CHEM 408 – Sp06 4/10/2006 1Worksheet #10 - Vibrations and Thermochemistry Team #: _______ Manager: _____________________________________ Class Period: Recorder: _____________________________________ Resources: Speaker: _____________________________________ Cramer Ch. 10, Leach 5.8 excerpt Analyst: _____________________________________ Electronic structure calculations and molecular mechanics both typically focus on the minima in the potential energy surface (PES) of a molecule. Nuclear motions affect the energies measured experimentally such that they differ from the energies of PES minima in two ways. First, the uncertainty principle requires that molecules are not localized to a single point on the PES but rather delocalized slightly, an effect that leads to vibrational zero-point energy. This effect is operative even at 0 K. In addition, a sample of molecules at non-zero temperatures will have a distribution of energies above the zero-point level as a result of nuclear motions. Both effects must be taken into account in order to make accurate comparisons to thermochemical data at finite temperatures. I. Separability of Nuclear “Degrees of Freedom”: The Born-Oppenheimer approximation enables us to separate nuclear and electronic motions. Nuclear motions are also separable to a reasonable approximation, so that the energy of a single molecule can be written: vibrottransnucnucelεεεεεεε++=+= (I-1) εnuc is a function of the spatial coordinates and momenta of all of the nuclei. The potential energy depends on the spatial coordinates and the kinetic energy on the velocities or momenta. Each pair of a spatial coordinate and its associated momentum is counted as one “degree of freedom”. Translational coordinates specify the position of the center of mass of a molecule relative to some laboratory coordinate system, rotational coordinates its orientation relative to this coordinate system. The vibrational coordinates are the “internal coordinates” specifying the relative positions of the nuclei with respect to one another. 1. How many nuclear degrees of freedom does a diatomic molecule possess? 2. Explain how these degrees of freedom partition into translational, rotational, and vibrational components.CHEM 408 – Sp06 4/10/2006 2 3. Using the diagram below, indicate what coordinates you might use to describe the translational, rotational, and vibrational degrees of freedom of a diatomic molecule. xzyxzy 4. A PES depicts the potential energy of a molecule as a function of its vibrational coordinates only. Why aren’t translational and rotational coordinates also included in the PES? 5. How is the PES of a diatomic molecule determined in molecular mechanics calculations? In electronic structure calculations?CHEM 408 – Sp06 4/10/2006 3 6. Enumerate the nuclear degrees of freedom present in the triatomic molecules H2O and CO2. (They should differ in the numbers of rotational and vibrational degrees of freedom.) 6. Why is there a difference between the H2O and CO2 cases, i.e. why is there a difference between linear and nonlinear molecules? 7. Fill out the following table, where fx denotes the number of degrees of freedom in nuclear motion x. Molecule ftrans frot fvib an atom an N-atom linear molecule an N-atom non-linear molecule acetylene (H-C≡C-H) dimethylacetylene benzeneCHEM 408 – Sp06 4/10/2006 4II. Molecular Vibrations: The description of molecular vibrations usually assumes a harmonic picture in which displacements of atoms about a minimum in the PES result in a linear restoring force (Hooke’s Law) or a quadratic change in the energy. Using this harmonic approximation, nuclear motions involving the vibrational degrees of freedom of a molecule can be viewed in terms of fvib independent normal modes of vibration, each having a characteristic frequency, energy, and pattern of atomic motions. 1. At the right is a cartoon of a molecule in which all nuclear motion is frozen except for a single bond stretching coordinate “x”, and below it is a plot of how the PES might change as a function of x. Explain why the vertical axis of this plot labeled “Eel” and the meaning of this energy. The general form of a Taylor expansion of a single variable x is: ...))((!31))((!21))(()()(32+−′′+−′′+−′+= axafaxafaxafafxf (II-1) where dxdfxf /)( =′, 22/)( dxfdxf =′′, etc. 2. Express Eel(x) as a Taylor expansion about the minimum in the PES at xe. 3. The harmonic approximation keeps only terms up to order x2 in such an expansion. Adopting the convention that the energy at the minimum of the PES is defined to be zero, and using the notation exxelxdxEdk=≡ )/(22, write an expression for the harmonic approximation to Eel(x). vvvxmatoms fixed vvvxmvvvxmatoms fixed Eelxxeharmonic fitEelxxeharmonic fitCHEM 408 – Sp06 4/10/2006 5 4. Under what conditions would you expect the harmonic approximation to be inadequate? The Classical Harmonic Oscillator: If nuclear motions could be treated classically, under the harmonic approximation, the atom under consideration would execute a trajectory x(t) consisting of sinusoidal motion of amplitude A about the equilibrium bond length xe: )2cos()(δπν++= tAxtxe (II-2) The vibrational frequency is given by: 2/121⎟⎠⎞⎜⎝⎛=mkxπν (II-3) 5. What is the repeat period of the vibration and what is the meaning of δ in Eq. II-2? 6. The total energy (potential + kinetic) is a constant for such a system and it is determined by the amplitude A. What is the potential energy at the classical turning points x=xe±A? Using the fact that at the classical turning points the velocity is zero, show that the total energy of the classical oscillator is given by 221Akxvib=ε (II-4)CHEM 408 – Sp06 4/10/2006 6The Quantal Harmonic Oscillator: It is not usually appropriate to think of molecular vibrations classically. Instead of thinking of a trajectory x(t) of the atom as described by Eq. II-2 it is better to think of molecules existing in stationary energy eigenstates ψn of the vibrator, whose energies are which are: )(21+= nhvibνε n = 0, 1, 2, … (II-5) where n is the vibrational quantum number. The frequency that determines the spacing between energy levels is identical to the classical vibrational frequency in