Ch120 - Study Guide 10 Adam Griffith October 17, 2005 In this guide: Symmetry; Diatomic Term Symbols; Molecular Term Symbols Last updated October 27, 2005. 1 The Origin of m l States and Symmetry We are familiar that there are three different ways we can arrange four electrons into three different p orbitals, and this denotes the 3 P states of O. (See Figure 1.) From quantum mechanics we know there are associated values of m l that describe the relative angular momentum of the three degenerate states. Assigning these m l values to the familiar labels p x, p y, and p z is not trivial, however. Recall that the spatial part for the wavefunction for any orbital can be expressed in terms of its radial and angluar components. I.e. the expression, ψ n,l,ml (r, θ, φ) =R n,l (r)y m l l (θ, φ) (1) corresponds to the wavefunction for a p orbital once we insert the quantum numbers associated with the state we wish to describe. The R term is the radial function (a function of distance), and the Y term is a spherical harmonic (a function of θ and φ). The spherical harmonics for p orbitals are complex since the m l = ±1 values correspond to functions with the form e ±iφ. (See McQuarrie or the Goddard book in the suggested reading section). Since we cannot see the complex plane, we cannot easily visualize what these orbitals look like, however it is perfectly acceptable to create linear combinations of these complex spherical harmonics to form real functions that we can visualize. As a result, we can make new expressions that describe the 3 degenerate states with the following real wavefunctions: ψ px = A sin θ cos φ (2) ψ py = A sin θ sin φ (3) ψ pz = A cos θ (4) Where A is a coefficient that arises with particular values of Z and quantum number n. 2 Diatomic Term Symbols Now that we have three real functions to describe the p orbitals, we can think of how each of the three states of O behaves under rotation about the z-axis. We start with the xyz 2 state and assign a wavefunction to this state. Since there is one electron in each of the p x and p y orbitals, and assuming that Hund s rules apply, 1
we can write its wavefunction as (xy yx)(αα). However, we know that both p x and p y have angular dependencies, and in order to see how these orbitals behave under rotation we substitute: x =cosφ and y =sinφ since these are the only φ-dependent parts of the p x and p y wavefunctions. Therefore: (xy yx)(αα) =(cosφ 1 sin φ 2 sin φ 1 cos φ 2 )(αα) =sin(φ 2 φ 1 )(αα) (5) which is invariant under rotation of the z-axis by φ. Since it is invariant, we do not expect it to have any angular momentum. We assign this xyz 2 state as m l =0. We then arbitrarily assing the x 2 yz and xy 2 z states to have m l values of -1 and +1, respectively. We can see that the x 2 yz state transforms in the same manner as a pure p y orbital. Even though the p x orbital does change with rotation, the doubly-occupied p x orbital does not change since the two electrons contribute to two sign-changes overall. The same argument can be used to say that the xy z state behaves like a pure p x orbital under rotation about the z-axis. The same technique we just used to assign the spatial functions of a p-orbital to values of m l can be used for d orbitals as well. Now that we know how to identify states by their characteristic orbitals, we can assign terms (Σ, Π, and ) for diatomic molecules that resemble the different types of atomic orbitals we encounter (σ, π, and δ). We use these capital Greek characters as terms for only diatomic molecules because the nuclei of any two atoms side by side will always have the same basic symmetry element: a C axis of rotation (i.e. it is invariant of rotation about its z-axis). A few additional symmetries must be specified when dealing with diatomic molecules. First, the symmetry of the wavefunction will be either symmetric or anti-symmetric with respect to reflection through a mirror plane parallel to the axis of rotation. We denote symmetry of the wavefunction by a superscript + and antisymmetric with a superscript -. Since Π states are never symmetric with respect to reflection we do not bother writing + or - for the Π states. We also need to consider the effect of inversion symmetry in diatomics. Recall that the effect of inversion is to take (x, y, z) (-x, -y, -z). If the wavefunction remains the same sign after application of the inversion operator, then it is said to be symmetric under inversion, and is denoted with a g. Likewise, if the sign of the wavefunction changes after application of the inversion operator, then it is antisymmetric under inversion and is denoted with a u. 3 Molecular Term Symbols When we think about molecules that consist of more than two atoms we need to devise different terms to identify states since these molecules do not have the same C element that all diatomics have. This new paradigm for molecular terms needs to account for the structural geometry of any molecule, and so again we invoke symmetry operations on the Hamiltonian to do so. Our new system for determining molecular terms depend on the fundamental symmetry elements that exist in the symmetry point group for the molecule. To illustrate, we use the water molecule. (See Figure 2.) Water is of the C 2v point group because it has certain characteristic symmetry operators. These operators operate on the wavefunction of water and are the following: 1. A C 2 axis of rotation A C n axis shows a completely identical representation of the molecule occurs n times during a rotation of 360 about a particular axis. In the case of water, rotation about the z-axis results in an identical representation after rotations of 180 and 360. Since two 2
representations of the molecule occur under this rotation, water has a C 2 axis of symmetry. (See Figure 3.) A Quick Excercise for the Reader: What are all of the C n s for: (a) a book with no distinguishable markings on its cover? (b) an OREO TM cookie with no markings on the cookie? (c) a four-sided die in the shape of a pyramid without any numbers? In the case of water, a C 2 operation rotates the water molecule 360 2 = 180. If we find that the wavefunction is symmetric upon application of this C 2 operator then the molecule is assigned the term A. If it is antisymmetric then it is assigned the term B. Application of the C 2 operator twice results in the exact same orientation we started with. This operation, the identity operator ( E for einheit) is also a symmetry element. 2. An identity operator, E All molecules have an identity operator. If the only symmetry operation that exists for a molecule is E, then it is said to be of the C 1 point group. Application of the E operator on a wavefunction will always result in the same wavefunction, so wavefunctions are always symmetric under the E operator. 3. One mirror plane that is parallel to the axis of rotation (σ v = σ xz ) The mirror plane along the xz-plane will take (x, y, z) (x, -y, z). If application of the σ xz operator results in the same wavefunction, then it is symmetric under σ xz.ifσ xz results in the negative of the wavefunction, then it is antisymmetric under σ xz. (See Figure 4.) A Quick Excercise for the Reader: (a) How many planes of symmetry are in an octagonal STOP sign, neglecting the word STOP? (b) How may are parallel to its highest order axis of rotation (C 8 in this case)? (c) How may are perpendicular to this axis of rotation? (Planes of symmetry parallel to the axis of rotation are σ v s. Planes perpendicular to the axis of rotation are σ h s.) 4. Another mirror plane that is parallel to the axis of rotation (σ v = σ yz ) The mirror plane along the yz-plane will take (x, y, z) (-x, y, z). If application of the σ yz operator results in the same wavefunction, then it is symmetric under σ yz.ifσ yz results in the negative of the wavefunction, then it is antisymmetric under σ yz. (See Figure 5.) These symmetry elements are not independent! One can test to see if any other symmetry elements exist in our point group by taking products of the operators to see if any new operations have not been accounted for. Taking the product of E with any other operator will always result in the operator itself. Additionally, in this point group, the product of any operator with the same operator also results in E. The product of σ xz C 2 = σ yz. Once we have all of the operators (also called generators), we can create a character table that contains all of the symmetry operations and their corresponding effect on the wavefunction (Figure 6). With this character table, by matching how the C 2 and σ xz operators affect H 2 O, we can identify its state as 1 A 1. 3
Depending on the term symbol of a molecule, we can identify its internal composition (i.e. whether the molecule has a pooched orbital or not, and thus how it prefers to bond to other atoms). NOTE: The C 2v point group has a unique property compared to other point groups. Generally, symmetry or antisymmetry of the wavefunction under reflection of a mirror plane is denoted by the subscripts of the A or B terms. The C 2v point group has two unique mirror planes, so therefore we need to decide which plane we choose to determine the unique terms. Standard convention is to define the primary mirror plane as the plane that intersects the most atoms, and so convention would be to place the H 2 O molecule in the xz plane. Professor Goddard prefers to define the primary mirror plane as the plane that reflects the most atoms onto other atoms, but in doing so he places the atoms in the yz plane. This is a subte detail that only arises in C 2v point groups, so be aware of the different conventions that are used. 4 Suggested Reading Atomic p Orbitals: McQuarrie: Hydrogen Atom Chapter 6.10 Term Symbols and Symmetry Operations for Diatomic Molecules: Goddard Book 10.1.1-10.1.3 General Symmetry Operations and Point Groups: Miessler and Tarr: Chapter 4 5 Figures Figure 1: The x 2 yz, xyz 2, xy 2 z, states of 3 Π oxygen Figure 2: The water molecule and our selected coordinate axes 4
Figure 3: The C 2 symmetry operator applied to water Figure 4: The σ xz symmetry operator applied to water Figure 5: The σ yz symmetry operator applied to water Figure 6: The symmetry character table for the C 2v point group 5