ps2_1987

ps2_1987 - MIT OpenCourseWare http://ocw.mit.edu 5.80...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 1987 Problem Set #2 Due March 9, 1987 Readings: J. I. Steinfeld, pp. 39-73, 77-79, 92-110, 113-142 Handouts: Anharmonic Oscillator, Vibration-Rotation Interaction Energy Levels of a Vibrating Rotor Construction of Potential Curves by the RKR Method Van Vleck Transformation (Lecture #3 supplement) (Lecture #23 supplement) (Lecture #4 supplement) (after Lecture #2) 1. An atom is in a (3d)2 3 P0 state. (a) List all L–S–J terms to which an electric dipole allowed transition might occur. (b) List all two-electron configurations into which electric dipole allowed transitions can occur from (3d)2 3 P0 . 2. J. I. Steinfeld, p. 111, #1. (a) Write out the electron configurations for the molecules O+ , O2 , O− , and O2− . 2 2 2 � � (b) Determine the ground state term symbols M Λ±,u for O+ , O2 , O− , and O2− . If there are two or g 2 2 2 more low-lying states, select one as that of the ground state and justify your selection. 3. J. I. Steinfeld, p. 111, #3. Use the “Gilmore diagrams” provided for O2 , N2 , NO, and H2 : (a) Sketch the expected low-resolution absorption spectrum to be expected for each of these four molecules between 1,000 and 10,000 Å. (b) A photoionization experiment is carried out by shining light on a gas sample placed between two electrodes. A positive ion current is measured between the cathode and ground. For the four molecules, what is the longest wavelength of light that would be effective in producing a measurable ion current (this is called the “photoionization threshold”)? � � (c) The first positive bands B3 Πg → A3 Σ+ of nitrogen are observed in an “active nitrogen” dis­ u charge, in which the active species are predominantly ground state N atoms. Suggest a mech­ anism for the population of the excited state. (d) When solutions of hydrogen peroxide and “Clorox” are mixed, a red chemiluminescence near 7,600 Å is observed. What might this be caused by? Problem Set #2 Spring, 1987 Page 2 4. J. I. Steinfeld, p. 143, #5. The squared “transition moment,” or the probability of transition between two rotational levels in a linear molecule, may be assumed to depend only on the permanent electric moment of the molecule and thus to be the same for all allowed pure rotational transitions. In the pure rotational emission spectrum of H35 Cl gas, lines at 106.0 and 233.2 cm−1 are observed to have equal intensities. What is the temperature of the gas? The rotational constant B for H35 Cl is known to be 10.6 cm−1 , and the ratio hc/k has the value 1.44 cm deg. 5. J. I. Steinfeld, p. 143, #6. The harmonic oscillator wave functions for the levels v = 0 and v = 1 are � � � a �1/4 −a(r − re )2 ψ0 = exp , π 2 � 3 �1/4 � � 4a −a(r − re )2 ψ1 = Δr exp , π 2 in which a is a constant. These functions are orthogonal and normalized to unity. (a) Show that ψ0 and ψ1 are orthogonal. (b) Calculate the average value of (r − re )−2 for the state v = 1. Also calculate Bv (in cm−1 ) for v = 0 and 1. Why is B1 > B0 for a harmonic oscillator? Why is B1 < B0 for almost all real molecules? 6. J. I. Steinfeld, p. 143, #7. What would happen to the Birge-Sponer extrapolation scheme for a molecular potential correlating with ionic states of the separated atoms? 7. You are going to use the Van Vleck Transformation (a fancy name for second-order quasi-degenerate perturbation theory) to solve a coupled harmonic oscillator problem. Consider H = H0 + H0 + ζ x2 y2 x y and let ω x = ωy . (a) Given that � � � n x | x2 |n x = (n x + 1/2) mω � � � n x | x 2 |n x ± 2 = [n2 + n x + 1 ± (2n x + 1)]1/2 2mω x ζ � ≡ b. 4m2 ω3 Problem Set #2 Spring, 1987 Page 3 Construct the H matrix through n = n x + ny = 4. Notice that the matrix factors nicely into an odd n and an even n block. (b) Apply the Van Vleck transformation to the n = 0, 1, and 2 blocks. Assume that �ω � b. Be sure to include corrections due to off-diagonal elements with n > 4 basis functions. Your bookkeeping will be simplified if you make use of “railroad” diagrams. For example, √ 3 6b 1, 1 � � � � � � � � 3,1 √ 3 6b 1,3 √ 3 6b √ 3 6b 6b 3,3 6b � � � � � � � � 1, 1 The diagram places the initial and final basis functions at left and right and in the middle all basis functions that can simultaneously have non-zero matrix elements with both. Above each line is the actual value of the matrix element. All that remains is to look up the relevant energy denominators. The correction to the 1, 1; 1, 1 matrix element is (neglecting b terms in denominators) 54b2 54b2 36b2 − − − . 2�ω 2�ω 4�ω (c) Now that you have effectively uncoupled the n = 0, 1, 2 blocks from all other blocks, you can focus your attention individually on these 1 × 1, 2 × 2, and 3 × 3 isolated effective Hamiltonians. Construct the (2, 0) ± (0, 2) and (1, 0) ± (0, 1) basis functions. This is called a “Wang Transfor­ mation”. Can you suggest any physical basis for this additional factorization? You should find that the n = 0 − 2 part of this Hamiltonian is now fully diagonal. Draw an energy level diagram for n = 0 through 2 which compares your eigenvalues against unperturbed (b = 0) levels. ...
View Full Document

This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

Ask a homework question - tutors are online