Shapes and sizes of molecular anions via topographical analysis of electrostaticpotentialShridhar R. Gadre and Indira H. Shrivastava Citation: The Journal of Chemical Physics 94, 4384 (1991); doi: 10.1063/1.460625 View online: http://dx.doi.org/10.1063/1.460625 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/94/6?ver=pdfcov Published by the AIP Publishing

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

93.180.53.211 On: Mon, 17 Feb 2014 18:25:55

Shapes and sizes of molecular anions via topographical analysis of electrostatic potential

Shridhar R. Gadre and Indira H. Shrivastava Department a/Chemistry, University 0/ Poona, Poona 411007, India

(Received 18 July 1990; accepted 7 December 1990)

The theorem proposed by Pathak and Gadre [J. Chern. Phys. 93, 1770 ( 1990) ], that the electrostatic potential (ESP) of negative ions must exhibit a directional negative valued minimum along any arbitrary direction has been verified for some small negative molecular ions, viz., OH - , CN - , N3- , N03- , and NH2- • Also, as predicted by Gadre and Pathak [Proc. Ind. Acad. Sci. (Chern. Sci.) 102, 18 (1989)], the molecular ESP (MESP) maps are found to be devoid of local maxima. As a consequence, these maps reveal rich topographical details in the form of several saddle points as well as point minima. From the location of these critical points, estimates of the sizes and shapes of the negatively charged molecular ions are obtained. For anions, there exists a surface on which VV·dS = 0 and which passes through all the negative valued critical points (VV = 0). The ionic size estimates from the location of the critical points of the MESP are found to be in good agreement with the corresponding (spherically averaged) literature values.

I. INTRODUCTION

The electrostatic potential (ESP), for atoms and mole­cules has found wide applications I in chemistry-oriented re­search fields. The ESP at a point r in space, for a molecular system of N nuclei, is given by (in Hartree units)

V(r) = A~I ZAlir - RA 1-J d 3r'p(r')/lr - r'l· (1)

In the above equation, Z A is the nuclear charge of a nucleus (NA being the total number of nuclei) located at RA and per') is the molecular electronic density. The first term in the above equation represents the nuclear potential and the second term is the electronic contribution to the potential.

Applications I of ESP in various branches of chemistry are well documented. For example, the reactivity of organic molecules toward electrophiles is determined by using mo­lecular ESP (MESP) contour maps to predict the site of electrophilic attack. I The Diels-Alder regiochemistry of cy­cloadditions has been modeled by matching the potential surfaces of dienes and dienophiles. 2 Kumar and Mishra3

have mapped MESPs using a dipole so as to gain in-depth knowledge about electric field directions in molecules. In the biological field4 also MESP has found varied applications. One such application is the investigation of mechanism of ion transport by ionophores. Many more biological applica­tions are enlisted in Ref. 1. The nature of bonding in high energy molecules and explosives has been explored by study­ing their MESP maps.5 Several other applications of MESP are listed in Ref. 1, but studies concerning rigorous results on MESP are very rare in the literature.

Weinstein et al.6 were the first to make a rigorous state­ment concerning the nonexistence of maxima in the ESP of spherically symmetric (atomic) systems. The absence of maxima in atomic ESP [putting NA = 1 in Eq. (1) gives the equation for atomic ESP] is easily verified from Poisson's equation:

V" + 2 V'lr = 41Tp(r), (2)

the prime denoting differentiation with respect to r. The nec­essary and sufficient condition for existence of a minimum at rare V'(r) = o and V"(r) <0, butsincep(r) is always non­negative, the occurrence of a local maximum is ruled out for atomic ESP. This theorem has been generalized and rigor­ously proved to be true for molecular electrostatic potential by Gadre and Pathak.7 For MESP, a set of necessary condi­tions for existence of a local maximum is Vx = Vy = Vz = 0 and Vxx < 0, Vyy < 0, Vzz < 0 (Vx denotes partial differenti­ation with respect to x, etc.). Thus, here also the occurrence of a local maximum in MESP is ruled out due to non-negati­vity of per) via the Poisson's equation, viz., v2 v = 41Tp(r) - ~AZA8(r - rA). Minima and saddle points are likely to occur in the MESP. Gadre and Pathak7

have illustrated this theorem for the cyclopropane molecule. Another theorem developed by Pathak and Gadre8 pertains to MESP of negative molecular ions. They proved that there has to be at least one negative-valued minimum in the MESP of negative molecular ions, along any arbitrary direction. From their theorem, it follows that whenever there are two or more local minima, they have to be joined smoothly by regions having negative valued saddle points.8

Several studies investigating the topographical features of atomic and molecular density maps are available in the Iiterature.9

-11 Comparisons between electron density and

bare nuclear potential (BNP) have also been made by Gadre and Bendale. 12 Rigorous studies on electrostatic potential for monoatomic negative ions have been done by Sen and Politzer, IJ but to date no study concerning the maximal! minimal characteristics of MESP has been done. From the theorems due to Pathak and Gadre8 mentioned above, the topography of an MESP map for a negative ion seems to contain a vast amount of information in the form of minima and saddle points. It is possible to relate the location of these points to the shape and size ofthese ions. The latter quantity,

4384 J. Chem. Phys. 94 (6),15 March 1991 0021-9606/91/064384-07$03.00 @ 1991 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

93.180.53.211 On: Mon, 17 Feb 2014 18:25:55

S. R. Gadre and I. H. Shrivastava: Shapes and sizes of molecular ions 4385

in turn, could be related to the literature ionic radii values. An introduction to the radii of anions is provided in the following Sec. II.

II. RADII OF NEGATIVE IONS

The sizes and shapes of atoms and ions have been a sub­ject of considerable interest to chemists for a long time. For example, in most organic reactions, the sizes of the reacting ions determine to some extent whether the reaction will be a favored one or not. Since the size of an ion varies with its environment (solid, solution, gaseous), it is rather difficult to obtain its exact dimensions by anyone method. A practi­cal way for the estimation of ionic radii is offered by x-ray crystallography, by measuring the distance of closest ap­proach. lot In crystals, this distance is considered to be the distance between the centers of two adjacent ions which are in mutual contact. But this method raises two problems, 14

viz., the constancy of the ionic radii so obtained, and the apportioning of the interionic distance between the two ions. Several chemists have tried to solve this problem by giving various defini tions of ionic radii. For a detailed and masterly description of mono atomic ionic sizes and radii refer to Paul­ing. ls

The first approximate values of ionic radii were ob­tained by Lande, 16 by assuming halide ions in lithium halo­genide crystals, to be in mutual contact. Wasastjerna l6 gave more accurate values, by dividing the observed interionic distances in crystals in the ratios obtained from their respec­tive mole refractions. These radii were further modified and extended by Goldschmidt, It> who, based on Wasastjerna's values of 1.33 A for F - and 1.32 A for 0 2

-, deduced em­pirical values for some 80 ions. For polyatomic ions, one encounters 17 experimental difficulties in estimating the ionic sizes. Also, the ion size varies from one compound to an­other, thus creating discrepancies in their measured sizes. Yatsimirskii 17 provided a method to estimate polyatomic ionic radii from Born-Haber calculations. Radii determined by this method are termed as thermochemical radii. For a comprehensive and detailed discussion see Ref. 18.

Recently, Sen and Politzer l3 have developed an electro­static potential (ESP) based model for ionic radii of mono at­omie negative ions. They showed that for negative atomic ions, there exists a minimum in the ESP at a distance r m such that the integrated charge in a sphere of radius r m centered at the nucleus equals the nuclear charge. This rm has been iden­tified with (monoatomic) anionic radius. The rm values for main group elements have been reported 13 which are in good agreement with literature values. However, to date, there is no discussion in the literature on polyatomic ions based on purely quantum mechanical investigations. The aim of this work is thus twofold: (i) To bring out the extremal charac­teristics of MESPs of negative ions, (ii) to correlate the de­tailed topography of these maps to the sizes and shapes of polyatomic anions. A comparison of these values with ear­lier literature values is then carried out. In Sec. III, we give the computational details and descriptions ofMESP features

of individual ionic systems.

III. MESP MAPS AND SIZES FOR MOLECULAR IONS

The systems studied are OH - , CN - , N03- , N 3- , and NH;- . The wave functions for the above systems were ob­tained by running an ab initio program MICROMOL (de­veloped by Colwell et al. 19

) with geometry optimization. The 4-31 G basis sets due to Colwell et al. 19 were used for the first four systems and a 6-31 G basis set for the last one. From the wave functions thus obtained, the MESP was calculated using the parallel version20 of the general program devel­oped by Gadre et a/. 21 The program to compute MESP has been made extremely efficient by judicious applications of rigorous bounds, further parallelized and run on a 32 node parallel computing system (Parsytec). MESP in a plane (10 000) points were generated in less than 100 s for any of the above systems. The exact timings are 39.6,50.48,97.57, 67.54, and 42.03 s, respectively, for OH - , CN -, N03-,

N 3- , and NH2- • The MESP contours were generated using a method in which the function F(x,y) is fitted to spline. The splining method is based on that given by Ahlberg et a/. 22

The graphics were developed on a Hewlett-Packard 9050 AM series computer. In the figures, the contour values, the coordinates of the critical points and the MESP values at these points are all reported in a.u. Along the axes, one unit corresponds to 0.5 a.u. The solid lines and the dashed ones indicate, respectively, the positive and negative MESP con­tours.

It may be observed that the ESPs of the molecular ions display very rich topographical details. In order to explore these, it is essential to locate the critical points of V(r), viz., the points at which VV(r) = O. The expressions for the first order partial derivatives av lax, av lay, and av laz for the various MESP integral types involving s- and p-type Gaus­sians are given in the Appendix. From the figures, the fol­lowing features are observed: There is always a positive val­ued (V(r) > 0) saddle point of type (3, - 1) (two of the diagonal elements of the Hessian matrix are negative) be­tween every pair of bonded nuclei. In addition, there are negative valued [V(r) < 0] saddle points [( 3, - 1) as well as (3, + 1) for which two of the diagonal elements of the Hessian matrix are positive] and minima such that every pair of negative minima is joined by a negative valued saddle point. A nrogram to locate critical points in MESP has been developed in our laboratory. This has been tested out on the systems mentioned above. Detailed description of the MESP features of the individual systems follows.

A.OH- ion

Consider the MESP ofOH - in a typical plane contain­ing the internuclear axis (Fig. 1). There is a positive valued saddle point (denoted by a solid circle) between oxygen and hydrogen nuclei. Further, two minima (shown by asterisk) and two negative valued saddle points (shown by a small cross) may be seen in Fig. 1. From this figure, one can also readily verify the theorem that, starting from any nucleus in the outward direction, there always exists a directional mini­mum. There exists a contour surface of value - 0.18 a.u. engulfing both the nuclei. In Fig. 1, the dotted curve corre-

sponds to a surface on which VV·dS = O. All the negative

J. Chem. Phys., Vol. 94, No.6, 15 March 1991 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

93.180.53.211 On: Mon, 17 Feb 2014 18:25:55

4386 S. R. Gadre and I. H. Shrivastava: Shapes and sizes of molecular ions

I I

I I

I I

/ I

I

I I I I I I I I J

Ii

1 \ \ \ \ \ \ \ \ \

\ \

\ ,

/

,

" " /

J \ \ \ \

/"

" ...-""

""

\ , , , , '" , ,

" " , , , , ....

_--- -0.16-- ___ ,

_-0·18 ......... -- ---- .................. __ -~O'20_ -.................... " -- --........... .....

"....... --....... ............ "

-- - ----' ..... ..... ------ -----

------

.... , " " ........ , ,

, , " " " \ , , \

\ \ \ \ \ \ \ \ \

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ I I ~ d \ I I r f ' I I

I I I

I I / I

I I I I

/ I

/ " I

I I

I I

I I

I I

I

/1 // / /

/" / .,,/ ,//

/ I

I I

I I

.... ....

.. Z

FIG. 1. MESP ofOH - in a plane contain­ing the molecular axis. H is at (0.0,0.0) and 0 is at (0.0, 1.843). The coordinates and MESP values at the critical points are e (0.0, 0.0, 0.659) 0.784, X (0.0, 0.0, 3.849) - 0.422, X' (0.0,0.0, - 2.132) - 0.180, * (1.752,0.0,2.70) - 0.459. In

this figure, the dotted line ... passing through the negative valued critical points is the surface on which VVdS = 0.

valued critical points lie on this surface. That such a surface is a neutral surface can be readily verified by using Gauss's theorem, viz., S V2 V dr = S V V' d S = O. The existence of such a surface gives a hint of the shape and size of molecular ions. This, in fact, is a generalization of Sen and Politzer's'3 atomic anions, i.e., our definition, when applied to a single center problem engenders that due to Sen and Politzer. The very definition of the surface implies that it passes through the critical points (VV = 0). For OH - such a surface is clearly seen to be an ellipsoid of revolution. The long axis of this ellipsoid is about 3.16 A, and the typical diameter being 1.85 A (the distance between two minima). Hence the "spherical" radius is expected to be greater than 0.93 A and smaller than 1.58 A, which indeed provide very good 'bounds' to the literature 'S thermochemical spherical ionic radius of 1.33 A.

a.u. which engulfs both the nuclei. There are two minima, one near the carbon nucleus and the other in the vicinity of

B. CN- ion

The MESP for eN - in the plane containing the nuclei is depicted in Fig. 2. Here also, a positive valued (3, - 1) type saddle point is found to occur between carbon and ni­trogen nuclei and one encounters a contour of value - 0.26

FIG. 2. MESPofCN - in a plane containing the molecular axis. C isat (0.0, - 0.(013) and N is at (0.0,2.195). The coordinates and MESP values at

the critical points are e (0.0,0.0,1.07) 1.260, X (2.67,0.0,0.0) - 0.260, * (0.0,0.0, 4.30) - 0.358, .' (0.0, 0.0, - 2.22) - 0.351.

J. Chem. Phys., Vol. 94, No. 6,15 March 1991 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

93.180.53.211 On: Mon, 17 Feb 2014 18:25:55

S. R. Gadre and I. H. Shrivastava: Shapes and sizes of molecular ions 4387

the nitrogen nucleus. The distance between these two mini­ma is 3.45 A. This distance corresponds to the long axis of ellipsoid of revolution for this ion, which has a cylindrical symmetry. The maximum diameter (distance between the two negative valued saddle points denoted by cross) is 2.83 A.. The "spherical" radius for this ion is thus estimated to be greater than 1.41 A and less than 1.72 A. The literature val­ues are 1.91 and 1.77 A corresponding to the thermochemi­cal 18 and crystal 17 radius, respectively, ofCN-. Thus, the MESP-based definition of CN - seems to yield a somewhat lower estimate than the typical literature values.

c. N; ion (azide)

Figure 3 displays the MESP contour map of N 3- ion in the plane containing the molecular axis. The molecule is lin­ear, with the three nitrogen nuclei lying along the x axis. As for the above ions, this molecule also shows the minimal characteristics as per Gadre and Pathak's8 theorem. Here, the contour which engulfs all the three nuclei is of value - 0.28 a.u. The long axis (distance between the negative

valued saddle points along the molecular axis) is4.64 A, and the maximum diameter (distance between negative valued saddle points along axis perpendicular to molecular axis) is 3.1 A.. These dimensions suggest that the radius of N 3- ion should be greater than 1.55 and less than 2.32 A. The ther­mochemical lR and crystal 17 radii ofN3- are 1.95 and 1.81 A, respectively. Thus for this ion also, the estimates offered by the topographical analysis are well comparable to literature values.

D.NOi ion

Figure 4 depicts the MESP map of N03- in the plane containing the molecular axis. This molecule is seen to have

z

D3h symmetry with the nitrogen at the origin. The minimum valued contour engulfing all the nuclei has value - 0.24 a.u. Since this molecule does not have ellipsoidal symmetry, the long axis and maximum diameter will not be a correct repre­sentation of ionic size for this case. Here the distances of the critical points from the center of the molecule (N nucleus in this case) provide a measure of its shape and size. The dis­tance of the origin from the negative valued critical points given in the figure caption of Fig. 4 are 2.43, 1.78, and 2.01 A, respectively. Along a direction perpendicular to the mo­lecular plane (Z axis) a critical point is found to occur at 3.56 a.u. giving a distance of 1.88 A from the center. Thus the molecule is seen to be slightly compressed from above and below. And as predicted in the above cases, the ionic radius of N03- should not be less than 1.78 and not exceed 2.43 A.. The literature values l7

•18 of ionic radius for N03-

are 1.65 and 1.79 A.. Once again the critical points in the MESP offer fairly good estimates of the ionic radii. Though from the MESP map, the N03- ion seems to be larger than N 3- ion, it is not really so, as seen from the ionic size esti­mates, which is in accordance with the literature ionic radius values.

E. NH2" ion

The MESP map ofNH2- ion, in the plane containing the molecular axes is displayed by Fig. 5. The distances from the center (origin in this case) from the negative valued critical points (as given in the figure caption) are 1.07, 1.47, and 1.90 A, respectively. Thus the ionic radius of this ion is ex­pected to be greater than 1.1 and less than 1.90 A which is comparable to its thermochemical radius 1.73 A.23 The esti­mates obtained by the critical points in the MESP are again in good agreement with the literature value.

FIG. 3. MESP ofNJ- in a plane containing the molecular axis. The nitrogens are at ( - 2.214, 0.0, 0,0), (0.0, 0.0, 0,0), and (2.214,0.0,0.0). The coordinates and MESP at the critical points are. (1.102, 0.0, 0.0) 1.403, X' (4.384, 0.0, 0,0) - 0.358, X (0.0, 0.0,2.93) - 0.280, * (3.85,0.0, 1.5) - 0.364.

J. Chern. Phys., Vol. 94, No.6, 15 March 1991 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

93.180.53.211 On: Mon, 17 Feb 2014 18:25:55

4388 S. R. Gadre and I. H. Shrivastava: Shapes and sizes of molecular ions

I I

I I

/ I

, , ,

FIG. 4. MESP ofNOJ~ in a plane containing the molecular axis. The nitro­gen is at (0.0, 0.0, 0,0), the oxygens are at (0.0, 2.373, 0.0), ( - 2.056, - 1.186, 0,0), and (2.056, 1.186, 0,0). The coordinates and MESP at the

critical points are. (0.0, 1.19,0.0) 1.191, X (0.0,4.592,0.0) - 0.249, X' (0.0, - 3.27, 0.0) - 0.250,. (3.8,0.0,0,0) - 0.285.

IV. CONCLUDING REMARKS

In the present work, characteristics of the topography of MESP of negative molecular ions have been studied. All

~--------

////..... -----­,,/ ----

// , .... " ,,/

" " .;' / ------

/ " " /

............. ", ----- .. ,

.;'..... , " / , / I I

I /

.;' "' ..... / ,,"

I I I I I I

I , I , I ...., I ~ I 0

/ '

/ " " / I ' / /

, I to!. , ",0 6<':) ,?

/ I

" /

I /' " 'I I

I \

I I I I \ \ \ \ \ \ \ \ \ \

\ " " " " "­, .....

"-' .... -

x

-- ... -------- -----

-- -------

these plots are found to be devoid of any local maximum whereas one encounters at least one directional minimum traversing outward along any arbitrary ray. These features are also noticed in the literature MESP maps24 which also conform to the behavior predicted by the theorem due to Pathak and Gadre. R The general maximal/minimal features of MESP of such systems have, however, escaped attention in the earlier works. The present study reveals the rich topo­graphical characteristics of ESP of negatively charged mo­lecular ions. Point minima do appear in these MESP maps. It has been found that there invariably occurs a (3, - 1) type saddle point in between a pair of nuclei of bonded atoms. Other saddle points of type (3, - 1) and (3, + I) are also seen in these maps. The topographical characteristics of MESP maps are much richer than those revealed by the re­spective electron-density ones. These yet remain to be ex­plored and we are in the process of development of a general package for such an analysis. It should be pointed out that the electric field gradients (which are related to the second partial derivatives of V) are natural ingredients of such an analysis.

From the location of the critical points in the MESP maps, the sizes and shapes of the ions can be estimated. The ionic radii 'bounds' as obtained from the critical points, in general, agree very well with the literature values for all the systems studied here. A surface on which VV·dS is zero is found to exist. Such a surface which is also a neutral surface is obtainable from the MESP of negatively charged ions. The shape and size of such a surface gives an approximate idea about the ionic dimensions. Thus the electrostatic potential has the potential to give us some knowledge about the an-

I..-V

FIG. 5. The MESP NH, in the plane containing the molecular axis. The nitrogen is at (0.0, 0.0, 0,0), and the two hydrogens are at (1.523, 1.241, 0,0) and ( - 1.523, 1.241, 0,0). The coordinates and MESP at the critical points are. (0.850, 1.043,0,0) 0.575, • (0.0, - 2.07,0.0) - 0.480, .' (0.0,2.78,0.0) - 0.282, X (2.99, - 1.999,0,0) - 0.221.

J. Chem. Phys., Vol. 94, No.6, 15 March 1991 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

93.180.53.211 On: Mon, 17 Feb 2014 18:25:55

S. R. Gadre and I. H. Shrivastava: Shapes and sizes of molecular ions 4389

ionic shapes and sizes which has eluded a rigorous definition in the earlier literature. Spherical ion is indeed an unrealistic idealization of the molecular anionic shapes. The present work thus provides a rigorous framework to describe the anisotropies and sizes of polyatomic anions via a detailed investigation of the topography of the MESP maps.

ACKNOWLEDGMENTS

S.R.G. would like to thank the Indian National Science Academy (INSA) (BS/RF/111916), the Center for the Development of Advanced Computing (C-DAC), PUne and the Department of Science and Technology (DST), New Delhi (SP /S 1IJOO/85) for financial support. I.H.S. ac­knowledges the award of Senior Research Fellowship by the University Grants Commission (UGC), New Delhi. We are grateful to Sangeeta Bapat and Sudhir Kulkarni for valuable discussions and computational assistance.

APPENDIX: ANALYTICAL EXPRESSIONS OF FIRST PARTIAL DERIVATIVES OF MESP

In this appendix, we give the analytical expressions of the first order partial derivatives of the molecular electro­static potential (MESP) VCr), as defined in Eq. (1) in Sec. I. In the individual type MESP integrals, for Gaussians cen­tered at A and B, there occurs an error function of the type

I

(sls)x= -4Ka(x-Cx )Fj ,

Fm(t) = f u2me- tu'du, (AI)

where t = ( - (a + b) [(x - Cx )2 + (y - Cy )2

+ (z - Cz )2]), a and b are the exponents of the Gaussians centered at A and B. Cx is defined as Cx = (a'A x + b· Bx)/ (a + b), Cy and Cz are defined similarly. The partial deriva­tive of Fm (t) with respect to x is

aFm(t)/ax= -2(a+b)(x-Cx )Fm + I (t). (A2)

The partial derivative with respect to x, of the nuclear contribution as in Eq. (1) in Sec. I is

a [ ZA] 2 - 2: =2: -ZA(x-Ax)/[(x-Ax) ax A II' - RA I A

+ (Y_Ay)2+ (Z_Ax)2]3I2, (A3)

where ZA is the charge of the nucleus centered at R A •

In the following analytical expressions of the first partial derivatives of the electronic contribution in Eq. (I),

K, a, and /3 are defined as 1T/(a + b)exp( - ab / (a + b) (A - B)2), (a + b), and (a - b), respectively. The following notation is used for denoting the first partial deriv­ative of Gaussians i andj with respect to x:

(A4)

The following are the analytical expressions of the partial derivatives involving s- and p-type primitive Gaussians:

(slpx)x = 2K [(1 - 2a' (Ax - Bx)(x - Cx»FI - 2a(x - Cx )2F2],

(slPx)y = 2K [ - 2a(x - Cx ) (y - Cy )F2 - 2a' (Ax - Bx) (Y - Cy )FI],

(Pxlpx)x =K[(2/3(Ax -Bx)/a-2(x-Cx )(1-2ab(Ax -Bx)2/a»FI

(AS)

(A6)

(A7)

- (2(x - Cx ) (2/3(Ax - Bx)(x - Cx) - 1) - 4(x - CX »F2 - 4a(x - Cx )3F3],

(PxlPx)y =K [2/3(Ax -Bx)(x - Cx )-1)

(A8)

X ( - 2(y - Cy )F2) - 4a(x - Cx )2(y - Cy )F3 - 2(y - Cy ) (1 - (2ab /a)(Ax - Bx )2)Fd,

(Px lPy) x = 2K [(a(Ay - By) + 2ab(x - Cx ) (Ax - Bx) (Ay - By) )F/a + «a(Ay - By) (x - Cx )

(A9)

- b(Ax - Bx)(Y - Cy»( - 2(x - Cx)) + (Y - Cy »F2 - 2a(x - Cx )2(y - Cy )F3], (AlO)

(Px Ipy)z = 2K [(2ab /a)(Ax - Bx )(Ay - By) (z - Cz)FI

+ (a(Ay - By) (x - Cx ) - b(Ax - Bx) (y - Cy » ( - 2(z - Cz )F2 - 2a(x - Cx )

X (y - Cy)(z - Cz )F3] (All)

'See, for example, P. Politzer and D. G. Truhlar, Chemical Applications of Atomic and Molecular Electrostatic Potentials (Plenum, New York, 1984).

's. D. Kahn, C. F. Pau, L. E. Overman, and W. J. Hehre, J. Chern. Soc. 108, 7381 (1986).

'A. Kumar and P. C. Mishra, Intern. J. Quantum Chern. 38,11 (1990); 32, 181 (1987); A. Kumar and P. C. Mishra, Proc. Ind. Acad. Sci. (Chern. Sci.) 101, 55 ( 1988) and references therein.

4N. Sreerama and S. Vishveshwara, Proc. Int. Symp. Biomol. Struct. Inter­actions, Suppl. J. Biosci. 8, 315 ( 1985); N. Sreerama and S. Vishveshwara, J. Biosci. 12,175 (1987). ~P. Politzer, N. Sukumar, K. Jayasuriya, and S. Ranganathan, J. Am. Chern. Soc. 110, 3427 (1988).

°H. Weinstein, P. Politzer, and S. Srebrenik, Theoret. Chim. Acta (Berl) 38,159 (1975).

7S. R. Gadre and R. K. Pathak, Proe. Indian Acad. Sei. (Chern. Sci.) 102, 18 (1990).

8R. K. Pathak and S. R. Gadre, J. Chern. Phys. 93,1770 (1990). 9K. Collard and G. G. Hall, Intern. J. Quantum Chern. 12,623 (1977). 'OR. F. W. Bader, T. T. Nguyen-Dang, and Y. Tal, Rept. Prog. Phys. 44,

893(1981). fly. Tal, R. F. W. Bader, and J. Erkku, Phys. Rev. A 21, I (1980). 12S. R. Gadre and R. D. Bendale, Chern. Phys. Lett. 130, 515 (1986). 13K. D. Sen and P. Politzer, J. Chern. Phys. 90, 4370 (1989). I4F. A. Cotton and G. W. Wilkinson Advanced Inorganic Chemistry (Wi­

ley, New York, 1969).

J. Chem. Phys., Vol. 94, No.6, 15 March 1991 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

93.180.53.211 On: Mon, 17 Feb 2014 18:25:55

4390 S. R. Gadre and I. H. Shrivastava: Shapes and sizes of molecular ions

ISL. Pauling, The Nature of the Chemical Bond (Cornell University, Ithaca, 1960).

16A. Lande, Z. Physik 1, 191 (1920); J. A. Wasastjerna, Soc. Sci. Fenn. Comm. Phys. Math. 38, 1 (1923); V. M. Goldschmidt, "Geochemische Verteilungsgestetze der Elemente", Skrifter Norske Videnskaps-Akad. Oslo. I. Mat.-Naturv. KI. (1926).

I7J. E. Huheey,lnorganicChemistry (Harper, New York, 1983); K. B. Yat­simirskii, Izvest. Akad. Nauk SSSR, Otdel, Khim. Nauk, 453 (1947); 398 (1948).

IHR. D. Shannon and C. T. Prewitt, Acta Cryst. B 25, 925 (1979); A. F. Kapustinskii, Quart. Rev. 10, 283 (1956); H. D. B. Jenkins and K. P. Thakur, J. Chern. Educ. 56,577 (1979).

19MICROMOL, an ab initio Quantum Chemistry package developed by S.

M. Colwell and N. C. Handy, University of Cambridge, U.K. See S. M. Colwell, A. R. Marshall, R. D. Amos, and N. C. Handy, Chern. Britain 21, 655 (1985).

20S. R. Gadre, S. Bapat, K. Sundararajan, and I. Shrivastava, Chern. Phys. Lett. 175, 307 (1990).

21S. R. Gadre, S. A. Kulkarni, and I. H. Shrivastava, Chern. Phys. Lett. 170, 271 (1990).

22H. J. Ahlberg, E. N. Nilson, and J. L. Walsh, in The theory of splines and their applications (Academic, New York, 1967).

2.lA. F. Wells Structural Inorganic Chemistry (Clarendon, Oxford, 1975). 24K. D. Sen, J. M. Seminario, and P. Politzer, J. Chern. Phys. 90, 4373

(1989).

J. Chem. Phys., Vol. 94, No.6, 15 March 1991 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

93.180.53.211 On: Mon, 17 Feb 2014 18:25:55