Physical Chemistry Laboratory II (CHEM 337) EXPT 9 3: Vibronic Spectrum of Iodine (I2) Obtaining fundamental information about the nature of molecular structure is one of the interesting aspects of molecular spectroscopy. In this experiment, you will obtain and analyze the vibronic spectrum of a diatomic molecule and determine information on the structure of the ground and excited state of the molecule. Introduction to Vibronic Spectra Traditional IR spectroscopy and UV/VIS spectroscopy normally provide very different types of molecular information due to their large differences in available light energy. The IR part of the optical spectrum only provides enough energy to look at the relatively low-energy transitions from one vibrational level. It does not have enough energy to induce an electronic vibration. In the case of UV/VIS spectroscopy, however, the incident photons are much higher in energy than in IR. Because of this additional energy there is now enough energy to excite the molecule from its ground electronic state to a higher electronic state (depending on the specific system). This gives rise to what is generally a broad absorption peak in the UV/VIS region of the spectrum. This transition from ground to excited electronic states does not occur in isolation however. One reason that the UV/VIS absorption spectrum is usually so broad is that in the process of inducing an electronic transition, there is still enough energy to also simultaneously excite various vibrational levels in that excited electronic state. The result is that there is a fine structure due to vibrations that can be sometimes be seen along with the electronic spectrum. We refer to this combination of vibrational and electronic transitions as generating a vibronic spectrum. We designate vibrational quantum numbers in the excited electronic state with a prime (ν') while vibrational quantum numbers in the ground electronic state are designated with a double prime (ν"). To understand the nature of the vibronic spectrum and to extract useful data from it, you need to be familiar with the Morse potential and Morse potential energy curves. The simplest quantum
mechanical model for the vibration of a diatomic molecule is the harmonic oscillator. In this model, the potential energy is parabolic, as shown below. The energy of a harmonic oscillator is: (1) E v = ω e (ν +1/2) where ν = 0, 1, 2, The fundamental vibrational frequency, ω e, is used to define the energy gap between each evenly-spaced vibrational level and is given by: (2) ω e = 1 2π k µ k is the force constant (in N/m) and µ is the reduced mass of the molecule. The harmonic oscillator is a reasonable model for molecular vibration at very low energies, but it does not account for a very important property the dissociation of the molecule at large distances (or energies). Morse Potential The simple harmonic oscillator has a parabolic potential and the molecule s energy can rise indefinitely; there is no upper limit to the parabola. A better model for the potential energy of a diatomic molecule was proposed by P. M. Morse in the later 1920s. The Morse potential is a superior model because in the Morse potential, the molecule can dissociate if its vibrations get too large. In the graph below you can see that at small vibrational energies the molecule remains constrained within the potential energy well. However at large values of r, the energy levels off at the dissociation energy (defined later).
The form of the Morse function is: (3) U β ( r re ) ( r) = D e (1 e ) 2 β = ω e 2 2π µ c D h e In this equation, D e is the dissociation energy of the molecule (measured from the bottom of the potential curve to where the curve levels off), r e is the equilibrium bond length of the molecule and β is the Morse parameter. The dissociation energy, D e, is defined as the energy necessary to dissociate the molecule when measured from the bottom of the potential energy well. It is slightly larger than the bond energy, D o, which is measured from the lowest vibrational energy. The ground state and each excited state have their own individual Morse curves. These curves have different dissociation energies, equilibrium bond lengths and curvature as you can see from the next figure.
Another difference between the Morse curve and the simple harmonic oscillator is the energy pattern. In the harmonic oscillator, the energy levels are equally spaced according to the equation 1, while for the Morse potential, the energy expression is: (4) E 2 v, Morse = ωe ( ν + 1/ 2) ωe xe ( ν + 1/ 2) The term ω e x e is the anharmonicity constant. It measures the deviation of the Morse curve from the ideal harmonic oscillator. Because of the quadratic anharmonicity term, the spacing between vibrational levels in the Morse potential are no longer evenly spaced like they are with the ideal harmonic oscillator. Analysis Assign the observed peaks to transitions from the ν = 0, 1 and 2 states to the different ν states in the electronically excited state. To help you assign the bands, see the following table 1. ν ν λ/nm ν ν λ/nm ν ν λ/nm 27 0 541.2 18 1 571.6 13 2 595.7 28 0 539.0 19 1 568.6 14 2 592.0 29 0 536.9 20 1 565.6 15 2 588.5 To make sure your assignments are correct it is best to produce a Deslandres table. Firstly convert the wavelengths of each peak to wavenumbers (cm -1 ). Construct the Deslandres table in the following way. ν ν 0 1 2 19 18 17 16
Place the corresponding wavenumbers into each cell. Between the cells containing the wavenumbers, calculate the difference between in each wavenumber. If your assignment is correct, then the differences down each row and across each column should be about the same. The observed transition frequency of an optical transition from the ground electronic state to a vibrational level in an excited electronic state is, (5) ν = E el + E v - E v (6) = E el + ω e (ν + 1/2) ω e x e (ν + 1/2) 2 ω e (ν + 1/2) + ω e x e (ν + 1/2) 2 A number of different analytical procedures are available in the literature including the commonly used Birge-Sponer method. However, with the linear regression algorithms included in Excel, a much simpler method can be used. While you are familiar with using Microsoft Excel's LINEST function to calculate simple linear regression lines, the LINEST function can actually calculate multiple linear regression lines. In simple linear regression, we are trying to fit the data to a simple linear function of only one independent variable - that is, we fit to the function: y = mx + b. Here, x is the independent variable, and y is the dependent variable. LINEST will perform a multiple linear regression to fit a single dependent variable to any number of independent variables. That is, it will fit data to the general equation: y = m n x n + m n-1 x n-1 +... + m 2 x 2 + m 1 x 1 + b Here, we have n independent variables, x n through x 1, and n coefficients, m n through m 1. The output of the LINEST function is an array of n columns and 5 rows with the general form: A B 1 m n m n-1... m 2 m 1 b 2 se n se n-1... se 2 se 1 se b 3 r 2 se y/x 4 F df 5 SS reg SS resid Here, the coefficients, m n through m 1 and the y-intercept, b, appear in row 1. The values se n through se 1 are the standard deviations of the coefficients, and se b is the standard deviation of the y- intercept. The value r 2 is the square of the correlation coefficient, r, and the value se y/x is the standard error of regression. df is the number of degrees of freedom associated with the determination of the standard deviations. SS reg is the regression sum of squares and SS resid is the sum of the squares of the y-residuals.
Arrange the data in Excel in the following way using all of your observed peaks. ν' ν" nm cm -1 ν' + 1/2 (ν + 1/2) 2 ν" + 1/2 (ν" + 1/2) 2 15 0 16 0 17 0 20 1 21 1 22 1 16 2 17 2 18 2 Using LINEST solve for E el, ω e, ω e x e, ω e and ω e x e. Calculate the energy difference, E 00, between the ν" = 0 and ν' = 0 states using the above values. Calculate the energy difference, E*, between the ν" = 0 and the top of the excited state potential well. Due to anharmonicity the spacing between adjacent vibrational levels approaches zero at higher values of ν'. Differentiate Eq 6 with respect to (ν' + 1/2) and set this to 0. This is the asymptopic limit where the vibrational spacing goes to zero at the point of dissociation. Express (ν' + 1/2) max in terms of ω e and ω e x e and substitute back into Eq 6 to calculate E*. D 0, the difference in energy from the ν' = 0 level to the top of the excited state potential well can now be calculated from E* and E 00. D e, the difference in energy in the excited state from the bottom of the potential well to the top of the potential well can be calculated from D 0 and the zero point energy from equation 4 = ω e /2 - ω e x e /4 D 0, the difference in energy from the ν = 0 level to the top of the ground state potential well can be calculated from E* and knowing that the energy difference between the dissociation point for upper and ground states is 7589 cm -1. 2 D e, the difference in energy in the excited state from the bottom of the potential well to the top of the potential well can be calculated from D 0 to the zero point energy from equation 4 = ω e 2 - ω e x e /4 In sophisticated spectroscopic analysis, data is sometimes evaluated using higher order polynomial terms. Recalculate your values of E el, ω e, ω e x e, ω e and ω e x e by including in Eq 6 the cubic term ω e y e (v + 1/2) 3. Comment on the accuracy and precision of the revised values.
Potential Energy Curves Calculate the Morse potential energy well using Eq. 3 at 0.1 Å intervals between 2.1 and 6.0 Å for both the ground and excited state I 2 molecules given that the equilibrium bond length of the ground state is 2.66 Å and for the excited state it is 2.98 Å. Ensure to convert all parameters to SI units before calculating the Morse potential. For the excited state potential you will need to add E el (in SI units!) to all of your calculated potential energy values in order to offset the excited state potential energy curve to its correct relative position above the ground state. References 1. McNaught, I. J., J. Chem. Ed., 57, 101-105 (1980) 2. Gaydon, A. G., Dissociation Energies, Chapman and Hall, London, 2 nd Ed., Rev., 1953.