- Email: [email protected]
Rotation of real spherical harmonics
Computer Physics Communications 52 (1989) 323—331 North-Holland, Amsterdam
323
ROTATION OF REAL SPHERICAL HARMONICS José Ramón ALVAREZ COLLADO, Jaime FERNANDEZ RICO, Rafael LOPEZ, Miguel PANIAGUA and Guillermo RAMIREZ Departamento de Quimica Fisica; Facultad de Ciencias C-XIV, Universidad Autónoma de Madri4 28049 Madri4
Spain
Received 1 May 1988; in final form 23 October 1988
In many physical problems, such as the calculation of interactions between multipoles, the rotation of spherical harmonics is necessary. In this paper, a computer program for the rotation of real spherical harmonics is reported. The program is designed for calculating sets of rotation matrices, for given Eulerian angles, and exploits some recurrence relations which make possible an important reduction of computational cost.
PROGRAM SUMMARY Title of program: ROTAR
ming real normalized spherical harmonics defined in the first system to those of the second one [1].
Catalogue number: ABHI Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computers for which theprogram is designed and others on which it is operable: IBM-4381/14, this program is operable on analogous computers Computer: IBM-4381/14; Installation: Centro de Cálculo, Umversidad Autónoma de Madrid Operating system or monitor under which the program is executed: VM/SP release 4 Programming language used: FORTRAN 77 High speed storage required: 48 Kwords No. of bits in a word: 32 No. of lines in combined programs and test deck: 673 Keywords: real spherical harmonics, multipolar expansions, molecular integrals with Slater basis, molecular electronic density Nature of physical problem Given a certain coordinate system and the Eulerian angles, the program provides the rotation matrices needed for transfor-
Method of solution The rotation is defined by the Eulerian angles a, $ and y. The rotation matrices for the real harmonics are obtained from those of the complex ones. The latter are obtained by using recurrence and symmetry relations. Restrictions on the complexity of the problem There are several definitions of real harmonics depending on the chosen phases and normalization factors. The rotation is referred to the harmonics as defined in text. Typical running time 0.5 s (IBM-4381/14) for the test deck (including input/output time). However, it is very important to consider the CPU time without the printing and test of the rotation matrices. In this 3 s for the calculation of all the case, we obtain 7.2x10 rotation matrices from a value of / = 0 up to a value / = 6 (test deck), or 4.4x102 s when calculating all the rotation matrices up to I = 12. Unusual features of the program Matrices are dimensioned from — 1 to + 1. This feature makes the code much simpler but must be taken into account when calling the subroutine from another program. References [1] J. Fernández Rico, J.R. Alvarez Collado and M. Paniagua, Mol. Phys. 56 (1985) 1145. J. Fernández Rico, J.R. Alvarez Collado, M. Paniagua and R. Lopez, Intern. J. Quantum Chem. 30 (1986) 671.
OO1O-4655/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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JR. Alvarez Collado et al.
/ Rotation of real spherical harmonics
LONG WRITE-UP 1. Theory There are many physical problems (calculations of molecular interactions, molecular integrals with Slater basis [1], the analysis of the molecular electronic density [2],...), where it is necessary to perform a rotation of spherical harmonics. Some of these problems are related to the calculation of the electrostatic interaction between two charge distributions placed at a long distance from one another. This interaction takes a simple form [3], in terms of the multipolar moments of the distributions, when these moments are referred to lined-up coordinate systems as those plotted in fig. 1. Nevertheless, the multipolar moments are often referred to non-aligned systems. In this case, the calculation of the interaction can be accomplished by transforming those multipolar moments to the lined-up systems. This transformation implies the rotation of spherical harmonics. Here, we are interested in the rotation when the multipoles appear in terms of normalized real spherical harmonics, defined as: m(O, ~) (_1)~m~ ~[(2l+ 1)(’- mi)!] =
point in the rotated system and in the original one, respectively. A similar, but reversed situation, occurs in the analysis of the molecular electronic density [2]. Now, the multipolar moments are calculated in the lined-up coordinate systems (fig. 1) associated with each bond, and then transformed to the molecular system in order to be added with those corresponding to the other bonds. To obtain the matrices corresponding to the rotation of normalized real harmonics, we start with those corresponding to the normalized cornplex harmonics: ym(g
~)
=
(_1)
+HD/2([(
[4 1
21+
1)(l— ml)!]
1/2
( + m i).i } x(—l)’~P”~1(cos9)
/t
e1m~~.
The rotation of complex spherical harmonics has been studied in detail by several authors [4,5]. It should be noted that there are different definitions of the associated Legendre functions employed by these authors. We use here the definition given by Gradshteyn and Ryzhik [61.
z, /[2~(1 +6mø)(1+
mI)!1}’~2
xpIm~i(cos9)cosImI~ mO,
ZA,;
(1) z~(O,~)=(_1)Im~{[(21+1)(l_
imi)!]
/[2~(l+lmi)!]}1/2 xPmIi(cos9)sinlml~
m
with plm as given in ref. [6]. This rotation can be expressed as: ~,
z~(8’, ~‘)
=
~z~’(O, ~)R’m’m(a,
/3,
y),
(3)
where a /3, y are the Eulerian angles defining the rotation (see fig. 2), R’ is the rotation matrix, and (9’, q’), (9, 4) are the angular coordinates of a
(4)
A
A
Fig. 1. The lined-up coordinate systems.
J.R. Alvarez Collado et a!. / Rotation of real spherical harmonics
~
325
the terms in the sum being all for which the factorials are defined. We transform from corn-
7
harmonics: plex normalized harmonics to real normalized
-
4,
(...z12, z[1, z,0,
z~,...)
y,—1 y 0 Y/ Y 2,...)xu’ by=(...~,—2, means of the matrix 1 1
(7)
U1 = 2~”~
ZI
i
1 i
1
X~
I
21/2 i 1
13
1 —1
—i
i
1
—1
I
z’ (8) The final transformation matrix is: RI’=U_lDtU,
(9)
where the elements of the matrix D’ are given by
~ Fig. 2.
Eulerian anglesydefining the rotation.
the elements eq. (5). of R’According are: to R’ m’,m =d’ —m’,—m cos(m’a +
eqs. my) (7) and (9), the
+(~1)md~’_m cos(m’a—my), 21~2df~m’ cos
=
The transformations under a general rotation of the axis system determined by the Eulenan angles a, /3, y are represented in the complex spherical harmonic basis by the matrices [5]. —
,~‘
=—-“
m’ ,- m +
(
(11)
sin(m’a+my) 1)tm d~’,_~sin(m’a
m’
~-
—
m’a,
—
(10)
—
my),
(12)
R~m = 2”2d~ =
m, m,m’. e_R~m’d1, ‘~$~ e~’
(5)
where d’~ m ,m\
=
(_1)m +m[(l_
0
cos my,
(13)
R’~
m)!(1+ m)!(1— m’)!
0—d~, 1”2d~ R~_m = —2 0sin my, R’ = ~ sin(m’a +
(14) (15) my)
/
md~’_~
sin(m’a—my),
(16)
+(—1) X
~(
i)~~ [cos(/3/2)]2/+m_m’_21c
R’.~’ 0
=
k
Dt
~‘—m’,—m
~ =
41
sin m’a, cos(m’a
m
—
k)!(l—
m’
—
my)
U_m’._m
+(_1)m~d~,_~
/[k!(l+
(17) +
cos(m’a
—
my), (18)
k)!
(6)
m, m’>O.
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J. R. Alvarez Collado et a!.
/
In many problems, such as the calculation of molecular electronic integrals or molecular calculations, the rotations can be chosen in such a way that the Eulerian angle y is equal to zero. In this case, the equations can be further simplified. An optimized version of the program for y 0 has been also developed and is available on request. =
2. Details of the calculation 2.1. Input The input is read in from the file ROTAR DATA Al, which contains only three records with a free format:
Rotation of real spherical harmonics
real normalized spherical harmonics z~ for the non-rotated system (original) and z’1 for the rotated system, and by calling the subroutine ROTAR we get the elements R’rn’ m of the rotation matrices needed for verifying the test. The subroutine ROTAR yields the rotation matrices R’ that are necessary to perform a coordinate transformation. This transformation converts each original real normalized spherical harmonic into a linear combination of the real normalized spherical harmonics with the same / and different m. The maximum value of / that we chose for the rotation matrices was 12, but this procedure can be applied to greater values. The resulting rotation matrices one for each / value are located in the array R, that is dimensioned to R( LMAX: LMAX, LMAX: LMAX, 0: LMAX). The subroutine ROTAR performs the computation of the four initial matrices d°(equalto 1), R°(equalto 1), d1 and R1, see eq. (5). The remaining rotation matrices are calculated by means of recurrence and symmetry relations using the previous ones. Thus, every d’ matrix requires the d~ and d~2matrices. The subroutine DLMN, called by ROTAR, —
-
record 1 datum: description:
LMAX maximum value of the quantum number / for the rotation matrices, in the present program this value must not be greater than 12.
record 2 data: description:
ALPHA, BETA, GAMMA the Eulerian angles in radians characterizing the rotated coordinate systems.
record 3
-
yields the d’ and R’ matrices corresponding to a given value of 1. In the construction of every d’ matrix, the elements d~rn’with m > 0 and m : rn’ ~ m are calculated by means of the following relations: —
=cos2(/3/2)d~
d1±1
1,
“1±1,1+1
data: description:
X, Y, Z coordinates of the P point related to the original system. These data are needed only for the test on the accuracy of the rotation matrices,
d’~ 1÷1 rn—I
—tg(/3/2)[(I+1+m) 1/2
/(I+ 2—rn)]
(20)
~
and 1
2.2. Calculations The MAIN program is intended only to test to correctness of the driver subroutine (ROTAR) and to verify the accuracy of the rotation matrices provided by it. The test is performed by comparison of both terms, computed independently, in the equality of the eq. (3) for all the possible rn’ values. In the main program, we only calculate the
=
(19)
rn ,rn
=
(1+ 1)/[(/+ m
+ 1)(1—
m+
1)
d’~
x (1+ m’
+
1)(/— m’
x ((2/ + 1) [cos /3
—
+
mm ‘/1(1 + 1)] ~
rn
_l~[(l+ m)(l— m)(l+ m’) x(l— m’)] ‘~2d~.’rn }.
(21)
JR. Alvarez Collado et a!.
/
Rotation of real spherical harmonics
The remaining elements of the matrix are obtained from these with the aid of two simple relations: =
Umrn’ U_,fl_pfl
~
m’
m,
327
generates the matrix dt for a fixed value of the 1 index in the spherical harmonics, and it needs the d’2 and d’’ matrices. The subroutine uses the
DLMN:
(22)
Um’rn =
d’m’,m
m’
m.
(23)
The corresponding R’ matrix is then obtained by using eqs. (10) to (18). The driver subroutine ROTAR calls the subroutine DLMN as many times as necessary for the construction of the required rotation matrices. 2.3. Output The output lists: all the rotation matrices up to the value of I LMAX, the Eulerian angles a, /3 and y, the coordinates of the point P relative to the original system, the coordinates of the point p’ (P in the rotated system), and finally, the comparison between the two members in eq. (3) for all the m’ values, that we use here to test the accuracy of the obtained rotation matrices. The discrepancies observed in the results between the two members of eq. (3) show that the error is located at the fifteenth decimal digit.
recurrence and symmetry relations given by the eqs. (19) to (23). Finally, computes the R’ matrix using the eqs. (10) to (18). MATPRT: is used to print out the matrices R’ with an appropriate format. This subroutine can be supressed in order to save paper (or disk space) and cornputing time. 3.2. Block data
=
SQR:
initializes the common SQROOT with the square roots of the first 25 integers and their reciprocals. When values of LMAX higher than 12 are required, values of the square roots of the natural numbers and their reciprocals up to 2 * LMAX + 1 must be included respectively in the arrays ROOT and ROOTI of the SQROOT common.
3. Description of the program 3.3. 3.1.
Common blocks
Subroutines
MAIN:
ROTAR:
makes a test on the accuracy of the rotation matrices provided by the driver subroutine ROTAR. The test is performed by verifying the equality of both members in the eq. (3). The discrepancies on these two terms give a criterium for evaluating the accuracy of the rotation matrices. yields the rotation matrices R’ up to a given value of I (LMAX). These matrices are necessary in order to perform a coordinate transformation. This transformation converts each original normalized real spherical harmonic into a linear combination of the normalized real spherical harmonics with the same value of 1.
SQROOT: contains the square roots of the first 25 natural numbers and their reciprocals.
4. Test run The test run presents the results for the rotation matrices and the accuracy test for LMAX 6. The Eulerian angles have the following values: ALPHA 0.78539816, BETA 0.78539816, GAMMA 0.39269908. The reference point P is located at (1.0, 2.0, 3.0) with respect to the original. system or (3.61626180, 0.73253780, 0.62132036) with respect to the rotated system. The error in the comparison of the two members of the eq. (3) is in the fifteenth decimal digit. =
=
=
=
—
—
328
JR. Alvarez Collado et al.
/
Rotation of real spherical harmonics
Acknowledgements Financial support from the Comisión Asesora para la Investigación CientIfica y Técnica (ref. PB-85-0211) is gratefully acknowledged. References [11 F.E. Harris and H.H. Michels, Advan. Chem. Phys. 13 (1967) 205. H.H. Michels, ETO Multicenter Molecular Integrals, eds. C. Weatherford and H.W. Jones (Reidel, Dordrecht, 1982). J. Fernández Rico, R. Lopez and G. Ramirez, J. Comput. Chem. 9 (1988) 790. [2] J. Fernández Rico, J.R. Alvarez Collado and M. Paniagua, Mol. Phys. 56 (1985) 1145.
J. Fernández Rico, JR. Alvarez Collado, M. Paniagua and R. Lopez, Intern. J. Quantum Chem. 30 (1986) 671. [3]
Fernández Rico, R. Lopez and G. Ramirez, to be published. JO. Hirschfelder, C.F. Curtis and R.B. Bird, Molecular Theory of Gases and Liquids (John Wiley, New York,
1954). [4] M.E. Rose, Elementary Theory of Angular Momentum (John Wiley, New York, 1957). ME. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, New Jersey, 1974). E.P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra (Academic Press, New York, 1959). [5] E.O. Steinborn, Advan. Quantum Chem. 7 (1973) 83. [6] l.S. Gradshteyn and tM. Ryahik, Table of Integrals, Series and Products, 4th ed. (Academic Press, New York, 1980) p. 1015, eq. (8.812).
JR. Alvarez Collado et a!. / Rotation of real spherical harmonics
TEST RUN INPUT 6
0.78539816
-0.78539816
0.39269908
1.00000000
2.00000000
3.00000000
TEST RUN OUTPUT THE ROTATION MATRICES FOR LMAX ROTATION MATRIX FOR L
=
O 1
To
6 ARE:
1
-1
-1
liP
0
O.4619397697D4-O0 -O.4999999966D+OO -0.2705980480D+00 -0.8446231974D+00
ROTATION MATRIX FOR L
0.70710678360+00 -0.50000000000+00 =
1
O.7325378165D+OO 0.6532814807D+00 0.1913417217D+00
2
—2
-1
-1 0 1 2
0.10355339350+00 0.4619397679D+00 -0.612372433611+00 0.19134172110+00 -0.3061862147D+O0 -0. 3314135727D+0O 0.250000005111+00 0.80010314580+00 -0.6035533894D+00 -0.4619397684D+00 -0.61237243780+00 -0. 19134171100+00 -0. 5000000070D+00 0. 6532814820D+00 0. 2942276548D-08 0.27059804490+00 2 0.5303300923D+0O 0.6035533894D+00 0.30618621680+00 -0. 1035533853D+00 -0.4999999964D+00
ROTATION MATRIX FOR 4 = 3 -3 -2 -3 -0.57332108280+00 0.6309457440D+00 -2 -0.60007735540+00 -0. 1250000025D+00 -1 -0.92632906870-01 0. 2555304602D+00 O -0.25823217650+00 -0.48412291500+00 1 -O.3546383452D+O0 -0.53503895850+00 2 -0.2343447925D+00 -0.56540797200-08 3 0.2350735065D+O0 -0. 18573299630-01 1 2 -3 0.92632914470-01 -0. 18573316780-01 -2 -0.25823218630+00 0.12500000340+00 -1 -0. 3O27230200D+00 0.53503895850+00 O 0.60007736410+00 0.48412291830+00 1 -0. 50567156180+00 -0. 25553045290+00 2 0.30253781430+00 -0. 39553554820-08 3 -0. 35463834240+00 0.63094574380+00
-3 -2
-1
0 -0. 19764235380+00 0.68465319460+00 -0.45927932890+00 -0. 17677668990+00 -0.45927933200+00 0.46521477140-08 0.19764234970+00
-4 -0.10823115920-07
-3 -2 0.23917715610+00 -0.33071891870+00
-1 0.30554443390+00
-0.43750000790+00
0.10557789530+00
=
4
-0.49607836860+00 -0.43210508580+00 -0. 16535945370+00 -0.26211566600-01 -0.48310816710+00
0.39921304850+00
-0.54780854180+00
0.88388343600-01 -0.6764951295D-O1 0.40088834690+00 0.294515765811+00
0
-0. 18487749070+00
1 2 3
-0. 16535945800+00 -0.40589351880+00 -0.22411165850+00 -0.65488231750-08 -0. 12656065580+00 0.37499999720+00 0.43749999300+00 -0.57742466740-01 -0.68494123090-01
4
O.5312500030D+OO -0.57162129320+00 0 1 0. 2512446891D-08 0. 1265606444D+00 -O,3697549877D+00 0.26211575840-01 0.6987712449D+00 0.16332036130+00 -0. 1397542548D+00 -0.54770357930+00 -0.40624999790+00 0.18259773420+00 -0. 13975425570+00 -0.59553901450+00 0.47480784070-08 0.23917714680+00 0.36975498020+00 -0.40589351590+00 -0. 18487749070+00 0.21605254270+00
-4 -3 -2 -1 0 1 2 3 4
-0.4619397684D+00
-1 -0.4089O14429D+00 0.10696326750+00 0. 4124045582D+00 -0.2485601815D+00 -0.7755655836D-01 0.73039090280+00 -0.22363563060+00 3 -0.24087691700+00 0.24856018710+00 0.40890144650+00 0.10696327130+00 -0. 22363562270+00 -0.56575835670+00 -0.57572493420+00
ROTATION MATRIX FOR I.
-4
0.43301269890+00
1
-0.53033008190+00
-2 -1 0 1 2
0. 1913417110D+0O
0
-2
-0.49410588410+00
0.35078037270+00 2 -0.33071890690+00 -0.684941339411-01 -0.88388349300-01 0.22411165850+00 0.49410588740+00 -0.40088834380+00 0.37499999350+00 0.39921304750+00 -0.35078038410+00
-0.75634457610-01
0.17903081980+00 0.57742471130+00 0. 36882476420+00 -0.89491888090-01 3 0.57742470380+00 0.23917711330-01 0.17898379190+00 0.54780854700+00 0.20010995780+00 -0.36882475400+00 -0. 30554443890+00 0.91567222640-01 0.236773304811+00
329
330
JR. Alvarez Collado eta!.
-4 -3 -2 -l O 1 2
4 -0.53033008890+00 -0.4419417311D+0O 0.6548823175D-08 0.23385358620+00 0.12562234520-08 -0.23385358240+00 -0.46770717180+00
3
-0.44194174600+00
4
-0. 10816865200-07
ROTATION MATRIX FOR L = 5 -5 -4 -5 0.41854490270+00 -0.41998999270+00 -4 O.2269033514D+O0 0.500196736211-08
-3 -2 -1 0 1
/
Rotation of real spherical harmonics
-3 -2 0.19711512300+00 -0. 10382787670-01 0.10147427500+00 -0.30618622610+00
-0.19711514260+00 -0.165728159211+00 0.40761195530+00 -0.33545344100+00 -0.53582588200+00 -0. 15001934410+00 -0. 1658668951D+OO -0. 21480823250+00 0.14503154170+00
-0.11457916180+00 -0.39218438340+00 -0.5978158599D+OO -0.32021721580+00 -0.43325078460-01 -0.21480823940+00 -0.23858504450+00 0. 1269575197D+00
2
0.1310027118D+OO -O.65831581O3D-O8
3
0.48074433280+00
4
0.54874338810+00 -O.243O679512D+OO -0.21653426930+00
5
0.46433351940-02 0.62499998940-01 0.45767643940+00
0.5858619251D-01
0.53033008690+00
0.16572815330+00 -0.21311959840+00 -0.15773643360+00
0.37888610930+00
-5 -4
0.17331543320+00 -0.41999001260+00 0.48074434090+00 -0.35270939110+00 -1 0 1 2 -0. 10459598930+00 0.87695O9202D-O1 -0. 1658668953D+O0 0.35270939080+00 0.5292184847D+00 0.53297047190-08 0.21920946830+00 -0.30618621370+00
-3 -2
-0.4824417373D+00 -O.457548456OD+O0 -0.1450315351D+OO -0.15773643370+00 -0. 22372973160+00 0.45285553100+00 0.54013134910+00 -0. 62500006580-01
-1
0.12411715120+00
0.18154608740+00 -0.47287271940+00 -0.12695751970+00
O 1 2 3
0.98252034410-01 -0.37565048090+00 -0.2372O13951D+O0 0.32021721800+00 0.2466074266D+00 0.18154608860+00 -0.42213558980+00 -0.45767644110+00 0.30554444050+00 0.30771065410-08 0. 1265606548D+O0 0.5303300825D+00 -0.44369061660+00 0.45754844670+00 -0. 23858504340+00 0.4643333051D-02
4
-0. 11625337190+00
5
0.19124613600+00 0.8769509500D-01 3 4 -0.48276003160+00 0.41926275670+00 0.24498055920+00 0.24999999490+00 0.10621531180+00 -0. 18750000080+00 0.6214004669D-O1 0.65831581050-08
-5 -4 -3 -2 -1
0.48244174240+00
-0.39218438340+00
0.40504629370+00
0.28066048880+00 -0.37888612020+00
-0.43325079420-01 5 -0. 17343989670+00 -0.54779318240+00 -0.48276002340+00 -0.13894935890+00
0. 1038280205D-O1
0.10459598780+00
O 1
0.24762344050+00 0.26648523730-08 -0.47460241640-01 -0.44369060840+00 -0.40504628810+00 -0. 19124613710+00
2 3 4 5
0.14143958140+00 -0.43301270480+00 -0.31626853760+00 0.36333074920+00 -0. 1875000069D+O0 -0.2O19814502D+OO 0.89691436270-01 0.5049O69883D-08 0.22729693940+00 0.20198143060+00 -0.41926273680+00 0.41859645720+00
ROTATION MATRIX FOR 4 = 6 -6 -5 -4 -3 -6 0.27345146120+00 -0.51256608540+00 0.53948692470+00 -0. 37470091520+00 -5 O.5548800241D+O0 -0.31724664620+00 -0.7773339628D-O1 0.14702528440+00 -4 0.38081429070+00 0.39663042820-01 0.8629972278D-08 -0.46316456220-01 -3 0.20587923960-02 -0. 14702529990+00 0. 15128$4087D+O0 0.36303027280+00 -2 -0. 18436093130+00 -0.46357216830+00 -0.36372201510+00 0.21653423560-01 -1 -0. 13269920100+00 -0. 28729538610+00 -0. 13395646990+00 0.33548873350+00
0 1
2 3 4 5 6 -6 -5 -4 -3 -2 -1
-0.59369859370-01 -0.26871195400+00 -0.55808816420+00 -0.52297993960+00 0.22767581460-01 -0.44409855530-01 -0. 13395647750+00 0.5185952437D-O1 0.17381714720+00 0.16293256660+00 -0.32938448740-08 0.21563282100+00 0.40763215310+00 0.37744674600+00 -0.15128840450+00 -0.24942111750+00 0.38147487190+00 0.10156358000+00 -0.41406250110+00 0.19766782810+00 -0.82482689880-04 -0. 13188437230+00 -0.77733399920-01 0.377446752311+00 -0.27343749390+00 0.21237494790+00 0.14656684010-07 -0. 15678195000+00 -2 0.18436092410+00 -0.38001772900-01 -0.85581659500-01 -0.29345660160+00 -0.12429611100+00 0. 3756960490D+OO
-1 0 1 -0.42069005750-01 -0.83961659460-01 0.10156357300+00 -0.22449038050+00 0.20566322150+00 -0.287295387011+00 0.63647815480+00 0.75843028130-08 0.26363787980+00 -O.2621482845D+O0 -0.40027152160+00 -0.335488732411+00 -0.27417489640+00 0.56606951570-01 0.66191675590+00 -0. 5031218583D-01 0.35441550370+00 -0. 16434628800+00
j.jt
0 1 2 3 4 5 6
-0.400271591811-01 0.34075748210+00 0.46093750450+00 -0.27001910900+00 0.25719030130+00 -0.46376511440+00 0.17381715460+00 2 -0. 18436093240+00 0.46376511340+00 -0.8558165193D-01 -0.2700191011D+00 0.12429610560+00 -0.34075748210+00 0.40027159450-01 -0.37569605360+00 0.46093750310+00 -0.29345660530+00 -0.25719030540+00 0.38001791810-01 0.17381714590+00 6 0.27345144450+00 -0.82512852700-04 -0. 38081430890+00 -0.40763215310+00 -0.18436092520+00 0.22767585070-01 -0.59369858160-01 -0. 13269920130+00 -0. 17381715330+00 -0.20588090150-02 0.38147485370+00 0.55488002410+00 0.27343751070+00
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Alvarez Collado et aL
0.19180828830+00 0.15178646190+00 0.3227902709D-O1 -0.42259319250+00 -0. 72496712200-01 0.36188717900+00 -0.143632581111+00 3 0.155206212211+00 -0.386765022311+00 -0. 11181780760+00 0.10295531480-01 -0.89691459840-02 0.26214828790+00 0.21662538700+00 -0.42259318930+00 0.52058367300+00 0.26398121050+00 -0.81876698180-01 0.16952157740+00 -0.37850508290+00
/ Rotation of real spherical harmonics
-0. 14843750780+00 0.35441550610+00 0.384638386911-09 0.40027151350+00 -0.55808816420+00 0.20566322850+00 -0.17115324000-08 4 -0. 14643988730-07 0.73287752750-01 0.441941738511+00 0.855816439311-01 0.329384488111-08 0.487139291311+00 0.379215142611-08 -0.487139286111+00 -0.121030738211+00 0.85581647730-01 0.881940881411-08 -0.73287749200-01 0.53855274750+00
THE ORIGIN OF COORDINATES IS LOCATED FOR THE “A” EULERIAN ANGLES (IN RADIANS): 0.78539816E+O0 POINT “P” COORDINATES: O.10000000E+01 “P’” (P IN THE ROTATED SYSTEM): O.36162618E+01
-0.4630661732D+00 -0.80634288590-01 0.13370416010-01 0.51859522680-01 0.17502257110+00 -0.44409856970-01 -0.59494560360-01 5 -0,21231180680+00 0.13073400110+00 -0.957550621211-01 -0.38676501570+00 -0.19201787150+00 0.2244903776D+00 -0.1113041330D+00 -0.361887181711+00 -0.39335403040+00 -0. 16952159310+00 0.42069009740-01 -0.31677014690+00 -0.51271852140+00
CENTRE AT THE POINT (0,0,0) -0.78539816E+00 O.392699O8E+O0 0.20000000E+01 0.30000000E+O1 -0.73253780E+00 0.62132036E+00
COMPARISON BETWEEN THE TWO MEMBERS IN EQUATION (3) M= M= M= M= M= M= M= M M= M= M= M= M=
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST
MEMBER=-. 14672108451268819E+01 MEMBER=-. 77260836316116457E+00 MEMBER=0.59739213610707670E+00 MEMBER=0.55953292739905587E+00 MEMBER=-. 23102710511818779E+OO MEMBER=-. 19930691523608743E+O0 MEMBER=-. 37276553506091320E+00 MEMBER=0.98390278472709068E+O0 MEMBER=0.54684736298212509E+O0 MEMBER=- . 81858733033658659E+OO MEMBER=- . 58083121497114518E+O0 MEMBER=O.49683241201166027E+O0 MEMBER=O.5718076482591769OE+00
2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND
MEMBER=-. 146721O8451268808E+01 MEMBER=-. 77260836316116374EfOO MEMBER=0.59739213610707612E+00 MEMBER=0.55953292739905514E+00 MEMBER=-. 23102710511818786E+O0 MEMBER=-. 19930691523608722E+00 MEMBER=-. 37276553506091351E+00 MEMBER=0.98390278472708878E+00 MEMBER=0.54684736298212412E+0O MEMBER-. 81858733033658504E+00 MEMBER=- . 58083121497114505E+00 MEMBER=0.49683241201165926E+0O MEMBER=O.57180764825917657E+0O
331
323
ROTATION OF REAL SPHERICAL HARMONICS José Ramón ALVAREZ COLLADO, Jaime FERNANDEZ RICO, Rafael LOPEZ, Miguel PANIAGUA and Guillermo RAMIREZ Departamento de Quimica Fisica; Facultad de Ciencias C-XIV, Universidad Autónoma de Madri4 28049 Madri4
Spain
Received 1 May 1988; in final form 23 October 1988
In many physical problems, such as the calculation of interactions between multipoles, the rotation of spherical harmonics is necessary. In this paper, a computer program for the rotation of real spherical harmonics is reported. The program is designed for calculating sets of rotation matrices, for given Eulerian angles, and exploits some recurrence relations which make possible an important reduction of computational cost.
PROGRAM SUMMARY Title of program: ROTAR
ming real normalized spherical harmonics defined in the first system to those of the second one [1].
Catalogue number: ABHI Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computers for which theprogram is designed and others on which it is operable: IBM-4381/14, this program is operable on analogous computers Computer: IBM-4381/14; Installation: Centro de Cálculo, Umversidad Autónoma de Madrid Operating system or monitor under which the program is executed: VM/SP release 4 Programming language used: FORTRAN 77 High speed storage required: 48 Kwords No. of bits in a word: 32 No. of lines in combined programs and test deck: 673 Keywords: real spherical harmonics, multipolar expansions, molecular integrals with Slater basis, molecular electronic density Nature of physical problem Given a certain coordinate system and the Eulerian angles, the program provides the rotation matrices needed for transfor-
Method of solution The rotation is defined by the Eulerian angles a, $ and y. The rotation matrices for the real harmonics are obtained from those of the complex ones. The latter are obtained by using recurrence and symmetry relations. Restrictions on the complexity of the problem There are several definitions of real harmonics depending on the chosen phases and normalization factors. The rotation is referred to the harmonics as defined in text. Typical running time 0.5 s (IBM-4381/14) for the test deck (including input/output time). However, it is very important to consider the CPU time without the printing and test of the rotation matrices. In this 3 s for the calculation of all the case, we obtain 7.2x10 rotation matrices from a value of / = 0 up to a value / = 6 (test deck), or 4.4x102 s when calculating all the rotation matrices up to I = 12. Unusual features of the program Matrices are dimensioned from — 1 to + 1. This feature makes the code much simpler but must be taken into account when calling the subroutine from another program. References [1] J. Fernández Rico, J.R. Alvarez Collado and M. Paniagua, Mol. Phys. 56 (1985) 1145. J. Fernández Rico, J.R. Alvarez Collado, M. Paniagua and R. Lopez, Intern. J. Quantum Chem. 30 (1986) 671.
OO1O-4655/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
324
JR. Alvarez Collado et al.
/ Rotation of real spherical harmonics
LONG WRITE-UP 1. Theory There are many physical problems (calculations of molecular interactions, molecular integrals with Slater basis [1], the analysis of the molecular electronic density [2],...), where it is necessary to perform a rotation of spherical harmonics. Some of these problems are related to the calculation of the electrostatic interaction between two charge distributions placed at a long distance from one another. This interaction takes a simple form [3], in terms of the multipolar moments of the distributions, when these moments are referred to lined-up coordinate systems as those plotted in fig. 1. Nevertheless, the multipolar moments are often referred to non-aligned systems. In this case, the calculation of the interaction can be accomplished by transforming those multipolar moments to the lined-up systems. This transformation implies the rotation of spherical harmonics. Here, we are interested in the rotation when the multipoles appear in terms of normalized real spherical harmonics, defined as: m(O, ~) (_1)~m~ ~[(2l+ 1)(’- mi)!] =
point in the rotated system and in the original one, respectively. A similar, but reversed situation, occurs in the analysis of the molecular electronic density [2]. Now, the multipolar moments are calculated in the lined-up coordinate systems (fig. 1) associated with each bond, and then transformed to the molecular system in order to be added with those corresponding to the other bonds. To obtain the matrices corresponding to the rotation of normalized real harmonics, we start with those corresponding to the normalized cornplex harmonics: ym(g
~)
=
(_1)
+HD/2([(
[4 1
21+
1)(l— ml)!]
1/2
( + m i).i } x(—l)’~P”~1(cos9)
/t
e1m~~.
The rotation of complex spherical harmonics has been studied in detail by several authors [4,5]. It should be noted that there are different definitions of the associated Legendre functions employed by these authors. We use here the definition given by Gradshteyn and Ryzhik [61.
z, /[2~(1 +6mø)(1+
mI)!1}’~2
xpIm~i(cos9)cosImI~ mO,
ZA,;
(1) z~(O,~)=(_1)Im~{[(21+1)(l_
imi)!]
/[2~(l+lmi)!]}1/2 xPmIi(cos9)sinlml~
m
with plm as given in ref. [6]. This rotation can be expressed as: ~,
z~(8’, ~‘)
=
~z~’(O, ~)R’m’m(a,
/3,
y),
(3)
where a /3, y are the Eulerian angles defining the rotation (see fig. 2), R’ is the rotation matrix, and (9’, q’), (9, 4) are the angular coordinates of a
(4)
A
A
Fig. 1. The lined-up coordinate systems.
J.R. Alvarez Collado et a!. / Rotation of real spherical harmonics
~
325
the terms in the sum being all for which the factorials are defined. We transform from corn-
7
harmonics: plex normalized harmonics to real normalized
-
4,
(...z12, z[1, z,0,
z~,...)
y,—1 y 0 Y/ Y 2,...)xu’ by=(...~,—2, means of the matrix 1 1
(7)
U1 = 2~”~
ZI
i
1 i
1
X~
I
21/2 i 1
13
1 —1
—i
i
1
—1
I
z’ (8) The final transformation matrix is: RI’=U_lDtU,
(9)
where the elements of the matrix D’ are given by
~ Fig. 2.
Eulerian anglesydefining the rotation.
the elements eq. (5). of R’According are: to R’ m’,m =d’ —m’,—m cos(m’a +
eqs. my) (7) and (9), the
+(~1)md~’_m cos(m’a—my), 21~2df~m’ cos
=
The transformations under a general rotation of the axis system determined by the Eulenan angles a, /3, y are represented in the complex spherical harmonic basis by the matrices [5]. —
,~‘
=—-“
m’ ,- m +
(
(11)
sin(m’a+my) 1)tm d~’,_~sin(m’a
m’
~-
—
m’a,
—
(10)
—
my),
(12)
R~m = 2”2d~ =
m, m,m’. e_R~m’d1, ‘~$~ e~’
(5)
where d’~ m ,m\
=
(_1)m +m[(l_
0
cos my,
(13)
R’~
m)!(1+ m)!(1— m’)!
0—d~, 1”2d~ R~_m = —2 0sin my, R’ = ~ sin(m’a +
(14) (15) my)
/
md~’_~
sin(m’a—my),
(16)
+(—1) X
~(
i)~~ [cos(/3/2)]2/+m_m’_21c
R’.~’ 0
=
k
Dt
~‘—m’,—m
~ =
41
sin m’a, cos(m’a
m
—
k)!(l—
m’
—
my)
U_m’._m
+(_1)m~d~,_~
/[k!(l+
(17) +
cos(m’a
—
my), (18)
k)!
(6)
m, m’>O.
326
J. R. Alvarez Collado et a!.
/
In many problems, such as the calculation of molecular electronic integrals or molecular calculations, the rotations can be chosen in such a way that the Eulerian angle y is equal to zero. In this case, the equations can be further simplified. An optimized version of the program for y 0 has been also developed and is available on request. =
2. Details of the calculation 2.1. Input The input is read in from the file ROTAR DATA Al, which contains only three records with a free format:
Rotation of real spherical harmonics
real normalized spherical harmonics z~ for the non-rotated system (original) and z’1 for the rotated system, and by calling the subroutine ROTAR we get the elements R’rn’ m of the rotation matrices needed for verifying the test. The subroutine ROTAR yields the rotation matrices R’ that are necessary to perform a coordinate transformation. This transformation converts each original real normalized spherical harmonic into a linear combination of the real normalized spherical harmonics with the same / and different m. The maximum value of / that we chose for the rotation matrices was 12, but this procedure can be applied to greater values. The resulting rotation matrices one for each / value are located in the array R, that is dimensioned to R( LMAX: LMAX, LMAX: LMAX, 0: LMAX). The subroutine ROTAR performs the computation of the four initial matrices d°(equalto 1), R°(equalto 1), d1 and R1, see eq. (5). The remaining rotation matrices are calculated by means of recurrence and symmetry relations using the previous ones. Thus, every d’ matrix requires the d~ and d~2matrices. The subroutine DLMN, called by ROTAR, —
-
record 1 datum: description:
LMAX maximum value of the quantum number / for the rotation matrices, in the present program this value must not be greater than 12.
record 2 data: description:
ALPHA, BETA, GAMMA the Eulerian angles in radians characterizing the rotated coordinate systems.
record 3
-
yields the d’ and R’ matrices corresponding to a given value of 1. In the construction of every d’ matrix, the elements d~rn’with m > 0 and m : rn’ ~ m are calculated by means of the following relations: —
=cos2(/3/2)d~
d1±1
1,
“1±1,1+1
data: description:
X, Y, Z coordinates of the P point related to the original system. These data are needed only for the test on the accuracy of the rotation matrices,
d’~ 1÷1 rn—I
—tg(/3/2)[(I+1+m) 1/2
/(I+ 2—rn)]
(20)
~
and 1
2.2. Calculations The MAIN program is intended only to test to correctness of the driver subroutine (ROTAR) and to verify the accuracy of the rotation matrices provided by it. The test is performed by comparison of both terms, computed independently, in the equality of the eq. (3) for all the possible rn’ values. In the main program, we only calculate the
=
(19)
rn ,rn
=
(1+ 1)/[(/+ m
+ 1)(1—
m+
1)
d’~
x (1+ m’
+
1)(/— m’
x ((2/ + 1) [cos /3
—
+
mm ‘/1(1 + 1)] ~
rn
_l~[(l+ m)(l— m)(l+ m’) x(l— m’)] ‘~2d~.’rn }.
(21)
JR. Alvarez Collado et a!.
/
Rotation of real spherical harmonics
The remaining elements of the matrix are obtained from these with the aid of two simple relations: =
Umrn’ U_,fl_pfl
~
m’
m,
327
generates the matrix dt for a fixed value of the 1 index in the spherical harmonics, and it needs the d’2 and d’’ matrices. The subroutine uses the
DLMN:
(22)
Um’rn =
d’m’,m
m’
m.
(23)
The corresponding R’ matrix is then obtained by using eqs. (10) to (18). The driver subroutine ROTAR calls the subroutine DLMN as many times as necessary for the construction of the required rotation matrices. 2.3. Output The output lists: all the rotation matrices up to the value of I LMAX, the Eulerian angles a, /3 and y, the coordinates of the point P relative to the original system, the coordinates of the point p’ (P in the rotated system), and finally, the comparison between the two members in eq. (3) for all the m’ values, that we use here to test the accuracy of the obtained rotation matrices. The discrepancies observed in the results between the two members of eq. (3) show that the error is located at the fifteenth decimal digit.
recurrence and symmetry relations given by the eqs. (19) to (23). Finally, computes the R’ matrix using the eqs. (10) to (18). MATPRT: is used to print out the matrices R’ with an appropriate format. This subroutine can be supressed in order to save paper (or disk space) and cornputing time. 3.2. Block data
=
SQR:
initializes the common SQROOT with the square roots of the first 25 integers and their reciprocals. When values of LMAX higher than 12 are required, values of the square roots of the natural numbers and their reciprocals up to 2 * LMAX + 1 must be included respectively in the arrays ROOT and ROOTI of the SQROOT common.
3. Description of the program 3.3. 3.1.
Common blocks
Subroutines
MAIN:
ROTAR:
makes a test on the accuracy of the rotation matrices provided by the driver subroutine ROTAR. The test is performed by verifying the equality of both members in the eq. (3). The discrepancies on these two terms give a criterium for evaluating the accuracy of the rotation matrices. yields the rotation matrices R’ up to a given value of I (LMAX). These matrices are necessary in order to perform a coordinate transformation. This transformation converts each original normalized real spherical harmonic into a linear combination of the normalized real spherical harmonics with the same value of 1.
SQROOT: contains the square roots of the first 25 natural numbers and their reciprocals.
4. Test run The test run presents the results for the rotation matrices and the accuracy test for LMAX 6. The Eulerian angles have the following values: ALPHA 0.78539816, BETA 0.78539816, GAMMA 0.39269908. The reference point P is located at (1.0, 2.0, 3.0) with respect to the original. system or (3.61626180, 0.73253780, 0.62132036) with respect to the rotated system. The error in the comparison of the two members of the eq. (3) is in the fifteenth decimal digit. =
=
=
=
—
—
328
JR. Alvarez Collado et al.
/
Rotation of real spherical harmonics
Acknowledgements Financial support from the Comisión Asesora para la Investigación CientIfica y Técnica (ref. PB-85-0211) is gratefully acknowledged. References [11 F.E. Harris and H.H. Michels, Advan. Chem. Phys. 13 (1967) 205. H.H. Michels, ETO Multicenter Molecular Integrals, eds. C. Weatherford and H.W. Jones (Reidel, Dordrecht, 1982). J. Fernández Rico, R. Lopez and G. Ramirez, J. Comput. Chem. 9 (1988) 790. [2] J. Fernández Rico, J.R. Alvarez Collado and M. Paniagua, Mol. Phys. 56 (1985) 1145.
J. Fernández Rico, JR. Alvarez Collado, M. Paniagua and R. Lopez, Intern. J. Quantum Chem. 30 (1986) 671. [3]
Fernández Rico, R. Lopez and G. Ramirez, to be published. JO. Hirschfelder, C.F. Curtis and R.B. Bird, Molecular Theory of Gases and Liquids (John Wiley, New York,
1954). [4] M.E. Rose, Elementary Theory of Angular Momentum (John Wiley, New York, 1957). ME. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, New Jersey, 1974). E.P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra (Academic Press, New York, 1959). [5] E.O. Steinborn, Advan. Quantum Chem. 7 (1973) 83. [6] l.S. Gradshteyn and tM. Ryahik, Table of Integrals, Series and Products, 4th ed. (Academic Press, New York, 1980) p. 1015, eq. (8.812).
JR. Alvarez Collado et a!. / Rotation of real spherical harmonics
TEST RUN INPUT 6
0.78539816
-0.78539816
0.39269908
1.00000000
2.00000000
3.00000000
TEST RUN OUTPUT THE ROTATION MATRICES FOR LMAX ROTATION MATRIX FOR L
=
O 1
To
6 ARE:
1
-1
-1
liP
0
O.4619397697D4-O0 -O.4999999966D+OO -0.2705980480D+00 -0.8446231974D+00
ROTATION MATRIX FOR L
0.70710678360+00 -0.50000000000+00 =
1
O.7325378165D+OO 0.6532814807D+00 0.1913417217D+00
2
—2
-1
-1 0 1 2
0.10355339350+00 0.4619397679D+00 -0.612372433611+00 0.19134172110+00 -0.3061862147D+O0 -0. 3314135727D+0O 0.250000005111+00 0.80010314580+00 -0.6035533894D+00 -0.4619397684D+00 -0.61237243780+00 -0. 19134171100+00 -0. 5000000070D+00 0. 6532814820D+00 0. 2942276548D-08 0.27059804490+00 2 0.5303300923D+0O 0.6035533894D+00 0.30618621680+00 -0. 1035533853D+00 -0.4999999964D+00
ROTATION MATRIX FOR 4 = 3 -3 -2 -3 -0.57332108280+00 0.6309457440D+00 -2 -0.60007735540+00 -0. 1250000025D+00 -1 -0.92632906870-01 0. 2555304602D+00 O -0.25823217650+00 -0.48412291500+00 1 -O.3546383452D+O0 -0.53503895850+00 2 -0.2343447925D+00 -0.56540797200-08 3 0.2350735065D+O0 -0. 18573299630-01 1 2 -3 0.92632914470-01 -0. 18573316780-01 -2 -0.25823218630+00 0.12500000340+00 -1 -0. 3O27230200D+00 0.53503895850+00 O 0.60007736410+00 0.48412291830+00 1 -0. 50567156180+00 -0. 25553045290+00 2 0.30253781430+00 -0. 39553554820-08 3 -0. 35463834240+00 0.63094574380+00
-3 -2
-1
0 -0. 19764235380+00 0.68465319460+00 -0.45927932890+00 -0. 17677668990+00 -0.45927933200+00 0.46521477140-08 0.19764234970+00
-4 -0.10823115920-07
-3 -2 0.23917715610+00 -0.33071891870+00
-1 0.30554443390+00
-0.43750000790+00
0.10557789530+00
=
4
-0.49607836860+00 -0.43210508580+00 -0. 16535945370+00 -0.26211566600-01 -0.48310816710+00
0.39921304850+00
-0.54780854180+00
0.88388343600-01 -0.6764951295D-O1 0.40088834690+00 0.294515765811+00
0
-0. 18487749070+00
1 2 3
-0. 16535945800+00 -0.40589351880+00 -0.22411165850+00 -0.65488231750-08 -0. 12656065580+00 0.37499999720+00 0.43749999300+00 -0.57742466740-01 -0.68494123090-01
4
O.5312500030D+OO -0.57162129320+00 0 1 0. 2512446891D-08 0. 1265606444D+00 -O,3697549877D+00 0.26211575840-01 0.6987712449D+00 0.16332036130+00 -0. 1397542548D+00 -0.54770357930+00 -0.40624999790+00 0.18259773420+00 -0. 13975425570+00 -0.59553901450+00 0.47480784070-08 0.23917714680+00 0.36975498020+00 -0.40589351590+00 -0. 18487749070+00 0.21605254270+00
-4 -3 -2 -1 0 1 2 3 4
-0.4619397684D+00
-1 -0.4089O14429D+00 0.10696326750+00 0. 4124045582D+00 -0.2485601815D+00 -0.7755655836D-01 0.73039090280+00 -0.22363563060+00 3 -0.24087691700+00 0.24856018710+00 0.40890144650+00 0.10696327130+00 -0. 22363562270+00 -0.56575835670+00 -0.57572493420+00
ROTATION MATRIX FOR I.
-4
0.43301269890+00
1
-0.53033008190+00
-2 -1 0 1 2
0. 1913417110D+0O
0
-2
-0.49410588410+00
0.35078037270+00 2 -0.33071890690+00 -0.684941339411-01 -0.88388349300-01 0.22411165850+00 0.49410588740+00 -0.40088834380+00 0.37499999350+00 0.39921304750+00 -0.35078038410+00
-0.75634457610-01
0.17903081980+00 0.57742471130+00 0. 36882476420+00 -0.89491888090-01 3 0.57742470380+00 0.23917711330-01 0.17898379190+00 0.54780854700+00 0.20010995780+00 -0.36882475400+00 -0. 30554443890+00 0.91567222640-01 0.236773304811+00
329
330
JR. Alvarez Collado eta!.
-4 -3 -2 -l O 1 2
4 -0.53033008890+00 -0.4419417311D+0O 0.6548823175D-08 0.23385358620+00 0.12562234520-08 -0.23385358240+00 -0.46770717180+00
3
-0.44194174600+00
4
-0. 10816865200-07
ROTATION MATRIX FOR L = 5 -5 -4 -5 0.41854490270+00 -0.41998999270+00 -4 O.2269033514D+O0 0.500196736211-08
-3 -2 -1 0 1
/
Rotation of real spherical harmonics
-3 -2 0.19711512300+00 -0. 10382787670-01 0.10147427500+00 -0.30618622610+00
-0.19711514260+00 -0.165728159211+00 0.40761195530+00 -0.33545344100+00 -0.53582588200+00 -0. 15001934410+00 -0. 1658668951D+OO -0. 21480823250+00 0.14503154170+00
-0.11457916180+00 -0.39218438340+00 -0.5978158599D+OO -0.32021721580+00 -0.43325078460-01 -0.21480823940+00 -0.23858504450+00 0. 1269575197D+00
2
0.1310027118D+OO -O.65831581O3D-O8
3
0.48074433280+00
4
0.54874338810+00 -O.243O679512D+OO -0.21653426930+00
5
0.46433351940-02 0.62499998940-01 0.45767643940+00
0.5858619251D-01
0.53033008690+00
0.16572815330+00 -0.21311959840+00 -0.15773643360+00
0.37888610930+00
-5 -4
0.17331543320+00 -0.41999001260+00 0.48074434090+00 -0.35270939110+00 -1 0 1 2 -0. 10459598930+00 0.87695O9202D-O1 -0. 1658668953D+O0 0.35270939080+00 0.5292184847D+00 0.53297047190-08 0.21920946830+00 -0.30618621370+00
-3 -2
-0.4824417373D+00 -O.457548456OD+O0 -0.1450315351D+OO -0.15773643370+00 -0. 22372973160+00 0.45285553100+00 0.54013134910+00 -0. 62500006580-01
-1
0.12411715120+00
0.18154608740+00 -0.47287271940+00 -0.12695751970+00
O 1 2 3
0.98252034410-01 -0.37565048090+00 -0.2372O13951D+O0 0.32021721800+00 0.2466074266D+00 0.18154608860+00 -0.42213558980+00 -0.45767644110+00 0.30554444050+00 0.30771065410-08 0. 1265606548D+O0 0.5303300825D+00 -0.44369061660+00 0.45754844670+00 -0. 23858504340+00 0.4643333051D-02
4
-0. 11625337190+00
5
0.19124613600+00 0.8769509500D-01 3 4 -0.48276003160+00 0.41926275670+00 0.24498055920+00 0.24999999490+00 0.10621531180+00 -0. 18750000080+00 0.6214004669D-O1 0.65831581050-08
-5 -4 -3 -2 -1
0.48244174240+00
-0.39218438340+00
0.40504629370+00
0.28066048880+00 -0.37888612020+00
-0.43325079420-01 5 -0. 17343989670+00 -0.54779318240+00 -0.48276002340+00 -0.13894935890+00
0. 1038280205D-O1
0.10459598780+00
O 1
0.24762344050+00 0.26648523730-08 -0.47460241640-01 -0.44369060840+00 -0.40504628810+00 -0. 19124613710+00
2 3 4 5
0.14143958140+00 -0.43301270480+00 -0.31626853760+00 0.36333074920+00 -0. 1875000069D+O0 -0.2O19814502D+OO 0.89691436270-01 0.5049O69883D-08 0.22729693940+00 0.20198143060+00 -0.41926273680+00 0.41859645720+00
ROTATION MATRIX FOR 4 = 6 -6 -5 -4 -3 -6 0.27345146120+00 -0.51256608540+00 0.53948692470+00 -0. 37470091520+00 -5 O.5548800241D+O0 -0.31724664620+00 -0.7773339628D-O1 0.14702528440+00 -4 0.38081429070+00 0.39663042820-01 0.8629972278D-08 -0.46316456220-01 -3 0.20587923960-02 -0. 14702529990+00 0. 15128$4087D+O0 0.36303027280+00 -2 -0. 18436093130+00 -0.46357216830+00 -0.36372201510+00 0.21653423560-01 -1 -0. 13269920100+00 -0. 28729538610+00 -0. 13395646990+00 0.33548873350+00
0 1
2 3 4 5 6 -6 -5 -4 -3 -2 -1
-0.59369859370-01 -0.26871195400+00 -0.55808816420+00 -0.52297993960+00 0.22767581460-01 -0.44409855530-01 -0. 13395647750+00 0.5185952437D-O1 0.17381714720+00 0.16293256660+00 -0.32938448740-08 0.21563282100+00 0.40763215310+00 0.37744674600+00 -0.15128840450+00 -0.24942111750+00 0.38147487190+00 0.10156358000+00 -0.41406250110+00 0.19766782810+00 -0.82482689880-04 -0. 13188437230+00 -0.77733399920-01 0.377446752311+00 -0.27343749390+00 0.21237494790+00 0.14656684010-07 -0. 15678195000+00 -2 0.18436092410+00 -0.38001772900-01 -0.85581659500-01 -0.29345660160+00 -0.12429611100+00 0. 3756960490D+OO
-1 0 1 -0.42069005750-01 -0.83961659460-01 0.10156357300+00 -0.22449038050+00 0.20566322150+00 -0.287295387011+00 0.63647815480+00 0.75843028130-08 0.26363787980+00 -O.2621482845D+O0 -0.40027152160+00 -0.335488732411+00 -0.27417489640+00 0.56606951570-01 0.66191675590+00 -0. 5031218583D-01 0.35441550370+00 -0. 16434628800+00
j.jt
0 1 2 3 4 5 6
-0.400271591811-01 0.34075748210+00 0.46093750450+00 -0.27001910900+00 0.25719030130+00 -0.46376511440+00 0.17381715460+00 2 -0. 18436093240+00 0.46376511340+00 -0.8558165193D-01 -0.2700191011D+00 0.12429610560+00 -0.34075748210+00 0.40027159450-01 -0.37569605360+00 0.46093750310+00 -0.29345660530+00 -0.25719030540+00 0.38001791810-01 0.17381714590+00 6 0.27345144450+00 -0.82512852700-04 -0. 38081430890+00 -0.40763215310+00 -0.18436092520+00 0.22767585070-01 -0.59369858160-01 -0. 13269920130+00 -0. 17381715330+00 -0.20588090150-02 0.38147485370+00 0.55488002410+00 0.27343751070+00
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Alvarez Collado et aL
0.19180828830+00 0.15178646190+00 0.3227902709D-O1 -0.42259319250+00 -0. 72496712200-01 0.36188717900+00 -0.143632581111+00 3 0.155206212211+00 -0.386765022311+00 -0. 11181780760+00 0.10295531480-01 -0.89691459840-02 0.26214828790+00 0.21662538700+00 -0.42259318930+00 0.52058367300+00 0.26398121050+00 -0.81876698180-01 0.16952157740+00 -0.37850508290+00
/ Rotation of real spherical harmonics
-0. 14843750780+00 0.35441550610+00 0.384638386911-09 0.40027151350+00 -0.55808816420+00 0.20566322850+00 -0.17115324000-08 4 -0. 14643988730-07 0.73287752750-01 0.441941738511+00 0.855816439311-01 0.329384488111-08 0.487139291311+00 0.379215142611-08 -0.487139286111+00 -0.121030738211+00 0.85581647730-01 0.881940881411-08 -0.73287749200-01 0.53855274750+00
THE ORIGIN OF COORDINATES IS LOCATED FOR THE “A” EULERIAN ANGLES (IN RADIANS): 0.78539816E+O0 POINT “P” COORDINATES: O.10000000E+01 “P’” (P IN THE ROTATED SYSTEM): O.36162618E+01
-0.4630661732D+00 -0.80634288590-01 0.13370416010-01 0.51859522680-01 0.17502257110+00 -0.44409856970-01 -0.59494560360-01 5 -0,21231180680+00 0.13073400110+00 -0.957550621211-01 -0.38676501570+00 -0.19201787150+00 0.2244903776D+00 -0.1113041330D+00 -0.361887181711+00 -0.39335403040+00 -0. 16952159310+00 0.42069009740-01 -0.31677014690+00 -0.51271852140+00
CENTRE AT THE POINT (0,0,0) -0.78539816E+00 O.392699O8E+O0 0.20000000E+01 0.30000000E+O1 -0.73253780E+00 0.62132036E+00
COMPARISON BETWEEN THE TWO MEMBERS IN EQUATION (3) M= M= M= M= M= M= M= M M= M= M= M= M=
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST 1ST
MEMBER=-. 14672108451268819E+01 MEMBER=-. 77260836316116457E+00 MEMBER=0.59739213610707670E+00 MEMBER=0.55953292739905587E+00 MEMBER=-. 23102710511818779E+OO MEMBER=-. 19930691523608743E+O0 MEMBER=-. 37276553506091320E+00 MEMBER=0.98390278472709068E+O0 MEMBER=0.54684736298212509E+O0 MEMBER=- . 81858733033658659E+OO MEMBER=- . 58083121497114518E+O0 MEMBER=O.49683241201166027E+O0 MEMBER=O.5718076482591769OE+00
2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND 2ND
MEMBER=-. 146721O8451268808E+01 MEMBER=-. 77260836316116374EfOO MEMBER=0.59739213610707612E+00 MEMBER=0.55953292739905514E+00 MEMBER=-. 23102710511818786E+O0 MEMBER=-. 19930691523608722E+00 MEMBER=-. 37276553506091351E+00 MEMBER=0.98390278472708878E+00 MEMBER=0.54684736298212412E+0O MEMBER-. 81858733033658504E+00 MEMBER=- . 58083121497114505E+00 MEMBER=0.49683241201165926E+0O MEMBER=O.57180764825917657E+0O
331