(2) E. 8.Wilson, Chem. Soc. Rev., 1, 293 (1972). (3) F. lnagaki, l, Harada, and T. Shimanauchi, J. Mol. Spectrosc., in press. (4) W. G. Fateleyand F. A . Miller, Spectrochim. Acta, 19, 611 (1963). (5) A. Murray, 1 1 1 , and B. L. Williams, Ed., “Organic Synthesis with Isotopes, Interscience. New York, N.Y., 1958, Part 1 1 , p 1341. (6) D. Smith, J, P. Devlin, and D. W. Scott, J. Mol. Spectrosc.. 25, 174 (1968). (7) 0 . W. Scott and G . A . Crowder. J. Moi. SpeCtrOSc., 26,477 (1968).
(8) K. Radciiffe and J . L, Wood, Trans. Faraday Soc., 62, 1678 (1966). (9) E. Hirota, J. Chem. Phys.. 37,283 (1962). (10) J. P. Lowe, Progr. Phys. Org. Chem., 6, 1 (1968). (11) C. S. Ewig and D. 0 . Harris,J. Chem. Phys.. 52. 6268 (1970). (12) S. Weiss and G. E.Leroi, d. Chem. Phys., 48, 962 (1968). (13) 8.Kirtrnan,J. Chem. Phys., 37, 2516 (1962). (14) T. Kojima and T. Nishikawa, d. Phys. SOC.Jap.. 92. 680 (1957) (15) N, Solimeneand R . P. Daiiey, J. Chem. Phys., 23, 124 (1955) (16) J. L, Binden, J . Chem. Phys.. 17,499 (1949).
Kunihiko lwasa Department of Physics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya 464, Japan (Received November 2, 1972; Revised Manuscript Received February 13, 1973)
A formulation considering most simply the size of low molecular weight ions in polyelectrolyte systems was given and was compared with the results of titration experiments of polyacrylic acid. It was shown that this new equation yields better agreement with experiment than the Poisson-Boltzmann equation does
Though the size effect of ions, especially of quaternary ammonium ions, in polyelectrolytes is known experimentally,l only little theoretical treatment has been reported. The difficulty lies in the complexity of the equation which describes the system. I wish to report here the effect of the ionic size considered on the basis of the two-body distribution function of low molecular weight ions. Let us consider the system of volume V in which a polyion occupies a fixed place and there are N small ions of different speeies. The number of the a t h species is No. The total potential energy is then written as the sum of the interaction potential v, between the polyion and the a t h ion species. The interaction potentials u,., and v, are low molecular weight ion species
lJ = XCua(xd>+ “
1
CCU,Y(X,L,
xr’)
(1)
UY I J
where x,J is the coordinate of the j t h ion particle of the a t h ion species. The interaction potentials u,, and vu are expressed by
u,,(x)
i=
j-dX’Z,Pp(X’)/(E/X
- x’l)
+ u;fx)
+
uar(x,x’) = Z,Z,/(ti x - x ‘ / ) ua7*(x,x‘) where z, i s the charge of the a t h ion species, pP i s the charge distribution in the polyion, the asterisk denotes the non-Coulombic part, and t is the dielectric constant of the solvent. The number density n, of the a t h species is expressed by the integral
ndn)
=
W,QAr).fdlN - 11 exp(-PU)IZ,l=x Qhr
=
SdWl exp(-PW
force potential of the a t h species
n,(x) = N,Q,
where /3 is equal to I / k T and d { N - 1) denotes the phase integral except for x o l , which is equal to x. The mean
exp[--PW,(x)l
(2’)
Similarly, the density of the a t h species at x and the r t h species at x’ is ~ , ? ( X , X ’ ) = lCr,NrQA~-lJd~N - 2 ) ~xP(-PU)IX,~=~
,-
,^?--*I
Substituting the mean force potential into (2) and taking the gradient yields
The size of the ions is expressed in terms of the hardsphere potential. The radial distribution function of the hard-sphere gas or liquid is described by YBG, HNC, or FY equations and the radial distribution is shown in Figure 1 . 2 In our problem, however, the electrostatic interaction and the charge asymmetry of small ions due to the existence of macroions make the problem much more complicated. In order to estimate the effect of the ionic size simply, let us assume that the pair distribution function for low molecular weight ions is a step function, in the first approximation. The part of the curve in Figure 1 where the pair distribution function is greater than unity should be lowered by the existence of Coulomb repulsion. Therefore this assumption will be valid, at least to the first order of estimation. Thus we have
n,r(x,x’) ( 21
W,(x) is defined by
=
n,(x>n,(x’)O(lx - x’/-raY>
(4)
where 8 is the step function defined by
0 - 1
Q(2) =
for 2 < 0 forzP0
The Journal of Physical Chemistry, Vol. 77, No. 16, 1973
Kunihiko lwasa
1982
6 distonca
Figure 1. Radial distribution function. Hard-core radius is denoted by ro. Broken line is radial disrribution of hard-sphere gas, and full line represents present approximation. and ray is the sum of the hard-core radii of the species a and y. Taking the divergence of eq 3, we obtain
o”w,(x>
=
-”-[ 4 . r r P p b ) E
ny(xY(Ix - x’l - r a y (x - X)* (5)
4- CZ,”b’
1
where the second term of the right-hand side is the surface integral. If the integration region has spherical symmetry, by transforming the variables, we obtain 2TJdpny(X’)l,.=,ay
(6)
where p denotes the angle against the axis connecting the small ion of the a t h species a t x and the macroion. If the point x is distant from the polymer, the integral tends to 47rn,(x) and eq 5 is reduced to the Poisson-Boltzmann equation. The equilibrium condition for the dissociation of weak polyelectrolytes is given by3
Figure
2.
Henderson-Hasselbach plot. Full lines express the re-
sult d u e to the new equation, and broken lines show PoissonBoltzmann result: A , ionic radius of counterions is 6 . 0 A, which Corresponds to Bu4N+; B, 3.5 A, corresponds to Et4”; C, 2.25 A,
MedN+.
Parameters in Henderson-Hasselbaeh Plot
TABLE I: Comparison of
Experimental valuea Radius,
A
Potential at a=OS
Theoretical value
Slope
Potential at t u = 0 . 5
Slope
2.26 2.72 2.93 3.62
2.64 2.71 2.79 2.93
1.95 2.11 2.33 2.64
pH = pK, - log [ c ~ / ( l - a)J 4- 0.4343PWH where the change in free energy per ionized acid group which is usually denoted by AF has been replaced in this equation by W H , the mean force potential for H+ in the neighborhood of the residue. The potential WH is calculated by solving the simultaneous equations of the type such as ( 5 ) with respect to all the ion species existing in the system. In order to know the result of these equations briefly, numerical calculation was performed for the salt-free condition by using the rod model of polyelectrolytes due to Fuoss, Lifson, and Katchalsky.4 Of course, even if a low molecular weight salt exists in the system, we can solve the equations with only increased computation time needed by the computer. The boundary condition is given by d W L y ( R=) 0; -WW,(R) = 0 dr where R is the radius of the model system employed. This boundary condition is similar to that for the PoissonBoltzmann equation because the former condition gives the zero point of the potential and the latter implies the electroneutrality of the system for both cases. In order to simplify the surface integral in the righthand side of eq 5, the polymer was regarded to be an array of spheres, whose radius was the same as that of the polymer rod, only for calculating this surface integral. Thus the integral was replaced by (6). This assumption implies some reduction of the geometrical restriction near the polymer. The Journal of Physicai Chemistry, Vol. 77, No. 16, 1973
Kf
Me4N + Et4N +-
Bu4N +
1.33 2.25 3.50 6.00
2.54
2.50 2.70 2.99
ref 1.
Calculation Procedure As the first approximation, n,(r) is replaced by the Poisson-Boltzmann distribution, and eq 5 is solved. ‘This zero-order W,(r) is substituted in eq 2’ to get na(r), and this is iterated till self-consistency is obtained. The variable r is limited within rp ra 5 r 5 R for the equation with respect to the a t h ion species, where r p is the polymer radius and r a is the hard-core radius of the a t h ion species. For each given value of n,(R), we thus obtain N , and W&). Then the degree of dissociation is evaluated, if the polymer Concentration and chain length of monomer unit are given. The numerical values of parameters used are dielectric constant, t = 78.5; chain length of monomer unit, 2.55 A; polymer radius rp, 4.0 A; polymer concentration, 0.0128 N . These values are taken to correspond to polyacrylic acid.l The results are shown in Figure 2 and Table I. In order to show the effect of the new equation, the result using the Poisson-Boltzmann equation instead of the new equation in solving for the potential Wlr is also shown in Table 1.5 If the size of the counterions and the degree of disso-
+
Comparison of 3 d Polarization and B o n d Functions ciation a are both large, the difference between the two equations is appreciable. In comparing the result with the experiment, the following points are to be noted: (1)the value of the potential of mean force WH at half-neutralization ( i e . , log a / ( l - a ) is equal to zero) is in good agreement with experiment; (2) the linearity, which is observed in the experiment is broken; and (3) the slopc of the linear part of the curve calculated is smaller than that obtained from the experiment. It is also to be noted that the difference of the slope is smaller when the counterions are small. Though polyacrylic acid cannot be supposed to be rodlike a t this polymer concentration and in this range of the degree of dissociation, it is a fact that a n improvement was accomplished by employing the new equation. This improvement suggests that the introduction of the simple pair distribution function is valid even in the frame of the Lif-
1983
son-Katchalsky model. If we do not simplify the surface integral of eq 5 into eq 6, then the effect of the geometrical factor is larger and the calculated value is expected to be closer to the experimental value.
Acknowledgment. Numerical calculations were performed on the Facom 230-60 a t the Nagoya University Computation Center. References and Notes H. P. Gregor and M. Frederick, J. Polym. Sci., 23,451 (1954). E. Waisman and J. L. Lebowitz, J. Chem. Phys., 52, 4307 (1970). A. Katchalsky, N. Shavit, and H. Eisenberg. J. Polym. Sci., 13, 69 (1954). R. M. Fuoss, A. Katchalsky, and S. Lifson, Proc. Nat. Acad. Sci.. 37,579 (1951). This treatment of the Poisson-Boltzmann equation corresponds to that of I. Kagawa and H. P. Gregor, J. Polym. Sci., 23, 467 (1957). Though their analytical expressions are correct, their numerical calculations include systematic errors.
Comparison1 of the Use of 3d Polarization Functions and Bond Functions in Gaussian Hartree-Fock Calculations T. Vladimiroff Propellants Division, feltman Research Laboratory, Picatinny Arsenal, Dover, NewJersey 07807 (Received A p r i l 72, 7973) Publication costs assisted by Picafinny Arsenal
The use of “bond functions” (Gaussian-type functions placed in the molecular bonding region) is examined for the nitrogen and oxygen molecules. On the basis of the total, ground-state energy and certain one-electron properties it is found that three optimized bond functions perform about as well as six optimized 3d functions. The incorporation of these bond functions reduces the computational time by a factor of 2 for the nitrogen molecule, when compared to the calculation using 3d functions. The use of 3d functions in conjunction with bond functions is also investigated and it is not found to be very encouragmg.
Introduction Recent advances in the capabilities of high-speed digital computers have made it possible to perform ab initio calculations on species of chemical interest. However, since the required computer time goes up rapidly as the number of atomic orbitals increases, picking the correct basis set is still a problem. For atoms, Slater-type orbitals seem to be the correct choice due to the infinite potential a t the nuclus. However, multiple-center, two-electron integrals are difficult to evaluate if Slater orbitals are used so that Gaussian-type orbitals are becoming more popular for nonlinear polyatomic molecules. It is, nevertheless, recognized that Slater-type orbitals are the superior choice so that Gaussian orbitals are usually used in such a way as to imitate the Slater calculations. For example, only nuclear centers are employed or Slater-type orbitals are expanded in sets of Gaussians. In the bonding regions of a molecule the electrons no longer experience a n infinite potential thus in these regions Slater-type orbitals are no longer clearly superior. For this reason, it has been sug-
gestedl that Gaussian orbitals should not necessarily be placed on nuclear centers. Many authors2 have adopted this idea to their own particular point of view. However, a systematic study of the best exponents and positions to be used has only recently begun,3 and this study makes no mention of the savings of computer time involved. In order to give this trend some impetus, we have studied the use of Gaussian orbitals placed at the center of the nitrogen and oxygen molecules and contrasted this study with a similar one reported by Dunning4 using 3d polarization functions. Upon the formation of a molecule from the individual atoms, we expect atomic orbitals to become distorted by the molecular fields. In order to describe this distortion, polarization-type functions are included in the calculation. A more elementary concept of a covalent bond is the piling up of negative charge between two positive nuclei so as to hold the molecule together by the classical electrostatic forces in agreement with the Hellman-Feyman theorem.5 In accordance with this idea, it seems reasonThe Journalof Physical Chemistry, Vol. 77, No. 16, 1973
