A semi-empirical method for estimating molecular quadrupole polarizabilities

Volume 50, number 2

CHEMICAL

A SEMI-EMPIRICAL

PHYSICS

LET1 FRS

1

1377

Scptcrr1bcr

METHOD FOR ESTIMATING MOLECULAR QUADRUPOLE POLARlZABILlTlES

Eric A. GISLASON and Malini S. RAJAN Dcparrntetzr of Clttvttistry. University of Illitlois Chicugo. Iilitrors

Received

at Ciucago C~tde.

60680. USA

20 May 1977

It IC demonstrated “chern~celly smlthr”

that a lo:+1c.g plot of (ticld-gradient) quadrupolc poldrizablhrlcs agm~s~ dqxAc pOi.lrizabllltle:b of atoms can be well fit with n str.+t lint. The tiieorctical Justific2! tmn for thlb 1s dlscusscti. ‘I Ile hype-

thesis is advanced that the average qundrupole Lability using the be% tit to the atomic data

quadrupole

polari~hility This method

The ion-induced

quadrupole

term in the intermolec-

[ 11.

V(R) = --e2q,/2R6,

(1)

plays an important role in the low energy scattering of ions by atoms and molecules [2]. The quadrupolc polarizability @q (also called the field-gradient quadrupole polarizability [3] has been calculated for several atoms and one molcculc, but no direct experimental determinations are available. Several different notations have been used for quadrupole polarizabihtics. In particular, the quadrupolc polarizability for a spherically symmetric charge distribution (in general, this polariLability is a fourth-rank tensor) is referred to as (Ye by McDaniel and Mason [I 1 and Dalgarno [4], as C by Ruckingham [5], and as C,z_.,z by McT_can and Yoshimine [G]. The numerical relation between them is given by Q9 = 2c=

can be estimated

from

the avcrdgc

d~polc

with ;tn accurately

polnri-

known

polanzablhty.

1. Introduction

ular potential

for a molecule

works well for II 2. the only molecule

3c,,,,,.

(2)

In this paper WC will use crd to refer to dipole polarizabilitics and a(1 for quadrupolc polariLabilitics. For the analysis of ion scattering experirneuts it is only necessary to know the average atomic or molccular quadrupole polarizability [ 1 J _ The anisotropy in (Ye will have a negligible effect on the cross section. Similarly, any anisotropy in the dipole polarizability

at1 can usually be ignored [?I. Accurate values of (Y,I and (Ye have been computed for many atoms and atomic ions using the coupled Ilartrec-Fock method. By comparison, accurate calculation of average molecular values of a9 are only available for 11, 171.

Calculations have been carried out for several linear molecules [8- 101, but results were only obtained for tht axial components of ad and oq_ The average molecular values are probably V~lUC!S.

quite

different

fiorn

the axial

For example, the average value of~r,~ for iIz IS

14% smaller than the axial value [7]. In addition, the results in rcfs. [g-lo] for the dipole polarizabilities often differ markedly from the cxpcrimcntal values. A case in point is the HCI molecule; the computed value is 2.34 R3 whcrcas the experimental value is 3.13 A5 [9]. Th c errors in the quadrupole polariza-

bilitics arc presumably even larger. It would be very useful to have some simple semiempirical method to estimate average molecular quadrupole polarizabilities. Experimental values of the average dipole polarizabilities arc usually available [ 1 1,121 and can be used in the estimation process. Several approximation methods have been put forward in the past [13-IS], but they are not sufficiently accurate for atoms to offer much hope for molecules. In the process of compiling various atomic polarizabilitics WC have observed a fairly simple relationship between ad and aq which should cxtcnd to molcculcs. This relationship is described and developed in the remainder 251

Volurn~

CffEMICAI. PHYSICS LEl-Tf~KS

50, nurllhcr 2

of the paper. In what

follows

the units of ad arc always

Ti~blc1 Poldriabllity --_-

a3 and of o!q are always A5.

Group -_-

2. Theory Exact

1 Scprclltber 1977

_

-

data used in paper --_--. Atoms or molecules _-__

I

A 9 Q‘i = 0 .667 - AS/Y4

LYE= 0.622 As/Z”_

II

(3)

This then gives the relation

Kcferenccs

rdrc g&es Mu isoelectronic SqucnLc Ne lsoclcctronic sequence alkali metal eons alkali atoms

values of CQ and CY~are known for one-eIec-

tron atoms with nuclear charge Z [4]

-_-

1161

1171 IfSI 1191 (2OI 120-221 1231

Li icocIcctronIc sequence III

DCIsoclcotronic bcquenaz

IV

cxcltcd Ilc

isoclcctronic

sequence

= 1.143 @. % An cxpanslon in powers of Z’-l for two-electron atoms and ions gives the result 141

(4)

-.-

(5) The value of a9 calculated for He frorn this relation is in error by 27%, and the results for Ions with larger .Z arc succcssivcly rnorc accurate. Pack Il.51 has described a simple oscillator model which should hold approximately for any atoln 01 ~nolecule. His results for an Isotropic electron distribution cn-n be written PI

I71

-

-

-

lineax

ITlolcculcs -_-

--_--

-

-_--

[S-101

(groups I--W in table 1) are shown as the top curve in fig. 1. The data, which span nine orders of magnitude in CY~,do approximate a straight line with a moderate amount of scatter. WC have obtained a Icastsquares fit to all of the datn using a curve of the form P

log cl4 =

c b,,(Iog

a&

11=0

(6)

All of these results suggest that a log-log plot of crC1 versus ad for chemically “similar” atoms or molecules should give a straight line with a slope of roughly I .5 ar,d a fairly sm:111intercept which wrll vary with the particular system. To cheek this we have plottcd log a4 versus log ad for n wide variety of atoms and posit& alornic ions. The calculation of polarizabilities of negative ions is more difficult [41, and WChave excluded them from our consideration_ Care has been taken to only include systems for which the dipole polari/abMies agree well with experimcn tal values or w-ith very accurate calculations. In general, coupled liartree-Fock computations meet this requirement. The data used in this paper and the sources arc summarized in table I_ Often two or Irlore Iaboratorics have computed with comparable accuncy the polanzabihties of the same sequence. In this case we selected the results which containect the most number of atoms in the sequence. Thus, no claim is made for completeness. We do expect, however, th3t the values used here arc accumtc. The results for all of the atoms and atomic ions 252

1201

bf2 misccllmcou.4

Crq = 0.808 aSJ’.

CY<~ = 0.546 @.

V

Here the logarithms are taken to the base IO, and in most cases p = 1. The goodness of fit was dctcrmined from the standard deviation

C7=

c

(IV - P -

-log

I)--’ 8

c@exact)]

’

[log +(calc)

l/2 1 .

(8)

The best fit straight lint to all of the data is shown in fig. 1, and the fitting parameters are given in table 2. The calculated slope of 1.4179 is close to 1.5 as predicted by eqs. (4)-(6). However, the standard deviation of 0.357 corresponds to an average error of a factor of 2.27. An uncertainty of this magnitude is unacceptable for most work, but may be useful for rough estimates of Qq. The polarizability data for atoms summarized in table 1 has been divided into four chcmiczllly similar groups. Thus group I corresponds to closed shell atoms and atomic ions, group II atoms have one electron beyond a closed shell, group III atoms have two elec-

Volume

SO, number

CtIEhlICAL

2

PIIYSICS

LETWRS

1 Scptcmher

Table 2 Lea\t-squat5 fit p”rm~ctcr5 ------------_-Group

h

----------

II 111

IV I, II, 111. IV ---------

_ _____

bl ----

I

a)

-0.059 -0.072 -0.665 -0.356 --1.125 -0.361 -----

-----_

1977

82 __--

-

”

_________

1.4843 0.105 0.077 1.3581 -0.04102 1.8170 0.042 1.8361 0.013 1.7894 0.158 1.4179 0.357 -----_-._.--_____

“) Tbc various groups dre gn!cn ui t&h! 1. ‘The coefficients b,-,, h 1, and 62 .irc detmed in cq. (7), zmd the stnud.lrd devl;ltlon u was computed from cq. (8).

DIPOLE

POLARIZABILITY

ad (P)

I-I& 1. Quadrupolc polarilabdities in AS plotted against dipole polai/abllitics m A3. Group I atom5 (we t.lblc 1) we shown a5 open cnclcs, group II atoms a5 squares. group II1 atoms us trinnglcs, and group IV atoms as clowd circles. Tlw upper set of data (scale shown to the left) Howe all four groups and the best-tit stmight hue (see table 2). I’hc other two sets of data have been dlrplaccd downward by four and tight powers of ten for clarity. The middle set of data shows the group I atoms and group II atoms and the best-fit straight lint for cacb group. TI:c closed square is the result for 112 from ref. 171. T!lc bottom set of data (scale shown to the right) shows groups III and IV and the best-fit straight hnes for cacb. beyond a closed shell, and group IV atoms are isoelectronic to the He 2% and 23S excited states. trons

Each of these groups is pfottcd separately in fig. 1, and the least squares fit for each group is shown. The least-squares parameters are collected in table 2. In each case the fit is remarkably good. Of particular interest are the results for the closcdshell atoms in group I. The linear fit to the data has a u value of 0.105, corresponding to an average 27% error in aq. Inspection of the data in fig. 1 shows a

definite curvature, so a quadratic least-square tit was also carried out (see table 2). The CTvaiuc of 0.077 corresponds to an average error in 4, of 19%. The poorest lit in the four groups occurs in ‘roup IV. This occurs hec;u.~sc the CX~values for the 2 $ S states arc about 90% larger than the values for the 2 ‘S StateS (normalired t0 tIlC Same “d V&X?). If the 2 3S results and the 2 IS results are fit separately, the u values for both fits arc Iess than 0.03. Ag& this shows the very simple dependence of 0~~on ad for chemically similar atoms. Based on the results obtained for atoms WCexpect that moIecuIcs will bcbave in a similar manner. In particular, we advance the hypothcsls that CX~for molecules with closed subshells urtd no low-lying excited stutes cat1 he accurately cakularcd from qi using tile fit for group I utoms. Similarly, we expect tllilt c~,~ for a molecule such as Lit can bc estimated from CY,]usmg the tit for group II atoms.

3. Discussion The onIy molecule for which the hypothesis can be tested is I~,. The linear and quadratic fits to the group 1 data both predict atI = 0.63 A5 for H, , which compiircs well with the accurntc value of 0.7 1 A5 [7] _ McLean and Yoshimine [8 -101 have computed the axial components of CQ and
I

253

CHEbfiCAL PHYSICS LIZ-I-TERS

Volume SO, number 2

1 September

1977

CH, would provide excellent tests of our hypothesis. Until these calculations are available, however, we suggest that quadrupole polarizabilities for molecules such as these bc estimated from the results for group I atoms using the: formulas surrmiarized in eq. (7) and table 2. These should give results accurate tp 30%. A similar procedure was used in ref. [2].

References

DIPOLE PoLARlzABlLlTY

ad,

22

(x3>

rig. 2. The axial component of the quadrupolc polarizabdity for various hnear mofcculcs in A5 plotted against tile avial component of the dipole polanlabllity in A3. Alkali halide rnolccules arc shown ds squares. The closed circle is the result for HE from ref. 171. 1%~ other data WBS taken from rcfs. [ 8- 101. Ah shown is the best-fit str&llt line for group I (see table 2).

sarily inconsistent

with our hypothesis for two reasons First, the average molecular values of o!d and aq may be quite different from the axial values. This can shift the vaiue of 01 relative to the value of q_ Ijowcver, we note that txi s is not the case for 112 (see fig. 2). More importantly, there is a serious question us to how accurate the numbers computed by McLean and Yoshimine [8-IO] ZIiC. They obtained values of ad

which varied by 20% for CO depending on the particular basis set used. The values of cys varied by 40% for these same basis sets. Their best value of ad for CO is still 20% smaller than the experimental value. We conclude that the basis sets may be inadequate for several of the molecules studied. The problem is probably most severe for those molecules which contain halogen ions. It is clear that accurate

calculations of moiccular quadrupole polarizbililies for molecules other than Ii,, are badly needed. Molecules such as NZ, HF, and

254

[ I] E.W. fifcDanicI and E.A. Mason, The mobdtty and diffusion of ions in gases (Wiley-InterscIence, New York, 1973). l’.IZ. Budenholrcr, IZ.A. Cislason, A.D. Jorgcnccn and J-C;. Sachs, Chcm. Phy\. Letters 47 (1977) 429. AD. Buckin@:nn. Quart. Rev. I3 (I 959) 183. A. D&:lrno, Advan. Whys. I 1 (1962) 281. A.D. Buckingham, Advan. Chcm. Phys. 12 (1967) 107. A.D. McLean and hf. Yoshiminc, J. Chem. Phy\. 47 (1967) 1927. W. Meyer, Chern. Phys. I7 (I 976) 27. A.D. McLean and hf. Yoihirnine, J. Chem. Phys. 64 (1967) 3682. hf. Yoshimine, J. Chcm. Phys. 47 (19G7) 3256. A.D. McLc,ui and hl. Yoshimine as reported III M. Krauss. Cornpcndium of ab rnitlo Calculations of Molecular Energcs and Propertics, Natl. Bur. Std. US lkcir. Note 438 (1967)p. 71. [II] A.A. hlaryott and I . BuLkIcy. Nat]. Bur. Std. US Circular 537 (1963). 1121 I.andolt-BorrIbtcln. Znhlcnwcrtc und I’unktmncr: (Springer, Bcrhn, 1961) p. 510. [I31 H. Margcnau. Phil. Sci. 8 (1941) 603. 1141 A. Dafgarno and J.T. Lewis, Proc. Roy. Sot. A240 (1957) 284. (151 R-T Pack. J. Chcrn. Phys. 64 (1976) 1659. 1161 M.B. Dornn, J. Phys. B 7 (1974) 558. (171 R-F. Stew‘trt and B.C. Wcbstcr, J. Chcm. Sot. Faraday

Tran\. II 69 (1973) 1685. 1181 J. Lahiri and A. hlukhcrji. Phys. Rev. 153 (1967) 386. I’91 R.M. Stcmhcimer, Phys. Rev. 115 (1959) 1198, 183 (1969) 112; A 1 (1970) 321.

[ZO] S.A. Adclman and A. Szdbo, J. Chcrn. Phys. 58 (1973) 687. 1211 M.R. Flnnnery and A.L. Stewart, Proc. Phys. Sot. (London) 82 (1963) 188. [22] P-W. Langboif. M. Karplur and R.P. Hurst, J. Chcrn. Phys. 44 (1966) 505. 123) J. Lahlri and A. Mukhcji, Phys. Rev. 141 (1966) 428; J. Phys. Sot. Japan 21 (1966) 1178.