of the brackets. Two guidelines will thereby be used: when dealing with coupling coefficients it is important to bear in mind that the coupling is based on the formation of a product, as we have illustrated in the preceding section, and, secondly, that we should treat the coupling coefficients as far as possible as ordinary brackets.

A direct consequence of the latter viewpoint is that the rules for complex conjugation of brackets apply:

Being expansion coefficients of SALCs, the coupling coefficients also obey two orthogonality rules. Column-wise orthonormality results from the orthonormal properties of the coupled states. In addition, the scalar products along rows are orthonormal, because of the orthonormal properties of the basic kets. Note that the summation runs over the irreps of the entire product space: Γ∈Γ _a ×Γ _b .

The permutational properties of the CG-coefficients refer to interchange of the bra and ket irreps. If Γ _a and Γ _b are not equivalent, their ordering will not affect the symmetry of the coupled state, since the factors in the direct product commute: We can therefore define the coupling coefficients in such a way that interchange of the coupled irreps leaves the coefficient invariant:

If the coupled electrons are equivalent, i.e. belong to the same shell, the situation is different. In this case, Γ _a =Γ _b , and the direct product becomes a direct square. For non-equivalent irreps, exchange of the components γ _a and γ _b was not possible, because they refer to different irreps. However, when they are components of the same irrep, this exchange is an important symmetry of the product space. Indeed, the squared space can be split into two separate blocks, one block which contains product functions that are symmetric under exchange of the component labels and one block which is antisymmetric. This implies that we can define two separate sets of direct square coupling coefficients, which are either symmetric or antisymmetric under exchange of the labels, i.e.: 〈Γ _a γ _a Γ _a γ _b |Γγ〉=±〈Γ _a γ _b Γ _a γ _a |Γγ〉. The symmetrized part of the direct square is denoted as [Γ _a ]². For n=dim(Γ _a ), the dimension of this subspace is equal to the number of symmetric combinations: On the other hand, if the coupling coefficients are antisymmetric under exchange of the labels, the coupled state belongs to the antisymmetrized direct square, denoted as {Γ _a }². This product space is restricted to combinations with γ _a ≠γ _b ; its dimension is equal to n(n−1)/2. The characters for either part of the square can be determined separately. For the character of the {Γ _a }² part the derivation runs as follows: one first applies a symmetry operator to an arbitrary antisymmetric function. The ket product |Γ _a γ _a (1)〉|Γ _a γ _b (2)〉 will be abbreviated here as: γ _a (1)γ _b (2). Taking the trace then yields: The trace for the symmetrized product is then found by subtracting the trace in Eq. (6.21) from the total trace for the direct product. In Table 6.3 these quantities are given for the direct product H _g ×H _g in icosahedral symmetry. The product resolution is as follows:

Note that this product contains one totally-symmetric irrep, notably in the symmetrized part. In general, for irreps with real characters the totally-symmetric irrep, Γ ₀, appears in a direct square only once. This can easily be derived from Eq. (6.8). When Γ _a is an irrep with real characters, one has: In the case of irreps that can be represented by real transformation matrices, it is possible to show that this totally-symmetric irrep will belong to the symmetrized part. In order to apply the character theorem to Eq. (6.22), the following intermediate result is needed: In order to arrive at this result we have made use of the GOT, on the assumption that the matrices are real. Combining the results of Eqs. (6.24) and (6.25) with Eq. (6.22) then leads to the conclusion that the unique totally-symmetric irrep belongs to the symmetrized part of the direct square: For irreps with real characters, but transformation matrices that cannot be all real, the unique totally-symmetric product appears in the antisymmetrized part. This is the case for spin representations, which will be dealt with in Chap. 7. In summary, as far as complex-conjugation properties are concerned, we have three kinds of irreps:

Equation (6.23) further exemplifies a case of product multiplicity. This is when an irrep occurs more than once in the decomposition of a direct product. Both the H _g and G _g irreps appear twice in the direct product H _g ×H _g . In the point groups product multiplicity is quite rare. It occurs only in the icosahedral group for the products G×H and H×H, as well for spin representations in cubic and icosahedral symmetries. Product multiplicity means that there are different coupling schemes for arriving at the product states. Each of these “channels” corresponds to a separate set of CG coupling coefficients. There are several ways of obtaining linearly-independent sets of coupling coefficients. For the separation of the two G _g irreps in Eq. (6.23) symmetrization of the product space is sufficient, since the symmetrized and antisymmetrized parts each contain one G _g . This strategy does not work for the two H _g irreps, which are both the result of symmetrized coupling. In this case, more elaborate splitting schemes have been constructed, based inter alia on higher symmetries [4, 5].

Last but not least, we should consider the relationship between coupling coefficients where irreps from bra and ket parts are interchanged. We shall limit the discussion here to the simplified case in which all ingredients of the coupling are taken to be real. A case with complex irreps will be treated in Chap. 7. Consider two related couplings: Γ _a ×Γ _b =Γ and Γ×Γ _b =Γ _a . The corresponding expansion coefficients are scalar matrix elements and are thus invariant under the group action. By importing the group action inside the brackets, as we have frequently done before, we obtain a set of equations in the CG-coefficients: These equations form a system of homogeneous linear equations from which the coupling coefficients can be obtained. At present, we shall use this result only in the simplified case where all components have been chosen to be real, so that Eqs. (6.27) and (6.28) form the same system of equations.¹ From this it follows that the corresponding coupling coefficients will be proportional to each other, independent of the components; hence: The proportionality constant can be determined by summing the square of the coefficients over all components and using the normalization result from Eq. (6.15). The permutation of irreps between bra and ket in the CG-coefficients thus requires a uniform dimensional renormalization: The renormalization leaves a phase factor undetermined. This phase factor is the same for the entire coupling table, and thus can be chosen in arbitrarity. As an example, in the group O (see Appendix F), the coefficients 〈T ₂ ξT ₁ x|Eθ〉 and 〈EθT ₁ x|T ₂ ξ〉 are related as follows: Here the phase was chosen to be +1.

In principle, the T _1g and T _2g coupled two-electron states, which we obtained in Table 6.2 of the previous section, could apply to the case of the (t _2g )¹(e _g )¹ excited states of a d ² transition-metal ion in an octahedral ligand field, which splits the d orbitals into a t _2g and an e _g shell. However, these coupled descriptions are not yet sufficient, since they make a distinction between electron 1, which resides in the t _2g orbital, and electron 2, which was promoted to the e _g level. The fundamental symmetry requirement that electrons must be indistinguishable is thus not fulfilled. The operator that permutes the two electrons is represented as : The |Γγ(1,2)〉 and |Γγ(2,1)〉 states will have exactly the same symmetries, since the factors in the direct product commute: As a result, commutes with the spatial symmetry operators, and we can symmetrize the coupled states with respect to the electron permutation. The permutation operator is the generator of the symmetric group, S ₂, which has only two irreps, one symmetric and one antisymmetric, corresponding, respectively, to the plus and minus combination in Eq. (6.35). These states have distinct permutation symmetries, and spatial symmetry operators cannot mix + and − states. This is a very general property of multi-particle states, to which no exceptions are known.

On the other hand the permutation symmetry of multi-electron wavefunctions is restricted by the Pauli principle.

Until now we have limited ourselves to the spatial part of the wavefunction. So far, only the antisymmetrized part obeys the Pauli principle. However, the principle places a requirement only on the total wavefunction. This also involves a spin part, which should be multiplied by the orbital part. Anticipating the results of Chap. 7, we here provide the spin functions for a two-electron system. Spin functions are characterized by a spin quantum number, S, and a component, M _S , in the range {−S,−S+1,−S+2,…,S−1,S}. The total number of components, hence the dimension of the spin-space for a given S, is equal to 2S+1. This number is called the spin multiplicity. For a two-electron system, S can be 0 or 1; hence, there is one singlet state, and there are three components belonging to a triplet state. In a |SM _S 〉 notation they are given by: These functions also exhibit permutation symmetry: the triplet functions are symmetric under exchange of the two particles, while the singlet function is antisymmetric under such an exchange. The total wavefunction can thus always be put in line with the Pauli principle by combining the coupled orbital states with spin states of opposite permutation symmetry. Altogether we can thus construct four states: ¹ T _1g ,¹ T _2g ,³ T _1g ,³ T _2g . This set of four states, totalling 24 wavefunctions, forms a manifold, representing all the coupled states resulting from the (t _2g )¹(e _g )¹ configuration. The dimension of the manifold is equal to the product of the six possible t _2g substates (including spin), and the four possible e _g substates. In this case, where the coupling involves electrons belonging to different shells, the Pauli principle does not restrict the total dimension of the manifold, since all combinations remain possible. All states can be written as linear combinations of Slater determinants. As an example, for the |¹ T _1g z〉 state, one writes:

The situation is different when coupling two equivalent electrons; these are electrons that belong to the same shell. In this case, the coupled states are already eigenfunctions of the exchange operator as a result of the special symmetrization properties of the coupling coefficients for direct squares. Equation (6.10) will take the following form: Now, if the product representation belongs to the symmetrized square, Γ∈[Γ _a ]², this result simplifies to: Hence, this function is symmetric under the operator. It also obeys the normalization condition of Eq. (6.15). It should always be multiplied by an antisymmetric singlet spin-function in order to obey the Pauli exclusion principle. On the other hand, if the product representation belongs to the antisymmetrized square, Γ∈{Γ _a }², the coupled state is given by: This function is antisymmetric under the operator, and should be multiplied by a symmetric triplet spin-function in order to obey the Pauli principle. As an example, for the (e _g )² configuration the allowed states are: For equivalent electrons the Pauli principle thus really does function as an exclusion principle, since the coupled states are either triplets or singlets, depending on their symmetrization. The dimension of the manifold is given by the binomial coefficient, where q is the number of equivalent substates (including spin), and n is the number of electrons: For the (e _g )² problem, one has n=2 and q=4; there are thus six two-electron states in this configuration (see Eq. (6.41)).

As a special result, we examine the symmetry of the maximal spin-multiplicity ground state of a system with a half-filled shell. The shell consists of the components |f ₁〉⋯|f _n 〉, transforming according to the irrep Γ _a , and each will be occupied by one electron with α spin. The ground state corresponds to a single determinant: Here, σ∈S _n is an element of the permutation group of the n electron labels, and sgn(σ) is its parity.² Equation (6.43) indicates that this permutation can equally well be applied to the component labels, since the determinant is invariant under matrix transposition. We can now calculate the matrix element in the symmetry operator: In this equation we have used the result from Eq. (2.​8), which identified matrix elements over symmetry operators as elements of the representation matrix. The double summation over permutations covers the permutation group twice and could be reduced to a single sum. The result indicates that the spatial symmetry of the half-filled shell ground state transforms as the determinant of the irrep of the shell. This is also called the determinantal representation. For the e _g shell the determinantal irrep is A _2g . The shell ground state is thus a ³ A _2g .

A general interaction element is a bracket around an operator. Each of the three ingredients, bra, ket, and operator, can be put in symmetry-adapted form, so that it transforms according to a given irrep. Moreover, provided that the symmetry adaptation is done properly, not only the irrep itself but also the subrepresentation is well defined. Altogether, the matrix element will thus be characterized by six symmetry labels, as: . The labels imply that the symmetry behaviour of each of these ingredients is fully known: Note that the general form of the operator refers to a component of an irreducible set. Such a set of operators is usually referred to as a tensor operator. Obvious examples are the components of the electric or magnetic dipole-moment operators. The Wigner–Eckart theorem introduces a symmetry factorization, which simplifies the evaluation of matrix elements.

The scalar constant is denoted by 〈ψ ^Ω ∥O ^Λ ∥ϕ ^Γ 〉, and is called the reduced matrix element. To prove this theorem, one first considers the coupling of two ingredients of the matrix element, and then compares the result with the third one. We thus first consider the coupling of the operator and the ket. The transformation of their product does indeed correspond to the super matrix which is due to the direct product Λ×Γ. This means that we couple the tensor operator and the ket to form product entities: We now invert this equation, using the unitary properties of the coupling coefficients, to yield: Then we combine this expression with the bra. The matrix elements on the right-hand side are now in fact reduced to an overlap integral where the direct product irreps are compared with the irrep of the bra. Hence, the selection rules for the overlap integrals apply: The trace summation in this equation is identified as a scalar interaction constant, which is represented by the reduced matrix element.

Combination of Eqs. (6.50) and (6.51) then yields the Wigner–Eckart theorem of Eq. (6.46), where the total interaction is the product of a scalar interaction constant and a CG coupling coefficient. The former refers to the interaction itself, the latter extracts the transformation properties. In case of product multiplicity, there will be one reduced matrix element for every coupling channel, and the matrix element is decomposed into a sum over the channels. The Wigner–Eckart theorem or matrix-element theorem is at the heart of most chemical applications of group theory. It provides an elegant method for separating interactions into an intrinsic part and a part that depends only on the symmetry of the problem under consideration.

An important consequence of the matrix element theorem concerns the definition of selection rules. An interaction will be forbidden if the corresponding coupling coefficient in the Wigner–Eckart theorem is zero. The conditions that control the zero values of the coupling coefficients are called triangular conditions, since they involve the combination of three irreps. Two kinds of triangular conditions must be taken into account:

In 1937 Jahn and Teller made the claim that degenerate states of molecules are intrinsically unstable [6, 7].

The theorem is based on a perturbation of the Hamiltonian by small displacements of the nuclei. A high-symmetry geometry is chosen as the origin, and the nuclear displacements are described by normal modes which transform as irreps of the point group. The nuclear positions are parameters in the electronic Hamiltonian. One has, to second-order: The partial derivatives with respect to the normal modes will affect only the electrostatic V _Ne term in the Hamiltonian. These operators are thus electrostatic one-electron operators. At the coordinate origin, the electronic state is degenerate, and is described by a set of wavefunctions, |Γ _a γ _a 〉, where Γ _a is a degenerate irrep. The energies as functions of the coordinates are obtained by diagonalizing the Hamiltonian matrix, , with elements: The matrix in is also called the Jahn–Teller (JT) matrix. The linear terms in this matrix are of type: We have used the fact that the integration in this matrix element runs over electronic coordinates, and does not affect the nuclear coordinates. The Wigner–Eckart theorem can be applied to derive the selection rules. Since the Hamiltonian is invariant under the elements of the symmetry group, the transformation properties of the operator part in this matrix element will be determined by the partial derivatives, ∂/∂Q _Γγ . As we have seen in Sect. 1.​3, a partial derivative in a variable has the same transformation properties as the variable itself.³ The operator part is thus given by: The coupling coefficient on the right-hand side of Eq. (6.57) restricts the symmetry of the nuclear displacements to the direct square of the irrep of the electronic wavefunction. This selection rule is made even more stringent by time-reversal symmetry. The Hamiltonian is based on displacement of nuclear charges, and not on momenta, so as an operator it is time-even or real.⁴ For spatially-degenerate irreps, which are of the first kind, i.e. can be represented by real functions, JT matrix elements can thus be chosen to be entirely real, which implies: Combining this result with Eq. (6.57) implies that the coupling coefficients, to first-order, should obey: In view of Eq. (6.31) this condition can be rewritten as: The JT distortion modes are thus restricted to the symmetrized square of the degenerate irrep of the electronic state, minus the totally-symmetric modes, since these cannot lower the symmetry: Modes that obey this selection rule, are said to be JT active. The evaluation of the second-order matrix elements requires two steps. One first couples the two distortion modes to a composite tensor operator: |Ωω|. The second-order matrix element then becomes: The second-order elements thus are related to a product of two 3Γ symbols.⁵ A special element arises when Ω is totally symmetric. In this case, the coupling coefficients are given by: The second-order expressions then are reduced to a diagonal matrix element: K _Γ in this equation is the harmonic force-constant. It gives rise to a constant diagonal term which provides an attractive potential around the minimum and keeps the surface bound at larger distances from the origin. The general expression for the potential-energy surface then becomes: Here, ε _k (Q) represents the kth root of the Hamiltonian matrix. This equation describes a surface with multiple sheets, one for each root, which cross in the high-symmetry origin. In its simplest form the Hamiltonian can be restricted to the linear terms only. In the second-order approximation non-totally-symmetric second-order terms will also be included.

The prototype of the JT surface is the celebrated Mexican hat potential, which describes the effect of the twofold-degenerate cubic or trigonal E state. A typical example is the ² E _g ground state of octahedral Cu²⁺ complexes, with (t _2g )⁶(e _g )³ configuration. The JT-active mode in this case is restricted to an e _g mode, corresponding to the symmetrized square. This distortion mode consists of the tetragonal and orthorhombic stretchings, which we already encountered as vibrational modes of UF₆, and are depicted in Fig. 6.1. By use of the appropriate 〈Ei|EjEk〉 coupling coefficients the JT matrix can easily be derived. The force element is defined as: The matrix then becomes: To diagonalize this Hamiltonian, it is convenient to transform to cylindrical coordinates {ρ,φ}: Then the secular equation of the force element matrix in Eq. (6.69) becomes: Two roots are found, which are independent of the angular coordinate. The corresponding eigenfunctions are: The surface consists of two sheets and exhibits rotational symmetry.

A cross section of this surface looks like a two-well potential, with two displaced parabolæ. The depth of the well is called the JT stabilization energy: In the 2D space of the active modes these parabolæ revolve around the centre, giving rise to the Mexican hat appearance. At the origin this surface has the shape of a conical intersection, indicating that the high-symmetry point is unstable, and will spontaneously relax to the circular trough surrounding the degeneracy [8]. The distorted system in the trough orbits around the origin. This motion is a pseudo-rotation, i.e. it is not a rotation of the molecular frame, but a gradual redistribution of the distortions between the Cartesian directions. We illustrate this in Fig. 6.2. The starting point is at φ=0, in the direction of the Q _θ mode. In this mode the z-axis is elongated, and the xy-plane is contracted. A counterclockwise rotation activates the orthorhombic Q _ϵ mode, while the tetragonal Q _θ mode is receding. This introduces a difference between the x- and y-axes: the distortion in the x-direction becomes more pronounced, while the y-axis contracts further. At the same time the elongation of the z-axis diminishes. At an angle of 60^∘ the x- and z-axes are both elongated to an equal extent, giving rise to a weak xz-plane and a strong y-axis. At the 90^∘ point the Q _θ contribution vanishes, and the distortion is orthorhombic, with a short y-axis, an elongated x-axis, and an undistorted z-axis. At an angle of 120^∘ we reach the point where the x-axis is weak, and the perpendicular yz-plane is strong. We thus have regained an elongated tetragonal configuration, but the elongation has been rotated from the z-axis to the x-axis. Continuing now at 240^∘ we shall have travelled another third of the trough and reoriented the tetragonal axis along the y-direction. We can also follow the wavevector along the trough. If it is assumed that F _E <0, the lower eigenfunction will be the |ψ ₁〉 eigenfunction of Eq. (6.72). In the starting elongated tetragonal configuration the ground state coincides with the |Eθ〉 basis function. By the time we have reached the orthorhombic configuration at φ=90^∘, the ground state has rotated by only half that angle and equals . At 180^∘, we reach a structure which is tetragonally compressed along the z-direction. Accordingly, the Q _θ mode has changed sign, in contrast to the eigenfunction, where |Eθ〉 is replaced by |Eϵ〉. The observation of rotational symmetry is an unexpected feature, which is not related to the point group, but which stems from the limitation of the JT Hamiltonian to linear terms. To describe this symmetry we first reformulate the force element Hamiltonian in Eq. (6.69) in Dirac notation:

The angular momentum operator, corresponding to a rotation in coordinate space, is given by: The partial derivatives were obtained from Eq. (6.70). The commutator of this operator with the Hamiltonian is: Surprisingly, this commutator does not vanish. This is an important observation, which directly points to the vibrational-electronic or vibronic coupling between the distortion modes and the electronic wavevector. When the system rotates around the origin in coordinate space, not only are the coordinates changing, but the wavevector is also rotating simultaneously, so we must also provide an angular momentum operator for a rotation in the function space (see [9]). We can construct this by analogy with Eq. (6.76), but with an important amendment: as we have argued while discussing Fig. 6.2, the coordinates rotate twice as quickly as the wavevector, and hence a prefactor of 1/2 is required! Only in this case does the total momentum operator commute with the Hamiltonian: As the reader will have noticed, we have made use of the standard spectroscopic symbols for orbital angular momentum, spin momentum, and total momentum. Vibronic coupling is indeed analogous to coupling of spin and orbit momenta in cylindrical molecules. To form the vibronic wavefunction, describing the dynamics of the Mexican hat system, the electronic state has to be combined with nuclear wavefunctions. If the JT effect is pronounced, the vibronic levels take the form of a radial oscillator, describing transverse oscillations in the bottom of the through, and pseudo-rotational levels, describing the longitudinal motion along the bottom of the trough. The total vibronic wavefunction should of course be single-valued after a full turn around the trough, which takes the system back to the starting point. Hence, since the electronic part changes sign after a full turn, the vibrational part should also show a compensating sign change. This is indeed the case: the pseudo-rotational levels are characterized by half-integral angular momentum [10].

Pseudo-JT interactions (PJT) refer to the second-order vibronic coupling between electronic states which are separated by a gap [11]. In this section we describe the case of a non-degenerate ground state, |ψ _Σσ 〉, which is coupled to an excited manifold. The Hamiltonian is identical to the expression in Eq. (6.54). Application of the selection rules shows that the diagonal contribution, , is limited to the totally-symmetric operator associated with the harmonic restoring-force, as in Eq. (6.66). Perturbation theory further provides interactions between the ground and excited states. These interactions are usually limited to first-order contributions, which give rise to a quadratic coordinate dependence. Hence one has, to second-order in the displacements: where we have used the property that the Hamiltonian matrix is hermitian. The selection rule in this process resides with the matrix elements in the enumerator of the bilinear term. The vibronic operator must couple ground and excited states; hence, it is required that their triple direct product contains the totally-symmetric irrep: Applying the Wigner–Eckart theorem to the matrix element yields: The sum over the λ components of the excited state, transforming as the Λ irrep, can be simplified by using the orthonormality property of the coupling coefficients from Eq. (6.16). Note that in case of a non-degenerate ground state the product Γ×Σ yields only one irrep, since the norm of the product character string equals the order of the group. The summation over λ∈Λ thus covers the entire product space of Γ×Σ. Combining the sum rule of Eq. (6.83) with the total expression for the PJT, one finds: Hence, when the ground state is non-degenerate, the first-order dependence of the energy on symmetry-lowering displacement vanishes, and the second-order term contains two contributions: the diagonal harmonic force constant, which is always positive, and the bilinear relaxation term, which is always negative. If the excited states are close in energy to the ground state, and if the vibronic coupling is strong, the relaxation term may be dominant, and a second-order symmetry-breaking effect will result. This is known as the pseudo-JT effect. There are two main applications of this effect: in geometry optimization, and in reaction dynamics.

In reaction dynamics the PJT may be responsible for stereoselectivity, because of the selection rules for vibronic coupling matrix elements. Via these relaxation matrix elements the Wigner–Eckart theorem is at the basis of the Woodward–Hoffmann rules [12]. We shall not discuss these rules in general, but consider some simple illustrations, related to electrocyclic reactions.⁶ Take as a simple example the ring closure of cis-butadiene, as illustrated in Fig. 6.3. The relevant occupied orbitals are the π-bonds in the reagent, and the remaining π- and newly formed σ-bonds in the product. As the diagram shows, in the common C _2v point group there is a mismatch between the symmetries. In order for the reaction to occur, the reaction coordinate has to reduce the symmetry so that the a ₂-orbital can interchange with an a ₁-orbital. This interchange is taking place via a PJT mechanism which couples the a ₂ occupied orbital to an a ₁ virtual orbital in the reagent. As the reaction coordinate proceeds, this coupling is intensified and leads to an interchange of both. The relevant matrix element is thus an orbital vibronic coupling element: Hence, a distortion coordinate is required which transforms as a ₂×a ₁=a ₂. The coordinate with this symmetry is the one that destroys the symmetry planes but keeps the twofold axis. This is typically a conrotatory reaction, where the extremal carbon atoms rotate simultaneously in the same sense, to form the σ-bond. Ring closure of substituted butadienes thus follows a conrotatory reaction stereochemistry, at least if the reaction is concerted.

This ring-closure selection rule is further confirmed by the closure reaction for the cis-1,3,5 hexatriene to 1,3-cyclohexadiene, as illustrated in Fig. 6.4. Here, a b ₁-orbital has to interchange with a virtual orbital of a ₁ symmetry. The selection takes thus place at the level of the orbital matrix element: Clearly, the distortion coordinate should now be of b ₁×a ₁=b ₁ symmetry, and this corresponds to the disrotatory mode, which destroys the axis but keeps the vertical reflection plane.

Selection rules are of primary importance in spectroscopy, where they provide direct evidence concerning the nature of excited states. As an application, we study the linear and circular dichroism of tris-chelate transition-metal complexes [14]. The prototype is a divalent ruthenium complex with three 2,2′-bipyridyl ligands, which is an important chromophore for energy conversion. In this section we shall describe the charge transfer and intra-ligand transitions of this type of complex. The linear dichroism (LD) spectrum measures the absorption of the chromophore under plane-polarized incident light for different orientations of the polarization with respect to the molecular frame. This requires that the molecules should be embedded in an oriented phase, such as a crystalline host. Circular dichroism (CD) measures the difference in absorption between left and right circularly polarized light. Since this is based on the intrinsic helicity of the molecule, it can be performed in non-oriented medium, such as a solution.

As always, we start the treatment by making a simple sketch of the structure. Two sets of Cartesian axes are relevant. In the usual octahedral coordinate system the x,y,z-axes coincide with the metal-ligand bond directions, assuming that the ligator atoms form a perfect octahedron. In addition, in Fig. 3.​6 of Chap. 3 a primed x′,y′,z′-coordinate system was introduced, which is adapted to the tris-chelate geometry. The z′-axis is along the threefold direction, and the x′ axis is oriented along a twofold axis, coinciding with the bisector of the positive x and the negative y axes. Next, we determine the point group, which in the present case is D ₃. This is a rotational group, which implies that the molecule is chiral. The figure shows the Δ-enantiomer.⁷ Thirdly, we define the functional basis. The relevant orbitals are the metal t _2g orbitals, which are fully occupied in the Ru²⁺ ground state, and the frontier orbitals on the ligand. For conjugated bidentate ligands, such as 2,2′-bipyridyl (bipy) or 2,4-pentanedionate (also named “acetylacetonate”, acac⁻), the frontier orbitals are of π-type. The essential parts of these orbitals are the contributions on the ligator atoms. These are either symmetric or antisymmetric with respect to the twofold axis through the bidentate ligand, as shown in Fig. 6.5. Following Orgel, we denote them as χ- or ψ-type, respectively [15]. The standard techniques of characters and projection operators yield SALCs for all these basis sets. The results are shown in Table 6.4. For the e-irrep the components are labelled as e _θ and e _ϵ , following the standard canonical format. As a splitting field we use the twofold axis along x′. Finally, we also include in the table the symmetries of the transition dipoles, which are the operators for the optical transitions. This completes the groundwork for the symmetry analysis.

In Chap. 4 we left induction after the proof of the Frobenius reciprocity theorem. In that proof the important concept of the positional representation was introduced. This described the permutation of the sites under the action of the group elements. Further, we defined local functions on the sites which transformed as irreps of the site symmetry. As an example, if we want to describe the displacement of a cluster atom in a polyhedron, two local functions are required: a totally-symmetric one for the radial displacement and a twofold-degenerate one for the tangential displacements. In cylindrical symmetry, these are labelled σ and π, respectively. The mechanical representation, i.e. the representation of the cluster displacements, is then the sum of the two induced representations: As an example using the induction tables in Sect. C.2 for an octahedron, we have: This is precisely the set of fluorine displacements that we constructed in Sect. 4.​8 in order to describe the vibrational modes of UF₆. One remarkable result of induction theory is that the mechanical representation can also be obtained as the direct product of the positional representation and the translational representation, T _1u ; this is the representation of the three displacements of the centre of the cluster. It is as if the displacements of the central point of the octahedron were relocated to every ligand site. The elementary function space of the displacements of the central atom, which transforms as the translational irrep, T _1u , is called the standard fibre. This fibre is attached to every site of the cluster, and the set of these fibres is the fibre bundle. The action of the group permutes fibres of the bundle. The following induction theorem holds:

For being the representation of the standard fibre, T _1u in our example, and the positional representation of the set of equivalent sites in the molecule, one has for the induced representation: For a proof of this theorem, we refer to the literature [19, 20]. The theorem is not only applicable to molecular vibrations but is also directly in line with the LCAO method in molecular quantum chemistry. In this method the molecular orbitals (MOs) are constructed from atomic basis sets that are defined on the constituent atoms. An atomic basis set, such as 3d or 4f, corresponds to a fibre, emanating, as it were, from the atomic centre. Usually, such basis sets obey spherical symmetry, since they are defined for the isolated atoms. As such, they are also invariant under the molecular point group [21]. As an example, a set of 4f polarisation functions on a chlorine ligand in a complex is itself adapted to octahedral symmetry as a _2u +t _1u +t _2u . This representation thus corresponds to . In the C _4v site symmetry these irreps subduce: a ₁+b ₁+b ₂+2e. According to the theorem, the LCAOs based on the 4f orbitals thus will transform as: In this LCAO space several irreps occur multiple times, but they can all be distinguished by the specific direct product from which they originated.

Leonhard Euler dominated the mathematics of the 18th century. One of his famous discoveries was the polyhedral theorem, which marks the beginning of topology. A polyhedron has three structural elements: vertices, edges, and faces.⁹ The numbers of these will be represented as v, e, and f, respectively. Then, for a polyhedron, the following theorem holds:

As an example, in a cube one has v=8,e=12,f=6, and hence 8−12+6=2. The 2 in the right-hand side of Eq. (6.128) is called the Euler invariant. It is a topological characteristic. Topology draws attention to properties of surfaces, which are not affected when surfaces are stretched or deformed, as one can do with objects made of rubber or clay. Topology is thus not concerned with regular shapes, and in this sense seems to be completely outside our subject of symmetry; yet, as we intend to show in this section, there is in fact a deep connection, which also carries over to molecular properties. The surface to which the 2 in the theorem refers is the surface of a sphere. A convex polyhedron is indeed a polyhedron which can be embedded or mapped on the surface of a sphere. Group theory, and in particular the induction of representations, provides the tools to understand this invariant. To this end, each of the terms in the Euler equation is replaced by an induced representation, which is based on the particular nature of the corresponding structural element. In Fig. 6.8 we illustrate the results for the case of the tetrahedron. The following theorem [22] applies:

The Euler theorem may be considered as the dimensional form of this theorem, which states that the alternating sum of the characters of the induced representations under the unit element, , is equal to 2, but the present theorem extends this character equality to all the operations of the group. The theorem silently implies that irreps can be added and subtracted. In the example of the tetrahedron, the theorem is expressed as: A straightforward interpretation of the theorem is possible in terms of fluid flow on the surface of a polyhedron.¹⁰ Suppose observers are positioned on the vertices, edge centres and face centres, and register the local fluid flow. When the incoming and outgoing currents at a node are not in balance, the observers located on these nodes will report piling up or depletion of the local fluid level. This is the scalar property represented by the vertex term. The corresponding connection between edge flow and vertex density is expressed by the boundary operation, indicated by δ in Fig. 6.8. Taking the boundary of an edge arrow means replacing the arrow by the difference of two vertex-localized scalars: a positive one (indicated by a white circle in the figure) at the node to which the arrow’s head is pointing, and a negative one (indicated by a black circle) at the node facing the arrow’s tail. This projection from edge to vertex will not change the symmetry. Hence, in this way, the boundary of the T ₂ edge irrep is the T ₂ vertex SALC, as illustrated in the figure. Similarly, observers in face centres will notice the net current that is circulating around the face. Such a circular current through the edges does not give rise to changes at the nodes (indeed the incoming flow at a node is also leaving again), but is observable from the centre of the face around which the current is circulating. The boundaries of circular currents around face centres are thus chains of arrows on the edges, which again conserve the symmetry. In Fig. 6.8 the boundary of the T ₁ face term is thus the T ₁ edge term. Clearly, the sum of the vertex and face observations should account for all currents going through the edges, except for two additional terms which escape edge observations. These are the two Euler invariants: the totally-symmetric Γ ₀ component corresponds to a uniform change of fluid amplitude at all vertex basins. This does not give rise to edge currents, since it creates no gradients over the edges. The other is the Γ _ϵ component. It corresponds to a simultaneous rotation around all faces in the same sense. Again, such rotor flows do not create net flows through the edges, because two opposite currents are flowing through every edge. The Euler invariant thus points to two invariant characteristic modes of the sphere. They are not boundaries of a mode at a higher level, nor are they bounded by a mode at a lower level. The phenomena, that these two terms describe, might also be referred to in a topological context as the electric and magnetic monopoles.

Because of this connection to density and current, this theorem may be applied in various ways to describe chemical bonding, frontier orbital structure, and vibrational properties. The applications of this theorem can be greatly extended by introducing fibre representations, as is shown below.

This result can be cast in the language of induced representations. The stretchings of the edges correspond to scalar changes of edge lengths and transform as σ-type objects, and hence will correspond to Γ _σ (e). On the other hand, the internal vibrations span the mechanical representation, which can be written as a bundle of the translation, minus the spurious modes of translation and rotation. The symmetries of these will be denoted as Γ _T and Γ _R , respectively. One thus has: