Stark Perturbed Rotational Energy Levels of the Rigid Linear Top RotorP. J. Bertoncini, R. H. Land, and S. A. Marshall Citation: The Journal of Chemical Physics 52, 5964 (1970); doi: 10.1063/1.1672884 View online: http://dx.doi.org/10.1063/1.1672884 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/52/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A semiclassical determination of the energy levels of a rigid asymmetric rotor J. Chem. Phys. 68, 745 (1978); 10.1063/1.435747 Fourth order perturbation theoretic Stark energy levels in linear rotators J. Chem. Phys. 60, 4632 (1974); 10.1063/1.1680955 Stark Energy Levels of SymmetricTop Molecules J. Chem. Phys. 38, 2896 (1963); 10.1063/1.1733618 The Stark Effect for a Rigid Asymmetric Rotor J. Chem. Phys. 16, 669 (1948); 10.1063/1.1746974 An Asymptotic Expression for the Energy Levels of the Rigid Asymmetric Rotor J. Chem. Phys. 16, 78 (1948); 10.1063/1.1746662

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5964 LETTERS TO THE EDITOR J. CHEM. PHYS., VOL. 52, 1970

TABLE 1. Substance and reducing parameters.

Temperature B C Substance (0C) (bars) (amagats)

Ar -73 1922 222.0 Ar 35 1271 250.1 Ar 100 1025 255.0 Ar 200 1060 272.4 Ar 300 941 281.1 Ar 400 850 288.0

Nz 35 1486 235.3 N2 100 1395 241.9 Nz 200 1270 250.9 N2 300 1265 260.1 N2 400 1254 264.5

CH. 35 2069 201.6 CH. 101 1554 199.5 C~ 200 1163 202.3

CaH, 35 3314 90.54 CaH, 100 2553 88.52 CaH, 200 2165 93.42

C2~ 150 1836 137.8 Ne 0 2195 726.1 Kr 150 1304 205.0 H2 125 3297 777.9

stances; the temperatures and Band C values used and substances are listed in Table 1. The isotherm data is from the literature.2

To prepare the graph the modified Tait equation was force fitted to density values at 6, 8, and 10 kbar where possible. The derivatives were generally calculated by computing chords between adjacent experimental points and the Band C obtained as above were then used to reduce the derivatives to the graph shown. For those substances whose isotherms do not extend to 10 kbar, Band C values were obtained by simple forced fitting at two points within the range of available data.

It should be emphasized that the parameter values given in Table I do not represent an optimized set and are poorly known. It is clear that substance and temperature-independent functions could be fitted to the curve and integrated to give an equation of state which may be in turn fitted to the original experimental data with least-squarcs analysis. These subjects will be discussed in a more extended paper.

The purpose of this note is to point out the existence of this graph. This scheme of reduction is capable of reducing (at least) several isotherms of diverse sub­stances to a single curve which is a function of pressure alone. That such a curve should exist does not appear obvious, nor does an explanation of the meaning of the constants spring readily from any theoretical discussion of dense gases with which the author is familiar. Thus this remains a strictly empirical observation at the present.

Less empirical discussions of reduced equations of state at these pressures have been given by Holleran3

and Verbecke,4 but they are quite different in their approach, and are not equivalent to that presented here.

This scheme of reduction cannot work at low pres­sures. If the extended isotherms are plotted, the reduced curve splits at low pressures to a curve for each isotherm. Each curve passes through a minimum, the pressure of which rises with increasing temperature. Even at temperatures as high as 1000D C, recent data for N2 5 indicates that this individuality of behavior disappears at a pressure below 3000 bar. It is also worth noting that apparently there do not exist Band C values which will reduce the hard sphere isotherm to this curve.6

Phenomenologically it is tempting to concludc that this curve represents the compression of the material once local order has been established, and that the deviations from it at low pressure are occasioned by the formation of this local order. Clearly further work is needed to confirm or deny this crude interpretation.

* This work supported in part by the National Science Founda­tion, and in part by the Faculty Research Fund of the University of Oklahoma.

1 S. L. Robertson, S. E. Babb, Jr., and G. J. Scott, J. Chern. Phys. 50,2160 (1969).

2 Argon: Ref. 1 but see Erratum O. Chern. Phys. 51, 3152 (1969)], and the 2000K isotherm is from R. K. Crawford, thesis, Princeton, 1968; N2 : S. L. Robertson and S. E. Babb, Jr., J. Chern. Phys. SO, 4560 (1969); CH. and CaH,: ibid. 51, 1357 (1969); C2H,: A. Michels and M. Geldermans, Physic a 9, 967 (1942) and A. Michels, M. Geldemans, and S. R. de Groot, ibid. 12, 105 (1946); Hz: A. Michels, W. de Graaff, T. Wassenaar, J. M. H. Levelt, and P. Louwerse, ibid. 25,25 (1959); Ne: A. Michels, T. Wassenaar, and P. Louwerse, ibid. 26, 539 (1960); Kr: N. J. Trappeniers, T. Wassenaar, and G. J. Wolkers, Physics 32, 1503 (1966).

a E. M. Holleran, J. Phys. Chern. 72, 1230 (1968), and references cited therein.

4 O. B. Verbeke (private communication). 5 P. Malhrunot and B. Vodar, Compt. Rend. 268B, 1327

(1969). 'Data abstracted from an article by F. H. Ree and W. G.

Hoover, J. Chern. Phys. 46, 4184 (1967).

Stark Perturbed Rotational Energy Levels of the Rigid Linear Top Rotor*

P. J. BERTONCINI, R. H. LAND, AND S. A. MARSHALL

Argonne National Laboratory, Argonne, Illinois 60439

(Received 11 February 1970)

The most successfulmcthods used to obtain molecular electric dipole moments are those of molecular beams and gas-phase microwave spectroscopy. To obtain such information, an analysis must be performed correlating the strength of an externally applied electric field and the subsequent displacement of spectral components. Of the various classes of polyatomic systems, the simplest to so analyze is the rigid linear top rotor. Eigenvalues to its rotational energy operator are obtainable in closed form and shifts in these eigenvalues

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J. CHEM. PHYS., VOL. 52, 1970 LETTERS TO THE EDITOR 5965

brought about by the application of an externally applied electric field may be obtained by standard methods of perturbation theory. The principal short­coming of the perturbation technique is that its useful­ness is generally limited to problems which do not demand a high degree of accuracy in determining spectral line positions. For this reason, a computer generated table has been prepared for the Stark perturbed eigenvalues of the rigid linear top rotor. The entries in the table were obtained by variationally solving the following eigenvalue equation:

Q\II+[W-A cosO-!p cos2o}It=0, where

Q= (sinO)-l(a/aO) [sinO (a/aO) ]+ (sin20)-I(a2jatJ>2),

10.= (JJ,E/hB) , p= [Ceq -al.).E2/hB], /-L is the molecular electric dipole moment, E is the strength of the exter­nally applied electric field, B = (h2/87r2I) , I is the molecular moment of inertia, h is Planck's constant, and a D and OIL are the molecular electric polarizabilities parallel and perpendicular to the molecular axis of symmetry, respectively. Exact eigenfunctions 'l' were obtained as a series expansion of spherical harmonics. The secular matrix resulting from the variation pro­cedure was diagonalized using the Givens-Householder method. The resulting eigenvalues were found to be accurate to better than 10 significant figures for an expansion in terms of the first 25 spherical harmonics. Eigenvalues for various values of A and p were compared with those obtained by second order perturbation theory and the errors involved in applying the approxi­mation are tabulated. Eigenvalue solutions are given for the quantum numbers j = 0, 1, 2, 3, and 4 and are listed for a sequence of the Stark parameter ~A2=0.05 in which 10.2 is taken from 10 to 0.05. Supplementary tables and instructions are given for extending the usefulness of the principal table to any value of 10.2 and p over the ranges 10.2::; 10 and p::; 10-3•1

* This work performed under the auspices of the U.S. Atomic Energy Commission.

1 For a detailed paper, order document NAPS-00831 from ASIS National Auxilliary Publication Service c/o CCM Informa­tion services, Inc. 22 West 34th Street N;w York New York 10001; remitting $1.00 for microfiche or '$3.00 for photocopies.

Analysis of Flash Desorption Data for Activation Energy Linearly Dependent on Coverage*

ROBERT S. HANSEN AND KUN-IcHI MATSUSHITA

Institute for Atomic Research and Department of Chemistry, Iowa State University, Ames, Iowa 50010

(Received 15 January 1970)

The principles involved in the determination of kinetic orders and activation energies in desorption proces~es have been extensively developed by Ehrlichl.2

In particular and the method and applications have been

reviewed by Hansen and Mimeault.3 For a desorption process satisfying the Polanyi-Wigner (PW) equation, -dn/dt=pnk exp[ -~H/RT], with the flash tem­perature programmed so that dr- 1/ dt= -b, a constant, we obtain

If ~H is constant we obtain immediately

In(F(no) -F(n)} = - (~/RT) +In(Rp/MH)

+In(1-exp[ -(~/R)(To-LT-I)]I. (2)

In most cases of interest the last term on the right is negligible over most of the desorption range. Inter­pretation of data then involves choosing the kinetic order k so that a plot of the left side of Eq. (2) against r- l is linear, and then obtaining ~H from the slope of this plot.

Frequently the activation energy ~ varies with coverage4; the linear variation ~H = ~o -an is the simplest example of this sort. Mimeault and Hansen5

used the linear variation form to represent the second order ,a-hydrogen desorption from tungsten, iridium, and rhodium. To do this, the parameters p and ~Ho were established, by very low coverage desorption experiments using the appropriate form of Eq. (2), and then the right side of Eq. (1) was integrated numeri­cally, with ~=~o-an, for a number of values of a until agreement with experiment was obtained. This process was handled by a computer using a minimum search routine. Th~ parameters ~Ho and a can also be obtained by

graphlCal methods, and the argument involved has, we believe, rathtx significant implications for the evaluation of activation energies from flash desorption data. It will be illustrated for a second-order process with using the hydrogen-tungsten data of Mimeault and Hansen [Figs. 4 and 6 of Ref. (5)].

For a second-order process with ~=~Ho-an, the PW equation can readily be put in the form dn/ dT-I = b- Ipn2 exp[ - (~O-an)RT]. The form of the left side of Eq. (2) appropriate for a second-order kinetics plot is In[n-I-no-l] or In[(no-n)/nno], and points of inflection in plots of this function against T-I will correspond to cent rs of substantially linear sections of such plots. We have d2[ln{ (no-n) /nnol ]/d(r-I)2=0 at point of inflection. Hence at a point of inflection

_n_o_-_2_n __ d_n_ = d [In _d_n_J / dr-I n(no-n) dT-I dr-I

= _ ~Ho-an + (~ ~) dn RT n + RT dr-I' (3)

from which

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