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 1976 Problem Set #3 1. (a) Write out the electron configurations for the molecules O+ 2, O2, O2−, and O22−. (b) Determine the ground-state term symbols MΛg,u± for O+, O2, O−, and O22−. If there are two or more 2 2low-lying states, select one as that of the ground state and justify your selection. 2. (a) Using symbols appropriate to the separated-atom approximation to a molecular orbital, write the elec-tronic configuration of lowest energy for the diatomic species C2, NO+, and S2. (b) Write the term symbols MΛg,u± for all the electronic states derivable from the configuration of lowest energy for all three species. Which of these states will have non-zero magnetic moments? 3. Gaseous HCl is normally a 3 : 1 mixture of H35Cl and H37Cl. To a high approximation, the rotational energy levels of such diatomic rotators are E(J)(in cm−1) = J(J + 1)B − J2(J + 1)2D where B, the rotational constant, is larger by a factor 1.0015 for H35Cl than for H37Cl, and the centrifugal distortion constant D is the same for both molecules within the error of its measurement. (a) Derive an expression for the separation of the pure rotational absorption lines of H35Cl and H37Cl as a function of J�, the J–value for the upper state (ΔJ = J�− J�� = +1). (b) What is the spacing in cm−1 of the two lines for which J� = 10? 4. The “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 dipole moment of the molecule and thus to be the same for all allowed pure-rotational transitions. In the pure-rotational emission spectrum of H35Cl gas, lines at 106.0 cm−1 and 233.2 cm−1are observed to have equal intensities. What is the temperature of the gas? The rotational constant B for H35Cl is known to be 10.6 cm−1, and the ratio hc/k has the value 1.44 cm deg. ·5. What would happen to the Birge-Sponer extrapolation scheme for a molecular potential correlating with ionic states of the separated atoms?5.76 Problem Set #3 Spring, 1976 page 2 6. The ground state and a low-lying excited electronic state of the BeO molecule have the following properties Term symbol 1Σ+ 1Π Electronic energy, Te/cm−1 0 9405.6 ωe/cm−1 1487.3 1144.2 ωe xe/cm−1 11.8 8.4 re/10−8cm 1.33 1.46 (Note that the electronic energy Te is the energy from the minimum of one curve to the minimum of the other; this is not equal to the vibrational origin of the 0 − 0 band.) (a) Construct a Deslandres table of the vibrational band origins of the 1Π − 1Σ+ system, for v�� = 0 through 3 and v� = 0 through 5. Which of these vibrational bands would you expect to be the most intense, when the system is observed in absorption? Comment on the relative intensities that you would expect for the other bands in your table. (b) In the rotational structure of the individual vibronic bands in this system, what branches would you expect to observe? In which branch would you expect to observe a band head? Calculate the transition in J that will give rise to a line at the band head, and the distance in cm−1from the band head to the vibrational band origin. (c) What would you guess about the MO configurations corresponding to these two states? (HINT: Note that BeO is isoelectronic with C2, so that the MO’s may be expected to be somewhat similar, except that the g-u property will be lost, and the orbitals will be distorted toward the higher nuclear charge of the O-atom.) Would you suspect the presence of any other excited electronic states below the 1Π state? If so, what would its term symbol be? 7. The following bands are observed in the second positive system of nitrogen (units are reciprocal centimeters corrected to vacuum): 35,522 cm−1 29,940 cm−1 25,913 cm−1 35,453 29,654 25,669 33,852 29,010 25,354 33,751 28,819 25,003 33,583 28,559 24,627 32,207 28,267 24,414 32,076 27,949 24,137 31,878 27,451 23,800 31,643 27,226 23,414 30,590 26,942 23,016 30,438 26,621 30,212 26,274 Arrange these in a Deslandres table, and find values for ω��e , ωex��e , ω�e, and ωex�e. Important Suggestion: Look at the pattern of bands first, before doing anything else. Do any natural groupings seem to suggest� � � � 5.76 Problem Set #3 Spring, 1976 page 3 themselves? It may help to draw a “stick spectrum” of the band origins, to scale, in order to pick out these patterns. Remember that bands having the same Δv fall along diagonals on the Deslandres table.) Is there any suggestion of a cubic term in either state? If so, derive an expression for the third difference,� �3 Δ3Gv+1/2, including terms in ωeye v + 21 in Gv+1/2, and estimate ωeye. 8. The first strong electronic band system of carbon monoxide (the ground-state vibrational frequency of which is observed at 2140 cm−1 in the infrared) appears in absorption at room temperature at about 1550 Å in the vacuum ultraviolet. The system shows a progression with a spacing of 1480 cm−1. The vibronic bands show a single set of unperturbed P–, Q –, and R–branches degraded to the red. Analysis by combination differences of these branches gives B�e = 1.61 cm−1, B��e = 1.93 cm−1. In each band, the lines nearest the origin are P(2), Q(1), and R(0). (a) Deduce all you can about the two electronic states involved in the transition from these data and your general knowledge of the properties of carbon monoxide. (b) Sketch the lower portions of the potential curves in cm−1 for CO, roughly, to scale,