Group theory#

The utilization of the mathematical field of group theory (mainly representation theory, to be precise) is presented here as applied to quantum chemistry, where the methods of quantum mechanics are used to solve chemical problems. Using the concept from group theory for molecular symmetries, it is possible to both reach fundamental understanding of the physical processes involved in chemistry and to speed up quantum chemical computations.

Symmetries operations#

Disregarding the possibility of translation actions (which are of relevance for crystals), a molecule can be manipulated in a series of ways that may leave the molecule invariant. A symmetry operation is such an action that leaves the molecules invariant, and therefore cannot change the physical properties of the molecules. This can be illustrated by considering a linear molecule, say carbon dioxide, and rotating this by any degrees along the axis intersecting the atoms. This action will move no atom of the molecules, and it is impossible this new geometry with the previous one.

Corresponding to each such symmetry operation there exists an element, a geometrical entity, along which this action is taken: the symmetry element. This element may be an axis, a plane, the entire molecule or a point. In the previous example this would be the axis intersecting all atoms.

Possible symmetry operations for a molecule includes:

  • The identity operator \(I\). This operation consists of doing nothing, and is thus possible for all molecules.

  • The rotation operator \(C_n\). This operation consists of rotating the molecule \((2\pi/n)\) along some axis, this axis is then the \(n\)-fold axis of rotation. If the molecules possesses several \(C_n\)-axes, the one with the largest \(n\) is referred to as the principal axis.

  • The reflection operator \(\sigma\). This operation consists of reflecting the molecule in a plane, leaving only the elements in the plane unaffected. If the molecule also have axes of rotational symmetries, the reflection plane is called a vertical reflection, designated \(\sigma_v\), if any such axes are contained in the plane. If the reflection/mirror plane is instead orthogonal to the axes it is called an horizontal mirror plane, designated \(\sigma_h\). Finally, if the plane bisects the angle between two \(C_2\) axes, it is called a dihedral mirror plane, \(\sigma_d\).

  • The inversion operator \(i\). This operation consists of inverting all point in the molecule through a single point. If this symmetry element is chosen as origin, this corresponds to \((x,y,z) \to (-x,-y,-z)\).

  • The improper rotation operator \(S_n\). This operation corresponds to rotating of molecule by \((2\pi/n)\), followed by a reflection through the plane orthogonal to this rotational axis.

It should be noted that several symmetry operations are equivalent, e.g. \(S_1\) is the same as \(\sigma\), \(C_1\) the same as \(I\).

As examples, consider water and ammonia:

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Water contains one \(C_2\)-axis of rotation, passing through the oxygen and between the hydrogen atoms, as well as two planes of reflection – one in the plane of the molecule and the other orthogonal to this plane but still containing the \(C_2\)-axis. Both planes contains the axis or rotation, and they are thus designated \(\sigma_v'\) and \(\sigma_v''\). Further, \(I\) is included, as for all molecules.

Ammonia possess three different \(C_3\)-axes, as well as three \(\sigma_v\) planes containing each such axes.

Point groups#

Not all combinations of symmetry elements can be present in a molecule, and we may thus categorize the molecules depending on the symmetry element present into different groups. These groups are called point groups, as they leave at least one point unaffected irregardless of the symmetry operation acted upon the molecule (of those allowed for the group). The notation of the group classification will here follow in accordance to the Shoenflies system, but we note that a different system, the Hermann–Mauguin system is also used (this last system also contains translation symmetries, for use in periodic crystals).

Identifying the point group for an arbitrary molecule can be done by consulting a flow chart constructed for this determination:

../../_images/flowchart.png

As an example, water:

  • is nonlinear

  • contains no \(C_{n}\) with \(n \ge 3\)

  • contains \(C_{2}\)

  • does not have Two \(C_{2}\) perpendicular to principal axis

  • has no \(\sigma_h\)

  • has two \(\sigma_v\)

…and thus belongs to the group \(C_{2v}\).

These point groups can immediately give information regarding a molecules dipole moment and possible chiral features, e.g.:

  • As a (static) dipole moment is a result of an asymmetric charge distribution, it is not possible to have a dipole moment along any other axis than a \(C_n\)-axis. Further, this \(C_n\)-axis cannot be affected by any operations such as reflection perpendicular to the axis, as this would interchange the ends. As a result, only molecules belonging to the point groups \(C_n\), \(C_{nv}\) and \(C_s\) can have a dipole moment.

  • A molecule that cannot be superimposed on its mirror image is said to be chiral. This is exemplified by the inability of the left hand to be superimposed on the right hand, and it has the physical effect of rotating the plane of polarized light in opposite directions for the two different specimen. For a molecule to be chiral, it cannot contain a \(S_n\)-axis, an axis often implied by the presence of other symmetry elements (e.g. \(\sigma\) is equivalent to \(S_1\)).

Mathematical groups#

In the mathematical field of group theory, a group is defined as a set of elements with an operation of combination obeying the following rules:

  • The combination of two elements in the group as by the operation must produce an element of the group, i.e. if \(A\) and \(B\) belongs to the group, \(C=AB\) belongs to the group.

  • The group must contain the identity element, such that \(AI=IA=A\).

  • The associative law must apply, i.e. \(A(BC)=(AB)C\).

  • An inverse element must exist for all members of the group, such that \(A^{-1}A=AA^{-1}=I\).

For the case of molecular symmetries, the group elements of interest are the symmetry operations above, with a combination rule consisting of one operation followed by another. This explains the seemingly redundant inclusion of the identity operator \(I\) for the set of symmetry operations, included in order to obey the aforementioned second rule.

Groups can have a number of different properties, some of which will be discussed here. If two different groups \(G\) and \(G'\) have a one-to-one correspondence such that

\[ AB=C \in G \qquad \mathrm{implies} \qquad A'B'=C' \in G' \]

the groups are isomorphic. Jumping somewhat ahead, the group tables of isomorphic groups have the same structure and form, even as the details may differ.

If three elements of the groups can be combined such that

\[ A=T^{-1}BT \]

then \(A\) is the transform of \(B\) by \(T\), and \(A\) and \(B\) are conjugate elements. All elements that are conjugate to each other forms a class \(i\) in the group, where the number of elements is designated \(g_i\). If all elements in the group commutes, the group is Abelian and all elements form its own class.

Group tables#

In the case of symmetry groups in chemistry, the combination rule is given above as one operation followed by another. The effects of different combinations of symmetry operations can be gathered in group tables, with each table being unique for one point group. For \(C_{2v}\), the group table is:

\(I\)

\(C_2\)

\(\sigma_v'\)

\(\sigma_v''\)

\(I\)

\(I\)

\(C_2\)

\(\sigma_v'\)

\(\sigma_v''\)

\(C_2\)

\(C_2\)

\(I\)

\(\sigma_v''\)

\(\sigma_v'\)

\(\sigma_v'\)

\(\sigma_v'\)

\(\sigma_v''\)

\(I\)

\(C_2\)

\(\sigma_v''\)

\(\sigma_v''\)

\(\sigma_v'\)

\(C_2\)

\(I\)

In those tables, the row operation should be carried out first, followed by the column operation. For non-Abelian groups the order of combinations matter, but not for Abelian groups (such as \(C_{2v}\)). Further, it is a general property of the group tables that each row and column contains each symmetry elements once and only once.

Matrix representations#

The symmetry operations of a group an be expressed as a matrix, operating on a vector of predetermined basis functions. Each such transformation matrix \(D (g)\), corresponding to an individual symmetry operations \(g\), is called the representative of the symmetry operation, and the full set of matrices is then the matrix representation of the group. Note that the matrices can be formed such that they operate on spatial vectors rather than basis function.

As a consequence of the requirement of all representatives to have an inverse, it can be immediately understood that the matrices must be square and must have a non-zero determinant. Further, the identity operation can be identified with the identity matrix, i.e. \(D(I)=\textbf{I}\).

The basis set chosen for a specific molecule is by no way unique, and group theoretical considerations open up the possibility of generating basis sets that contains symmetries inherent in the molecules nuclear framework. When constructing two different basis sets we must first know how to transform one into the other.

Consider a basis set \((x_1',\dots,x_n')\) constructed such that each basis function \(x_i'\) is a linear combination of basis functions from the original basis \((x_1,\dots,x_n)\). This transformation can be achieved with a transformation matrix \(\textbf{C}\), such that \(\bar{x}'=\bar{x} \textbf{C}\). As a consequence, the matrix representations of the point group in the two different bases are related by a similarity transform

\[ D'(g)=\textbf{C}^{-1}D(g)\textbf{C} \]

The trace of any matrix representation is called the character of said representation, and it can be shown that the character is invariant under similarity transforms. Further, the character of representatives belonging to the same class are equal.

Consider the symmetry operations \(I\), \(C_3^+\) and \(C_3^-\) of ammonia. If we choose a basis set consisting of the valence \(s\) orbital of nitrogen and core \(s\) orbital of hydrogen, i.e. \((s_N,S_{H1},S_{H2},S_{H3})\), we see that the symmetry operations have the following effects

\[\begin{align*} I \qquad (s_N,s_{H1},s_{H2},s_{H3}) \to(s_N,s_{H1},s_{H2},s_{H3}),\\ C_3^+\qquad (s_N,s_{H1},s_{H2},s_{H3}) \to (s_N,s_{H2},s_{H3},s_{H1}),\\ C_3^- \qquad(s_N,s_{H1},s_{H2},s_{H3}) \to(s_N,s_{H3},s_{H1},s_{H2}) \end{align*}\]

This can be achieved with the following matrices

\[\begin{split} D(E)=\left( \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right), \quad D(C_3^+)=\left( \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array} \right), \quad D(C_3^-)=\left( \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right) \end{split}\]

with this, for example

\[\begin{split} (s_N,s_{H1},s_{H2},s_{H3})\left( \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right)=(s_N,s_{H3},s_{H1},s_{H2}) \end{split}\]

These three matrices have a block-diagonal form, as can be written as a direct product of a one- and a three-dimensional square block, i.e.

\[ D^{(4)}(g)=D^{(1)}(g) \otimes D^{(3)}(g) \]

Now, if one considers all representatives for \(C_{3v}\) using this basis, it is clear that this reduction is possible for all matrices, and it is also clear that the matrices of lower dimensions also form a group. With these reduced representations we now explore the possibility of reducing the matrices to even lower dimensions. By a similarity transform this can be achieved, resulting in the following normalized basis functions

\[\begin{align*} s_N&=s_N\\ s_1&=\frac{1}{\sqrt{3}}(s_{H1}+s_{H2}+s_{H3})\label{salc2} \\ s_2&=\frac{1}{\sqrt{6}}(2s_{H1}-s_{H2}-s_{H3})\label{salc3} \\ s_3&=\frac{1}{\sqrt{2}}(s_{H2}-s_{H3}) \label{salc4} \end{align*}\]

By carrying out the matrix operations, this new basis yields a block-diagonal form such that

\[ D^{(4)}(g)=D^{(1)}(g) \otimes D^{(1)}(g) \otimes D^{(2)}(g) \]

This representation cannot be reduced any further, and we thus refer to it as the irreducible representation, or the irrep, of the point group using this specific set of basis functions. The previous representation that was reduced is an example of a reducible representation. Each irrep is a member of a symmetry species, i.e. they transform under a specific way for the symmetry operations and form a basis for the same matrix representation. These symmetry species are labeled according to a set of rules

  • 1D representations are labeled A if symmetric under rotation about the symmetry axis, or else B.

  • 2D representations are labeled E, 3D are labeled T

  • If the group contains inversion, subscript \(g\) for symmetric or \(u\) for antisymmetric behaviour (i.e. do they change the sign of the orbital).

  • If the group contains no inversion but a horizontal mirror plane, prime for symmetric and double prime for antisymmetric.

  • The group may have additional subscript 1 or 2 to denote the character of the \(C_2\) rotation perpendicular to the principal axis, or the character of a vertical reflection if no \(C_2\) rotations.

Note that the same function may transform as different irreps for different point groups. Further, the dimension of the irreps describes the correspondence between the irreps and functions: if the irrep is 1D, there is a one-to-one correspondence. Else there are several degenerate functions that transforms jointly as the irrep.

Character tables#

With a character table we summarize the behaviour of the irreps with respect to the symmetry operations of the point group. In this, we include the character of the irrep under each possible symmetry operation, as well as functions that transforms as the irreps and the order of the group.

Consider again ammonia, or more generally the point group \(C_{3v}\). With the reduced representation from above, we have two 1D irreps and one 2D irrep. Both 1D irreps behaves equivalently under symmetry operations and belongs to \(A_1\). Meanwhile, the 2D irrep belongs to \(I\), and with a different choice of basis functions it would also be possible to find an irrep that belongs to \(A_2\). In total, the number of irreps present in a point group equals the number of classes, and this can be used to determine if the chosen basis set spans all the irreps of the point group.

Symmetry adapted linear combinations#

Now we are ready to find a linear combination of basis functions that transforms the matrix representatives to a block-diagonal form, and thus transforms the irreps of the system. These functions are referred to as the \emph{symmetry adapted linear combinations}, or SALCs, and they are constructed by the use projection operators, by which the basis functions \(x_i\) are transformed to \(x_i'\) as

\[ x_i'=\sum_g \chi_k(g)gx_i \]

The number of SALCs constructed in such a manner should match the original number of basis functions. If too many SALCs are found, some of them are linearly dependent and the number of functions can thus be reduced.

Applications#

Let us now discuss some applications of group theory in chemistry. This list will by no means be complete, but rather a brief discussion of some properties that are of great interest for the quantum chemist.

Determination of integrals#

In many instances in quantum chemistry one wishes to determine a number of different integrals, such as overlap integrals, expectation values of some operators, etc. Using group theory it is possible to predict if a specific integral must be, by necessity, zero.

Consider a one-dimensional function, integrated from \(-a\) to \(a\). It is trivial to see that this integral must be equal to zero if the function (integrand) is odd. In more general terms we may have more then one dimension, thus making the discussion less intuitive, but we can safely say that the integral (being a number) must be invariant under any symmetry operation. Thus, in order for the integral to be non-zero, the integrand must transform as the totally symmetric irrep in the point group. Note that the function does not always transform as a single irrep, but then it must include the totally symmetric irrep. Note further that this only determines whether an integral is necessarily zero: it may, however, be zero for any other reason.

Molecular bonds#

In the case of molecular bonds, the construction of such by use of two specific atomic orbitals is only possible if the overlap matrix between the orbitals is non-zero, i.e. if

\[ S_{ij}= \langle \phi_i | \phi_j \rangle \ne 0. \]

In above we saw that the integrand must belong to the totally symmetric irrep, and in the case of the overlap between two different atomic orbitals this means that the orbitals must belong to the same symmetry species of the point group, i.e.

\[ D_{ij}(g)=D_i(g) \otimes D_j(g) \]

must include the totally symmetric irrep. In the case of polyatomic molecules the situation is somewhat more complicated, but by use of group theory we can again simplify the situation. We should then construct the SALCs from a set of basis functions (in the case of molecular bonds, only the basis functions of the valence electrons are of interest) and take linear combination of those SALCs to find the bonds. Note that this only gives the form of the functions, the coefficients of the linear combinations cannot be determined by group theoretical consideration. The SALCs will in this case simplify the calculations by yielding a subproblem for each irrep, in which the molecular orbitals can be formed, rather determining all molecular orbitals of the molecules simultaneously.

Character tables#

For the \(C_{2h}\) point group.#

Irrep

\(\hat{I}\)

\(\hat{C}_2(z)\)

\(\hat{i}\)

\(\hat{\sigma}_h\)

Operation

\(A_g\)

1

1

1

1

\(R_z\), \(x^2\), \(y^2\), \(z^2\)

\(B_{g}\)

1

-1

1

-1

\(R_x\), \(R_y\)

\(A_{u}\)

1

1

-1

-1

\(z\)

\(B_{u}\)

1

-1

-1

1

\(x\), \(y\)

For the \(D_{2h}\) point group.#

Irrep

\(\hat{I}\)

\(\hat{C}_2(z)\)

\(\hat{C}_2(y)\)

\(\hat{C}_2(x)\)

\(\hat{i}\)

\(\hat{\sigma}(xy)\)

\(\hat{\sigma}(xz)\)

\(\hat{\sigma}(yz)\)

Operation

\(A_g\)

1

1

1

1

1

1

1

1

\(x^2\), \(y^2\), \(z^2\)

\(B_{1g}\)

1

1

-1

-1

1

1

-1

-1

\(R_z\), \(xy\)

\(B_{2g}\)

1

-1

1

-1

1

-1

1

-1

\(R_y\), \(xz\)

\(B_{3g}\)

1

-1

-1

1

1

-1

-1

1

\(R_x\), \(yz\)

\(A_{u}\)

1

1

1

1

-1

-1

-1

-1

\(B_{1u}\)

1

1

-1

-1

-1

-1

1

1

\(z\)

\(B_{2u}\)

1

-1

1

-1

-1

1

-1

1

\(y\)

\(B_{3u}\)

1

-1

-1

1

-1

1

1

-1

\(x\)