S . W. Harrison, H.-J. Nolte, and D. L. Beveridge
2580
curate equation of state, eq 34. An improved theory of mixtures must begin by finding a better equation of state. This raises the important question of the nature and source of the imperfections in the present theory: are they inherent to the lattice model used to describe the fluid state or are they largely due to the mean field (random mixing) approximation used to evaluate the partition function? Our opinion is that i t is the latter and not the former. This view has never been (and probably never will be) unanimous among theorists. The lattice model has been criticized and rejected by many as a viable model of a mixture. Its demise may have been premature. The simple generalization of the lattice model presented here to incorporate what can be called free volume yields a unified theory of liquid and gaseous mixtures which is richer in predictive power than many would have ever thought possible. References and Notes (1) I.C. Sanchez and R. H. Lacombe, J. Phys. Chem., 80,2352 (1976); Nature (London), 252, 381 (1974). In the latter reference an expression for the free energy of a binary mixture was given for the special case v,* = V2'.
(2) P. J. Flory, "Principles of Polymer Chemistry", Cornell University Press, Ithaca, N.Y., 1953, Chapters 12 and 13. (3) P. J. Flory, R. A. Orwoll, and A. Vrij, J. Am. Chem. Soc., 86, 3507, 3515 (1964). (4) P. J. Flory, J. Am. Chem. SOC.,67, 1833 (1965); A. Abe and P. J. Flory. ibid., 87, 1638 (1965). (5) B. E. Eichinger and P. J. Flory, Trans. Faraday SOC.,2035 (1968).
(6) R. L. Scott, "Physical Chemistry: An Advanced Treatise", Vol. Vlll A, H. Eyring, D. Henderson, and W. Jost, Ed., Academic Press, New York, N.Y., 1971, Chapter 1. (7) E. A. Guggenheim, Proc. R. SOC.London, Ser. A, 183, 203 (1944). (8) E. A. Guaaenheim, "Mixtures", Oxford Universitv Press, London, 1952, Chapters% and XI. (9) E. A. Guggenheim, "Applications of Statistical Mechanics", Oxford University Piess, London, 1966, Chapters 4 and 7. (10) R. L. Scott and P. H. van Konynenberg, Discuss. Faraday SOC., 49, 87 (1970). (1 1) D. Patterson, Macromolecules, 2, 672 (1969). (12) P. I. Freeman and J. S. Rowlinson, Polymer, 1, 20 (1960). (13) S. Saeki, N. Kuwahara, and M. Kaneko, Macromolecules, 9, 101 (1976). (14) S. Krause, J. MacromolSci. Rev. Macromol. Chem., C7, 251 (1972). (15) L. P. McMaster, Macromolecules, 8, 760 (1973). (16) J. Winnick, Chem. Techno/., 5, 177 (1975). (17) D. Henderson and P. J. Leonard, "Physical Chemistry: An Advanced Treatise", Vol. Vlll B, H. Eyring, D. Henderson, and W. Jost, Ed., Academic Press, New York, N.Y., 1971, Chapter 7. (18) J. C. Chu, R. J. Getty, L. F. Brennecke, and R. Paul, "Distillation Equilibrium Data", Reinhold Publication, New York, N.Y., 1950. (19) J. Timmermans, "Physico-Chemical Constants of Binary Systems In Concentrated Solutions", Vol. I, Interscience, New York, N.Y., 1959. (20) J. G. Roof and N. W. Crawford, J. Phys. Chem., 62, 1138 (1958). (21) V. Mathot and A. Desmyter, J. Chem. Phys., 21, 782 (1953). (22) C. P. Brown, A. R. Mathieson, and J. C. J. Thynne, J. Chem. SOC., 4141 (1956). (23) A. R. Mathieson and J. C. J. Thynne, J. Chem. SOC., 3708 (1956). (24) A. W. Francis, Adv. Chem. Ser., No. 31 (1961). (25) A. W. Hixson and A. N. Hixson, Trans. Am. lnst. Chem. Eng.. 37, 927 (1941). (26) D. F. Othmer and S. A. Savltt, lnd. Eng. Chem., 40, 168 (1948). (27) M. B. King, "Phase Equilibrium in Mixtures", Pergamon Press, Oxford, 1969, Chapter 2. (28) A. R. Mathieson and J. C. J. Thynne, J. Chem. SOC.,3713 (1956). (29) J. M. Bardin and D. Patterson, Polymer, I O , 247 (1969).
Free Energy of a Charge Distribution in a Spheroidal Cavity in a Polarizable Dielectric Continuum Shirley W. Harrison, Science Department, Borough of Manhaftan Community College of the City University of New York, New York, New York 100 19
Hans-Jurgen Nolte, and David L. Beveridge' Chemistry Department, Hunter College of the City University of New York, New York, New York 70027 (Received April 22, 7976) Publication costs assisted by the CUNY Faculty Research A ward Program
General expressions for the free energy of an arbitrary discrete electrostatic charge distribution in a prolate or oblate spheroidal cavity in a homogeneous polarizable dielectric continuum are derived and discussed. The derivation is checked by correspondence with expressions for special cases previously reported and by correspondence with the analogous expression for the spherical case as the cavity dimensions approach those of a sphere. The results are useful for elementary theoretical treatments of environmental effects on molecular structure and properties.
I. Introduction for The classical contiuum model is a convenient reducing the dimensionality of theoretical treatments of molecul& in condensed phases. The dissolved system is represented as a set of discrete charges q k a t positions rk or as a continuous charge distribution p(r) in a cavity imbedded in a polarizable dielectric continuum. The charges induce a POlarization in the dielectric, giving rise to a reaction potential @R(rk)which acts back on the dissolved charges. The energy The Journal of Physical Chemistry, Vol. 80. No. 23, 7976
of this interaction is just the difference between the reversible work of assembling the charge distribution in the presence of the and under vacuum and is given by 1 qk@R(rk) (1) A = ;k for a discrete charge distribution and by
p ( r ) h ( r )dr
(2)
Charge Distribution in a Polarizable Dielectric Continuum
2581
for a continuous charge distribution; A can be identified as a Helmholtz free energy. The free energy of a charge distribution in a spherical cavity in a polarizable dielectric has been fully treated by Kirkwood,l with the energy of an ion in its reaction potential (Born charging term2) and.the energy of a dipole in its reaction field (Onsager term3) coming out as special cases. The spherical cavity continuum model has been used extensively in diverse application^,^ including a recent theoretical study of ion hydration using the statistical thermodynamic supermoleculecontinuum model from this laboratory.5 To facilitate treatment of dielectric saturation, we have also extended the spherical cavity formalism to a charge distribution in a cavity imbedded in concentric dielectric continuaa6 The spherical cavity formalism has serious limitations when applied to polyatomic molecules of diverse shapes. Thus Kirkwood and Westheimer7 in their treatment of electrostatic effects on the dissociation constants of organic acids extended the continuum model to a prolate spheroidal cavity. Their treatment was restricted however to dipole polarization for the case in which the dipole is situated on the axis of the spheroidal cavity. The dipole energy for a molecule in a ellipsoidal cavity was also introduced into the theory of the dielectric constant of polar liquids by S c h ~ l t eand ~ , ~later by Abbott and Bolton.lo A general consideration of the ellipsoidal cavity case including polarization due to all electric moments of the charge distribution has not yet to our knowledge been given. In the course of extending the statistical thermodynamic supermolecule-continuum methodology to dissolved molecules, the need for general cavity formalism arose and was developed forthwith. In view of the numerous other possible applications of this formalism we present herein the derivation of the general cases of the polarization energy of a charge distribution in prolate and oblate spheroidal cavities imbedded in a polarizable dielectric continuum. The method involved, completely parallel to Kirkwood's general treatment of the spherical cases, is briefly reviewed in section 11. The derivations of te energy for the prolate and oblate spheroidal cavity cases follow in sections I11 and IV, respectively. The derivation is discussed in section V. 11. Background
The derivation of the energy in its reaction potential for a charge distribution in a prolate spheroidal cavity, formed by rotating an ellipse about the major axis, is based on Laplace's equation in prolate spheroidal coordinatesll (A, p, 4) V2(A, II., 4 ) m , P, 4) = 0
(3)
where V2 is the Laplacian operator in this coordinate system and 9 is the potential at all points other than the sites of point charges in a discrete charge distribution or at all points outside a continuous distribution. The potential inside the cavity @i and the potential outside the cavity a0are formulated separately, and related by the boundary condition
@i(h A ~)A=XO =
P, ~ ) X = A ,
(4)
applied to assure the potential is continuous across the cavity surface X = XO and the boundary condition ti[a@i(A, P, + ) / ~ X ] X = A = ~ to[a@o(X, p ,
4)/a~]x=A~ (5)
applied to assure that the normal component of the dielectric displacement vector is continuous across the boundary ellipsoid. These conditions are sufficient to form an expression for 9 ifrom which the reaction potential 9~can be extracted.
The corresponding energy is then calculated from eq 1or 2. Analogous considerations hold for a cavity formed by rotating an ellipse about the minor axis except that the Laplacian operator in oblate spheroidal coordinates is used. The general derivation of the free energy for a charge distribution in a dielectric is reviewed by Beveridge and Schnuelle in a spherical cavity in ref 7 . The notation and development of the spheroidal cases discussed herein are organized so as to correspond as fully as possible to this treatment, and the expressions developed for spheroidal reaction potentials go smoothly over to those for the spherical reaction potential as the spheroid becomes a sphere. 111. T h e Prolate Case
Laplace's equation in prolate spheroidal coordinates is
where A and p are defined as A = (rA p
+ rg)/2d
15 A
= (rA- r ~ ) / 2 d
-< m
-1 5 p 5 1
(7) (8)
and + indicates the azimuthal angle around the rotation axis, 0 _< 5 27r. Here r A and rB are the distances of a point from the two foci a distance 2d apart. The general solution to Laplace's equation appropriate for the problem under consideration my be written as
+
@(A, P , 4) =
E
5
n=O m=-n
[Bnmdnpnm(A)
+ 6nmQn n(A)/dn+']Pnm ( p ) e L m @ (9) where the P, and Qn are associated Legendre polynomials of the first and second kind, respectively. Due to variance in the definitions of these quantities in the literature, we have included explicit considerations on the conventions followed herein in the Appendix. The powers of d have been introduced to keep the dimensions of the constants identical with those in the spherical case.6 The quantities Bn, and G,, in eq 9 will be determined by the boundary conditions. However, in order that the formulas derived for the spheroidal case reduce properly to formulas derived previously for the spherical case as the cavity dimensions approach those of a sphere, we define S n m and G n , in terms of the constants E,, and E,,, according to snrn
=
2 n n ! ( n - m)! Brim (2n)!
and
and determine E,, and E,, by application of the boundary conditions. The formulation of 9 i and a0follows from eq 9. Inside the cavity, the potential is taken as Cpi
=
ti-l
2
+n
2 [Bn,dnPnm(A)
n=O m=-n
+ G,
Qn
(A) / d n+ l]P, ( p )e im@ (12 )
where ti is the dielectric constant inside the cavity, taken as unity in most applications but included here for generality. The Journal of Physical Chemistry, Vol. 80, No. 23, 1976
S.W. Harrison, H.-J. Nolte, and D. L. Beveridge
2582
The terms involving Q n m ( A ) / d n + lcan be identified with a multipolar expansion of the potential due to the central charge distribution
a general expression for the reaction potential of an arbitrary discrete charge distribution in a prolate spheroidal cavity QR=
-
f n
C C y n m s P n m ( A ) P n m ( p ) e i m (24) @
ti-1
n=O m=-n
where G ,
The free energy of the system in its reaction potential follows from eq 1using eq 24 for QR. Substituting and collecting terms in n results in
is defined as in eq 11 with
(14) The terms involving d n P n m ( A )in eq 12 define the reaction potential inside the cavity Using eq 11for G,,
and eq 14 for E,, results in
Outside the cavity, the potential is
where analogous to eq 11
+ +
(2n I)! (17) Cnm 2"n!(n m)! In 9, the coefficients of terms in d nPn (A) must be zero in order for the potential to vanish properly at infinity. The constants B,,, E,,, and ,C , are related by the boundary conditions of the problem. Since the Legendre polynomials constitute a linearly independent set of functions, the boundary conditions of eq 3 and 4 can be applied term by term to Qi and Go. The cavity surface is defined by A0 = a/d, where a is the semimajor axis of the prolate spheroid. Equation 3 leads to @nm = (-1)m
A first check on the derivation up to this point can be effected in terms of the reduction of the expression for the spheroidal cavity reaction potential to the expression for the spherical cavity reaction potential as the cavity dimensions approach those of a sphere. The eccentricity e of an ellipsoid is given by dla. A sphere has e = 0, hence d = 0. For the limiting case we havell r lim X = d
rA
= rB = r
d-0
lim
p = cos 6
(27) (28)
d -0
and eq 4 leads to lim P n m ( p )= Pnm(cos6 ) d-0
-
(31)
where
Using eq 30 and 31, eq 13 reduces as d
0 to
and similarly for Qnm(A). Eliminating the e,, from eq 18 and 19, we have
which is identical with eq 5 of ref 6; using eq 29 and 31, eq 14 reduces as d 0 to
where
i.e., eq 6 of ref 6. The limiting behavior of .fin,,, can be examined by using eq 23 as d 0. The limiting expressions for P, (A) and Qn (A) can be found analogously to those given for Pnm(A)and Qnm(A).ll
-
-
lim
with t = co/ti. A simple rearrangement of eq 22 leads to
P , ~ ( A=)
d -0
(2n)! An-1 2 q n - l)!(n - m)!
(34)
and showing that the quantity ynm is a generalization of the quantity y introduced by Abbott and Bolton. With the 9 n m thus defined in terms of the,,&, the basic characteristics of the charge distribution, we use eq 11,14,21,22,and 15 to find The Journal of Physical Chemistry, Vol. 80, No. 23, 1976
-
Substituting eq 29,30,34, and 35 into eq 23 we obtain for the limiting case as d 0
Charge Distribution in a Polarizable Dielectric Continuum lim ynm=
d-0
22"(n!)2(n - m ) ! (2n)!(2n l)! ( n 1)(1+6 ) X (n l ) E n
+
(-l)m
+
+ +
2583
lim
@R=
-
+n
n=O m=-n
(44)
(1- p2)1/2= (rA - r ~ ) / 2 d
(45)
and 1
(36)
Xo2n+l
'From eq 36 and 2 1 and using once again eq 11 and 33, we find d-0
+ 1)ll2 = (rA + r ~ ) / 2 d
(Az
+ + +
( n 1)(1- E ) ( n 1)t n
where rA, r B , and 4 are defined as in the prolate case, but the rotation axis is now the minor axis of the cavity. The general solution to Laplace's equation for the oblate case is
(f)"'Q,
t ,&, identical with the expression for the spherical cavity reaction potential obtained from eq 7 and 13 in ref 6, where a is now the cavity radius. Also
where Qn
=
ti
= 1is assumed, and
cM
M
k = l I=1
qkqlrknrkl
Ol)eim(4i-Qk)
where once again we follow the defnitions of Legendre funtions as given by Smythe.ll The entire derivation of the free energy for the oblate case follows an identical line with that given explicitly above for the prolate case, except that X is everywhere replaced by iX and d is everywhere replaced by d/i. The result is
(39)
Here M
+ l ) d n IC q&'n(h)Pn(pl) = 1
where
-
This expression is found to pass smoothly over to the expression for the spherical case as d 0.
V. Discussion In view of the extensive application of the Born and Onsager terms in problems where the spherical cavity applies, we discuss here the analogous terms for the spheroidal case in more detail. For n = 0, we have from eq 23, A9, and 33 Yo0
M
= (2n + l ) d n C q P n b 1 ) I= 1
(41)
since Pn(Xl)= 1for a charge 41 at a point pl on the line joining the foci. Thus for this case @R
=
-
1-€ 2n+1 =cl n=O
(46)
(n+ m)!
so that the individual terms given in eq 26 reduce to those for the spherical cavity polarization free energy from ref 6. A second check on the derivation comes from reproducing the expressions for special cases of the prolate spheroidal cavity problem taken up elsewhere in the literature. The formalism specific for charge disposed on a line joining the foci of such a cavity has been described by Westheimer and K i r k ~ o o dThis . ~ problem is an application to eq 24 with m = 0, hence
= (2n
( p ) eim$
(n- m)!
+n m=-n
x Pnm(cos e h ) P n m ( C O S
&no
(i A ) ] Pn
=
(e)
+
Xo 1 Xo-1
- In 2
(49)
and
Qn(Xo)Qn(Xo)
1 Pn(Xo)Qn(Ao) - - P n ( X d Q n ( X o ) M
X
C
1=1
41Pn( p l ) P n(X)Pn(/J)
(42)
identical with the reaction potential contained in the combined eq 13 and 8 of ref 7 . Here also is contained eq 6.6 derived by Abbott and Bolton in ref 10.
IV. The Oblate Case Laplace's equation in oblate spheroidal coordinates is
where X and p are defined in terms of the expressions
The potential due to a uniform line charge of total charge Q and length 2d is given by
Thus eq 51 corresponds to the reaction potential for a line charge equivalent to the net charge of the charge distribution, uniformly distributed between the foci of a cavity defined by X = Xo. The n = 0 term from eq 26
is the energy of this line of charge in its reaction potential and reduces to the Born charging energy when the dimensions of the cavity go over to a sphere. The dipole terms emerge from the terms with n = 1and m The Journal of Physlcal Chemistry, VoL 80, No. 23, 1976
S. W. Harrison, H.-J. Nolte, and D. L. Beveridge
2504
= -1, 0 , l in eq 25. For a general ellipsoidal cavity with semiprinciple axes a, b, and c , surrounded by a dielectric, the reaction field R due to a homogeneous dipole density in the cavity is given by9
R =F*p
(54)
where p is the total dipole moment and F is a tensor in a coordinate system with the axes of the ellipsoidal cavity as principle axes. The diagonal elements of F a r e given by
F,
3 A,(1 - A,)(t - 1) abc 6 (1 - € ) A ,
-
+
a = a, b, c
(55)
where c = €,/ti and ci = 1has been assumed, and (56)
with t 2 = (s
+ a2)(s+ b 2 ) ( s+ c 2 )
and, since A,
+1 + -21 ho(Xo2- 1) In xo - 1 A0
(58)
+ Ab + A , = 1
1 1 Ab = A , = - (1 - A,) = - A02 2 2
ho + 1 - -41ho(ho2- 1) In A0 - 1
A'=--
1 2 6
1
[qkZkFbqLZI
-t qkPkFaq1PL cos (4l - 4 k ) l where the two terms are the z and p components of
(65)
and p is the dipole moment of the arbitrary charge distribution. For the spherical cavity, a = b = c and A, = Ab = A , = l/3. In this case F , = -1 2(€ - 1) (66) a3 2 t + 1 and the Onsager reaction field contribution to the energy can be obtained from
(57)
For a prolate or oblate spheroid where b = c or a = c, the integrations in eq 57 can be carried out and the results expressed in terms of A0 = ald (prolate) and A0 = b/d (oblate). For the prolate case
A, = 1 - Xo2
Thus the dipole contribution to the energy in eq 25 is
(59)
where the coordinate system has been chosen in such a way that p is along one of the axes. In the course of the application of the formalism developed in sections111 and IV to certain problems dealing wth environmental effets, we have prepared a digital computer program in Fortran for the IBM 370/168. Copies are available on request. We conclude this section with a discussion of special numerical problems encountered in developing this program. The associated Legendre functions of the first kind, P, m, are calculated by upward recursion which is stable for any real argument; once given the elements P&) and Pl(x) we can create the whole set of functions, using the relations
For the oblate case, A, can be obtained from Ab prolate by replacing ho by iho. From eq 22 and the expressions for Fa, Fb, and F, found by using eq 58 and 59 with eq 55, it can be shown that n
1 y 1,c= - - d 3 F , 3 2 (60) %,I = - d3Fb 3 With the values of Prim for n = 1, m = -1,O, 1 and eq 11 and 14, it also follows that
&l,O
=3
cM qkzk
k=l
(61)
where z = dhg is the axis of symmetry in the prolate (or oblate) spheroidal coordinate system and, e.g., for the prolate case p = d(X2
- 1)ll2(1- g2)'/2 = ( x 2 + y2)1/2
+
M
=
- k = 1 4kPkFaP COS (4 - 4 k )
The Journal of Physical Chemistry, Vol. 80, No. 23, 1976
- (n + m - l)P,-zm(x)](70) The recurrence relations for the associated Legendre functions of the second kind are stable only if these functions are calculated by backward recursion. I n first applications we used an algorithm, published by Gautschi12that was translated into a Fortran procedure. Values for the functions and their derivatives could be checked against published v a l u e ~up~ to ~ the order n = 10. For calculating the associated Legendre functions of purely imaginary argument, we used algorithms, published by Herndon,14J5which were translated into Fortran procedures and slightly modified to yield i-"P, (ix) and in+2m+1Qn m ( i x ) , respectively.
(62)
is the perpendicular distance from the axis of symmetry. Using eq 60 and 61 in the appropriate terms in eq 24 gives ( @ ~ ) i , - i (@R)1,1
(69)
(63)
Acknowledgment. Helpful discussions with Professor Gary W. Schnuelle of Colorado State University are gratefully acknowledged. This study was supported by NIH Grant No. NS-12149-01 from the National Institute of Neurology and Stroke and a CUNY Faculty Research Award. D.L.B. acknowledges a U S . Public Health Service Research Career Development Award No. 6K04-GM21281 from the National Institute of General Medical Studies.
Charge Distribution in a Polarizable Dielectric Continuum
2585
Appendix The Legendre polynomials of the first kind for any argument z in the complex plane are defined following Smythe's accountll by Rodrigues' formula 1 dn P n ( z ) = --( 2 2 - l), 2nn! dzn In this work only arguments z of the complex plane excluding those on the real axis between -1 and +1appear as arguments of the Lengendre polynomials of the second kind; these polynomials are then defined a d 6
here K represents both P and Q and the relation is valid for any argument in the complex plane. In this connection it should be noted that Hobson's equation for the inverse of the distance in oblate spheroidal coordinates (p 430 in ref 18) has to be multiplied by the imaginary unit i in order to give real values; this missing factor can be traced back to the previous equation of that derivation. The derivatives of the associated Legendre functions, used in eq 23 are calculated from the relations, given by Smythe." We have for arguments XO > 1 or for imaginary arguments 1 1 - A02
k , m ( X o ) = -[(n
+ m)Kn-im(Ao) - nA&nrn(Xo)l (AS)
(A2) and it can be shown that these polynomials can also be written in the form given by Morse-Feshbachll
and k,m(Ao)
=
1
[(n + l)X&nm(Ao) 1 -A02 - (n - m + 1)Kn+lm(Xo)1 (-49)
Here K represents both P and Q and the dot denotes the derivative with respect to the argument (cf. eq 20). For complex arguments z , excepting the points x on the axis between -1 and +1 we use the following definitions for the associated Legendre functions
whereas for the points x we have dn P , ( x ) = (1 - x 2 ) m ' 2 - - P n ( x ) (A@ dx In contracting the formulas given by Hobson's for the inverse of the distance between two points in prolate or oblate spheroidal coordinates, respectively, we have made use of the definitions for the associated Legendre functions with negative index m (n - m)! K,-m = Kn rn (n m)!
+
References and Notes (1) J. G. Kirkwood J. Chem. Phys., 2, 351 (1934). (2) M. Born, Z.Phys., I,45 (1920). (3) L. Onsager, J. Am. Chem. SOC., 58, 1486 (1936). (4) T. Halicioglu and 0. Sinanogiu, Ann. N. Y. Acad. Sci., 158,308 (1969); D. L. Beveridge, M. M. Kelley, and R. J. Radna, J. Am. Chem. Soc., 96,3769 (1974); B. Linder, Adv. Chem. Phys., 12, 225 (1965). (5) G. W. Schnuelle and D.L. Beveridge, J. Phys. Chem., 79, 2566 (1975). (6) D.L. Beveridge and G. W. Schnuelle, J. Phys. Chem., 79, 2562 (1975). (7) J. G. Kirkwood and F. H. Westheimer, J. Chem. Phys., 6, 508, 513 (1938). (8) Th. H. Scholte, Physlca, 15, 437 (1949). (9) C. J. F. Bottcher, "Theory of Electric Polarizability", Elsevier, Amsterdam, 1978. (10) J. A. Abbott and H. C. Bolton, Trans. Faraday Soc., 48, 422 (1952). (11) W. R. Smythe, "Static and Dynamic Electricity", McGraw-Hill, New York, N.Y., 1950. (12) W. Gautschi, Commun. ACM, 8, 488 (1965). (13) "Tables of Associated Legendre Functions", Columbia University Press, New York, N.Y., 1945. (14) J. A. Herndon, Commun. ACM, 178 (1961). (15) J. A. Herndon, Commun. ACM, 320 (1961). (16) E. Jahnke and F. Emde, "Tables of Functions", Dover Publications, New York, N.Y., 1945. (17) P. M. Morse and H. Feshbach, "Methods of Theoretical Physics", McGraw-Hill, New York, N.Y., 1953. (18) E. W. Hobson, "Spherical and Ellipsoidal Harmonlcs", Cambridge University Press, New York, N.Y., 1931.
The Journal of Physical Chemistry, Vol. 80, No. 23, 1976
