GeometricMean Variation Function for H2 +Stanley R. Radel, Ronnie Gorman, Carol Cutler, and Luis Kahn Citation: The Journal of Chemical Physics 50, 3642 (1969); doi: 10.1063/1.1671605 View online: http://dx.doi.org/10.1063/1.1671605 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/50/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A geometric interpretation of prolongation by means of connections J. Math. Phys. 51, 113502 (2010); 10.1063/1.3504172 Effects of Variations on Geometrical Parameters on a 2.4 GHz Printed Dipole Antenna architecture AIP Conf. Proc. 1203, 427 (2010); 10.1063/1.3322481 Geometric phase effects in H+O2 scattering. I. Surface function solutions in the presence of a conicalintersection J. Chem. Phys. 104, 7475 (1996); 10.1063/1.471460 Erratum: Variational wavefunctions for H+ 2 J. Chem. Phys. 63, 5509 (1975); 10.1063/1.431906 Relation of Variational Functions to Exact Function for H2 + J. Chem. Phys. 41, 820 (1964); 10.1063/1.1725966

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3642 LEITERS TO THE EDITOR J. CHEM. PHYS., VOL. 50, 1969

TOK 1500 1000

1.0 0.8

0.6

E 0.4 -c (.

•

, , • • • •

• • • • • • •

•

• HBr -Hel

U 0.2~~----~----~-----L ____ ~ ____ ~ ______ L-~ FIG. 1. Landau-Teller plots

of vibrational relaxation times at 1 atm pressure for HBr and HI. The solid line represents similar data for HCI from Ref. 1.

Q) en 2.0 ::t.. -... a. 1.0

0.8

0.6 •

0.4

similar to that found for HCl. The HI data, however, show no departure from linearity within the experi­mental scatter throughout this temperature range. The Pr values for HI are in general agreement with those found by Chow and Greene.2

The fair agreement of the HCI data with predictions from Moore's rotation-vibration transfer theory3 was pointed out in Ref. 1. However, this theory [Eq. (4), Ref. 3J predicts a monotonic decrease in vibrational relaxation times through the series HCI, HBr, HI, with the values for HCI and HI differing by a factor of 5-6 in the range lOOOo-2000°K. The disagreement of these predictions with the data presented here is perhaps not surprising, since Moore uses a single value of the repulsive potential parameter a to fit experimental data for a wide range of molecules. This parameter does in fact vary significantly within this series of hydrogen halides; values of a at 10000 K calculated from the Lennard-Jones potential parameters4 by Method B of de Wette and Slawsk~ range from 5.81-1 for HCI to 4.61-1 for HI. The success of Moore's theory in pre­dicting the relative relaxation rates of HCI and DCI found in Ref. 1 suggests, however, that it does correctly account for the effects of isotopic substitution. We plan

,

• HI -HCl

••

to investigate this point further by measuring vibra­tional relaxation in DBr and DI.

* Work supported by the U.S. Atomic Energy Commission. 1 W. D. Breshears and P. F. Bird, J. Chem. Phys. 50, 333

(1969). 2 C. C. Chow and E. F. Greene, J. Chem. Phys. 43, 324 (1965). 3 C. B. Moore, J. Chem. Phys. 43, 2979 (1965). 4 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular

Theory of Gases and Liquids (John Wiley & Sons, Inc., New York, 1954), p. 1112.

6 K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press Inc., New York, 1959), p.281.

Geometric-Mean Variation Function for H2+*

STANLEY R. RADEL, RONNIE GORMAN, CAROL CUTLER, AND

LUIS KAHN

Department of Chemistry, The City College of the City University of New York, New York, New York 10031

(Received 9 December 1968)

This report gives energy results obtained from a new two-parameter variation function for the 1sO' ground

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J. CHEM. PHYS., VOL. 50, 1969 LEITERS TO THE EDITOR 3643

TABLE I. Energy and parameters for the 1s., ground state of the hydrogen molecular ion.-

Geometric-mean variation functiono

R -E(Exact)b -E(g.m.) Z a f3 2a+f3

0.00 2.0000000 2.000000 2.000000 1

0.50 1.7349879 1.734715 1.792302 0.190253 0.671315 1.051821

1.00 1.4517863 1.451485 1.594820 0.223164 0.705338 1.151667

1.50 1.2489899 1.248748 1.455522 0.258868 0.748557 1.266294

2.00 1.1026342 1.102447 1.353889 0.296975 0.792074 1.386024

2.50 0.9938235 0.993672 1.277151 0.337279 0.831231 1.505789

3.00 0.9108962 0.910765 1.217847 0.379263 0.862885 1.621412

3.50 0.8465698 0.846449 1.171380 0.422074 0.884629 1.728778

4.00 0.7960849 0.795967 1.134735 0.464510 0.894936 1.823957

4.50 0.7561623 0.756043 1.105809 0.505135 0.893524 1.903795

5.00 0.7244203 0.724298 1.083046 0.542519 0.881581 1.966619

6.00 0.6786357 0.678509 1.051337 0.603582 0.836573 2.043737

7.00 0.6484511 0.648327 1.032209 0.644886 0.784880 2.074652

8.00 0.6275704 0.627456 1.020680 0.670136 0.746574 2.086845

9.00 0.6123066 0.612206 1.013619 0.684825 0.731630 2.101281

10.00 0.6005787 0.600493 1.009193 0.693276 0.744523 2.131074

12.00 0.5835016 0.583443 1.004481 0.701173 0.864322 2.266669

14.00 0.5714975 0.571459 1.002371 0.704280 1.169614 2.578174

16.00 0.5625364 0.562511 1.001351 0.705506 1.700473 3.111485

18.00 0.5555775 0.555560 1.000821 0.706155 2.727249 4.139560

20.00 0.5500143 0.550002 1.000527 0.706498 4.712813 6.125809

00 0.5000000 0.500000 1.000000

a All distances are in Bohr radii and all energies are in hartrees. The total energy may be obtained by adding the nuclear-nuclear repulsive energy I/R.

b Exact from Ref. I. o Determined such that (oE/oZ)B<IO .....

state of the hydrogen molecular ion. The function, which builds up the probability density at intermediate distances by the addition of a geometric-mean term to the well-known Finkelstein-Horowitz function, may be expressed as follows:

= (za/-rr) 112 exp( -!ZRX) [2a cosh(!ZR~) +f3],

where a and f3 are coefficients related by normalization; Z is a scale factor (effective charge); X and ~ are the usual confocal elliptic coordinates, XR=Ta+Tb and ~R=Ta-Tb, where R is the internuclear separation and T Q and Tb are the respective distances of the electron from nuclei A and B. The function, which we believe to be the first containing a geometric-mean term, satisfies the minimum criteria for a valid ground-state H2+ wavefunction.

The Rayleigh-Ritz variation method was employed to find the energy as a function of Z and R. To ensure that the energy is minimized with respect to Z for any R, we required that (fJE/fJZ) R=O. Then for each R, Z, and E were determined by the Newton-Raphson method. The coefficients were obtained from the secular equations and the normalization requirement.

For various separations, Table J compares our calcu­lated energies with the exact energies.1 The results agree to within 0.0003 a.u. for the full range of inter­nuclear separations. Table I also includes the effective charge experienced by the electron at each separation as well as the coefficients. The limiting values of Z, a, and 2a+f3 are as expected. We would like to emphasize that our function preserves the concept of an "effective charge," and to the best of our knowledge, gives better energies than any other simple variation function including the very accurate Guillemin and Zener wavefunction.2

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3644 LEITERS TO THE EDITOR J. CHEM. PHYS., VOL. 50, 1969

In addition to further studies on the lsO' state of H2+, we are extending the work along the following lines: The excited states of H2+ are being studied; Rayleigh­Schrodinger perturbation calculations are being carried out for the ground state of H2+ using the geometric­mean function as the zeroth-order wavefunction; geometric-mean valence bond and LCAO functions are being studied for the hydrogen molecule.

This research was supported in part by the National Science Foundation Undergraduate Research Par­ticipation Program. The calculations were performed on the IBM 7040 at the City College of New York, the use of which is hereby gratefully acknowledged.

* Supported in part by the National Science Foundation Under­graduate Research Participation Program.

1 H. Wind, J. Chern. Phys. 42, 2371 (1965). 2 S. Kim, T. Y. Chang, and J. O. Hirschfelder, J. Chern. Phys.

43, 1092 (1965).

Rate Constant for the Reaction 02(td g ) +03~202 +0*

R. P. WAYNEt AND J. N. PITTS, JR.

Department of Chemistry, University of California, Riverside, California 92502

(Received 16 December 1968)

The rate constant for the reaction

(1)

is required in any assessment of the contribution that singlet molecular oxygen, 02(ldg), may make to the chemistry of the upper or lower atmospheres. In particular, 02(I.1g) is implicated in certain processes occurring in polluted urban atmospheres of the "Los Angeles" typel and the stationary concentrations of both 02(1dg) and 0 3 may be determined in part by the rate of (1). McNeal and Cook2 have given a value of kl = 1.5±0.3X 1()6 liters mole-l·seeI, which is at least an order of magnitude smaller than earlier estimates.3

McNeal and Cook determine [02 (1dg ) J by a photo­ionization technique which is possibly subject to interference from vibrationally excited oxygen. Clark and Wayne4 find kl =2.1±0.6X 106 liters mole-I·seci

using a modified version of the photoionization method, in which vibrationally excited oxygen is unlikely to survive to the detection point. However, it seemed desirable to obtain confirmation of the essentially identical "photoionization" values by a different experimental technique. The present report describes a determination of ki from intensity measurements of the l.1g~S];g- emission band at 1.27 p..

A conventional discharge-flow system was used: ozone could be added to the main gas stream at anyone of four jets situated at different distances downstream.

Flow velocities were of the order of 0.3-1 m seeI, and total concentrations lay in the range 1.9X 10--4 moles literl to 6X10-4 moles literi. Ozone concentrations in ozonized oxygen were determined from the optical absorption at ~=2500 A using the extinction coefficient of Vigroux.5

The intensity of the A = 1.27 -p. emission was measured by a germanium photoconductive ce1l6; light was chopped at 360 Hz and the cell output was amplified and phase detected with a Princeton Model 120 lock-in amplifier. A Bausch & Lomb monochromator (675-lines mm-I grating blazed at 1.0 p.) was used to isolate the emission band.

A microwave discharge was excited in molecular oxygen, and atomic oxygen removed from the discharge products by a film of HgO. In the absence of ozone, no emission of the air afterglow could be detected by eye when nitric oxide was introduced to the gas-flow down­stream of the HgO, although addition of ozone at once led to the appearance of a green glow even when nitric oxide was not intentionally present. However, this emission from residual NO was not sufficiently intense at A = 1.17 p. to produce a signal from the germanium detector, and it was assumed that it made no con­tribution to the measured signal at 1.27 p..

Absolute intensity measurements are not required if the kinetics are pseudo-first order in [02 (1.1g) ]. Fortu­nately, this condition holds in the O2 (l.1g)-02-0S system under the experimental conditions employed even when [OaJ is not in excess of [02 (1.1g) J, since the recombination reaction

(2)

is the major fate of atomic oxygen produced in (1). At all times, [02J> 1.9X 10-4 mole literl, while [03J<5X 10-7 mole literI, so that reaction of 0 with 0 3, or loss at the walls, cannot compete effectively with

7.5~-------------------------------,

=- 5.0 I () CIJ !II

.::.:. o - 2.5

234

107 [03] mole Iiter-'

FIG. 1. Variation of first-order rate constant, k, with ozone concentration: 107 [08J mole liter-I (abscissa) and 10k (seconds-I) (ordinate) .

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