Continuous Charge Distribution Models of Ions in Polar Media Part 2.-Self and Interaction Energies for Soft Charged Ring Systems Dissolved in a Polar Medium BY PARsURY P. SCHMIDT? Department of Chemistry, The University, Southampton SO9 5NH Received 20th March, 1975 This paper is concerned with the calculation of self and interaction energies for ring type charge distributions as representations of a class of ions (namely, radical ions of the aromatic ring type). The ionic systems are considered to be dissolved in a suitable polar solvent medium. The energy quantities are useful in the calculation of solvation energies. In particular, the interaction energy enters into the consideration of the repolarization energy for an electron transfer transition between donor and acceptor species.The Born model of a solvated ion has been most often used in attempts to calcu- late ionic self and interaction energies in polar media. Apart from its conceptual disadvantages, i.e., a hard metallic shell enclosing a point charge source, it is not particularly well suited to use for the analysis of more complicated charge distribution geometries. This arises partly from the difficulty in carrying out integrations for non-simple charge distributions, even where spherical symmetry can be applied, and partly from the metallic-sphere character of the model. In contrast, in a previous paper an alternate, but still phenomenological, ionic model was introduced which is based on the use of soft ionic charge distributions which are continuous to the coordinate origin.2 The results found on the examina- tion of simple monatomic ionic systems are encouraging.Fairly good agreement with experimental self energies follows with relatively little computational effort. In this paper the previous treatment of soft charge distribution systems is extended to cover cases of more complicated molecular ionic geometry. In particular, simple charged ring systems are considered. Such systems are frequently encountered in connexion with radical ions.3 The self energy figures in the solvation energy, and both the solvation and interaction energies enter the repolarization energy, a quantity which is an important component of the activation energy for an electron transfer transition between a donor and acceptor.CHARGE DISTRIBUTION The form of the charge distribution for the ring system studied in this work is p(v> = P5 sin2 or2exp(-p2r2)n2 where P = J2Ia t permanent address : Department of Chemistry, Oakland University, Rochester, Michigan 48063, U.S.A. ;Fulbright Senior Research Scholar. 171 172 CHARGED RINGS IN POLAR MEDIA and a is an effective Bohr radius.2 It is not convenient to work with the charge distribution expressed in terms of trigonometric functions. Hence, we convert to a form expressed in terms of spherical harmonic functions, Yl,m(P): A further conversion is useful in considering the interaction energy expressions. Specifically, it is convenient to refer the ring system to a space fixed axis system.In this case one most easily does this with the use of the spherical harmonic addition theorem : Thus, another form of p(r) is where f refers to the space fixed axis system ;it is the unit vector in that system. In order to evaluate the self and interaction energies in a polar medium we need the Fourier transforms of the electrostatic fields. These we get from the transforms of the charge distributions through the use of the transformed Gauss law : 1%E(k) = -p(k). (6)k The transform of p(r) in momentum space is given by p(k) =Jd3rp(r) exp(ik .r) (7)with exp(ik r) =471 i'j,(kr)Yz1,(3)Y,,,(ft)l,m andjl(x) is the spherical Bessel function of the first kind.4 Thus, with eqn (3), we find and with eqn (5) p(k) =4712 Y&(f)Yoo(k) (1--k2 -c,y~m~)y~m(1%)exp(-k2/4p2)* (10){The equation for the field transform using eqn (9) is :;2)+ E(k) = -(4z)*Zi-{ Yoo(k)(1-')+(1/5)'Y20(ft)~} exp(-k2/4p2) (11)ft k 6P2 6P2 and with eqn (10) it is P. P.SCHMIDT SELF ENERGY The self energy of an ion immersed in a polar medium can be derived most simply in the momentum space representation 2* ws= --/d3k C,(k)E*(k) 4(2~)~ .E(k) (13) where 2 n=O CN(k) = D, exp(-k2Li). (14) In eqn (14) the A, are correlation lengths associated with spatially disperse fundamental excitations in the polar s01vent.~ The D,depend on dielectric constants associated with each mode of the solvent : D,= l/En+l+l/&n* (15) The above expression gives the self energy with account taken of medium dispersion effect^.^^ In the dispersion-free limit this reduces to W,= -(1-l/&st)-4 d3k E"(k) .E(k) (16)4(24's and ESt is the static dielectric constant.In order to obtain the self energy we evaluate eqn (13) with the use of eqn (11). We find Y&(L)Yoo(&)(l- k2/3p2 +k4/36p4))+-1 k4 -Y* (L)Y,,(L) +180p4 2o J D, fudk(1-k2/3p2+k4/30p4)exp Dn(4/n)*(A-A2/3+A5/5) where and XI =p&. In the dispersion-free limit A = 1, and z2 W, = --(1-1/~,~)(26/15Jn). (19)2a The above results may be compared to the simple spherical case for which one finds z2 w;Ph = --CDn(4/z)'(A-A3/3+A5/12).2a n CHARGED RINGS IN POLAR MEDIA The ring system self energy is larger than that of the simple spherical system by an amount Aw~= -(Z2/2a)CDn(1/~)'(7/30)A5 n which can be about 20% greater.In the dispersion-free limit the difference is Aw:') = -(Z2/2a)(1-1/~,t)(7/30Jn). Thus, one readily sees that as far as self energy quantities are concerned, more com- plicated ionic geometries only change slightly the result one would obtain using a spherical model representation. This result, however, is strictly valid only for vacuum charge distributions. It is expected to be approximately true in polar systems -see, the discussion section. ELECTROSTATIC INTERACTION BETWEEN THE CHARGED RING AND A SPHERICAL CHARGE DISTRIBUTION The formula for the interaction energy for two charges is '(k)E"(l, k) .E(2, k) exp(ik .R)wi = 2(244 d3k~,t~ where E(1, k) and E(2,k) are the Fourier transforms of the fields arising from the electrostatic charge distributions at sites R, and R, : R = R2-R1.(22) We let E(1, k) be given by eqn (11) and E(2,k) by &E(2, k) = -(4n)%-YOO(k)Zsphexp(-k2/4p2) (23)k which is the transform of the field of a spherical charge distribution of the form p(r) = Zsph(q3/7+) exp(-q2r2) (24) normalized to the net charge in the distribution : Sd3rp(r) = Zsph. (25) In eqn (21) ~,t'(k)is the Fourier transform of the medium permittivity : 2-5 E, '(k) = 1-C,(k). (26) The specific expression for the interaction energy is ~wi = ""Jd3k ~,t'(k)(Y~~(k)Y~~(k)(2- k2/3p2)+2(2n)4 (1/ 15)) Ygo(f)Yz'( &)(k2/p2) k -ex p( -k2(p2+q2))exp(ik .R).4p2q2 (27) In order to evaluate this expression we use the Rayleigh expansion exp(ik .R) = 4n iyl(kR)qtlt(l?)Y&(&)1,m together with the spherical harmonic integral relation /4z d~ky:rn,<k>K,nt,(k>~linl(~ where C(Zl, Z2, Z3 ;m,, m,, m,) is the CIebsch-Gordon c~efficient.~ P.P. SCHMIDT Carrying out the angular integrations in eqn (27),we get the radial integral for wi : k2(P2+q2>(k2/p2)j,( kR)( 1-n Although there are six integrals in (30), only three of these need to be evaluated. If we write I-J dt t2"j(tR/a(x,, z))e-", (s = 0, 1); (Z = 0,2) (31)a(x,, z)2s+i 0 where t = a(x,, z)k, z = p/q, x,,= pi,, and then three of the integrals follow with x, = 0. The last definition, eqn (34),is useful in the final expression of the interaction energy. Define the function F(xn,Z) = +(p2R2A2(xn, 4PRz)>--A3(x,, z) x 3Jn ~XP(-P2R2A2(xn, ~ ) ) -1~ ~ 2 cos e-j1) x ~~2 ~-(3 {$4(p2R2A2(xn, a))-FA(,,,,z)J2 The above function F(x,, z) follows naturally from the integrals in eqn (30).The interaction energy can now be written as ZZ ~i = "-'{F(O, z)-C D,F(x,, z)]. (37)R n In the dispersion-free limit this reduces to CHARGED RINGS IN POLAR MEDIA The first two terms in F(x,,, z) constitute the contribution to the interaction from a Is and 2s type Gaussian radial charge distribution.2 The last term gives the angular contribution due to the distortion of the charge from a sphere into a ring. INTERACTION BETWEEN TWO CHARGED RINGS In this section the interaction between two charged rings separated by a distance R is considered.The form of the interaction energy again is given by eqn (21). However, now the field transforms given by eqn (12) are used for both charge distributions. Consider first the problem of the angular integrations : dokE*(k).E(k) exp(ik .R). 14n The product of the field quantities is E*( 1, k).E(2, k) = (~Tc)~Z,Z~~-~{ Yoo(I)Y$o(2)Y&(ft)Yoo(E)x (1 -k2/6p2)(1-k2/6q2)+ Yoo(I) Ygo(k)iYtI(2)Yzm(k) x -2 With the use of the integral (29) together with eqn (28), one finds the interaction energy reduces to the k-space radial integral (1 -k2/6p2)(1- P. P. SCHMIDT I77 The quantities I,,,(x,,,z) in eqn (41) and (42) are defined by the following (with a(x,, z) defined as before by eqn (33) and (34)) : 7c Il(x,,,z) = dt jo(tR/a) e-" = -4(R/2a) (43)a0 2R -Sm Jndt t2j,(tR/a) e-f2 = -exp(-R2/4a2) (44)4a ne-'2 = T(3-R2/2a2) exp( -R2/4a2) (45)8a 14(xn, 2) = -fodt t2j2(tR/a) e-I2 a3 0 = 4R25{6@(R/2a) -(R/a)(6 +R/a)exp(-R2/4a2)) (46) 15(xn, 2) = -{a dt t"j,(tR/a) e-" = @ exp(- R2/4a2) (47)as 0 16 a7 = ~{105,/?~(R/2a)-(R/a)[IO5+Y(R/a)' + 2~5 i(R/a)6]exp(-R2/4a2) (48) (The arguments of a(x,, z) have been suppressed for convenience.) In the dispersion-free limit the interaction energy can be expressed as 1-[f2(3 COS~8, -i)+(3 COS~e2-I)]I~(o,12p2 2)- CHARGED RINGS IN POLAR MEDIA Z2-4(sin2 sin2 e2cos[2(+, -42)][13(0, Z)++QI~(O, Z)++I,(O, z)] -240p sin 28, sin B2cos(q51-(p2)[13(0, z)-+I5(O, z)-?I,(O, z)] + The multipole moments of the charge distribution are defined by qlm= Jd’y’r’’ Y~(~’)P(Y’) (50) and the components of the quadrupole tensor by Qfj = Jd3x‘(3xtx;-r’26ij)~(~’).(51) Thus, with the use of the charge distribution, eqn (5), it is possible to write the function G(xn,z) as This expression clearly indicates the dependence of the interaction energy on the mono- pole and qzadrupole components of the individual ring charge distributions. More-over, it reveals a marginal dependence on the quadrupole components as compared to the monopole terms. DISCUSSION It is essential to point out that strictly speaking the theory for the solvation and interaction energies derived in this paper is invalid.Nevertheless, it is expected to be reasonably accurate under certain conditions. The lack of validity of the theory is a result of the fact that for a homogeneous, isotropic dielectric the self energy can only be calculated for a uniform charged sphere or spherical charge distribution. In connexion with the interaction energy quantities, all theories based on Maxwell’s static equations and the constitutive equations for isotropic dielectrics are invalid. For large inter-ionic separations, however, one reasonably anticipates that the inter- action energy quantities will differ from whatever the true quantities are by a very small amount. The homogeneous, isotropic dielectric is defined only for systems devoid of external field source charge distributions embedded within the medium.Thus, as soon as a test charge is introduced into the dielectric medium, the condition of isotropy is violated.” This causes no problems for the calculation of the self energy for spherical charge distributions. However, for the ring systems considered here there will be an error. In terms of the k-space integrations used in the derivation of the self energy, * I am indebted to A. A. Kornyshev for a discussion of this problem. P. P. SCHMIDT eqn (13), the expression may be valid in the range 0 < k < kmax is the inverse of the distance, measured from the centre of the charge distribution, less than which the isotropy of the medium ceases to be a reasonable approximation. If we limit the range of integration as indicated, the contribution to the self energy which can be expected to be accurate is where Y Yh4, B3A(2m-1, x,,,a: amax)= (I +2x3"-3 (54) and y(m-3, B:) is the incomplete gamma function ?(in -3,B,f)= (nz-3/2) y (m-3/2, Bi)-Bim-3exp( -Bi) (55)m# 1 and finally, B, is a 1+2x3'B, = amax( (57) The inner region possibly can be treated discretely, that is, in the manner of Bernal and Fowler or Muirhead-Gould and Laidler lo to fill in the remaining contribution.The equipotential energy surfaces of the charge sphere are concentric shells (an infinity of them) around the source charge sphere. For a non-spherical charge distribution the equipotential surfaces are neither concentric around the source nor are they even spherical themselves.Thus, it is conceivable that the value of amax = necessary TABLEVA VALUES OF THE FUNCTION G(0,l) (WHICH IS A FUNCTION OF R)FOR THREE RING-RING CONFIGURATIONS : A : (~,O)(O,O)B : (90,0)(0,0) C : (90,0)(90,0) WHERE THE NOTATION IMPLIES (e,,~l)(e,,~,)FOR RINGS 1 AND 2. G(0,l) x 10-1 RIA A B C 4 8.67 9.37 10.01 5 9.21 9.74 10.34 6 9.46 9.85 10.30 7 9.60 9.89 10.21 8 9.70 9.92 10.16 9 9.76 9.94 10.13 to approximate the distance from the origin at which the equipotential surface is essentially closed may be extremely large. Such an enclosed spherical region easily can contain such a large number of molecular species and ions as to preclude any economic and reasonable computer analyses.We know that the results we have obtained are accurate in vacuum. It is also suspected that the correction for dielectric anisotropy and inhomogeneity may be reasonably small. Therefore, we proceed to make several observations concerning the results as they apply to the electron transfer problem mindful of the restrictions which must be placed on the credibility of the numbers obtained. CHARGED RINGS IN POLAR MEDIA In table 1 is found a representative list of the values of the G-function for three distinct angular configurations calculated in the dispersion-free limit. It is immediately clear that the quadrupolar component contribution to the interaction energy over that of the monopole contribution is marginal.The three configurations for which data are given are illustrated in fig. 1. Rotations of the individual rings about the x-axes by their angles q51 and 42 do not produce any configurations with energies higher or lower than the limiting forms given in the table. This result may seem somewhat surprising. However, a similar situation exists with respect to the interaction between two dipoles. FIG.1.-The three limiting orientations for the ring-ring interaction energy. (a) (0,O)(O, 0); (b)(90,0)(0,0); and (c) (90,0)(90,0). Concerning the electron transfer reaction, the repolarization energy, which enters into the activation energy, can be written as ’ Jd3k Cn(k)E*(l, k). E(1,k)+E*(2, k). E(2, k)- Er = 404 2(2n)4 d3kCn(k)E*(l, k) .E(2, k)exp(ik.R).--s This expression applies to the radical anion-neutral ring system electron transfer. Using the results of the previous sections, this expression can be evaluated. In the dispersion-free limit (which is used here merely for ease in illustration) one finds The example envisaged is benzene and its radical anion. If one assumes that the effec- tive Bohr radius is 2A, and that the separation distance is 7 A, then the repolarization energy in hexamethylphosphoramide with a refractive index of n = 1.55 and a static dielectric constant of 30 would be (0, O)(O, 0) ;E, = 4.86 eV (90,0)(0,0) ;E, = 4.85 eV (90,0)(90,0) ;E, = 4.83 eV for the three distinct configurations.* As the difference between these energies is * These repolarization energies are somewhat high and probably reflect the fact that no account has been taken of ion pair formation between the radical anion and alkali (or other) counter cations in the solution.P. P. SCHMIDT very small, it is reasonable to expect that electrostatic factors alone play only a mar- ginal role in the determination of the inter-ring configuration, or orientation, in the electron transfer transition state. Rather, this work suggests that ring-ring quantum overlap factors will predominate. The electronic symmetry representations of the initial and final states must be the same according to the von Neumann-Wigner rule.’ This will remain true even with the modifications of Herzberg and L~nguet-Higgins.’~ Thus, the conformation of the transition state should be determined by examining both the ring-ring overlap quantities and the electrostatic matrix elements between the initial and final states as functions of the relative ring-ring orientations.This should enable one to decide the nature of the transition state for the electron transfer. .85 I I I I I I 4 5 6 7 8 9 A FIG.2.-The behaviour of the function G(0, 1) for the ring-ring orientations (a), (0, O)(O, 0); (b), (90,0)(0,0) ; and (4, (90,0)(90,0>. There is, however, one instance in which the electrostatic interactions can play a predominant part in the determination of the activation energy and hence the transi- tion state. In the case in which the reacting species must approach sufficiently close together that it is possible to consider only a vacuum interaction, then the inter-ionic interaction becomes iniportant.The repolarization contribution due to the medium degrees of freedom will be accounted for in terms of the charge density of the aggre- gate. This term may not be greatly dependent on the ring quadrupole moments. Obviously, these considerations, including the inter-species attraction or repulsion energy, will not be very important for radical anion-neutral species transfers. Hovv-ever, for disproportionation reactions involving two radical anions, for example, the inter-ionic repulsion will be a major contribution to the activation energy if the species approach sufficiently close together at the instant electron transfer takes place.It is a relatively simple matter to extend these calculations to include naphthalenic molecules. Such a system would be represented as a charged ellipse. Although it would seem that because of the smallness of the quadrupole components of the poten- CHARGED RINGS IN POLAR MEDIA tials and fields, the value of this analysis is limited, this is not the case. The equili- brium charge distribution of an ionic system is important in the determination of the modes of excitation of the ionic inner sphere.14 It is quite likely, therefore, that the nature of the ionic charge distribution, which includes a distribution over inner sphere solvent molecules, will be important with respect to inner sphere reactions. This work was supported in part by a grant from the S.R.C.I wish to thank Prof. M. Fleischmann for his hospitality and support given to me. I have derived considerable stimulation and guidance from conversations and correspondence with Dr. A. A. Kornyshev for which I am thankful. 'M. Born, 2.Phys., 1920, 1,45. P. P. Schmidt and J. M. McKinley, J.C.S. Faraday ZI, 1976, 72, 143. N. Hirota, in Radical Ions (Interscience, New York, 1968). G. Arfkin, Mathematical Methods for Physists (Academic Press, New York, 2nd edn., 1970). R. R. Dogonadze and A. A. Kornyshev, Phys. stat. solid (b), 1972,53,439 ;1973,55, 843. R. R. Dogonadze and A. A. Kornyshev, Doklady Akad. Nauk S.S.S.R., 1972, 207, 896; English translation, Proc. Acad. Sci. U.S.S.R.,207, 983.'M.E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957). J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962). J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1933,1, 515. lo J. S. Muirhead-Could and K. J. Laidler, Trans. Faraday Soc., 1967, 63, 944. A. A. Kornyshev, personal communication. l2 J. von Neumann and E. Wigner, Phys. Z., 1929,30,467. l3 G. Herzberg and H. C. Longuet-Higgins, Disc. Faraday Soc., 1963, 35, 46. l4 P. P. Schmidt, J.C.S. Faraday ZZ,submitted. (PAPER 5/546)