Additions and Corrections
J. Phys. Chem. A, Vol. 102, No. 45, 1998 9099
ADDITIONS AND CORRECTIONS 1996, Volume 100 Joseph Ivanic and Klaus Ruedenberg*: Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion Page 6342. Several misprints have been found in the original article. In eqs 6.7, 7.9b, and 7.9c a minus sign should be changed into a plus sign. In eq 7.9a an index should be changed from 1 to -1. The correct equations are as follows: l+1 l+1 l l+1 l R-mm′ ) amm′ R00R-mm′ + bmm′ [(1 + δm1)1/2R-10Rm-1,m′ + l l+1 l (1 - δm1)R10R-m+1,m′ ]/2 + b-mm′ [R-10Rm+1,m′ l ]/2 (6.7) R10R-m-1,m′ l+1 l+1 l l ) c0,m′+1 (R0,-1R0m′ + R01R0,-m′ )R0,-m′-1 l+1 l l l [R1,-1R1m′ + R11R1,-m′ + R-1,-1R-1m′ + d 0,m′+1 l ]/x2 (7.9a) R-11R-1,-m′
l+1 l+1 l l l+1 ) cm,m′+1 (R0,-1Rmm′ + R01Rm,-m′ ) + d m,m′+1 [(1 + Rm,-m′-1 l l + R11Rm-1,-m′ ) - (1 - δm1) × δm1)1/2(R1,-1Rm-1,m′ l l + R-11R-m+1,-m′ )]/2 (R-1,-1R-m+1,m′ l+1 l l [(R1,-1Rm+1,m′ + R11Rm+1,-m′ )+ d -m,m′+1 l l + R-11R-m-1,-m′ )]/2 (7.9b) (R-1,-1R-m-1,m′
l+1 l+l l l ) cm,m′+1 (R0,-1R-mm′ + R01R-m,-m′ )+ R-m,-m′-1 l+1 l l [(1 + δm1)1/2(R-1,-1Rm-1,m′ + R-1,1Rm-1,-m′ )+ d m,m′+1 l l + R11R-m+1,-m′ )]/2 + (1 - δm1)(R1,-1R-m+1,m′ l+1 l l [(R-1,-1Rm+1,m′ + R-11Rm+1,-m′ )d -m,m′+1 l l + R11R-m-1,-m′ )]/2 (7.9c) (R1,-1R-m-1,m′
The possibility of errors was mentioned to us by Mr. Hannes Edvardson of Uppsala University. The correct Tables 1 and 2 are as follows: l l l TABLE 1: Definitions of the Numerical Coefficients u mm′ , W mm′ , and w mm′ Occurring in Eqs 8.1
|m′| < l u
l V mm′
w
[
l mm′
l mm′
(l + m)(l - m) (l + m′)(l - m′)
[
|m′| ) l
]
]
1 (1 + δm0)(l + |m| - 1)(l + |m|) 2 (l + m′)(l - m′) -
[
[
]
1 (l - |m| - 1)(l - |m|) 2 (l + m′)(l - m′)
1/2
1/2
(1 - 2δm0)
(1 - δm0)
]
(l + m)(l - m) (2l)(2l - 1)
1/2
[
1/2
]
1 (1 + δm0)(l + |m| - 1)(l + |m|) 2 (2l)(2l - 1) -
[
]
1 (l - |m| - 1)(l - |m|) 2 (2l)(2l - 1)
10.1021/jp9833340 CCC: $15.00 © 1998 American Chemical Society Published on Web 11/05/1998
1/2
1/2
(1 - 2δm0)
(1 - δm0)
9100 J. Phys. Chem. A, Vol. 102, No. 45, 1998
Additions and Corrections
l l l TABLE 2: Definitions of the Functions U mm′ , V mm′ , and W mm′ Occurring in Eqs 8.1
m)0 l U mm′ l V mm′ l W mm′
l 0P0m′ l 1P1m′ +
l -1P-1m′
m>0 l 0Pmm′ 1/2 l 1Pm-1,m′(1 + δm1) l l 1Pm+1,m′ + -1P-m-1,m′
m
