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 ) t (r is the coordinate of the oscillator) deof ( ~ ( t ) vs.
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
R. DUBROW, D. HATZENBUHLER, W. MARX,E. ZAHORIAN,AND D. J. WILSON
2490
Table I: Exact and WKB Energies for Various One-Dimensional Oscillators Potential
WKB energy
Exaot energy
2D[$+$] ,-
D(1
- exp[--P(r
-
r
(n
+
l/d
pi? 2r
- - (n
+
m,ra Same as WKB energy
vo
(g - $
X vo cot2 Ta
+
~ )=. 8pV0a2/fi2;B = f i z / a A = 2Dpre2/fi2. b S, is tabulated by Watson10 and is the nth root of J i / , ( 2 ~ ’ / ~ / 3 ~ /J~i )/ , ( 2 ~ ~ / ~ / 3 ~ / A 2 ~ ~ 2d .A = 8pVoa2/Ti2;B = (2fi/a)(2V0/p)1/2. A = 2pVoa2/n2fi2;B = n2fiz/2pa2.
the expansion coefficients of the initial wave packet are not subject to a variational principle (as are the energies) ; hence these matrix elements may introduce significant errors in ( r ( t ) ) even when the energies are calculated rather accurately by the linear-variation method. The WKB method7r8 yields extremely good values for the energy levels of a variety of anharmonic oscillators. We compared the exact and the WKB energies for the harmonic oscillator and for seven anharmonic oscillators; we use well depth, range, and mass parameters appropriate for hydrogen. (Most of these formulas may be found in the problem book by Go1’dmnn and his collaborators.s) See Table I.l0 The agreement is extremely good in all cases and, as one would expect, improves at higher energies, in contrast to the linear-variation method. These results prompted US to investigate the use of WKB eigenfunctions for calculating the matrix elemerits and expansion coefficients needed to investigate the dynamical behavior of
(4t)
>*
The Fues potential (see the first expression in column 1 of Table I) yields a coulombic force a t large distances and hence is useful for molecules which dissociate into ions. The Fues oscillator can be solved quantum mechanically,“ and the exact and WKB energies are in excellent agreement for parameters chosen to simulate real molecules. The Fues potential is a slowly increasing potential for large r; hence the Fues potential provides one with a rather severe test for the linearvariation method with harmonic-oscillator basis functions. The Journal of Physical Chemistry
We expanded the Fues potential in positive powers of x = r - re, yielding
V
=
D
[nl2
]
(-l)”(n - 1)(x/r.Jn
(1)
The matrix elements of the Hamiltonian in the harmonic-oscillator basis set (for VHO= D x2re2)are
where b, = (- l P ( p - l ) D and V H O = (2D/pre2))”’/27r. We note that some convergence questions arise if one extends the summations in eq 1 and 2 to infinity, owing to the singularity in V at x = --re; we evade this issue by redefining V as a truncated series similar to that in eq 1 and we choose our termination such that the final results are insensitive to the number of terms retained. This restricts us to wave packets, the amplitudes of which are negligible for 1x1 2 re and precludes the study of wave packets of really high energy (ie., (237) > 0.25D), as mill become apparent. The initial wave packets were chosen to be gaussian minimum wave packets centered at various values, TO, (7) L. I. Schiff, “Quantum Mechanics,” McGraw-Hill Book Co., Inc., New York, N. Y., 1955. (8) L. D. Landau and E. M. Lifshitz, “Quantum Mechanics,” Addison-Wesley Publishing Co., Reading, Mass., 1958. (9) I. I. Gol’dman, V. D. Krivchenkov, V. I. Kogan, and V. M. Galitakii, “Problems in Quantum Mechanics,” Academic Press Inc., New York, N. Y., 1960. (10) G. N. Watson, “Theory of Bessel Functions,” Cambridge University Press, London, 1922. (11) E. Fues, Ann. Phys., 80, 367 (1926).
249 1
THEDYNAMICS OF ANHARMONIC OBCILLATORS I
a.
b. 90 I
.b
Ixlo'"ssc
2x10-"
t
Figure 1. Expectation value of the coordinate as a function of time: (a) results obtained by the linear-variation method; (b) results obtained by the WKB method with the same parameters. Data given for the Fues oscillator: = 0.74 A; ro (the center of the initial wave packet) = 0.85 A; D = 0.7120 X 10-11 erg/molecule; 1.1 = 0.83 X g; the width parameter in the initial wave packet = 0.1507 A; @),nitis, = -0.6781 X 10-11 erg/molecule; (E)Ho = -0.6778 x 10-11 erg/molecule; (E)WKB= -0.6782 X 1 O - I l erg/molecule; N ~ o 2= 1.0000; N W K B = ~0.9998.
of the coordinate. The expansion coefficients for such wave packets in terms of the basis set used are well known.I2 The matrix of the Hamiltonian was diagonalized by the Jacobi method; the matrix of the coordinate was transformed into the new basis set of the approximate energy eigenfunctions ; and the expansion coefficients of the initial wave packet were transformed into this new basis set. Then the expectation value of the coordinate as a function of time was calculated. The procedure used for these operations was essentially the same as that described in part A of ref 4. The calculations outlined above were carried out on the IBM 7074 computer of the University of Rochester; this facility was also used to do the other computations described later in this paper. The results of some representative runs are given in Figures la-3a. In all the computations presented, the parameters D, p , and re (and p for the Morse oscillator) were chosen to simulate Hz. The dynamical behavior of the Fues oscillator was then investigated by the WKB method. The WKB energies were calculated by means of the appropriate formula in Table I, and the corresponding WKB wave functions were computed by use of the connection formula^'^ in the vicinities of the classical turning points and by means of numerical integration of Schiff's formula (Unsatisfactory results were obtained by terminating the wave functions in the classically allowed region and by neglecting the connection formulas.) Each wave function in the regions of the
80
rtt)
I
IX IO%c
1 2x104'
t
Figure 2. Expectation value of the coordinate as a function of time: (a) results obtained by the linear-variation method; (b) results obtained by the WKB method with the same parameters. Data given for the Fues oscillator: re = 0.74 A; TO = 0.95 A; D = 0.7120 X 10-11 erg/molecule; fi = 0.83 x 10-24 g; the width parameter in the initial wave packet = 0.1507 R ; (E)i,itial = -0.6595 x 10-11 erg/molecule; (E)Ho= -0.6571 X lo-" erg/molecule; (E)WKB = -0.6600 X 10-11 erg/molecule; NHO*= 1.0000; N W K B = ~0.9999.
turning points was then rescaled to make it continuous at the junctures of the regions of the turning points with the region including the bulk of the classically allowed values of the coordinate. The precise locations of the junctures did not significantly affect the results, provided that the junctures were sufficiently removed from the singularities at the classical turning points. The resulting wave functions were then normalized. It was found that the functions were not quite orthogonal, so the Graham-Schmidt orthogonalization process15was used to force orthogonality and, thereby, to facilitate computation of the expansion coefficients of the initial wave packets. The functions were orthogonalized in order of increasing energy; $% is orthogonalized with respect to $0, $1,. . .$%-I. Some of the resulting functions are plotted in Figure 4. The wave functions showed the appropriate number of nodes in (12) See ref 7 , p 67. (13) See ref 7 , p 188.
(14)See ref 7, p 191. (15), R. Courant and D. Hilbert, "Methods of Mathematical Physics, Vol. I, Interscience Publishers, Inc., New York, N. Y., 1953, p 50.
Volume 78, Number 7
July 1968
2492
R. DUBROW, D. HATZENBUHLER, W. MARX,E. ZAHORIAN, AND D. J. WILSON
(E) = 1.2
f #*@,O)H#(r, 0) dr
(3)
and
1.1
00
(E) = nC cn*cnEn -0
(4)
( E ) calculated by eq 3 is denoted as (E)initial. It was found that disagreement between the results of eq 3 and those of eq 4 occurred for the linear-variation
.9
7
method with harmonic-oscillator basis functions even when normalization and agreement between (r(0)) and ro were quite good. The results of eq 3 and those of eq 4 were in good agreement for all computations carried out by the WKB method. The discrepancies
n
t
1X I P e r c
2x10'"
Figure 3. Expectation value of the coordinate as a function of time: (a) results obtained by the linear-variation method; (b) results obtained by the WKB method with the same parameters. Data given for the Fues oscillator: re = 0.74 A; ro = 1.25 A; D = 0.7120 X lo-" erg/molecule; p = 0.83 x 10-24 g; the width parameter in the initial wave packet = 0.1507 A; (E)initial = -0.5793 X lo-" erg/molecule; (E)Ho= -0.2767 X 10-11 erg/molecule; (E)WXB = -0.5786 X 10-11 2 = 0.9999. ~ ~ ~ erg/molecule; NHO*= 0.9991; N
all cases, and their graphs did not appear significantly different from plots of the functions before orthogonalization. All integrations were done numerically with the use of the simple trapezoidal rule; for the runs presented here, it was found that the results did not vary significantly with the spacing of the points when 200 or more points were used. Initial gaussian minimum wave packets (as mentioned above) were employed in all runs; expansion coefficients were then calculated by numerical integration. The accuracy of the expansion is indicated by the closeness of the sum of the squares of the coefficients to unity; this sum is given for each of the runs plotted and was generally of the order of 0,9999. Another test of the adequacy of the approximations ) used is the agreement obtained between ( ~ ( 0 )and YO, the location of the maximum of the initial gaussian wave packet. This agreement was to substantially better than 1% in all cases studied, both WKB and linear variation. I t should be noted that neither good normalization nor accurate agreement of (~(0))and yo provides a very sensitive test of the validity of the computations; it was, therefore, felt advisable to calculate (E), the expectation value of the total energy, by means of the two formulas The Journal of Physical Chemistry
Figure 4. Plots of some WKB wave functions for the Fues oscillator: re = 0.74 A; D = 0.7120 X 10-ll erg/molecule; = 0.83 x 10-24 g.
observed with the linear-variation method are due to poor values of the energy levels, which in turn are due to (1) basis functions ill adapted to the Fues potential and (2) divergence of the series expansion used for the Fues potential in evaluating the matrix elements of the Hamiltonian. The exact, WKB, and linear-variation energies are given in Table I1 and constitute an impressive caveat against blind use of the linear-variation method. The failure of the linear-variation method to produce good values of the higher energies is not significantly improved by increasing either the number of basis functions or the number of terms retained in the power-series expansion of V ( r ) , although the lower energies are improved. Plots of ( ~ ( 2 ) ) os. t were calculated with the WKB method by computing the matrix elements of r in the basis of the WKB wave functions and then using
where the c's are the expansion coefficients of the initial wave packet and the E's are the WKB energies. Re-
2493
THEDYNAMICS OF ANHARMONIC OSCILLATORS
Table I1 : Energy Levels of the Fues Oscillator" Energy (WKB), X 10-11 erg
Energy (exact), X 10-11 erg
- 0.683328 - 0.631091 - 0.584579 - 0.543026
0.683389 -0.631109 - 0,584600 - 0.543053 0.505790 - 0.472223 -0.441900 - 0.414403 - 0.389394 -0.366583
-
-0.505751 -0.472186 - 0.441856 - 0.414355 - 0.389345 -0.366533 P =
0.83 X
g;
T,
=
Energy (linear variation), X 10-1' erg
- 0.683071 -0.630510 0.580883 - 0,526483 -0.462627 -0.308343 0.389505 - 0,219938 - 0.123958 -0.018130
-
packets having ( E ) 5 3fiw, where w is the angular frequency of the corresponding harmonic oscillator. The availability of results for the Morse oscillator obtained by two independent methods (yielding good agreement) suggested that we test the WKB method by applying it to the same system. The WKB method gives the exact energies of the Morse oscillator, so the frequency of oscillation of ( r ( t ) ) and the frequency of oscillation of the envelope are of necessity correct. This test, therefore, checks the ability of the WKB wave functions to approximate the true values of the matrix elements and the expansion coefficients of the initial wave packet. Exact and WKB expansion coefficients for two runs are compared in Table 111; these
0.74 A; D = 0.712 X 10-11 erg.
Table I11 : Exact and WKB Expansion Coefficients of Initial Wave Packets
sults are shown in Figures lb-3bJ which differ from the results plotted in Figures la-3a only in that the WMB method was used rather than the linear-variation method. The initial wave packet relevent to Figure 1 is of quite low energy, and we find that the two methods give essentially the same frequency of oscillation of ( r ( t ) ) . However, even for this most favorable case, the frequencies of oscillation of the envelopes of the two plots are quite different, owing to the magnification of small differences in the energy levels yielded by the two methods when one calculates the second differences of the energies which determine the behavior of the envelopes. Figures 2 and 3 compare runs made with initial wave packets with progressively increasing ( E ) . In ) Figure 3 the frequencies of oscillation of ( ~ ( t ) are quite different and the envelopes of the curves bear no resemblance. The value of ( E ) calculated by the linear-variation method is grossly in error, while that obtained by the WKB method is in good agreement with the energy calculated by eq 3. The failure of the linear-variation method on this run is due to the failure of the series expansion for V(r) to converge for SUEciently large r.
The Morse Oscillator The Morse potential (see the second expression in column 1 of Table I) is a convenient and reasonably accurate approximation to a variety of bond-stretching potentials. We previously studied the dynamics of Morse oscillators by means of expansion in terms of the exact Morse eigenfunctions and by means of the linearvariation method with a harmonic-oscillator basis.4 Expansion in terms of Morse eigenfunctions and the exact energies gives excellent results, provided that the initial wave packet can be adequately approximated by the bound-state eigenfunctions. The linear-variation method also gives good results, but only for wave
Expansion coef ( T O = 1.25 A), Exaot WKB
n
0 1 2 3 4 5 6
7 8 9 E a
0.0579 0.1848 0.3800 0.5573 0,5803 0.3929 0.1222 - 0.0294 -0.0216 0.0106 0.2974a
Expansion coef (TO
0.0594 0.1849 0.3773 0.5556 0.5821 0.3945 0.1257 - 0.0247 -0.0164 0.0123 0.2972a
= 0.85
k),
Exact
WKB
0.9051 0.3984 0.1480 -0.0142 - 0.0026 - 0.0041 0.0049 -0.0039 0.0022 - 0.0024 0.0585a
0.8989 0.4233 0.1107
- 0.oooi -0.0041
- 0.0007 0.0046 - 0.0045 0.0021 - 0.0002 0.0589
E = value X 10-11 erg.
runs are plotted in Figures 5 and 6, one being of low energy (ro = 0.85 A) and the other of high energy (ro = 1.25 8). The agreement between corresponding expansion coefficients calculated by the two methods is rather good, and the agreement between the values of ( E ) is excellent. It is of particular interest to note that the WKB approximation appears to be at least as adequate at high energies as it is at lower energies, which is what one would expect; this conclusion was borne out by other runs not presented here. The results of three representative runs (with parameters simulating Hz and with the usual initial gaussian minimum wave packet) are shown in Figures 5-7. The figures labeled a pertain to the WKB method; those labeled b pertain to the expansion in exact energy eigenfunctions. In all three cases the plots are very similar, even down to rather small details. We note one discrepancy; the minimum values of ( r ( t ) ) for the WKB plots in Figures 5a and 7a are approximately 0.01 less than the corresponding minima for the exact eigenfunction plots in Figures 5b and 7b. This is due to the fact that the value of the magnitude of the slope at the left classical turning point is both rather small and rapidly increasing with decreasing r. This Volume 78, Number 7 July 1068
R. DUBROW, D. HATZENBUHLER, W. MARX,E. ZAHORIAN,AND D. J. WILSON
2494 a.
I
ltti
...-
I.
b.
81 .a0
rlil 75
10 1XlO'"rrc
ZXIO-'
t
Figures 5. Expectation value of the coordinate as a function of time: (a) results obtained by the WKB method; (b) results obtained by expansion in terms of the exact Morse eigenfunctions (ME). p (the range parameter in the Morse potential) = 0.1963 x 109 cm-1; r e = 0.74 R; ro = 0.85 A; D = 0.7120 X 10-11 erg/molecule; p = 0.83 X loez4g; the width parameter in the initial wave packet = 0.1507 A; (@initial = 0.0585 X 10-11 erg/molecule; (E)WKB = 0.0586 X 10-11 erg/molecule; (@ME = 0.0585 X 10-11 erg/molecule; NWKB = ~0.9995; N M E 0.9999. ~
means that our use of the WKB connection formulas for calculating the first few wave functions in the left classically forbidden region must yield wave functions whose amplitudes are too large in this region of small r . The magnitude of the error greatly decreases as we consider wave functions of progressively higher energy, since the magnitude of the slope at the left side of the well becomes very large and the penetration of both exact and approximate wave functions into the classically forbidden region becomes less and less. This explains the fact that the discrepancy between the two methods a t small (r(t)) is greatly reduced for the runs plotted in Figure 6. As is evident from Table 111, the
Table IV : Morse Oscillator Energies Energy (exact and WKB), X
10-11
erg
0.0422 0.1228 0.1982 0.2685 0.3335 0.3955 0.4482 0.4978 0.5423 0.5816
Energy (integration) X 10-11 erg
Energy (diagonalization) X 10-11 erg
0.0547 0.1251 0.1993 0.2692 0.3346 0.3945 0.4498 0.5000 0.5460 0.5818
0.0539 0.1247 0.1994 0.2693 0.3346 0.3946 0,4499 0.5003 0.5462 0.5820
The Journal of Physical Chemistry
Figures 6. Expectation value of the coordinate as a function of time: (a) results obtained by the WKB method; (b) results obtained by the expansion in terms of the exact Morse eigenfunctions (ME). rB = 0.74 A; ro = 1.25 A; D = 0.7120 X 10-11 erg/molecule; p = 0.83 X 10-24 g; the width parameter in the initial wave packet = 0.1507 A; (E)initial = 0.2968 X 10-11 erg/molecule; (E)WKB = 0.2972 X 10-11 erg/molecule; @)YE = 0.2974 X 10-11 erg/molecule; N W K B=~1.0000; N M E=~0.9999,
contributions of the first few eigenfunctions are quite small, so that this inaccuracy of the first couple of WKB wave functions introduces an error in ( r ( t ) ) smaller than was seen in Figures 5 and 7. The accuracy of the WKB energies does not, unfortunately, imply similar accuracy of the WKB wave functions. We therefore tested the Morse WKB orthogonalized wave functions by using them as basis functions for calculating the matrix of the Hamiltonian. The diagonal matrix elements Bhould be approximately equal to the Morse energy levels, and the off-diagonal elements should be approximately aero. The diagonal elements are given in Table IV in the middle column; the off-diagonal elements ranged from 0.01 to 0.79 X 10-13 erg. We then diagonalized the matrix of the Hamiltonian by means of an orthogonal transformation; this did not make any great changes in the energies, and the wave functions were not significantly changed. The energies resulting from this operation are listed in Table IV in the last column. The energies calculated from the WKB wave functions are in rather good agreement with the exact ener-
THEDYNAMICS OF ANHARMONIC OSCILLATORS
2495
Figures 7. Expectation value of the coordinate as a function of time: (a) results obtained by the WKB method; (b) results obtained by expansion in terms of the e x p t Morse eigenfunctions (ME). re = 0.74 if; yo = 0.95 A; D = 0.7120 X lo-" erg/molecule; p = 0.83 X 10-24 g; the width parameter in the initial wave packet = 0.1607 if; (E)rnitial = 0.1037 X 10-llerg/molecule; (E)WKB= 0,1030 x 10-11 erg/molecule; @)ME = 0.1037 X 10-11 erg/molecule; NWKB = ~0.9997; NmZ = 0.9999.
harmonic-oscillator basis set to the Fues oscillator is fraught with some hazard, owing to the limited radius of convergence of the power-series expansion of the potential about the equilibrium internuclear distance. The situation may possibly be improved somewhat by use of the technique of Harris, Engerholm, and Gwinn;16 we are currently looking into this. On the other hand, the application of the WKB method to the Fues oscillator may be expected to yield quite good results for all initial wave packets which can be adequately approximated by the bound state energy eigenfunctions of the oscillator. This permits one to investigate initial wave packets of far greater energy than is possible using the linear-variation method we employed. The WKB method and the method of expansion in terms of exact-energy eigenfunctions yieId extremely similar results for wave packets of both high and low energy, even for H2,which is the most unfavorable case (the molecule for which the semiclassical approximation should be least adequate). For results of highest accuracy from the WKB method, it is necessary to take some care in calculating the wave functions in the classically forbidden region; for states of low energy, simply using the connection formulas throughout the classically forbidden region introduces small errors. We note that the accuracy with which the WKB method approximates the energy levels of eight different oscillators, even at high energies, makes it an excellent approximation method for studying the quantum dynamics of anharmonic oscillators. We hope to use this method in developing a computationally tractable quantum theory of unimolecular reactions and a theory of inelastic collisions of molecules at rather high energies.
gies, except for the ground state. Here, of course, is where one would expect a semiclassical approximation to be in greatest difficulty. (FORTRAN decks of the computer programs used in this investigation are available from D. J. W. on request.)
Acknowledgments. We are indebted to Dr. Paul Endres for carrying out the Xorse energy eigenfunction runs with his program (described in ref 4) and for helpful discussions. We are, as usual, grateful to the staff of the University of Rochester Computing Center for their valued assistance.
t
IXlO-"sec
2XIO-"
Conclusions The application of the linear-variation method with a
(16) D. Harris, G. Engerholm, and W. Gwinn, J . Chem. Phys., 43, 1515 (1965).
Voiume 78, Number 7 July 1968
