Rapid and Stable Determination of Rotation Matrices between Spherical Harmonics by Direct Recursion

Transcription

1 Chemistry Pubications Chemistry Rapid and Stabe Determination of Rotation Matrices between Spherica Harmonics by Direct Recursion Cheo Ho Choi Iowa State University Joseph Ivanic Iowa State University Mark S. Gordon Iowa State University, Kaus Ruedenberg Iowa State University, Foow this and additiona works at: Part of the Chemistry Commons The compete bibiographic information for this item can be found at chem_pubs/355. For information on how to cite this item, pease visit howtocite.htm. This Artice is brought to you for free and open access by the Chemistry at Iowa State University Digita Repository. It has been accepted for incusion in Chemistry Pubications by an authorized administrator of Iowa State University Digita Repository. For more information, pease contact

2 Rapid and Stabe Determination of Rotation Matrices between Spherica Harmonics by Direct Recursion Abstract Recurrence reations are derived for constructing rotation matrices between compex spherica harmonics directy as poynomias of the eements of the generating3 3 rotation matrix, bypassing the intermediary of any parameters such as Euer anges. The connection to the rotation matrices for rea spherica harmonics is made expicit. The recurrence formuas furnish a simpe, efficient, and numericay stabe evauation procedure for the rea and compex representations of the rotation group. The advantages over the Wigner formuas are documented. The resuts are reevant for directing atomic orbitas as we as mutipoes. Keywords Poynomias, Recurrence reations, Second harmonic generation Discipines Chemistry Comments The foowing artice appeared in Journa of Chemica Physics 111 (1999), 8825, and may be found at doi: / Rights Copyright 1999 American Institute of Physics. This artice may be downoaded for persona use ony. Any other use requires prior permission of the author and the American Institute of Physics. This artice is avaiabe at Iowa State University Digita Repository:

3 Rapid and stabe determination of rotation matrices between spherica harmonics by direct recursion Cheo Ho Choi, Joseph Ivanic, Mark S. Gordon, and Kaus Ruedenberg Citation: The Journa of Chemica Physics 111, 8825 (1999); doi: / View onine: View Tabe of Contents: Pubished by the AIP Pubishing Artices you may be interested in An aternative to Wigner d-matrices for rotating rea spherica harmonics AIP Advances 3, (2013); / Fast and accurate determination of the Wigner rotation matrices in the fast mutipoe method J. Chem. Phys. 124, (2006); / Generaizations of the Hohenberg-Kohn theorem: I. Legendre Transform Constructions of Variationa Principes for Density Matrices and Eectron Distribution Functions J. Chem. Phys. 124, (2006); / A recursive parametrization of unitary matrices J. Math. Phys. 46, (2005); / An efficient method for cacuating maxima of homogeneous functions of orthogona matrices: Appications to ocaized occupied orbitas J. Chem. Phys. 121, 9220 (2004); / This artice is copyrighted as indicated in the artice. Reuse of AIP content is subject to the terms at: Downoaded to IP:

4 JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER NOVEMBER 1999 Rapid and stabe determination of rotation matrices between spherica harmonics by direct recursion Cheo Ho Choi, Joseph Ivanic, Mark S. Gordon, and Kaus Ruedenberg Department of Chemistry and Ames Laboratory USDOE, Iowa State University, Ames, Iowa Received 19 Apri 1999; accepted 6 August 1999 Recurrence reations are derived for constructing rotation matrices between compex spherica harmonics directy as poynomias of the eements of the generating 33 rotation matrix, bypassing the intermediary of any parameters such as Euer anges. The connection to the rotation matrices for rea spherica harmonics is made expicit. The recurrence formuas furnish a simpe, efficient, and numericay stabe evauation procedure for the rea and compex representations of the rotation group. The advantages over the Wigner formuas are documented. The resuts are reevant for directing atomic orbitas as we as mutipoes American Institute of Physics. S I. INTRODUCTION Spherica harmonics pay important roes in many areas of theoretica and appied physics. In quantum chemistry, they occur for instance as factors of atomic orbitas and as factors in mutipoe expansions. Our current interest derives from their use in the identification of atoms in moecues and in the fast mutipoe method FMM 1 for the cacuation of Fock matrices. In this as in many other contexts, it is often necessary or expedient to rotate the spatia coordinate axes and there then arises the need to express the spherica harmonics defined with respect to one coordinate axis system in terms of the spherica harmonics defined with respect to the other. It foows from group theory 2 that the two sets of harmonics associated with the two axis systems are reated to each other by a transformation that is bock-diagona with respect to the azimutha quantum number, i.e., Ŷ m m1 Y m D mm, 1.1 for compex spherica harmonics Y m and Ŷ m m Y m R mm, 1.2 for rea spherica harmonics 3 Y m, where the summations do not go over. The D mm are compex unitary matrices and the R mm are rea orthogona matrices. These so-caed rotation matrices are determined by the 33 orthogona matrix R that defines the origina rotation between the basis vectors of the two axis systems, ê k i e i R ik. 1.3 Manifesty, Y m, Y m refer to the basis e i whereas Ŷ m, Ŷ m refer to the basis ê k. Wigner 2 has given expicit direct formuas for the eements of the compex rotation matrices in terms of the Euer anges of the matrix R. D mm In practica appications, one typicay deas simutaneousy with the spherica harmonics of a azimutha quantum numbers 1,2,3,...L, where L is the ength of some initiay presumed expansion. In such a context, the appication of Wigner s formuas is inefficient because it entais the independent cacuation of the rotation matrices for every, a procedure embodying a arge amount of dupication in the cacuation of factoria factors. It moreover oses significant figures for arger vaues of L. Recurrence reations with respect to m for fixed were given by Edmonds 4 and recenty impemented by White and Head-Gordon 5 for use in fast mutipoe method cacuations. They are unstabe in the vicinity of particuar poar anges and, athough the instabiity can be partiay remedied through aternative agorithms, probems remain with regard to the consistent cacuation of a terms to the same accuracy. Ivanic and Ruedenberg 3 have recenty shown that the rotation matrices between rea spherica harmonics obey a set of recurrence reations that aow for a much more efficient determination of the R mm. Their anaysis differs from the aforementioned approaches in two respects: i It is based on the recognition that the eements of the rotation matrices R mm can be directy expressed as poynomias of degree in terms of the matrix eements R jk of the origina 33 axis rotation. ii These poynomias can be obtained recursivey because the eements R mm can be represented as biinear expressions in terms of the eements R mm 1 and the eements R jk. Since the origina axis rotation R is typicay defined in terms of a number of interatomic distance vectors in a moecue, this approach aso avoids the detour over the Euer anges. The procedure has since been used for image anaysis at the University of Uppsaa, Sweden, and in eectronic engineering at the University of Auckand, New Zeaand. In the present artice, we estabish the anaogous system /99/111(19)/8825/7/$ American Institute of Physics This artice is copyrighted as indicated in the artice. Reuse of AIP content is subject to the terms at: Downoaded to IP:

5 8826 J. Chem. Phys., Vo. 111, No. 19, 15 November 1999 Choi et a. of recurrence reations for the compex rotation matrices D mm. The forma reasoning as we as the organization of the materia foow the paper of Ivanic and Ruedenberg. 3 The required background mathematics is assembed and aid out in Secs. II to V. The heart of the investigation is Sec. VI which presents three sets of recursion reations, anaogous to those in Ref. 3. The derivation is contained in Sec. VI B. For the execution of actua cacuations, the compex identities of Sec. VI are transformed into rea equations in Sec. VII. In Sec. VIII, the quantitative reations between the compex rotation matrices D mm of Eq. 1.1 and the rea rotation matrices R mm of Eq. 1.2 are formaized. The fina section provides information about the computationa impementation and a documentation of the advantages of the new approach as regards speed and accuracy. II. DEFINITION OF COMPLEX SPHERICAL HARMONICS Using the spherica coordinate definition x,y,zrsin cos,sin sin,cos, 2.1 and adopting the phase conventions of Condon and Shortey, 6 we define the compex spherica harmonics by the equations Y m,1 m P m cos e im / with the normaized Legendre functions as defined by Bethe 7 P m t 21m! 1/2 2m! P m t, 2.3 where tcos and the standard Legendre functions are given by P m t 1 dm 2! 1t2 m/2 dt m t By virtue of these definitions, it is readiy verified that P m t1 m P m t. 2.5 For exampe, for the azimutha quantum number 1, one has Y 11 4/3 1/2, Y 1,1 4/3 1/2, 2.10 Y 10 4/3 1/2 0. Because of the identity 2.9, the homogeneous form 2.6 can of course be converted into various nonhomogeneous forms, but we sha consider these as nonstandard. In what foows, we sha aways think of the spherica harmonics as the standard homogeneous functions 2.6 of the Cartesian coordinates in a given frame defined by basis vectors e 1,e 2,e 3 see Eq. 1.3, rather than as functions of the corresponding anges and as is conventionay done. III. RECURRENCE RELATIONS FOR COMPLEX SPHERICAL HARMONICS A subsequent derivations are deduced from the foowing three recurrence reations 7 for the normaized Legendre functions of Eq. 2.2: cos P m A m P m 1 A m 1 P m 1, 3.1 sin P m B m P m1 1 B m1 1 P m1 1, 3.2 sin P m B m P m1 1 B m1 1 P m1 1, 3.3 where A m 2121 mm 1/2, 3.4 B m mm1 1/ Mutipication of Eq. 3.1 by (1) m e im /2 yieds 0 Y m A m Y 1,m A m 1 Y 1,m, 3.6 and simiar mutipications for Eqs. 2.3 and 2.4 yied & Y m B m Y 1,m1 B m1 1 Y 1,m1, 3.7 Since the so-caed soid spherica harmonics r Y m are we known to be homogeneous poynomias of degree in the Cartesian coordinates x,y,z, the surface harmonics Y m can be expressed as homogenous poynomias Y m Y m, 2.6 in terms of the compex components of the rea unit vector in the basis e 1,e 2,e 3, & Y m B m Y 1,m1 B m1 1 Y 1,m Equations 3.1, 3.2, 3.3 as we as Eqs. 3.6, 3.7, 3.8 are vaid for m being positive, negative, or zero. In accordance with the remarks at the end of Sec. II, we perceive Eqs. 3.6 to 3.8 as identities between the standard homogeneous poynomia representations of the Y m in terms of, 0,., 0,, 2.7 IV. INTEGRAL FORMULAS where e i Expicit expressions wi be needed in the subsequent sin /&xiy/&r, sections for the transition moment integras Y 1,m i Y e i sin /&xiy/&r, 2.8 where the impied integration is defined to extend over the unit sphere in the Cartesian space. By virtue of the orthonormaity of the spherica harmonics, they are readiy derived 0 cos z/r, which satisfy from the recurrence reations of the previous section. From Eq. 3.6 one obtains x 2 y 2 z 2 /r Y 1,m 0 Y A 1 m, 4.1 This artice is copyrighted as indicated in the artice. Reuse of AIP content is subject to the terms at: Downoaded to IP:

6 J. Chem. Phys., Vo. 111, No. 19, 15 November 1999 Recurrence reations for rotation matrices 8827 and from Eqs. 3.7 and 3.8 foows that, for given and fixed, the ony nonvanishing integras invoving and are Y 1,m Y 1 B 1 1 m,1, and Y 1,m Y 1 B 1 1 m, Eqs. 4.1, 4.2, and 4.3 are vaid for m being positive, negative, or zero. V. ROTATION OF COMPLEX SPHERICAL HARMONICS In anaogy to Eq. 5.3, a compex rotation matrices of Eq. 1.1 can be resoved into their rea and compex parts, i.e., D mm F mm ig mm. 5.6 Since, from Eqs. 1.1, 2.2, and 2.5 one readiy deduces D m,m it foows that F m,m G m,m 1 mm D mm *, 1 mm F mm, 1 mm G mm These genera identities account in particuar for the reationships seen to exist between the matrix eements in the case of Eqs. 5.4, 5.5. In the present section, we coect some eementary reations regarding rotation matrices that wi be used in the subsequent sections. A. Decomposition into rea and imaginary parts The rotation of the basis vectors in rea threedimensiona space, introduced by Eq. 1.3, impies the foowing transformation between the coordinates of any one vector with respect to the two bases: xˆ,ŷ,ẑx,y,zr xx R xy R xz R yx R yy R yz R zx R xy R zzx,y,zr. 5.1 By virtue of the invariance of r(x 2 y 2 z 2 ) 1/2, the compex components of the unit vector with respect to the two bases transform therefore as foows: ˆ ˆ,ˆ 0,ˆ, 0, DD, 5.2 with DFiG 5.3 where the rea and imaginary parts of D are given by the foowing expressions in terms of the eements of R: F, F,0 F, F 0, F 0,0 F 0, F, F,0 F, B. Genera rotation matrices D mm as homogeneous poynomias of D mm In anaogy to Eq. 2.6, the transformed harmonics on the eft hand side of Eq. 1.1 are defined as the standard homogeneous poynomias in terms of the transformed compex coordinates (ˆ,ˆ 0,ˆ ), i.e., Ŷ m Y m ˆ. 5.9 The transformation 1.1 can therefore be determined by first substituting the transformation 5.2 into the poynomias given by Eq. 5.9 and, then, transforming back to the Y m () using one of the possibe inverses of Eq From this reasoning, it is apparent that the eements of the genera rotation matrices D can be expressed as homogenous poynomias of degree in terms of the eements of D. They can thus be cacuated directy from the eements of the rotation matrix R without the detour over the Euer anges. In particuar, Eq shows that the matrix for 1 is identica with the matrix D given by Eqs. 5.3 to 5.5, i.e., D 1 D C. Rotation matrices as integras By virtue of Eq. 1.1, the rotation matrices can aso be expressed as the integras D mm Y m Ŷ m, 5.11 where the integration goes over the invariant Cartesian unit yyr xx)/2 R xz /& (R yyr xx)/2 sphere and the arguments of Y m and Ŷ m, respectivey are (R R zx /& R zz R zx /& 5.4 the components of the same unit vector reative to the two (R yy R xx )/2 R xz /& (R yy R xx )/2, fixed Cartesian bases e 1,e 2,e 3 and ê 1,ê 2,ê 3 that are connected by R Eq. 13. G, G,0 G, VI. RECURRENCE RELATIONS FOR COMPLEX G 0, G 0,0 G 0, G, G,0 G, ROTATION MATRICES As seen in the preceding section, the rotation matrix eements D yxr xy/2 mm yxr xy/2 R yz /& R can be obtained as homogeneous poynomias of the eements of D. We sha now buid up these poynomias by recursion, starting with D 1 D. From the identities R R zy /& 0 R zy /&. 5.5 R yx R xy /2 R yz /& R xy R yx /2 in Sec. III, a variety of recurrence reations can be deduced This artice is copyrighted as indicated in the artice. Reuse of AIP content is subject to the terms at: Downoaded to IP:

7 8828 J. Chem. Phys., Vo. 111, No. 19, 15 November 1999 Choi et a. between the rotation matrices for the harmonics of order and those of order (1). Here, we derive three sets that prove usefu for the quantitative evauation of the matrices. A. Recurrence reation for 1 m 1 This recurrence reation is derived from the recurrence reation 3.6 between spherica harmonics. It yieds the recurrence reation D mm a m,m b m,m D 00 D m,m 1 b m,m 1 1 D 10 D m1,m D 10 D m1,m, 6.1 for the rotation matrices, where a mm A m /A m mm mm 1/2, 6.2 b mm B m /&A m mm1 1/ mm It shoud be noted that the cases m are not covered and that a mm 0 for m, 6.4 b mm 0 for m and m1 6.5 which, in certain cases, eiminates one or two terms in Eq where the coefficients are those defined by Eqs. 6.2 and 6.3. Repacement of by (1) in this equation yieds Eq C. Recurrence reation for m 2 This recurrence reation is derived from the recurrence reation 3.7 for spherica harmonics. Starting with Eq. 3.7, a derivation that is entirey anaogous to that just discussed yieds D m,m c m,m 1 D 0,1 D m,m 1 1 d m,m D 1,1 D m1,m 1 d m,m D 1,1 D m1,m 1, 6.9 where 1/2 c mm &A m /B m 2mm, 6.10 mm1 d mm B m /B m mm1 mm1 1/ It shoud be noted that the case m and m(1) are not covered by Eq. 6.8 and that 1 c mm 0 for m, 6.12 B. Proof of Eq. 6.1 The recurrence reation 3.6 appies to the spherica harmonics in the rotated coordinate frame as we as to those in the unrotated coordinate frame. Mutipying 3.6 in the rotated frame by the unrotated Y 1,m and integrating over the unit sphere in Cartesian space, as discussed at the end of the ast section, yieds Y 1,m ˆ 0Ŷ m A m Y 1,m Ŷ 1,m A m 1 Y 1,m Ŷ 1,m. 6.6 The rotated quantities on both sides of this equation are now expanded in terms of the unrotated quantities, using Eq. 1.1 for the harmonics and Eq. 5.2 for ˆ 0. It is noted that the first term on the right hand side vanishes, because the rotated Ŷ 1,m can be expressed as inear combinations of the unrotated Y 1,m which, in turn, are orthogona to the Y 1,m. One obtains therefore D 1 mm A 1 m 1 i Y 1,m i Y D i0 D,m, 6.7 which expresses the matrix D 1 in terms of the matrices D and D. Inserting now, for the moment integras occurring in this equation, the expicit expressions derived in Sec. IV, one finds, for any vaue of m, the formua D 1 1 mm b m,m a m,m D 10 D m1,m 1 D 10 D m1,m b m,m 1 D 00 D m,m, 6.8 d mm 0 for m and m D. Recurrence reation for 2 m This recurrence reation is derived from the recurrence reation 3.8 for spherica harmonics. Starting with Eq. 3.8, a derivation that is entirey anaogous to that used in Sec. VI B yieds D m,m c m,m d m,m 1 D 0,1 D m,m 1 where the coefficients c mm 1 d m,m D 11 D m,m 1 1 D 1,1 D m1,m 1, 6.14 and d mm are again those defined by Eqs and This recurrence reation does not cover the cases m and m(1). E. Comment A knowedgeabe referee has caed the author s attention to a very genera reationship between Wigner D-functions with arbitrary sub- and superscripts which is isted, for instance, as Eq. 5 in Sec of the comprehensive formua coection for the quantum theory of anguar momenta by Varshaovich et a. 8 Due to its compete generaity, this identity is compex, containing numerous Cebsch Gordan coefficients. For the case that three superscripts in this equation are chosen as j 1 1, j 2 j1, j 3 j, the genera equation coapses in fact into our reations 6.1, 6.9, It is because this case is so eementary, that we were abe to reach the recurrence reations given here by a ine of reasoning that is simper than the body of deri- This artice is copyrighted as indicated in the artice. Reuse of AIP content is subject to the terms at: Downoaded to IP:

8 J. Chem. Phys., Vo. 111, No. 19, 15 November 1999 Recurrence reations for rotation matrices 8829 vations required to estabish the genera theory of the Wigner D-functions. One has to assume that it is because of the arge number of genera formuas in that theory and because of the compexity of most of them that the practica usefuness of our particuar identities for evauating rotation matrices has so far escaped notice. VII. OPERATIONAL ALGORITHM In the context of the practica quantitative use of rotation matrices, evauations of rea quantities are utimatey required. For the numerica execution it is therefore advantageous to recast the compex recurrence scheme of the preceding section in terms of a rea recurrence scheme for the rea and imaginary components F and G introduced through Eq This is accompished by inserting this resoution as we as Eq. 5.3 into Eqs. 6.1, 6.9, 6.14 and then separating the rea and the imaginary parts. The resuting formuas become more transparent by use of the intermediary quantities H m.m i, jf ij F 1 m,m G ij G 1 m,m, 7.1 K m,m i, jf ij G 1 m,m G ij F 1 m,m, 7.2 where the F ij and G ij are defined in Eqs. 5.4 and 5.5. With these definitions, one deduces from Eqs. 6.1, 6.9, 6.14 the foowing recurrence reations for F mm and. G mm A. Recurrence reation for 1 m 1 F mm G mm a m,m H m,m b m,m a m,m K m,m b m,m 0,0b m,m H m1,m,0, 0,0b m,m K m1,m,0. H m1,m K m1,m B. Recurrence reation for m 2 F m,m c m,m H m,m 1 d m,m d m,m 0, H m1,m 1 H m1,m 1,,,,0, G m,m c m,m K m,m 1 d m,m d m,m 0, K m1,m 1 K m1,m 1,,. C. Recurrence reation for 2 m F m,m G m,m c m,m H m,m 1 d m,m c m,m K m,m 1 d m,m The coefficients a mm 0,d m,m H m1,m 1,, 0,d m,m K m1,m 1,b mm,c mm,.,d mm H m1,m 1 K m1,m 1 7.6, 7.7, 7.8 occurring in these equations are those defined by Eqs. 6.2 to 6.5 and 6.10 to We note that every one of them is of the form / where,,, are a square-roots of integers not arger than (2L1) with L being the argest azimutha quantum number considered. VIII. RELATION TO REAL SPHERICAL HARMONICS AND THEIR ROTATION MATRICES In agreement with most authors, 9 Ivanic and Ruedenberg 3 define the rea spherica harmonics as where Y m,p m cos m, m 2 1/2 m0, 8.1 m 1/2 cos m m0, 8.2 m 1/2 sin m m0. They are reated to the compex harmonics of Eq. 2.2 by Y, Y,m Y,0 Y,m Y, Y, Y,m Y,0 Y,m Y, W, 8.3 where m is presumed to be a positive integer and the unitary matrices W are W W, W m,m W m,m W 0,0 W m,m W m,m W,. 8.4 W, W, This artice is copyrighted as indicated in the artice. Reuse of AIP content is subject to the terms at: Downoaded to IP:

9 8830 J. Chem. Phys., Vo. 111, No. 19, 15 November 1999 Choi et a. TABLE I. Accuracy of various methods for cacuating matrix eements D 0,0 for the Euer ange Wigner, exact a Recursion, d.p. b Wigner, d.p. c,d Inaccessibe e a Cacuated by Eq. 9.1 using MATHEMATICA 10 with the specification of 15 or more significant figures for the resut. b Cacuated by the recursion procedure of Sec. VII of this paper in doube precision arithmetic. c Cacuated using a Wigner formua program in doube precision arithmetic see Acknowedgments. d Incorrect numbers due to oss of significant figures are indicated by underined itaics. e Because in excess of 16 significant figures are ost. As indicated, they contain nonzero eements ony on the two diagonas; a other eements vanish. The nonvanishing eements are given by W 00 1, 8.5 and for m0: W m,m W m,m W m,m W m,m i/& i1 m /& 1/& 1 m. 8.6 /& It foows that the compex rotation matrices D of the present investigation and the rotation matrices R for the rea spherica harmonics 3 are reated by the simiarity transformation D W R W, 8.7 where (W ) denotes the hermitian conjugate of W. Insertion of Eq. 5.8 into the eft hand side of Eq. 8.7 and of Eqs. 8.4, 8.5, 8.6 into the right hand side of Eq. 8.7 yieds, after separation of the rea and imaginary parts, the reations 2F mn 2G mn m n R m,n m n R m,n, 8.8 m n R m,n m n R m,n, 8.9 where m and n can be positive, zero, or negative and the factors m, m are given by m 1 m0 1 mm 1/2, m signm1 mo m. The inverse identities are R m,n R m,n 8.10 m n F m,n m n F m,n, 8.11 m n F m,n m n F m,n, 8.12 IX. IMPLEMENTATION AND ASSESSMENT In the impementation of the code, we have used Eqs. 7.3, 7.4 to cacuate a matrix eements with m. Equations 7.5, 7.6 were used to cacuate the matrix eements with m, and Eqs. 7.7, 7.8 were used to cacuate the matrix eements with m. The program input consists of the axis rotation matrix R defined by Eq. 5.1 and the highest quantum number L. The program then finds the rea and imaginary parts of the compex rotation matrices D mm as we as using Eqs to 8.14, the rea rotation matrices R mm, for a quantum numbers 0,1,2,3,...L. The repeated cacuation of square-roots is avoided by generating a square-root tabe for a integers up to (2L1) before beginning the recursion. The accuracy of the program was tested by comparing our quantitative resuts with those obtained with the Ivanic Ruedenberg program 3 for rotation matrices of rea spherica harmonics for a number of cases. The resuts found by the two methods agreed to 14 significant figures up to L40. Since the two procedures go through very different sequences of extended numerica arithmetic, it can be inferred that no significant figures are ost by either one of the two agorithms. We confirmed this inference by additionay cacuating the eements to sufficient accuracy using MATHEMATICA 10 and Wigner s formuas. We aso found that the identities 5.8 were satisfied to fu accuracy by the vaues of F mm and F m,m. Computations with the Wigner formuas have the probem that the D mn, a of which have absoute vaues ess than unity, are obtained as sums of very arge positive and negative numbers. A transparent exampe is the expression for mm0 which is independent of and : R m,n R m,n m n G m,n m n G m,n, 8.13 m n G m,n m n G m,n, 8.14 where m and n are presumed to denote nonnegative integers. It shoud be noted that the matrix R for 1 is not identica with the matrix R of Eq. 5.1 but differs from it by a permutation of rows and coumns as foows D 00,, 1 k!/k!k! 2 x k 1x k, k xsin 2 /2. Its examination using Stiring s formua shows that, for arge and near /2, a oss of approximatey og(2 1 /) significant figures where og denotes the decima ogarithm R1,1 R 1,0 R is to be expected, suggesting a oss of about 7, 10, 13, 16, 1,1 Ryy Ryz Ryx and 29 significant figures for 30, 40, 50, 60, and 100, R 0,1 R 0,0 R 0,1 R zy R zz R zx respectivey. These predictions are in fact confirmed by a R 1,1 R 1,0 R 1,1 R xy R xz R xx. comparison of the numerica resuts dispayed in the three 8.15 rows of Tabe I which ist D 00 vaues obtained in three dif- This artice is copyrighted as indicated in the artice. Reuse of AIP content is subject to the terms at: Downoaded to IP: 9.1

10 J. Chem. Phys., Vo. 111, No. 19, 15 November 1999 Recurrence reations for rotation matrices 8831 TABLE II. Execution times and oss of significant figures for Wigner formuas. Euer anges /4. L Highest Vaue of Ratio of CPU Times a Matrix eements F mm Magnitudes of Largest Errors b with argest errors for L Number of Eements b Magnitudes of Eements to to to to to to to to to to 0.1 a Execution time for Wigner method/execution time for present method. The time is that used for the cacuation of a rotation matrices up to L. b Ony the matrices for the argest vaues L are considered. Cacuations are executed with a machine accuracy of 15 significant figures. The errors quoted are the differences between the resuts obtained by using the Wigner formuas and the present method. ferent ways, viz: i From Eq. 9.1 with MATHEMATICA, 10 exact to 15 significant figures, ii with our recurrence procedure of Sec. VII executed in doube-precision arithmetic, and iii with a genera Wigner formua program executed in doube-precision arithmetic. Striking in particuar are the resuts for 100 inasmuch as they impy that no significant figures whatsoever are ost by our recursion in a case where, even with quadrupe precision arithmetic, a significance is ost by the Wigner representation. A comparison of the performance, regarding speed as we as accuracy, of the present method with that of the evauation by Wigner s formuas is provided by Tabe II which ists some statistics for the rotation matrices for L 5, 10, 20, 30, 40, deduced from the rotation R with the Euer anges /4. Dispayed are the ratios of the execution times for the evauation of a matrices from 1 to L taken by the two methods, as we as some information pertaining to those matrix eements for the highest vaue L which have the argest errors in the Wigner procedure. Listed are the orders-of-magnitude of the argest errors found, the number of eements having such an error, and the magnitude of the eements themseves. Discrepancies of the same order-of-magnitude were aso found when inserting matrix eements obtained by the Wigner method into the identities 5.8. The quoted quantitative resuts exhibit the advantages of the described recursion. ACKNOWLEDGMENTS C.H.C. and M.S.G. acknowedge support by a DoD CHSSI grant and a grant from the Air Force Office of Scientific Research. K.R. and J.I. acknowedge support by the- Division of Chemica Sciences, Office of Basic Energy Science, U.S. Department of Energy, Contract No. W Eng-82. The authors are indebted to Dr. Dimitri Fedorov for etting them use his conventiona Wigner formua code. They aso thank Dr. Michae Schmidt for hepfu discussions. 1 L. F. Greengard, The Rapid Evauation of Potentia Fieds in Partice Systems MIT, Cambridge, MA, E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren Vieweg, Wiesbaden, Germany, J. Ivanic and K. Ruedenberg, J. Phys. Chem. 100, ; 102, A. R. Edmonds, Anguar Momentum in Quantum Mechanics Princeton University Press, Princeton, NJ, C. A. White and M. Head-Gordon, J. Chem. Phys. 105, E. U. Condon and G. H. Shortey, The Theory of Atomic Spectra Cambridge University Press, Cambridge, H. Bethe, in Handbuch der Physik, edited by H. Geiger and K. Schee Springer-Verag, Berin, 1933, Vo. 24/1, Chap. 3, Eq D. A. Varshaovich, A. N. Moskaey, and V. K. Khersonskii, Quantum Theory of Anguar Momentum Word Scientific, Singapore, See e.g., L. Pauing and E. B. Wison, Introduction to Quantum Mechanics McGraw-Hi, New York, S. Wofram, Mathematica, A System for Doing Mathematics by Computer Addison-Wesey, New York, This artice is copyrighted as indicated in the artice. Reuse of AIP content is subject to the terms at: Downoaded to IP: