Communications to the Editor
2306
x
Calculated rates in the Computer B rows in Table I were obtained when Fzl was assigned a value of exp(-71.7 X lQ3/RT). To our satisfaction, this optimum value not only leads to good conversion agreement for the computer B results with experiment but, itself, also corresponds reasonably well to the 1014.41exp(-67 X lQ3/RT)and 1014.78. exp(-75 X 103/RT) values proposed for kl by Laidler et al.5 and Herriott et a1.,6 respectively.
D$1,((,7?,{) = (-1)”‘”
Acknowledgment. We gratefully acknowledge the support of this research by the National Science Uouncil of the Republic of China.
where PkP)(t)is Jacobi’s polynomial4of degree n in t. This formula is valid for positive and negative values of m’, m as the conditions a,fl > -1 are necessary only to ensure convergence at end points of the interval of orthogonality; integration over the group manifold is not vitiated since DgL,., = (-l)m’-mD~~~, If A, p, u are direction cosines of the axis of rotation and G the angle of rotation, the matrix representing R in terms of these parameted leads to
References and Notes (1) A. M. Benson, AIChEJ., 13, 903 (1967). (2) A. Lifshitz and M. Frenklach, J . Phys. Chem., 79, 686 (1975). (3) A. Llfshitz, K. Scheller, and A. Burcat, Proc. 7th Int. Shock Tube Symp., 690 (1973). (4) D. A. Leathard and J. H. Purnell, Proc. R . SOC.London, Ser. A , 305, 517 (1968). (5) K. J. Laldler, N. H. Sagert, and 6.W. Wojciechowski, Proc. R . SOC. London, Ser. A , 270, 242 (1962). (6) M. M. Papic and K. J. Laidler, Can. J. Chem., 49, 535 (1971). (7) G. E. Herriott, R. E. Eckert, and L. F. Albright, AIChEJ., 18, 84 (1972). (8) J. T. Cheng, Y. S. Lee, and C. T. Yeh, J. phys. Chsm.,81, 687 (1977). (9) J. T. Cheng, MS Dissertation, National Tsing Hua University, 1976; J. T. Cheng and C. T. Yeh, J . Phys. Chem., 81, 1982 (1977). Institute of Chemistry National Tsing Hua University Hsinchu, Taiwan, Republic of China
Wen-hong Kao Chuln-tlh Yeh”
Received April 28, 1977
0’ + m’)!(j-m.’)!112 0’ + m)!O’-m ) !
t x
D;; (A,
p , v , (a) = (-1)m’-
(p
+ ih)”‘(p - ih)” (1-v z y
X
where A2 + p2 + v2 = 1 and R33 = 1- 2(1 - v2) sin2 l l Z i P . A neater form results on introducing 8 , cp as polar angles of the rotation axis D ~ > , ( S cp, , CP)= (-l)m’(e-ip sin s sinl/z(a)m’-m x ( i cos 1 / z @ cos 6 sin 1 / 2 ~ ) ~ ’x+ ~
+
Parametrizations of the Rotation Group Pobilcatlon costs assisted by The Graduate School, University of Kentucky
Sir: The irreducible representations of the three-dimensional pure rotation group are usually given in terms of Euler’s parametrization, that is, successive rotations $, 8, 4 about the z , y, and z axes, respective1y.l For the representation of dimension 2 j + 1
D:,:
($,o,+ ) = (- lp’-me-im’$e-im$ x
+ m ) ! ( j- m)!U + m’)!(j- m’)]”2 X Y-’)’0’ + m - o ) ! -~ m’ - a)!a!(m’ - m + a)! (cos i / z ~ ) 2 j + m - m ’ - 2(sin ~ 11 2e ) * ‘ + m ’ - m
(1)
Other parametrizations are preferable in particular problems. For example, when working with finite subgroups of the rotation group a parametrization in terms of the axis and angle of rotation is especially convenient. The irreducible representations for this parametrization have been examined in considerable detail2 without recognizing that they may be obtained by an elementary calculation. Indeed, the irreducible representations for any desired parametrization may be obtained by the following argument. Let R denote a 3 X 3 orthogonal matrix with det R = +1,interpreted as a change of reference system: (xyz) (xIy’zI). The elements in the third row and third column implicitly define the three independent parameters 5, q , characterizing the rotation, as these five quantities are related by two conditions: R3i2 + R = R1S2 + R232 = 1 - R332. Now R33is the cosine of L zQ.? = 0, and, from the matrix for a rotation in terms of Euler’s angles? Rsl + iR32 = e,@sin 0, RI3 + iR23 = -e-,* sin 0. Straightforward eliminations in conjunction with (1) yield the general formula
-
r
The Journal of Physical Chemistry, Vol. 8 1, No. 24, 1977
with R33 = 1 - 2 sin2 S sin2 The irreducible representations for other parametrizations may be obtained from (2) and the appropriate matrix R with equal facility.
References and Notes (1) E. P. Wigner, “Group Theory”, Academic Press, New York, N.Y., 1959, p 167. Wigner’s formula gives the matrix elements of exp(larJ,) exp (ipJ ) exp (lyJ,); Equation 1 of the text follows on setting a
= - 4 , 8 = -6, y = -*.
(2) H. E. Moses, Nuovo Clmento, XL A, 1120 (1965); Ann. Phys., 37, 224 (1966); 42, 343 (1967). (3) See, for example, P. L. Corio, “Structure of High-Resolution NMR Spectra”, Academic Press, New York, N.Y., 1966, p 479. (4) 0. Szego, “Orthogonal Polynomials”, Amerlcan Mathematics Society, New York, N.Y., 1939, Chapter IV. (5) Reference 3, p 477. Department of Chemistry University of Kentucky Lexington, Kentucky 40506
P.
L. Corlo
Received Ju/y 20, 1977
Chain Expansion of Neutral Polymer Coils upon Catlon Binding
Sir: Alkali and alkaline earth cation binding to the peptide group is a phenomenon of interest in problems of physical biochemistry such as salt-induced conformational transitions in proteins and carrier-mediated transport. Several proteins and polypeptides have been shown to bind these cations and undergo conformational alterations.14 Model
