8565
J . Phys. Chem. 1991, 95, 8565-8574 cluster systems involving toluene$3 aniline?b and phenol6 seeded in solvent clusters (e.g., NH3, HzO, and CH30H). The mass spectra for DMS cluster ionization are consistent with the above trends. Very little dissociative hydride transfer to form (DMS)H+ is observed by photochemical excitation of the dimer cation (cf. Figures 7a and 8). The dimer cation reaction is a little more evident for E1 ionization than for MPI (compare parts a and b of Figure 7), perhaps because of the greater total energy of ionization by E1 (70 eV of electron energy vs about 9-13 eV of photon energy). Hydride transfer is observed to occur more extensively in larger clusters. Although the E1 mass spectrum in Figure 6a does not fully resolve unit masses, the shoulder corresponding to the protonated ion (DMS),H+ does increase considerably relative to the parent ion signal (DMS),+l+ with cluster size n. This result supports a reaction enthalpy that decreases with increasing cluster size. Summary and Conclusion The formation of a 3e,2c bond in the DMS dimer cation creates strongly absorbing bands in the visible region that are not present in the monomer cation. This has enabled us to study photochemistry in the cluster ions. The results of this work are summarized by the following: (43) Brutschy, B.: Janes, C.: Eggert, J. Ber. Bunsen-Ges. Phys. Chem. 1988, 92,74.
(1) E1 ionization/dissociation of DMS clusters promotes reactions that closely resemble those observed in gas-phase bimolecular ion-molecule reactions and collision-induced dissociation, namely, (i) dealkylation by S--C bond cleavage and (ii) hydride transfer to form protonated monomer and clusters. (2) REMPI excitation followed by cluster ion absorption leads to reactions that are quite different from the E1 results. The most distinct photochemistry occurred by exciting the dimer cation Q* state at 400 nm. The strongest observed photoproduct, C2H3S+, is believed to have the cyclic thiirenium structure, which has been implicated as an important intermediate in solution-phase sulfide chemistry. (3) The ionization/fragmentation pattern for monomer DMS is essentially invariant to mode of excitation (Le., E1 vs REMPI). This result is important for distinguishing intramolecular from intermolecular cluster reactions. We have shown that cluster chemistry can be very specific to the mode of excitation. The finite size of small molecular clusters has enabled us to selectively induce reactions associated with both the gas phase and the solution phase. The study of cluster ions has direct correspondence to solution-phase chemistry, particularly those involving nucleophilic displacement, for which the formation of c a r h a t i o n s and sulfur cations constitutes the central reactive site. The observation of analogous reactions in small clusters provides a bridge to understanding solution-phase chemistry on a molecular level.
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Resonantly Enhanced Multiphoton Ionization of N2 arr1Zgf( v’ = v”) XIZg+ ( v” = 0-2). 1. Determination of Rotational Populations and Virtual State Character Thomas F. Hanisco and Andrew C. Kummel* Department of Chemistry, University of California, San Diego, La Jolla, California 92093 (Received: May 30, 1991)
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The resonantly enhanced 2 + 1 multiphoton ionization spectrum (REMPI) of the (O,O), (1,1), and (2,2) transitions of the X’ES+ states of molecular nitrogen has been recorded at high resolution, and the rotational and vibrational constants a”1ZI of the excited states have been calculated. The sensitivity of the REMPI technique is 5 X lo5 molecules/(cm3 quantum state). The (0,O)transition was originally seen by Lyke and Kay in REMPI and by Dressler and Lutz in one-photon emission spectroscopy. Our measured spectroscopic constants are as follows (in cm-I): T, = 98938 f 5, de= 2167 f 3, = 1.938 +
f 0 . 0 0 1 , ~ ’ , = 0 . 0 4 7 f 0 . 0 0 1 , v ~ = 9 8 8 4 0 . 5 9 f 0 . 1 2 , B ~1.9143f0.0002,D:=o=6.6X ~o= 1 p f O . 2 X 1od,vll=98655.3 f 0.2, = 1.882 f 0.001, Y~~ = 98458.8 f 0.2, and = 1.820 f 0.001. In addition, some general questions concerning 2 n REMPI Z Z transitions have been addressed by using N2 a”’Z ‘XZg+ as a model. In two-photon Z Z absorption spectroscopy, the relative intensities of the different rotational branches (0,Q, and S) and the polarization line
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strengths of the Q branch depend very strongly upon the Z versus n character of the virtual state. Accurate measurement of the character of the virtual state was made using two methods: recording the relative two-photon absorption intensity of the 0-and Q-branch lines and recording the two-photon absorption intensity as a function of the elliptical polarization of the laser light. For nitrogen, from measurement of the relative intensities of the 0 and Q branch, we have determined the virtual state path ratio: $/p: = 12.5 f 5, where the path ratio is a function of the L: versus n character of the virtual state. While the Q-branch measurements are always sensitive to the rotational populations and quadrupole alignments moments, measurements of absorption via the S and 0 branches are critical to determining the angular momentum orientation of an anisotropic sample. Therefore, accurate assignment of 0-and S-branch lines and determination of the character of the virtual state are critical to using any two-photon Z Z transition for determination of populations, alignment, and orientation.
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( I ) Dresslcr. K.; Lutz, B. L. Phys. Reu. Len. 1967, 19, 1219. ( 2 ) Lcdbettcr, J. W. J . Mol. Specfrosc. 1972, 42, 100.
(3) Lykke, K. R.; Kay, B. D. J . Chem. Phys. 1989,90,7602. Lykke, K. R.; Kay, B. D. J . Chem. Phys. 1991, 95, 2252.
0022-3654/91/2095-8565%02.50/0 0 1991 American Chemical Society
8566 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991
Hanisco and Kummel
Figure 1. Schematic diagram of the laser setup used for Nz REMPI spectroscopy. An injection seeded Nd:YAG pumps a dye laser at 10 Hz. The dye laser output is doubled in KD*P, rotated 90°, and mixed with the residual fundamental in &BaBzO,. The resulting 202-nm light is separated and focused into an UHV test cell. The ions are detected with TOF-MS and the laser power is detected with a stainless steel photodiode. The ion signal and laser power are counted by a gated integrator and read by computer via GPIB.
possible by their ability to generate intense vacuum ultraviolet (VUV) light using a p-BaB204 crystal to frequency triple light from a dye laser. A considerable amount of information on the spectroscopy of molecular nitrogen (N2)has been acquired; a compendium of this information has been compiled by Lofthus and Krupenie.4 Until &BaB204 crystals were commercially available, quantitative REMPI spectroscopy of the X’Z,+ground state of N2had been XIZg+ 2 + 2 transition. This limited to the study of the aln, REMPI transition was first reported by Carelton et aL5 for determining the rotational populations of an unpolarized sample. This transition has also been used by Kummel et aL6 to determine the rotational populations, alignments, and orientations in a nonisotropic sample, by Randeniya et ale7to determine the temperature dependence of reactions involving N2,by Sitz and Farrod to study state-to-state rotational energy transfer rates in N2, and by Chu et al.9 to measure the internal energy distribution of N2 produced from the photodissociation of hydrazoic acid. The paucity of N2 REMPI transitions can be explained by the limitations on the allowed 2 + n REMPI pathways for a homonuclear diatomic molecule. These allowed transitions are essentially limited to those that are not allowed in one-photon spectroscopy; specifically, g g, u u, + +, and - -. In addition, for a Hund’s case (a) molecule, PA = 0, f l , f 2 and A 2 = 0. Thus, the resonant state for any two-photon allowed transition involving the N2 IZg+ ground state is restricted to k i n g a singlet state with gerade inversion and positive reflection symmetries. The allowed transitions must meet other criteria to be useful: the transition must be in reach of available laser wavelengths; the transition must not be congested with overlapped lines from multiple branches; the transition must have a sufficient Franck-Condon overlap for detection at low pressures; and the state must not predissociate so fast as to significantly broaden the
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P.H.J . Phys. Chem. Ref.Dura 1977,6, 113. (5) Carelton, K. L.; Welge. K.H.;Leone, S.R. Chem. Phys. Leu. 1985, 1 IS. 492. ..., ( 6 ) Stz, G. 0.;Kummel, A. C.; Zare, R. N. J. Vac.Sei. Technol. A 1987, 5, 513. Sitz, G. 0.; Kummel, A. C.; Zare, R. N. J. Chem. Phys. 1987, 87, 3247. Sitz, G. 0.;Kummel, A. C.; Zare, R. N.; Tully, J. C. J . Chem. Phys. 1988. 88, 7357. Sitz, G. 0.;Kummel, A. C.; Zare, R. N.; Tully, J. C. J . Chem. Phys. 1988.89.2572. Kummel, A. C.; Sitz, G. 0.;Zare,R.N.;Tully, J. C. J . Chem. Phys. 1989, 89, 6947. Kummel, A. C.; Sitz, 0.0.;Zare, R. N.; Tully, J. C. J . Chem. Phys. 1989, 91, 5793. (7) Randeniya, L. K.; Zeng, X. K.; Smith, R. S.;Smith, M. A. J . Phys. Chem. 1989, 93, 8031. (8) Sitz. G. 0.;Farrow, R. L. J . Chem. Phys. 1990. 93,7883. (9) Chu,J. J.; Marcus, P.;Dagdigian, P.J. J . Chem. Phys. 1990, 93,257. (4) Lofthus, A.; Krupenie,
line widths or diminish the REMPI signal. II-Z and A-Z: transitions will have at least four intense branches; thus, these transitions will exhibit considerable overlap between 0 and P branches as well as R and S branches. Conversely, 2-Z: transitions usually have only a very intense, condensed Q branch and weak, well-separated 0 and S branches. Since the P and R branches are absent, there is little interbranch congestion. Therefore, the primary requirement for determination of rotational populations when probing a two-photon 2-2 transition is a well-resolved Q branch. To minimize congestion within the Q branch, a 2-2 transition having a moderate difference between ground- and excited-state rotational constants is required. The a”I2,+ X’Z,+2 1 REMPI transition is a special type of two-photon absorption: a AA = 0 transition (AA = the difference in orbital angular momentum between the ground (initial) and resonant (final) states). One other P A = 0, 2 n REMPI transition has been used to determine rotational populations: H2 IZg+ XIZg+. In the aforementioned case, only the Q branch was employed to determine the rotational populations, and the line strengths were calculated assuming an infinite path ratio, p t >> p,2 (see Table 111). In this paper, we report the measurement of the virtual-state path ratio and the line positions of the Q branches for the (O,O), (1,1), and (2,2) bands as well as the 0 and S branches of the (0,O)band. In part 2 of this series, we report upon the use of the transition for the determination of alignment and orientation; in addition, we describe how to correct the measured line intensities for effects of angular momentum alignment when probing anisotropic samples. We have combined the use of /3-BaB204with a reliable narrow-bandwidth laser to generate narrow-band W V laser radiation to probe the a”lZ:,+ X’Z,+2 1 REMPI transition. A rotational analysis for the a”’Zg+ excited state is presented here. The analysis is based on our experimental line positions of the a”’Z,+ XIZ: (O,O), ( l , l ) , and (2,2) transitions and the literature values !or the X’Z: state. The advantages of this new transition include (1) the presence of a very intense Q branch that can be probed with a sensitivity for rotational populations and quadrupole alignment moments 10-20X greater than any of the rotational branches in the a’n, X’Z,+2 + 2 REMPI transition and (2) an S branch with no overlapped lines that can be probed with a sensitivity only 5-1OX less than any of the rotational branches in the aln, XIZ,+ 2 + 2 REMPI transition. This is important because the detection of the S and 0 branches is critical to the measurement of orientation moments and higher order alignment moments. The major disadvantage of employing the a“lZ,+ XIZg+ 2 1 REMPI transition is that the absence
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The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 8567
REMPI of N2 of the P and R rotational branches inhibits the accurate determination of higher order orientation moments and noncylindrically symmetric alignment moments. In the experimental and results sections of this paper we first describe the determination of line positions and rotational and vibrational constants for the a"lZg+ state. Second, we describe the determination of the character of the virtual state for the a"'?,+ XIZg+2 1 REMPI transition. In the Appendix we derive the equations employed to determine the line strength as a function of the laser polarization.
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+
Experimental Section
The experimental setup is shown schematically in Figure 1. The second harmonic from an injection-seeded, IO-Hz pulsed Nd:YAG laser (Quantel YG581C) pumps a dye laser equipped with an angle tuned, Littrow-type oscillator with an intracavity etalon (Lambda Physik FL3002E). Approximately 50 mJ of 607-nm light with 8-11s pulse width and 0.04-cm-' bandwidth is produced. The injection seeding of the Nd:YAG laser does not have a large effect on the dye laser bandwidth as measured with a 0.33-cm-I FSR solid etalon. Nor does it have a significant effect on an iodine absorption spectra measured at 607 nm. However, injection seeding of the Nd:YAG has a profound effect of the resolution of the MPI spectra. The best resolved spectra were taken with the Nd:YAG laser properly seeded. This requires that (1) the seed laser be spatially overlapped with the Nd:YAG beam and (2) the seed laser have its frequency set to the maximum of the Nd:YAG gain curve. The frequency was adjusted while observing fringe patterns of the laser beam projected through a 1.75-cm-l FSR solid etalon. The seed laser frequency was set so that the fringes of the seeded beam were centered in the fringes of the unseeded beam. The dye laser output is doubled in an angle-tuned KD*P crystal (Inrad Autotracker 11). The doubling crystal has an entrance window that is antireflection coated for visible light but has an uncoated UV-grade fused silica exit window. A standard UVcoated exit window induces etalon effects on the residual visible light, which results in power fluctuations as the wavelength is scanned. The resulting UV light is rotated 90' with a polarization rotator (Inrad) and then mixed with the residual visible light in an angle-tuned ( h a d Autotracker 11) 8-BaB204crystal (Cathay American). The O-BaB2O4crystal is housed in a cell with uncoated CaF2 windows. The mixing produces -1.5 mJ of 202-nm light, which is separated from the remaining fundamental and doubled light with dichroic mirrors (Virgo Optics). The tripled light is focused with a 25-cm LiF lens either into a table top ultrahigh-vacuum (UHV) time-of-flight mass spectroscopy (TOFMS) cell (Thermionics) or into a molecular beam apparatus, each equipped with UV-grade fused silica windows. The ion-pumped UHV cell has a background pressure of -2 X IO4 Torr and is filled to 1 X lO-' Torr with commercial grade N2 through a leak valve. All experiments in the UHV cell were done on room temperature N2. However, large quantities of vibrationally excited N2 are present due to production of metastable N2* by the ion pump. The molecular beam system is pumped by diffusion and sublimation pumps and hence does not produce any detectable amount of vibrationally excited N2. In both systems, the ion signal is collected with a CEMA detector (Galileo) and counted with a gated integrator (LeCroy 2249SG). The 202-nm light is detected at the output of the cell or molecular beam chamber with a stainless steel and nickel photodiode.'O The photodiode consists of a 2.5-cm-diameter stainless steel cathode and a 2.5-cm-diameter nickel grid (70 Ipi) anode separated by 0.15 cm. The assembly is housed in a vacuum cell at a pressure of 1 X Torr. The photodiode is maintained in vacuum for two reasons: (1) the signal is several orders of magnitude greater than in the atmosphere and (2) the signal is affected by the multiphoton ionization of atmospheric nitrogen at resonance energies. The photodiode is biased a t -48 V with a 1.S-A supply. The relatively large current is necessary to ensure
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(10) Rettner, C. T.; Bethunc. D.
S.Rev. Sci. Instrum. 1989, 60. 3824.
a linear response. The linearity of the nickel photodiode was checked by continuously varying the laser power with a variable attenuator (Newport 935-5) while monitoring power and referencing to a commercial joulemeter (Molectron 5-3). The power dependence of the ion signal intensity was determined by varying the power with the variable attenuator while monitoring ion signal and laser power. The ion and photodiode signals are read by an Apple Macintosh I1 via an IEEE-488 (GPIB) interface (National Instruments NI-488) and CAMAC controller. The Macintosh is also used to control the dye laser via the GPIB interface and to analyze the collected data. For a typical spectrum, 10 laser shots are taken at each wavelength as the dye laser is scanned over a 0.14.5-nm range. The ion signal is corected for power fluctuations by dividing by the square of the photodiode signal. The normalization is done on single shots or on bins of 10 shots, depending on noise levels. To obtain a more accurate wavelength calibration for the dye laser, the N 2 MPI spectrum was measured while simultaneously measuring an iodine absorption spectrum with the visible output of the dye laser by using a 3dcm iodine cell heated to 50 OC. The measured transition energies were corrected with published values" throughout the spectrum. Calibration with this technique yields absolute energy values accurate to f0.06 cm-' a t 98 800 cm-I. Circularly polarized light and a molecular beam were used to help identify overlapped peaks in the spectra and to measure the path ratio. Spectra taken with circularly polarized light were used to distinguish between the overlapped peaks of the Q and 0 branches. Molecular beams were used to distinguish between lowand high-J states. Measurements of Q-branch intensity versus polarization angle and comparisons of the relative intensities of Q and 0 branches were used to determine the path ratio. For the spectra taken with circularly polarized light, a quarter-wave plate (Optics For Research) or a Babinet-Solei1 compensator was inserted before the lens. In the determination of the path ratio, the polarization of the laser light was gradually changed by rotating the quarter-wave plate on a computer-controlled stage (Oriel) while keeping the laser tuned to a resonance. The data were taken at each position of the rotation stage as the angle was incremented. A second set of data was taken as the angle was incremented in the reverse direction, thus minimizing the effect of any drift in the experiment. The measurement was also done by repeatedly scanning the laser through a resonance at different polarization angles. Both techniques yield the same results. For measurements on the Q(0) and Q(l) line, a rotationally cold molecular beam (- 5 K) was used in place of background gas, thus minimizing effects from overlapped low-J lines. The experiments that required a molecular beam were done in a separate UHV chamber equipped for studies of surface dynamics. It will be described in detail in a later paper that deals specifically with surface dynamics. Briefly, it consists of a three-stage, differentially pumped source chamber and a UHV scattering chamber with a background pressure of 1 X torr. The beam is made with 20% N2 seeded in He expanded through a pulsed-valve nozzle (General Valve) with a 1.5" orifice. The beam passes through a 250-pm conical skimmer, a mechanical chopper, and a 0.1-cm collimator. The chopper operates a t 200 Hz and has two equally spaced 10-mm-wide slits on a diameter of 12.5 cm, producing pulses that are roughly 125 /IS long. The pulses travel approximately 12 cm before being ionized and extracted by a time-of-flight mass spectrometer. The molecular beam path, laser propagation direction, and TOF extraction are mutually orthogonal.
Results
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Determination of Line Positions and Spectral Constants. A low-resolution 2 1 REMPI spectrum of the N2 aNIZ: X I Zg (O,O), (1,1), and (2,2) bands is shown in Figure 2. The intracavity
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( I 1) Gerstenkorn, S.;Luc. P. Atlas Du Spectre D'Absorbrion De La Molecule Diode 14800-2oooO cm-I; Editions Du Centre De La RecheKx Scientifique: Orsay, France, 1978; pp 43-44.
8568 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991
Hanisco and Kummel
1 -
v=OOBmch
(a)
21
1
0
1.00
22
m
v = 1 QBranch
5
IO
,
I
I
I
I
202.72
202.74
202.77
202.79
~ , , ' , , " , , ' " ' 1 , ' ' ' 1 ' , ' ' ,
202.00
202.30
202.60
202.90
203.20
Wavelength (nm) 203.50
Wavelength (nm)
v=2
-
QBranch
Figure 2. Low-resolution 2 + 1 REMPI spectrum of the a"'Z,+ XIZp+(O,O), ( l , l ) , and (2,2) bands of room temperature N1. The intensity is plotted on a logarithmic scale. S branch IO
5
hr
10
0 branch
rim" IO IS 20
m[lilllillllilllilll 10 IS
0
5
0
(a)
Qbranch
as
30
I
203.10
"
"
I
"
"
I
203.15
203.20
Wavelength (nm) 202.00
202.25
202.50
Figwe 4. REMPI spectra of the N2a"lZp+ and (b) (2,2) Q branch.
202.75
Wavelength (nm)
+
XiZ:
(a) (1,l) Q branch
Q Branch QBranch
7 0
10
15
21
in
25
OBranch OBranch
7 5
25
1 8
5
6
21
h - Linear - Circular
202.33
202.36
202.39
h 202.42
Waveleng!h (nm)
Figure 3. High-resolution REMPI spectra of the N2 (0,O) transition: (a) S and 0 branches; (b) Q branch.
202.45
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XIZ,+
etalon was removed from the dye laser so that the entire spectrum could be measured with one continuous scan. With only the grating in the dye laser, the visible bandwidth is 0.2 cm-I. The (0,O)Q branch was measured in a separate scan under less Sensitive detection conditions and later added to the overall spectrum. The (0.0) band can be resolved into an S branch and an 0 branch in addition to the Q branch. Only Q branches can be resolved for the (1,l) and (2,2) bands. High-resolution spectra of the S,Q,and 0 branches of the (0,O) band are shown in Figure 3. Spectra of the (1,l) and (2.2) Q branches are shown in Figure 4. Line positions of the branches are listed in Table 1. The transition energies are accurate to 0.12 cm-l. The relative energies are accurate to 0.06 cm-l. The assignment of the various lines was facilitated by the use of a molecular beam and circularly polarized light. The positions of the J" = 0 and J" = 1 transitions of the S and Q branches were determined with a spectrum of a rotationally cold sample produced
202.40
202.44
202.48
Wavelength (nm)
Figure 5. Comparison of the (0,O)Q and 0 branch spectra taken with
linearly polarized light (solid line) and taken with circularly polarized light (dashed line).
in a molecular beam. At 5 K, 90% of the molecules are in J = 0 or J = 1; hence these lines are clearly resolved. The identification of the closely spaced or overlapped lines of the (0,O)and (1,l) Q and 0 branches were determined with circularly polarized light. The use of circularly polarized light was useful because the relative intensity of a Q line to an 0 or S line is roughly 175:l with linearly polarized light while only 5:1 with circularly polarized light (see Appendix). Figure 5 shows a region where the (0,O)0 branch and (1,l) Q branch overlap. The solid line represents data taken with linearly polarized light, and the dashed line shows the same region when scanned with circularly polarized light. Several lines that appear in the u = 1 Q branch spectra could not be identified. Two possible contaminants, NO and CO, were
REMPI of N2
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 8569
TABLE I: Wavenumbem of the Lirns of the N2a”lZl+ - X lZ1+ (O,O), (l,l),lad (29) Tnnsitioac (Vduw Accurate in Absolute E m to 0.06 cm-’ and in Relative Enem to 0.05 cm-9 u=o J
0 1
2 3 4 5 6 1 8 9
IO 11
12 13 14 15
16 11 18 19 20 21 22 23 24 24 25 26 26 21 28 29 30 31 32 33 34 35 36 37 38 39
v=l
Q(4 S(J) Q(4 98840.61” 98852.21e b 98 84O.6lb* 98 859.74‘ b 98828.80”s 98840.12’ 98861.10 98 654.30” 98 820.61’d 98 839.62’ 98 874.30 98 653.92” 98812.28” 98838.97’ 98881.33 98 653.50” 98 803.95 98 838.32’ 98 888.20 98 652.49“ 98195.6lod 98 831.33’ 98 895.00 98 651.59 98 186.63d 98 836.35” 98901.31 98 650.18 98111.80 98 835.21 98 901.16 98 649.01 98168.83 98 833.14 98914.30 98 641.46 98159.59 98 832.26 98 920.31 98 645.16 98150.38 98 830.73 98926.26 98643.14 98140.93 98 828.93 98 932.00 98 641.33 98131.25 98 826.98 98 931.12 98 638.82 98121.59 98 824.85 98 943.29 98711.68 98 822.51 98 948.11 98 953.18 98101.55 98 820.11 98691.30 98811.50 98 958.86 98680.94 98 814.12 98 963.18 98610.56 98 81 1.78 98 968.31 98659.98 98 808.84 98 912.95 98648.98 98 805.58 98911.48 98638.01 98 802.15 98 981.40 98198.11 98 985.65 98 987.93‘ 98 194.96 98 990.54‘ 98 191.I9 98 993.15 98 788.1w 98 991.01 98 185.64‘ 99000.26 98 183.20 99003.52 98 118.95 98 114.38 98 169.80 98 165.01 98 160.20 98 155.14 98 149.91 98 744.53 98 138.98 98 133.21 98 121.24 98121.36 O(4
u=2
Q(4 b b 98 458.3’ 98 451.I’ 98 456.0” 98 454.8’ 98 453.0 98 451.3 98 449.1 98 441.0 98 444.0 98441.1 98 438.0 98 434.4 98 430.1
‘Overlapped peak. Less than 15% of peak above baseline. bobscurd. ‘Perturbed. Resolved with circularly polarized light. eResolved with molecular beam. examined. NO has a strong multiphoton transition in this region, but no lines that match those in the spectra. CO has no transition in this region. The peaks are assumed to be the result of perturbations and ignored for the calculations. Molecular constants were determined by using published ground-state values and experimental line positions. The B‘and D’ constants were determined by fitting the rotational energies of the Q-branch transitions to the relation
M(J)
YO
+ (B: - B’:)J(J + 1) - (D: - D’:)P(J + 1)2
(1) The fitting was done with a multiple linear regression (Lionheart). The results and reference values are listed in Table 11. All errors listed are from the regression. The ground-state constants used for the calculation are B’i,o = 1.989 544 cm-I and D’.’,,o = 5.76 X 10” cm-I.l2 The calculation was done for each of the three branches. The rotational constants determined for the (0,O)band are uo = 98840.59 f 0.12 cm-’, B:-o = 1.9143 f 0.0002 cm-l, and D’,,lo = 6.6 X 10” f 0.2 X lo4 cm-l. The calculations for the (1,l) and (2,2) bands were done without the second-order term D, because of the small number of lines available. The ground-state constants were calculated according (1 2) Bendtsen, 1. Roman Spcrrosc. 1974, 2, 133.
TABLE II: Molecular Parameters for the N2a”%,+ X ‘2: Svstem (em-’) level constant present work ref values 98938 f 5 Te a”, 2351.84 2161 3 w’e 1.998232 O.OO0 12 B’: 0.017292 0.000003 allc 1.938 f 0.001 B: 0.047 0.001 a’, v = o Yo0 98 840.59 0.12 98 840.30f 0.07 1.989514f 0.000012 B” B’ 1.9143 f 0.0002 1.9133 f 0.0008 D” 5.16 X IOd f 0.03 X lod 6.6 X IOd 0.2 6.2 X IOd f 2.9 X IOd D’ x 10-6 98 655.3 f 0.2 u= 1 VI1 B” 1.9711 f 0.0006 1.882 0.001 B’ u=2 v22 98458.8 f 0.2 B” B’ 1.820 f 0.001 +
~
~~
*
*
ref (1 1)
(IO) (IO)
*
*
*
-’ -2
(2)
(IO) (2) (10) (2) (10)
1__11: 1 . 10
-’d
0 - 2 -
’
5
’
10 ” ’
15
J
(c) 1
0 1 -2
0
1
0
2
0
3
0
4
0
J
Figure 6. Difference plots of transition energies versus J”: (a) (0,O)0 branch, (b) (0,O)Q branch, (c) (0,O)S branch, (d) (1,l) Q branch, and (e) (2,2)Q branch. Note in b and c the perturbations on J ’ = 26
(marked by asterisks).
+
to B’: = B’i - 1 / 2 d ’ e (I f~ 2 ) using B‘L = 1.998 24 cm-l and a‘’, = 0.017292 cm-I. The constants determined for the (1,l) band are uI1 = 98655.3 f 0.2 cm-’ and B i = l = 1.882 f 0.001 cm-’. The (2,2) band constants are Y~~ = 98 458.8 f 0.2 cm-’ and BL12 = 1.820 f 0.001 cm-I. The errors presented for the (1,l) and (2,2) band constants are the standard errors from least-squares fits. They do not reflect the greater inaccuracy one would expect from having only 13 or 14 data points to fit a function that may require more. The values for B and Y along with o”,= 2357.84 cm-l 13 were used to determine the constants for the a”IZ,+ state: T, = 98938 f 5 cm-l, w’, = 2167 f 3 cm-l, B i = 1.938 f 0.001 cm-I, and a’, = 0.047 f 0.001 cm-l. The difference between the calculated and experimental line positions is shown in Figure 6. A perturbation of J’ = 26 (u’ = 0) in the a’’ state can be clearly seen in the difference plots of the S and Q branches, Figure 6b,c. In the spectra, the S(24) and Q(26) lines of the (0,O)band appear as asymmetrically split (13) Stoicheff,
B. P. Can. J . Phys. 1954, 32, 630.
8570 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 2.8 7 2.6
A
Hanisco and Kummel TABLE IIk Bny-Hwhburcr Aulytierl EqmW for tbe Liw Stremtb of I%-I2 Two-Photon T m i W
-
0 branch:
2.4 -
2.2
-
2.0
-
1.8
-
Q branch:
S branch:
2
K d ( J + l)(W+
Ko(W+ 1) 9
Ko(J
piz
+
+ 1 ) ( J + 2)
30(W + 3)
45(W - 1)(W
1)
+ 3)
*
where
I.”
I
1.8
1.9
2.0
2.1
2.2
.
2.3
,
2.4
log[Power] (Arbitrary Units) Figure 7. Least-quarts fit of the log on ion signal intensity of the (0,O) bandhead versus the log of laser power. 2 0
1 +
- 3 1 0
KoJ(J30(2J - I ) ”
,
, 100
,
, 200
* *
0-Branch: 318 22K Q-Branch: 289 f 14K S-Branch: 298 13K
+\
. 360
460
\
I 5 L
J(J+1) Figw 8. Boltzmann plots of (0,O) S-,Q-, and 0-branch peak intensities versus rotational state. The line indicates the slope correspondingto 298
K.
doublets. The splitting is due to an unknown perturbation of J’ = 26. There also appears to be considerable perturbation of the (1,l) Q branch, since the difference plot Figure 6d shows a systematic deviation from the zero line. The dependence of the ionization signal on the laser power was determined by using isolated lines in the S,Q, and 0 branches of the (0,O)band. The ion signal was measured while the laser power was varied continuously with a variable attenuator. The signal intensity was fit to the power using the relation I = P.A least-squares fit of the data with a slope of 2.00 f 0.03 is shown in Figure 7. This indicates that the two-photon step is not saturated, while the ionization step is saturated. No branch or rotational state dependence was observed. Quantitative rotational-state populations were determined for the S,Q, and 0 branches of the (0,O)band by using measured peak intensities. Because of the clean signal, integration of peak areas was not necessary for accurate population measurements. Boltzmann plots of the populations of the S,Q, and 0 branches with least-square fits are shown in Figure 8. The slopes from these plots yield temperatures of 298 f 13 K for the S branch, 289 i 14 K for the Q branch, and 3 18 f 22 K for the 0 branch. The S branch populations were measured from S(2) through S(19); the Q branch from Q(4) through Q( 16); and the 0 branch from O(8) through O(20). The peaks that were used in the calculation were chosen because they can be easily resolved in a single scan without changing any detection parameters. Determination of Virtual State Character and Path Ratios. In a two-photon 2-2 transition, the intermediate virtual state can be of either Z or II character. Therefore, there are two distinct pathways in a two-photon 2-Z: transition: one is a 2-2-2 transition and the other is a 2-n-2 transition. We will use the
‘The formulas for p t and p: are valid for linearly polarized light. For circularly polarized light, 1 : = 0, while k2increases by 1.5.
nomenclature of refs 14-16, hereafter referred to as KSZ1,14 KSZ2,I5 and BH;16 this nomenclature is explained in the Appendix. We denote the transition probability amplitude of these two respective channels as RepRf2and Rei+lRf;’. ReiMis the radial portion of the transition amplitude between the ground/initial state i and the excited/virtual state e, while RfeM is the corresponding quantity between the excited/virtual state e and the final/resonant state f. The explicit definitions are given by eqs A3a-c. As shown in the Appendix, because of interference between the 2-2-2 and 2-n-2 paths, the line intensity formulas cannot be factored into two independent terms: the first depending only upon R,:Rf,O and the second depending only upon &+lRf;l. However, the line intensity formulas can be decomposed into two terms, one of which is zero at circular polarization. During the factoring process, the radial portions of the transition amplitudes are grouped together into the quantities pi2and p z (see Table 111). For linear polarization, the Q-branch line strength depends upon both p: and p: while the S-and 0-branch line strengths are solely dependent upon p z ; therefore, the path ratio terms, pi2 and p:, can be determined by measurement of the relative intensities of several Q- and 0-branch lines at room temperature. With use of linearly polarized light, several Q-branch and 0-branch line intensities were measured in a single dye laser scan. With the analytical BH formulas in Table I11 and an assumed Boltzmann population distribution of the ground rotational states, a linear least-squares fit of the data was made to calculate the path ratio, p:/pz. If the temperature of the gas is assumed to be 298 K, then the least-squares fit gives p:/pz = 12.5 f 5.0; however, if the temperature of the gas is assumed to be 315 K, then the ratio is slightly lower, p:/c(z = 10.8 f 4.0. As shown earlier in this paper, when measuring the line strengths for a room-temperature sample, the Q branch is best fit by a temperature of 289 K, while the 0 branch is best fit by a temperature of 318 K. The resulting fit for 298 K is shown in Figure 9. The quality of the fit indicates that the virtual state character is constant over the range of J quantum states probed. To check the analytical formulas of Bray and Hochstrasser, the line strengths were calculated by using eq A4 as explained in the Appendix. This confirmed (a) the analytical dependence of the line strength upon rotational state, (b) the change in path ratios pi2 and p: as a function of the polarization of the excitation photons, and (c) the definitions of the path ratio terms, pi2 and k2,in terms of the character of the virtual state, ReiMand RfeM, and vice versa. To confirm the measurement of pt/p: = 12.5 f 5.0,elliptical polarization measurements were performed. At circular polarization, the pi2term for the line strength equals 0, while the p: term increases by 3/2. Therefore, measuring the intensity of the (14) Kummel, A. C.; Sitz, G. 0.;a r e , R. N . J . Chem. Phys. 1986,85, 6874. (15) Kummel, A. C.;Sitz, G. 0.;Zare, R. N. J . Chem. Phys. 1988.88, 6707. (16) Bray, R. G.; Hochstrasser, R. M. Mol. Phys. 1976, 31, 1199.
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 8571
REMPI of N2
1
lo00
100
10
Experimental Line Intensity Figure 9. Calculated and experimental line intensities for the (0,O) Q( 1!&21,23,25,27) and 0(9-15) lines. The calculated intensity resulted
from a linear least-squares fit of the experimental data to the formulas in Table I11 assuming T = 298 K. The linear least-squaresfit gave a path ratio of p t / p : = 12.5 5.0. Linear Circular
.-0
p, Ellipticity Figure 11. REMPI intensity for Q branch (0,O) lines versus elliptical polarization for isotropic samples. (a) Q(18), (b) Q(lO), and (c) Q(0) plus Q(1). The experimental data are depicted with open circles, while the theoretical fits of eq A10 are represented by solid lines.
U
d
b m c
.r(
0 e
c1
I
10
Path Ratio,
100
P2 3
m
Ps
W -
Figure 10. Comparison of the intensity ratios for J = 12 for linear/ circular polarization and Q branch/O branch versus the path ratio, p t / p : . The graph was calculated by using the equations in Table 111.
Q-branch lines as a function of elliptical polarization allows US to measure the path ratio, p:/p:. For the measured value of $ / k 2 = 12.5, the intensity ratio of a Q-branch line probed with linearly versus circularly polarized light should be about 175:l (see Figure 10). To confirm this expected ratio, we tuned the laser to several Q-branch lines and then varied the polarization of the laser light using a quarter-wave plate. The data were modeled by using the equations of KSZ2 (see eq A10) since the formalism of Bray and Hochstrasser as well as the 3-5 expression of eq A4 only allow the light to be circularly or linearly polarized. The polarization studies are shown in Figure 11 along with the associated fits. The KSZ equations are in reasonable agreement with the data. However, there is a systematic deviation near circular polarization, /3 = 45O. In all three cases, the experimental data points fall 50% below the theoretical fit at circular polarization. This may indicate a slight imperfection in the quarter-wave plate or its alignment (seebelow). If we ignore this small deviation, these polarization studies indicate that the path ratio should be p t / p s 2= 0.7. This is because the intensity at linear polarization
h Intensity + L i n e a r : Q B r a n c100 -C-
Circular Q Branch Intensity
8572 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 for J = 1 the ratio of intensity at circular versus linear polarization is slightly higher than at J > 1. This change in polarization ratio is due to the fact that the virtual state cannot be both J = 0 and a II state. Since J" = 0 should have double the population of J" = 1 due to nuclear spin statistics of the IZg+ ground state, we would expect that the blended J = 0 and J = 1 lines should have linear/circular polarization intensity ratio about 50% larger than the ratio at high J . In addition, the intensity at circular polarization should be zero at J = 0, thus possibly changing the shape of the blended J = 0, 1 line. As shown in Figure 1 IC, we observe that the dependence of the REMPI intensity upon elliptical polarization for J = 0, l is the same as the dependence at high J . In addition, we note that the peak shape of the overlapped J = 0, 1 line does not change perceptively with polarization. These discrepancies from the expected results cannot be explained by the excited state having partial ll-state character since the measurements were made at J = 0, 1 where the Z to ll coupling is either zero or nearly zero. In addition, Figure 1la,b show that the measured polarization ratio is the same for J = 10 and 18. Therefore, our data indicate that the excited state has very little ll-state character, and our low intensity ratio for linear/circular polarization cannot be explained by the excited state having partial II-state character. A second possible explanation of the low polarization ratio is imperfections in the quarter-wave plate and its alignment. As noted above, there is a small systematic deviation of the experimental data from the theoretical fit near p = 45' (see Figure 11). The "zero-order" quarter-wave plate is made from two multiple order plates differing slightly in thickness so as to produce a 90' phase delay. To function properly, the two multiple order plates must be parallel to less than 0.5' and must be azimuthally aligned to within 2'. In addition, the assembled quarter-wave plate must be aligned perpendicular with respect to the incident beam to within 0.5', and this alignment must be constant as the quarter-wave plate is rotated. The extreme sensitivity to alignment can be observed experimentally; for example, when our quarter-wave plate is misaligned by 12' with respect to the incident beam, the quarter-wave plate acts like a half-wave plate. Our measurements of the behavior of the quarter-wave plate indicate that the two multiple order plates have the correct azimuthal alignment because the quarter-wave plate, when rotated exactly 90°, can rotate linear polarized light by 90'. To check the parallelism of the quarter-wave plate and its constituent multiple order plates, we can look at the laser back-reflections as the plate is rotated. Our back-reflections precess by about 3-5' even when the wave-plate is carefully aligned with respect to the incident beam. This indicates that nonparallel alignment between the two multiple-order plates or between the quarter-wave plate and the rotation stage may be significant. We can model the imperfections in the quarter-wave plate and its alignment by assuming the phase shift is less than 90'. We denote the deviation of the phase shift from 90' by the symbol 5'; we denote the phase delay as [; therefore, [ = 90' - f'. As shown in Figure 13a, a deviation of the phase shift by 10-20' changes the relative intensity of the Q branch at linear versus circular polarization by a factor of 2-8. By assuming 5' = 17.5', we can fit the polarization data of the Q(10) line to a path ratio of fii2/p'Jz= 12.5 (see Figure 13b). The agreement between the experimental data and the model is just as good as that shown in Figure 1 l a where f' = 0. Therefore, it is quite possible that the low value of the path ratio, pi2/p: = 0.7, determined by the polarization studies may be due to imperfections in the quarter-wave plate and its alignment. A further check of the ratio of the Q branch at linear versus circular polarization was done using a single Fresnel rhomb to generate perfect circularly polarized light. The Q(O),(l) and S(0) and S(1) lines were measured in a molecular beam using pure linearly polarized light and again using pure circularly polarized light. The measured intensities were normalized using the 3/2 ratio of the S branch for circularly versus linearly polarized light (see Table 111). The ratio determined for linear versus circular polarization with this technique was roughly 80:1, still smaller
Hanisco and Kummel
-15
45
105
p, Ellipticity Figure 13. (a) Simulated Q(l0) REMPI intensity versus elliptical po-
larization from a variable-wave plate assuming p ? / =~ 12.5. ~ The phase delay of the variable-wave plate is denoted {, and { = 90° - E', thus 5' is the deviation of the variable-wave plate from a perfect quarter-wave plate. (b) Intensity versus elliptical polarization of Q(l0) assuming $, = 17.50.
-
than the expected ratio of about 1751. The 8O:l polarization ratio corresponds to pi2/p,2 6. The small difference in polarization ratios is probably due to a small birefringence in our vacuum window or a small misalignment of the single Fresnel rhomb. However, the 8O:l polarization ratio determined with the single Fresnel rhomb is greater than the 30:l ratio determined with the quarter-wave plate. This indicates that there are imperfections in the quarter-wave plate that are very significant to this type of measurement.
Discussion The rotational constants determined for the (0,O) band: yo = 98840.59 f 0.12 cm-l, B:,o = 1.9143 f 0.0002 cm-I, and Dil0 = 6.6 X 10" f 0.2 X lo6 cm-I, agree well with Ledbetter's one-photon emission spectroscopy results: uo = 98 840.30 f 0.07 cm-I, B'"=, = 1.9133 f 0.0008 cm-I, and D:,o = 6.2 X lod f 2.9 X 10" cm-'. The rotational constants for the (1,l) and (2.2) bands should be reasonably accurate. The techniques employed in our experiments are not as well suited to the determination of rotational and vibrational parameters as one-photon absorption and Raman spectroscopies, but they are sufficient for our purposes. We can convert the path ratio of pi2/p: = 12.5 measured by comparing the REMPI intensity of the Q and 0 branches into the transition amplitudes using eqs A7a,b. As explained in the Appendix, there are two roots: R,~Rf~/Rei+lRr;'= -1.494 or -0.6283. Only one of these roots is physically real and both are negative. Thus, these experiments would indicate that the transition amplitudes Re:Rf2 and R,:lRf;l are of opposite sign. In fact, for all cases where p?/p: > 1, the transition amplitudes are of opposite sign. This is somewhat counterintuitive, but it may be reasonable because there may be a phase shift in two-photon absorption when the virtual state is of Z versus ll character. The elliptical polarization studies gave a path ratio of p i 2 / p z = 0.7, which corresponds to R e ~ R r ~ / R c i + l R=~-5.5 ; ' or +0.12. The latter root is positive. However, we feel that the polarization studies are not accurate because of problems in aligning the quarter-wave plate, while the measurements of relative Q and 0 branch intensities at linear polarization are quite accurate. Therefore, the path ratio is w:/p: = 12.5 f 5.0, and there must
The Journal of Physical Chemistry, Vol. 95, NO.22, 1991 8573
REMPI of N2
be a phase shift for absorption via virtual states of Z versus If character. Further verification of the path ratio can be made by comparing the REMPI intensities of S,Q, and 0 lines corresponding to the ground-state J versus linear polarization for REMPI of anisotropic samples; these studies will be presented in part 2 of this series of papers. Conclusions The resonantly enhanced 2 1 multiphoton ionization spectrum (REMPI) of the (O,O), (1,1), and (2,2) transitions of the a”lZ8+ X’Z.8B+ states of molecular nitrogen has been recorded a t high resolution, and the rotational and vibrational constants of the excited states have been calculated. The sensitivity of the REMPI techniques is 5 X los molecules/(cm3 quantum state). Our measured spectroscopic constants are as follows (in cm-l): T,= 98938 f 5, w’, = 2167 f 3, B’, = 1.938 f 0.001, a’, = 0.047 f 0.001, YW = 98840.59 f 0.12, BL.0 = 1.9143 f 0.0002, DL-0 = 6.6 X IO4 f 0.2 X lo4, u I I = 98655.3 f 0.2, BLSI = 1.882 f 0.001, uZ2 = 98458.8 f 0.2, and BL12 = 1.820 f 0.001, Measurement of the character of the virtual state was made using two methods: recording the relative two-photon absorption intensity of the 0-and Q-branch lines and recording the two-photon absorption intensity as a function of the elliptical polarization of the laser light. The former technique is most reliable, and for nitrogen, we have determined the virtual state path ratio: p ? / k 2 = 12.5 f 5.0, where the path ratio is a function of the Z versus rI character of the virtual state. Assignment of the 0 and S branches as well as determination of the path ratio is essential for spectroscopic studies of anisotropic samples. This paper has focused upon spectroscopic studies of isotropic samples, and part 2 of this series of papers will focus upon anisotropic samples.
-
+
Appendix The analytical formulas for two-photon rotational population line strengths for symmetric tops in Hund‘s case (a) were derived by Bray and Hochstrasser (hereafter referred to as BH) and for multiplet states in other coupling schemes by Halpern et al.” as well as McClain and Harris.’” In addition, Bain et aI.I9 have made extensive calculations for Hund’s case (a) symmetric tops of two-photon line strengths for the determination of rotational populations and polarizations using linearly or circularly polarized light for the S and 0 branches (dl= f 2 ) . Dixit and McKoyZo (hereafter referred to as DM) have derived several general formulas that are quite useful and lucid for the two-photon excitation of isotropic samples by linearly and circularly polarized light. A more general formalism has been developed by Kummel, Sitz, and Zare (KSZl“ and KSZ2IS). This formalism allows the calculation of population, alignment, and orientation line strengths for Hund’s case (a) symmetric top molecules probed by elliptically polarized light. In KSZl and KSZZ explicit line strength values were computed only for Z-rI transitions, and no discussion was included concerning the effects of the character of the virtual state upon population, alignment, and orientation line strengths. In this paper we use three complementary methods of calculating the line-strength formulas: (I) The analytical BH formulas. We use the notation and normalization of the BH formulas shown in Table IX of KSZl simplified to the explicit case of a 2-2 transition. These analytical BH formulas are shown in Table 111 of this paper. They allow us to calculate only the population line strengths at linear and circular polarization. (11) The 3-J equations. The 3-J equation derived below may be employed to calculate the population line strengths with linearly and circularly polarized light. This permits a numerical verification of the analytical BH formulas and the relationships of the transition probability amplitudes ReiMand RIeMwith the path ratio w?/p,Z. (17) Halpern, J. B.; Zacharias, H.; Wallenstein, R. J . Mol. Specrrosc. 1980, 79, 1. (18) McClain, W. M.; Harris, R. A. In Excited Stores; Lim,E. C., Ed.; Academic: New York, 1977; Vol. 3, pp 1-56. (19) Bain, A. J.; McCaffery, A. J.; Proctor, M. J.; Whitaker, B. J. Chem. Phys. Lett. 1984, 110, 663. Bain, A. J.; McCaffery, A. J. J. Chem. Phys. 1985,83, 2621, 2632, 2641. (20) Dixit, S.N.; McCoy, V. J . Chem. Phys. 1985, 82, 3546.
In addition, the 3-5 equation allows us to examine the explicit dependence of the population line strength upon the magnetic quantum number, M. (111) The numerical KSZ formulas. This formalism allows the calculation of the line strengths for elliptically polarized light of arbitrary phase delay; this is important when fitting the data for the REMPI intensity versus elliptical polarization. The formulas used are the same as those in KSZZ for case 111 geometry, excitation along t h e y axis; here we simply review the notation. 3-5 Equation. We can model the absorption of light into a two-photon resonance using eq AI of KSZl. When the laser is tuned to a two-photon resonance
Ye
hv + V,/2)ll2) (AI) We note that the angular brackets indicate an averaging over all Mi states (Mi = magnetic quantum number of the ground/initial state). The subscripts on the quantum numbers indicate the electronic state: i, e, and f refer respectively to the initial/ground, excited/virtual, and final/resonant states. This results in our intensity expression differing from that of BH and many other authors by a factor of (24 + l), but ours is a more convenient formalism for modeling absorption of light by anisotropic samples. We can solve for the matrix elements using the same derivation employed in eq A25 of KSZl. However, in this paper we allow the laser light to be either linearly or circularly polarized. Using the techniques of DM eq 6-9 and allowing the laser light to be either linearly or circular polarized (r,,Je,Me,4l~,.r2lJii,Mi,~i)/[E,j-
(v2J~A21r~”lvlJ1MAl) = (-1)’-~~A3R:“~Al’[(~)(2/2 + 1) x
where rzI(A2-A1)
=
J dRx,(R)
( * Y ~ ) ( b i I ~ RY) ~ l ” ( j ~ ~ i ) * ~ l ( e ) ( ~) xU,(R) ri~;R)
(A3a) &(&-Ai)
= xrei(Ae-Ai)/Dei
(A3b)
7.
&,(Ark) = rf,(ArAJ (A3c) The Dei represent the energy denominators of eq A1 and Y and x are the vibrational and electronic state quantum numbers. Equation A2 differs from eq 9 of DM by only the phase convention. We use the KSZl convention to ensure that our three methods of calculating the line strengths are consistent. The RdM are the total radial parts of the transition dipole moments; note that they are not probabilities but rather probability amplitudes. There are three possible polarizations for which eq A2 is valid, and the polarization determines the value of the quantum number A: (i) For linearly polarized light propagating along t h e y axis, X = 0. (ii) For right circularly polarized light propagating along the z axis, X = 1 . (iii) For left circularly polarized light propagating along the z axis, X = -1. This is somewhat clumsy for modeling excitation for anisotropic samples because when comparing excitation by linearly versus circularly polarized light, you are also implicitly switching the propagation direction of the laser. However, for modeling the isotropic samples in this paper, the assumed switching of propagation axes can be ignored and thus presents no difficulty. Substituting eq A2 into eq AI, we derive
8574 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991
Note that the radial terms are inside the square in eq A4, and hence there is interference between absorption pathways via different intermediate states. This is unfortunate because it precludes us from factoring eq A4 into two separate terms, one of which describes absorption via a A A = 0 virtual state and a term that describes absorption via a AA = f l virtual state. We used eq A4 to check the BH analytical formulas shown in Table I11 for 2-L: transitions. The calculations were performed on a Macintosh 11 computer with a Motorola 68881 math coprocessor chip using Absoft MacFortran. The program took a few seconds of computation time. The routines for calculating the 3-5 symbols were checked by using the standard summation/normalization formulas for 3-3 symbols. The algorithm was assumed to be correct because of the agreement with the BH analytical formulas as well as the KSZ method of calculating the line strengths. Because the definition of a Il state is that the absolute value of the projection of J on the internuclear axis is unity, there are two paths from a ground 2 state to virtual II state. When calculating the path ratio from the radial terms, we set R,?'Rf;' = Re{lRfe+'.This is completely arbitrary, and we could just set the latter term equal to zero, which would simply change the reported values of the virtual state character by a factor of 2. With our chosen convention, the relationships are as follows:
Hanisco and Kummel substituted into either the 3-J formula of eq A4 or the KSZ expressions. Instead of reporting the absolute values of RdoRf,O and &+IRf;', we can just calculate the ratio RdoRl:/Rd+'Rf;'; unfortunately, for a given ratio of h2/p,z, there are still two p i b l e values of Rc?Rf,O/Rei+'Rf;'and only one of these is physically correct. The KSZ Equations. The KSZ method of calculating the line strength decomposes the intensity into a sum over real polarization moments, A!$, and their respective real line strengths, fiqIin the Hertel-Stole notation:
+
I = C det n(Ji)C[Pq+'k'(Ji,Ai,Jf,Af;52)A~~~~(Ji) kq
P,'k'(JilAi,Jf,Af;52)A,'"(Ji)] (A8) where k = 0-4 and q = 0-4. K S Z 2 contains all the formulas to calculate the polarization moments. The polarization moments, AB(Ji) are calculated by fitting eq A8 to the measured variation of the intensity with the polarization of the excitation light. In this paper, we are interested only in modeling the absorption of polarized light by isotropic samples. For isotropic samples all polarization moments are equal to zero except AfPL(Ji)= 1.0. For an isotropic sample
I = C det n(Ji)@J.(Ji,Ai,Jf,Af,52) (A91 In eq A9, 52 represents the detection geometry including the polarization. To calculate one normally must assume a pair of values for R,tRf: and R,i+'Rf;l. As stated previously, one cannot decompose p6pl into two independent terms one of which depend solely on RaoRl,O and the other depending solely on Rei+'Rf;'. To use the KSZ equations to fit the variation of intensity with elliptical polarization, we can combine two equations of the form (A9): I ( J i J i , J f J f )= C det n(Ji)[pFm(R,?Rf: = Rei+'Rf;l = p,2@J.(Rc?Rf,O = y3t Rei+'Rf[' = f/6)] (A10)
m,
v3,
or alternatively
Using eqs A6a,b, we checked the different terms in the BH analytical formulas. Next, using eqs A5a,b, for any arbitrary real values of Re?Rf2 and Rci+'Rf;l,we get agreement between the 3-J formulas of eq A4 and the BH analytical formulas. One would also like to check the formulas in the reverse direction in a twestep process. First, assume values of the path ratio, pi2and p:, then directly calculate the BH equations. Second, calculate the character of the virtual state, RdoRf: and &+'Rf;', then use these values in the 3-J equations. The problem with this technique is that when taking square roots of eqs A5a and A5b, there are four possible pairs of Re?Rl,O and R,i+'Rf;':
Rc?Rf,O = (pi
+ rs)/3
Rci+'RfL' = ( 0 . 5 ~- ~p i ) / 3
(A7a) (A7b)
where pi = f ( ~ ? ) ' and / ~ ps = * ( P : ) ' / ~ . Though only one pair of the roots represents physical reality, all four pairs give identical calculated line strengths when
m
-v3)+
In eq A10 we have the calculations in eqs A6a and A6b to create line strength terms that depend only upon either p t or p:. A simple linear least-squares fit of eq A10 to the recorded variation of the experimental REMPI signal with elliptical polarization allows us to determine the path ratio p?/p,Z. The advantage of the KSZ method is that we can readily calculate the moments of the line strength for arbitrary polarization not just for linear and circular polarization. In this paper we use Table I11 of KSZ2 with case I11 geometry to model polarization with a quarter-wave plate or even a variable-wave plate. Calculating the variation of with ellipticity for a given pair of k 0 R f : and &+'Rr[l takes about 30 s on a Macintosh I1 computer and with a Motorola 68881 math coprocessor chip using Absoft MacFortran.
flz
m
Acknowledgment. We thank R. Field, V. McKoy, D. A. V. Kliner, D. Chandler, and D. Rakestraw for useful discussions. We also thank R. Huang for much technical support. The project was supported by the National Science Foundation under Grant No. CHE 88-13805. T.F.H. gratefully acknowledges support from a National Science Foundation Graduate Fellowship.
