THEOXIDATION OF SODIUM,
POTASSIUM, AND
2483
CESIUM
Richard Rosenfeld and the vapor pressure osmometry was performed by Mr. Michael Sinensky. This study was supported in part by research grants from the University of Connecticut Research Foundation and the Public Health Service (National Heart Institute H E 8897) and a Public Health Service Train-
ing Grant (GM-317) from the National Institute of General Medical Sciences. The computational part of this work was carried out in the computer center of the University of Connecticut, which is supported in part by Grant GP-1819 of the National Science Foundation.
The Oxidation of Sodium, Potassium, and Cesium in Flames
by R. Carabetta Space Sciences Laboratory, General Electric Company, Philadelphia, Pennsylvania 19101
and W. E. Kaskan' Chemistry Department, State University of N e w York at Binghamton, Binghamton, N e w York 18901 (Received December 26, 1967)
The rates of oxidation of Na, K, and Cs have been measured in lean Hz-02-N~ flames in the pressure and temperature ranges 100-1520 torr and 142@-160O0K. In all experiments, the reactions proceed via the mechanism ka alkali metal O2 31 +alkali superoxide M. The rate constants, ICZ, have been measured; average and 2.1 X lowasan6 particle2 sec-l for sodium, potassium, and ce1.02 x values are 0.82 x sium, respectively.
+
+ +
I. Introduction The bulk of existing information on the behavior of alkali metals in fuel-rich flames is derived primarily from studies in which the metals were introduced into rich H2-02-N2 flames burned at atmospheric pressure.2-6 On the basis of this work, it appears that the metals are involved in fast, equilibrated reactions of the type A Hz0 AOH H (1)
+
+
where A is an alkali metal. I n fuel-lean H2-02-N2 ffames, however, the situation seems to be somewhat different. Maskan's study' of Na and K in such flames indicates that the balanced reaction 1 does not apply but rather that the reaction of alkali metal with flame gases involves the formation of an alkali metal oxide. The conclusions drawn in this work were that the oxide is possibly AO, but most probably A02, which was formed via the forward reaction in the process kz
A
+ + M 7ztAOz + M 0 2
k -2
(2)
All of the work in ref 7 was performed at a single pressure (1 atm) so that the tennolecular nature of eq 2 could not be proven but was assumed. McEwan and Phillips* have also reported that the kinetics for the re-
action of both sodium and potassium are explicable in terms of process 2. The rate constants for the formation of the superoxides, k ~ were , in good agreement, both for Na and K, with those reported by Kaskan.' I n addition, they also reported measurements of the equilibrium constants for eq 2, when A is Na, at a number of temperatures, and from these measurements, they (65 f: deduced the bond-dissociation energy, DON~-O$ 3 kcal mol-'). Because the work on process 2 was carried out at atmospheric pressure, essentially a t a constant (M), it was felt that the mechanism represented could be put on firmer ground if the termolecularity of the forward reaction could be demonstrated. I n the present study, (1) Consultant to the General Electric Space Sciences Laboratory, General Electric Co., Philadelphia, Pa. 19101. (2) C. G. James and T. M. Sugden, Nature, 171, 428 (1953). (3) C. G. James and T. M. Sugden, Proc. R o y . Soc., A227, 312 (1965). (4) E. M. Bulewioz, C.G. James, and T. M. Sugden, ibid., A235, 896 (1956). (5) D. E. Jensen and P. J. Padley, Trans. Faraday SOC.,62, 2132 (1966). (6) D. E. Jensen and P. J. Padley, ibid., 62, 2140 (1966). (7) W. E. Kaskan, Tenth Symposium (International) on Combustion, Cambridge, England, The Combustion Institute, Pittsburgh, Pa., 1965, p 41. (8) M. J. McEwan and L. E. Phillips, Trans. Faraday sbc., 62, 1717 (1966).
Volume 72, N u m b e r 7 July 1068
2484
R. CARABETTA AND W. E. KASKAN
the rates of reaction of sodium, potassium, and cesium have been measured in a number of lean H2-02-N2 flames burned at pressures of from 100 to 1520 torr. It has been found that the reaction rate of the metal is first order in (A), in (OZ), and also in (31) and that the reaction rate constants, k2, for both Na and K, are in good agreement with those reported in the two previous studies. I n addition, kz, for Cs, has been determined.
-10
VAC. PUMP .
,LIGHT SOURCE
11. Experimental Section The variable-pressure burner system is illustrated schematically jn Figure 1. The burner was a watercooled porous-metal type consisting of an inner circular part, 7.5 cm in diameter, surrounded by an annular part about 2.5 cm wide, each having a separate gas-inlet system. The burner, the characteristics of which have been described elsewhere,Bwas designed to provide a one-dimensional flow of unburned gas which would, in turn, provide a flat flame of diameter equal to that of the burner. The outer burner was used to support a shield flame, identical with that supported by the inner burner, but without alkali metal seeding. Metal was added to the inner flame by passing part of the gas stream to the inner burner through an atomizer bottle containing a solution of alkali metal salt, usually the chloride. Larger particles from the atomizer were separated from the smaller in a separate settling flask placed in the line between the atomizer and the burner. A wire-mesh screen placed above the burner head served to stabilize the column of hot flame gases so that after exiting from the reaction zone they would continue to flow perpendicular to the burner surface. With this arrangement, the vertical axis is also a time axis. The burner head was mounted on a pipe 5 cm in diameter which entered the lower portion of the chamber through an O-ring seal and both pipe and burner could be moved (in the vertical direction) with respect to the fixed axis of the optical system by means of a micrometer mechanism. Pressure inside the chamber was regulated by simultaneously admitting nitrogen into the system through a number of bleed valves and by adjusting a large gate valve leading to a vacuum pump. The gas flows to the burners were metered with calibrated critical flow orifices. Figure 1 also shows a schematic of the optical system. Light from the source S was passed over the burner, parallel to its surface, via the mirror M1 and was refocused on the monochromator slit with the mirror M2. A tungsten strip filament lamp (filament dimensions 1 X 13 mm) served as a background source for alkali metaI resonance-he absorption measurements. To measure OH concentrations and for OH rotational temperature determinations, the tungsten source was replaced by a water-discharge lamp.'O For a typical flame, the spatial resolution in the vertical direction afforded by this arrangement of optics was about 1 mm. T h e Journal of Physical Chemistry
FOR SEE0 SOLUTION
Figure 1. Schematic of burner and optical system.
Temperature profiles in the flames were measured with a silica-coated 0.0025-cm diameter Pt-Pt-lO% Rh thermocouple and were corrected for radiation losses.* Occasionally, the thermocouple measurements were supplemented by measurement of ground-state OH rotational temperature. Agreement to within 50°K was always obtained. Determination of the alkali metal atom concentrations in the A ame gases from resonance-light-absorption measurements required the use of a set of calibration curves, one for each metal. Fuel-rich H2-02-N2 flames having equivalence ratios of 1.3 were chosen as calibration flames. In these flames, the alkali metal concentration did not change perceptibly with distance from the burner surface. Furthermore, based on the data for hydroxide formation provided by the work of Jensen and Padley,j the fraction (AOH)/(A) for Cs (the most favored of the three hydroxides) at the points of measurement in each of the calibration flames was computed to be less than could be measured with our apparatus, ie., less than 0.02. Calibrations were carried out in a plane in each flame a t which the temperature was 15OO"II; for all measurements, the tungsten background source was operated a t a temperature of about 2 5 O O O I i so that no emission correction for flame background was required. The following procedure was used to establish the calibration curves. A concentrated solution containing equimolar quantities of NaCl, KC1, and CsCl was made up and diluted so as to produce a number of solutions, each less concentrated than the next by a factor of 2. Each of these solutions was used, in turn, to seed the calibration flame, and for each, the absorbance (Io- I ) / I owas measured, where Iois the intensity of the background lamp and I is the intensity of the light transmitted through the gas at the wavelength of peak absorption. This was done for both the DI (9) W. E. Kaskan, Sixth Symposium (International) on Combustion, Yale University, Williams and Wilkins Go., Baltimore, Md.,1953, p 68. (10) W. E. Kaskan, Combust. Flame, 2, 229 (1958).
2485
THEOXIDATION OF SODIUM, POTASSIUM, AND CESIUM and D~ lines of Na and K and the D~ line of Cs. The D~ line for Cs could not, however, be conveniently measured because it lies just to the red side of the long wavelength cutoff of the monochromator. The absolute sodium atom concentration produced by each solution in each flame was deduced from the Van der Held curve of growth,l’ the a parameters suggested by the work of Sobolev, et a1.,12 the known f number of this line,’* and the experimentally measured area under the absorption line. From this a calibration curve [log (absorbance) os. log (concentration) ] for sodium was established for the series of mixtures. Since the conditions of our experiments were such that a t the planes of measurement equal quantities of the three metals were expected to be present, calibration curves based on the absorbance of the D~ lines of Cs and K were obtained by plotting the measured absorbance of each against the absolute sodium concentration appropriate to the equimolar solution employed. Using this procedure, three smooth curves, one for each metal, were obtained at each pressure. Two checks were then carried out. First the absorbances of the Na DI and K ~1 lines for each solution were plotted a t half the concentration appropriate to the corresponding D~ lines (this, of course, was not done for Cs); for each metal, these points fell on the smooth curve defined by the absorbance measurements on the D~ line, as would be expected. The second check made use of the fact that the absorption in the optically thin regime13
(3) where L is the path length and N is the number density of the absorbing species (m, c, e, and f have their usual meanings). Since the left-hand side of eq 3 can be experimentally measured for the three species, it is possible to solve for N . Using the smallest measurable values of absorbance for each species and eq 3, it was found that the absolute concentrations of the three metals were equal-as would be expected if virtually all of the alkali metal were converted to free metal in the calibration flame. To measure the oxidation rates of the alkali metals, a solution containing equimolar quantities of the metals was used to introduce the alkalies into a number of H2-02-NZ flames having different equivalence ratios. Flames were burned a t 5 pressures: 150, 250, 380, 760, and 1520 torr. I n each experiment, the temperature profile through the flame was recorded, as was the alkali metal concentration profile. I n seeding flames, solutions producing initial concentrations of ca. 1Olo to loll alkali atoms/cm3 in the flame gases were used; thus the measurements of alkali metal profiles almost always involved the use of the nearly linear (and more sensitive) portions of the calibration curves.
E
100 P
20
I 0
1
2
3
4
5
6
1
t(MILLISEC0NDSI
Figure 2. First-order reaction of Na, K, and Cs in flame 4: 0, Na; 0,K ; A, Cs.
:[ 20
Figure 3. First-order reaction of Na, K, and Cs in flame 5: 0, Na; 0, K ; A, Cs.
111. Results and Discussion The flames employed to study the oxidation are listed in Table I. The gas compositions shown are nominal in that they were simply computed from the flame stoichiometry, assuming all Hz was burned to H20. I n each experiment, the kinetic measurements were limited to the burned-gas region, downstream of the reaction zone, where the temperature changes little with distance. I n Table I the temperature shown (T) designates the average temperature in this region; similarly, Pcorresponds to the mean hot gas velocity in this region. Some general statements can be made about the observations. I n all cases, alkali atom concentrations are a t a maximum in or near the reaction zone. I n rich flames these concentrations remain constant downstream of the reaction zone, but in lean flames they (11) E. F. M. Van der Held, 2.Phys., 70, 508 (1931). (12) N. N. Sobolev, E. M. Mezhericher, and G. M. Rodin, Zh. Eksperim. Teor. F k , 21, 350 (1951). (13) A. C. G. Mitchell and W. Zemansky, “Resonance Radiation and Excited Atoms,” Cambridge University Press, London, 1961, Chapter 3.
Volume 78, Number 7 July 1968
R. CARABETTA AND W. E. KASKAN
2486
decay downstream. At each pressure, these decays were found to be first order in the concentrations alkali, 0 2 , and M, and were found to be independent of the nature of the salt used (nitrates gave the same results as chlorides). Some typical data are shown in Figures 2 and 3. Since the decay was found to be first order with respect to the metal, the metal concentrations shown are given in arbitrary units in order to facilitate the presentation of the data. In each of these plots, the first measurement (corresponding to that at t = 0) was made just downstream of the reaction zone where the temperature is within about 25'K of its peak value. Assuming that the decay of alkali metals proceeds via the forward process 2, then
-d(A) - k2(A)(02)(M)
(4)
dt For constant (M) and (02)
-d[ln (A)1 = kz(O2)(M) dt 0
The measured values for -d[ln (A)]/ldt are contained in columns 9-11 in Table I. Columns 12-14 of this table contain the deduced values for k2 as obtained from eq 5, uncorrected for the effects of diffusion. Since the reaction rate is first order with respect to the metal, the following modified form of the diffusion equation was applied to our datal4
.+;i
D -d2(A) dX
(A) P d- kz'(A)(02)(M) dX
=0
(6)
where D is the diffusion coefficient of A, X is the distance in the direction of gas flow in the flame, and k2' is the true diffusion corrected third-order rate constant for the forward reaction in process 2. Integration of eq 6 for the boundary conditions, (A) = 0 at X = and (A) = (A)oat X = X o = 0 yields
~
Values for k' are given in the last three columns of Table I. I n applying the diffusion correction (eq 7) to our results, binary diffusion coefficients for the diffusion of the metal into N2 were computed using the simple kinetic-theory approach16 ( D N ~ - - (STP) N~ = 0.191 cm2 (STP) = 0.134 cm2 sec-l, Dc+-N%(STP) sec-I, DK-N~ = 0.098 om2 sec-l). As a check, k2 was measured for each of the metals in a number of flames differing primarily in p. Flames 8 and 9 in Table I are two such flames. The fact that the kz values for the three metals, respectively, are comparable for both flames (14) R. Friedman and J. Cyphers, J. Chem. Phys., 23, 1875 (1955). (15) W. Jost, "Diffusion in Solids, Liquids and Gases," Academic Press Inc., New York, N. Y.,1952, Chapter X.
2487
THEOXIDATION OF SODIUM, POTASSIUM, AND CESIUM indicates that diffusion effects are relatively unimportant over the range of conditions in our experiments. This result is consistent with what would be predicted using the computed D values. The flames chosen in this study were in fact designed so as to minimize diffusion effects by experimentally minimizing the first term in the brackets of the right-hand side of eq 7. The k2' values for Na and for M listed in Table I are in excellent agreement with those reported by Raskan' and are about a factor of 2 less than those reported by McEwan and Phillips.8 I n this work the reproducibility of a rate constant from a given experiment was about 10%. The experimental scatter in the results shown in Table I may well reflect differences in thirdbody efficiencies (see below). The dependence of the rate of the forward reaction in process 2 on a third body is shown in Figure 4. The fact that the data fall on a fairly good straight line having a slope of 1 shows that the reaction is first order in (M). I n fuel-rich H2-02-N2 flames, the behavior of alkali metals has been interpreted in terms of process 1, involving alkali hydroxides. Kaskan has pointed out7 then that since eq 1 is postulated to be eq~ilibrated,d-~
where K1 is the equilibrium constant for eq 1. Thus since the species H2,OH, O,OZ,and H have been shown viz. to be involved in fast equilibrated reactions4~16
+ OH J_ HzO + H H+ OH + 0 0 + Hz OH + €I
Hz
0 2
(9) (10) (11)
then
so that
I n Kaskan's work, it was possible to check relation 13, since both (OH) and the ratio (A)/(AOH) could be determined experimentally in those planes in the flame where the temperature and, therefore, the equilibrium constants in eq 13 changed but very slightly. The result was that (A)/(AOH) for Na and for K varied more nearly as the second power of (OH) than as the third power predicted by eq 13. It was pointed out7 that this result is formally explicable in terms of the process
A
+ OH Z
A O
+H
(14)
and as such, this possibility, rather than AOz formation, could not be ruled out.
500
400 300
200
1,.
1.0
2
10
20
100
200
1000
Figure 4. The pressure dependence of the reaction rate of A on (M): 0, Na; 0, K ; A, Cs.
I n the work carried out by Kaskan, there was no large variation in (HZO). If the fair correlation of (A)/(AOH) with (OH)2were fortuitous, this might more clearly be shown from measurements in CO-02-K2 flames containing a little added H2, since in such flames (H20) is very much smaller than in H2-02-N2 flames. Such experiments were carried out and the data were treated in the following manner. Using the procedure previously de~cribed,~ the quantities (OH) and (A)/ (AOH) were first measured, as a check, in some atmospheric pressure H2-02-Nz flames. I n computing (OH), the more recently reported valueI7 for the oscillator strength was used; the line strengths used are those given by Dieke and Crosswhite18as corrected by Learner.lg The concentration of AOH at a given location in the flame was deduced by difference from measurements of (A) and (A)o. The quantity (A)owas deduced from measurements in similar rich flames having the same alkali salt delivery rate but in which no metal oxide or hydroxide was formed. For flames 18 and 11 in Table I, the results mere found to be in agreement with Kaskan's, in that (A)/(AOH) seemed to for both sodium and potascorrelate with (OH)1*8-2.0 sium. I n Figure 5 are shown the results obtained in an atmospheric pressure flame of unburned composition : Hz:CO:02:N* = 0.033:0.160:0.161:0.645; T = 1665°K. Clearly, in these flames, (A)/(AOH) does not correlate with any power of (OH). On this basis, it would seem reasonable to conclude that the observed correlation of this ratio with (OH)zin Hz-0,-N2 flames is fortuitous and that the equilibrated reaction 1 is not operative. The data from this flame were also (16) W. E. Kaskan, Combust. Flame, 3, 49 (1959). (17) R. G. Bennett and F. W. Dalby, J. Cherr~.Phys., 40, 1414 (1964). (18) G. Dieke and H. M. Crosswhite, J . Quant. Spectrosc. Radiat. Transfer., 2, 97 (1962). (19) R. C. M. Learner, Proc. Roy. Soc., A269, 311 (1962).
V o l u m e Y2, N u m b e r 7 J u l g 1968
2488
R. CARABETTA AND W. E. KASKAN produced. Indeed, if one assumes that the dissociation energy of NaOz is that reported by McEwan and Phillips,* then (based on JANAF data) the following exothermic and, therefore, conceivably, fast reactions could proceed
t
1
+ 0 +KaO + Oz XaOz + OH +NaOH + O2 NaOz + H +NaOH + 0 NaOz
a’ol
0.01 -
1
1.0
2
4
6 8 10
20
40 60 100
OH/lO1’
Figure 5. Correlation of (A)/[(A)o - (A)] us. (OH) in H2-CO-02-N2 flame: 0, Na; 0, K ; A, Cs.
interpreted in terms of the superoxide formation; the third-order rate constants obtained for the three metals were a factor of 2 lower than the average for each listed in Table I. Although the difference could easily be explained in terms of the differences in efficiencies of HzO and COz as third bodies, this question was not pursued further (previous work7 has indeed shown HzO to be more effective than Nz for the forward reaction 2). The data from all of our experiments indicate that the mechanism responsible for the decay of alkali metals in lean flames is reaction 2 operating in the forward direction. It is interesting to note that in the atmospheric flame work reported by 1lcEwan and Phillips8 and K a ~ k a n it , ~ was found that the compoundforming reaction did not go to completion; instead, following the initial first-order decay of (A), there was an apparent fall-off of the rate with time. I n the work in ref 8, the fall-off is attributed to the reverse reaction in process 2, which makes its effect felt as the superoxide concentration increases. In the present work, the fall-off of the over-all oxidation rate was also observed in flames burned a t pressures greater than 360 torr but not in the lower pressure flames. This result is consistent with the idea (but by no means a positive indication) that the reverse reaction is a function of (M). I n our experiments, the 0 2 concentration was adjusted in such a manner that the forward rate of reaction 2, kz(Oz)(AI)(A), did not change much with pressure; thus a reverse reaction having a rate proportional to (a!)X (concentration of fihal oxidation products) would be expected t o become less favored relative to the forward reaction as the pressure and, therefore, (11) is reduced. It is difficult to believe that MO, is the final species The Journal of Physical Chemistry
(15) (16) (17)
At equilibrium, in the highest temperature flame used ( T ,2030°K; by McEwan and Phillips to measure DN*-O~ unburnt gas composition, HZ:0z:N2 = 1.5:l: 1.3; pressure, 1 atm) the concentrations of 0, OH, and H are, respectively, 8.85 X 1014,1.12 X 10l6,and 1.58 X 1014particles/cma. Since the dissociation reaction requires a 65 kcal/mol activation energy, then a t equilibrium in the flame in question, the concentration of H, 0, and OH are certainly high enough so that if reactions 15-17 are operative, the final products would probably be NaO and/or NaOH rather than NaOz. In an effort to help clarify this issue, some ancillary experiments involving the use of a mass spectrometer were carried out. Available thermodynamic dataz0suggest that of the three superoxides, NaOz KOz, and CsOz, the last is the most stable (we note here that the heats of formation of these species are available only for the solid state and that their heats of sublimation are not known). Since the flame results show that CsOz is the most readily formed of the three in the gas phase, it seemed worthwhile to attempt to detect the presence of this species in the gas phase directly. To this end, crystalline cesium superoxide was placed into a platinum-lined Knudsen cell which would be heated by radiation and electron bombardment from a tungsten filament. The cell was also provided with an inlet system so that a pressure of about 1 torr of 0, could be maintained over the superoxide. The temperature of the cell and its contents was measured with a thermocouple. After heating the cell, the gaseous products were allowed to effuse from the cell into the ion source of a time-offlight mass spectrometer (Bendix Model 12). I n these experiments the cell was heated at temperatures of from 600 to 1300°K in approximately 100” steps. I n each experiment, the electron-beam energy was varied from 0 to 70 eV. The cesium species, Cs+, CSZ+,CszO+, and CszOz+ were observed. However, CsOz+could not be identified even at low energies where fragmentation is minimal. On the basis of the foregoing, it would seem that although the forward reaction 2 is the rate-determining step in the oxidation of alkali metal atoms in lean flames, the superoxides may not be the final product. (20) J. M . Mellor, Supplement to “Comprehensive Treatise on Inorganic and Theoretical Chemistry,” Vol. 11,Suppl 111, Part 11, John Wiley and Sons, Ino., New York, N. Y., 1961.
2489
THEDYNAMICS OF ANHARMONIC OSCILLATORS Finally, it must be stated that the results of these experiments are not in disagreement with the work in ref 2-6 in which process 1 has been postulated as operative. I n the present study, the work was not only carried out in lean rather than rich flames but was carried out also a t lower temperatures. I n fact, the existenoe of a process such as eq 1 is demanded even in lean flames by the fact that the reverse of this reaction is most probably the type reaction responsible for the reduction of alkali metal compounds to free alkali atoms in the reaction zone. Thus although the reaction of alkali metals at relatively high 0 2 concentrations and low temperatures has been shown to take place via AOz as a kinetically important intermediate, it is quite possible that an increase in temperature and/or a decrease in (02)(as with HO2 in hydrocarbon oxidation) leads to conditions where process 1 is dominant. Based on the data of McEwan and Phillips,* it would appear that in a lean flame the change in kinetics would occur only a t temperatures in excess of about 2000°K.
IV. Conclusions The rates of reaction of sodium, potassium, and ce-
sium in lean H2-02-N2flames, burned at pressures of from 100 to 1520 torr, scale with pressure in a manner such that the rate-determining step is shown to be
The rate constants computed for the reaction on this basis, for sodium and for potassium, are in excellent agreement with those measured at atmospheric pressure by Kaskan’ and McEwan and Phillips.s It is questioned, however, whether the species AOz is the final product of the reaction. Acknowledgments. The authors are indebted to Mr. R. Everett, mho carried out the experimental work, and to Drs. R. Porter and AI. Linevsky of the Space Sciences Laboratory, who offered helpful discussions and comments throughout the course of this work. Special thanks are extended to Dr. P. Zavitsanos of the Space Sciences Laboratory both for the use of his mass spectrometer and also for the technical assistance rendered. This work received financial support from ARPA under Contract No. DA31-124-ARO-D-214 and from the Air Force under Contract No. AFO4(694)-916.
Application of the WKB Method to the Dynamics of Anharmonic Oscillators* by Robert Dubrow, Douglas Hatzenbuhler, William Marx, Eva Zahorian, and David J. Wilson Department oj Chemistry, University o j Rochester, Rochester, N e w York 14627
(Received December 26, 1067)
The dynamical behavior of the Morse oscillator and the Fues oscillator is investigated by the WKB method. The results are compared with those obtained by using the exact-energy eigenfunctions (Morse oscillator) and with those obtained by using the linear-variation method with harmonic-oscillatorbasis functions (Fues oscillator). The WKB method yields results superior to those of the linear-variation method with harmonicoscillator basis functions, and the results are in excellent agreement with those obtained using the exact-energy eigenfunctions as the basis set.
Introduction The quantum dynamics of several anharmonic oscillators have been explored in earlier papers in this series.2-6 The work of both Endres and Smyser indicated that the linear-variation method with harmonicoscillator basis functions is not well adapted to dealing with wave packets having high vibrational energies. Neither the energies nor the approximate wave functions obtained by this method are of adequate accuracy to provide good results unless a very large number of functions are used. The form of the envelope to plots of ( ~ ( t ) vs. ) t (r is the coordinate of the oscillator) de-
pends upon the second differences of the energies. It is therefore necessary to have energies of quite high accuracy in order to obtain envelopes of even modest accuracy. The matrix elements of the coordinate and (1) This work was supported by the National Science Foundation. (2) E. Alterman, C. Tahk, and D. J. Wilson, J . Chem. Phys., 44, 461 (1966). See this paper for earlier references. (3) R. Baetzold, C. Tahk, and D. J. Wilson, ibid., 45, 4209 (1966). (4) P. F. Endres and D. J. Wilson, {bid., 46, 425 (1967). ( 5 ) W. Smyser, Doctoral Dissertation, University of Rochester, Rochester, N. Y . , 1967. (6) P. F. Endres, J . Chem. Phys., 47, 798 (1967).
Volume 72, Number 7
J u l y 1968
