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A topological geometric method for the obtention of symmetry-adapted functions for point groups IV. The dihedral groups
Computers Math. Applic. Vol. 29, No. 10, pp. 41-44, 1995 Copyright©1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/95 $9.50 + 0.00
Pergamon
0898-1221(95)00043-7
A Topological Geometric M e t h o d for the Obtention of S y m m e t r y - A d a p t e d Functions for Point Groups IV. The Dihedral Groups E. MARTINEZ-TORRES Departamento de Qufmica-Ffsica E.U. Profesorado de E.G.B. Universidad de Castilla-La Mancha 13071 Ciudad Real, Spain J . J . L O P E Z - G O N Z A L E Z AND M . F E R N A N D E Z - G O M E Z Departamento de Quimica-Ffsica y Analitica Facultad de Ciencias Experimentales Universidad de Ja~n 23071 Jadn, Spain
(Received April 1994; accepted November 1994) A b s t r a c t w A new mathematical method for the obtention of symmetry-adapted functions of the axial finite point groups (Cn, Cnh, Cnv, Dn, Dnh, Dnd, Sn), based on the topological geometric properties of spatial arrangements of sets of charged equivalent particles, has been used. Since in this method the explicit matrix representations of the different symmetry species of the point groups are not necessary, it has proved to be more simple than the projection operator classical method. K e y w o r d s - - D i h e d r a l , Eigenfunctions, Topological, Symmetry-adapted, Point groups.
1. I N T R O D U C T I O N With this paper, we would like to complete the application of the topological geometric method [1] to all the s y m m e t r y finite point groups in order to obtain s y m m e t r y - a d a p t e d functions. In previous papers, we have dealt with the case of the icosahedral [2] and cubic [3] groups. Here, we apply the method to those groups of lower symmetry, e.g., groups having an unique order n finite main rotation axis, namely axial or dihedral groups (Cn, Cnh, Cnv, Dn, Dnh, Dnd, Sn). S y m m e t r y - a d a p t e d functions for these last groups appear tabulated in the literature [4-6]. We will first study the case of the Dn group to any n value in a simple way and then the obtained results can be applied to the remaining finite axial point groups by using the mathematical relationships among them. Since Dn is a subgroup of K, we can obtain s y m m e t r y - a d a p t e d functions to the first taking suitable linear combinations of surface spherical harmonics (SALC's) [7].
2. S A L C ' S F O R G R O U P S
Dn
In order to apply the topological geometric method, let's build a set, C(n), of equivalent particles t h a t generates a representation of Dn in which all its irreducible representations are contained. This can be accomplished by applying the s y m m e t r y operations of Dn to a particle in Typeset by ~4A/~S-TEX 41
E. MARTINEZ-TOItRESet al.
42
an a r b i t r a r y spatial position. We thus obtain a set of 2n particles whose Cartesian coordinates are:
xg=r
cos(~r(2~ -1)
yg = r sin ( 7 r ( 2 ~ - 1 ) Z~ = (--1)'Z,
+ (-1)"¢) , (i)
+ (-i)"¢),
# = 1, 2, 3 , . . . , 2n,
where ¢ and z are arbitrary constants. T h e set C(n) generates the representation regular A n of the Dn group, whose reduction is the following: A n = A i q~ A2 ~ B1 @ B2 (~ 2E1 ~ 2E2 E~... ~ 2E(n/2)_1,
for even n,
A n = A1 (B A2 E~ 2Ei ~ 2 E - . . (B 2E(n-i)/2,
for odd n.
(2)
From equation (29) of [1], the elements of the adjacency matrix of the set C(n) are:
M(n)ij
[" 1,
if [ i - j l ( m o d 2 n -
I
otherwise.
0,
3) = 1,2, (3)
T h e eigenvalues and the eigenvectors' coefficients of M(n) are shown in Table 1. T h e general expressions of the S A L C ' s for groups Dn, which are shown in Table 2, have been obtained by substituting the spatial coordinates of the particles of the set C(n) from equation (1) and the coefficients S(F)~i from Table 1 in equation (4) of [3]. Table 1. Eigenvalue and coefficients for the eigenvectors of M(n). F eigenvalue i
S(F)~i
A1 A2
4 0
B1
-2
1 (-1)~/2 (for # even) (-1) (~-U/2 (for # odd)
B2
-2
1 (-1) (/~-2)/2 (for # even) (-1) (~-1)/2 (for # odd)
Ek
0
1 1 1 (-1) (~-1)
Table 2. SeAL's for groups D,~. (x axis coincident with C~ axis.) F
n
_n 2
1
i
¢~l)r
A1
even even odd odd
even/odd even/odd -
even odd even odd
1 1 1 1
~ cvYlnp/2'c ~ cpY~r'p/2'8 ~%Yz np'c
~cpYl np's
A2
even even odd odd
even/odd even/odd
even odd even odd
1
V nl p / 2 , /_..cp.
1
~'~CpYlnp/2'c
1 1
~ cpYlnp's ~-~cpYlnp'c
even even even even
even even even even
even odd even odd
1
~ opYln(2p+l)/4'c
1
~ CpYln(2p+l)/4'8
1
Y~C~pYln(2p+l)/4's ~ CpYln(2p+l)/4"c
1 ei~rk(~-l)/n 2 e -i~rk(~-l)/n B1 B2
1
s
Ek
odd even/odd 1 ~ [cpYta + (cpyta) *] odd even/odd 2 ~ [cpylb + (c~ylb)*] where p=0,1,2,... ; a= n(2p+l)+k]/2, b--[n(2p+l)-k]/2 for k odd; and a----(2pn+k)/2, b=[2n(p+l)-k]/2 for k even.
3. S A L C ' S
FOR THE
OTHERS
DIHEDRAL
GROUPS
According to the s y m m e t r y descent from Dn to Cn displayed in Table 3 [8], the S A L C ' s for Cn groups are the same as for Dn. O n the other hand, for the Dnh and Sn groups, we m u s t take into account the direct products shown in Table 4 [8]; thus, bearing in mind the s y m m e t r y properties of the spherical harmonics with respect to inversion, i, and the reflection a,~ [6]: i[Yzm(0, ¢)] = Ylm(Tr - 0, ¢ + 7r) = (-1)tYtm(0, ¢), ah[ylm(0, ¢)] = Ylm(Tr - 0, ¢) = (--1)l+mylm(0, ¢),
(4)
The Dihedral Groups
43
the function ~/~(0r ~i of Dn must be the ith basis function of the irreducible representation Fg or r~ (F ~ or r " ) of the group Ci®D,~ (Cs®Dn) if 1 (l+m) is even or odd, respectively. Finally, as shown in Table 5 for groups Dnd (with even values of n) and C~v, we can make the symmetry descent from D2nh and Dnh, respectively. Table 3. Symmetry descent from Dn to Cn. Dn A1 A2 B~ B2
Cn A A B B
Ek
Eka~]~ Ekb
Table 5. Symmetry descents from Dnh and D2nh. Dnh (odd n) Cnv A~ AI
A~ E~ A~ A~ Eg
D2nh Dnd Alg A2g Big B2g
A2 E~ A2 A1 Ek
Ekg Alu A2u Blu B2u Eku
Table 4. Direct products. even n odd n
Cnh = Ci® Cn Dnh = Ci® Dn $2, = Ci® C, C,h = Cs® C, Dnh = C8® Dn Dnd =- Ci® Dn
(even n) C2nv (even n) C2nv (odd n) AI AI A1
A2 B1 B2 E~ ) B1 B2 A1 A2
A2 B1 B2 Ek A2 AI B2 B1 Ek
A2 B1 B2 Ek A2 AI B1 B2 Ek
(*)see the Appendix
APPENDIX
In making the symmetry descent from D2nh to Dnh, the irreps Ekg and Eku are transformed into Er and Et, respectively. In this Appendix, we will obtain the relationship between the parameters k, r and t. The irreducible representations E8 of the Dnd group is that where the character of the symmetry operation S~n (=i c~n+l~ "~2, J is 2cos(2rs/2n). On the other hand, the character of this transformation is 2cos[27r(n+l)k/2n] and - 2 cos[21r(n+l)k/2n] in the irreducible representations Ek9 and Eku of the D2nh group, respectively [8]. Therefore, the representations Ekg and Ek~ of the D2nh reduce to Er and Et of the Dnd group respectively, where r and t are such that: cos
~(n + 1)k ~r = cos-n n
and
-cos
~(n + 1)k ~t - cos--. n n
(5)
Since Dnd has n - 1 twofold irreducible representations, we conclude from equation (5) that v, r
=
ifv
(w, t = ~
-
2n-v,
ifv>n-1,
2n-w,
ifw
ifw>n-1,
(6)
where v = k(n + 1)(mod 2n)
and
w = [k(n + 1) + n ] ( m o d 2 n ) .
(7)
REFERENCES
1. E. Martinez Torres, J.J. L6pez Gonz~lez and M. Fernandez G6mez, A topological geometric method for the obtention of symmetry-adapted functions for point groups--I. General theory, Computers Math. Applic. 26 (2), 79-85, (1993). 2. E. Martfnez Torres, J.J. L6pez Gonz~lez and M. Fern~.ndez G6mez, A topological geometric method for the obtention of symmetry-adapted functions for point groups--II. The icosahedral group, Computers Math. Applic. 26 (7), 67-77, (1993). 3. E. Martlnez Torres, J.J. L6pez Gonz~lez and M. Fernandez G6mez, A topological geometric method for the obtention of symmetry-adapted functions for point groups--III. Cubic group, Computers Math. Applic. (to appear).
44
E. MARTINEZ-TORRESet al.
4. S.L. Altmann, On the symmetries of spherical harmonics, Proc. Cam. Phil. Soc. 53, 343-367, (1957). 5. S.L. Altmann and C.J. Bradley, On the symmetrics of spherical harmonics, Phil. Trans. Royal Soc. A255, 199-215, (1963). 6. C.J. Bradley and A.P. Cracknell, The Mathematical Theory of Symmetry in Solids, Clarendon Press, London, (1972). 7. A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, (1957). 8. J.A. Salthouse and M.J. Ware, Point Group Character Tables, Cambridge University Press, Cambridge, (1972).
Pergamon
0898-1221(95)00043-7
A Topological Geometric M e t h o d for the Obtention of S y m m e t r y - A d a p t e d Functions for Point Groups IV. The Dihedral Groups E. MARTINEZ-TORRES Departamento de Qufmica-Ffsica E.U. Profesorado de E.G.B. Universidad de Castilla-La Mancha 13071 Ciudad Real, Spain J . J . L O P E Z - G O N Z A L E Z AND M . F E R N A N D E Z - G O M E Z Departamento de Quimica-Ffsica y Analitica Facultad de Ciencias Experimentales Universidad de Ja~n 23071 Jadn, Spain
(Received April 1994; accepted November 1994) A b s t r a c t w A new mathematical method for the obtention of symmetry-adapted functions of the axial finite point groups (Cn, Cnh, Cnv, Dn, Dnh, Dnd, Sn), based on the topological geometric properties of spatial arrangements of sets of charged equivalent particles, has been used. Since in this method the explicit matrix representations of the different symmetry species of the point groups are not necessary, it has proved to be more simple than the projection operator classical method. K e y w o r d s - - D i h e d r a l , Eigenfunctions, Topological, Symmetry-adapted, Point groups.
1. I N T R O D U C T I O N With this paper, we would like to complete the application of the topological geometric method [1] to all the s y m m e t r y finite point groups in order to obtain s y m m e t r y - a d a p t e d functions. In previous papers, we have dealt with the case of the icosahedral [2] and cubic [3] groups. Here, we apply the method to those groups of lower symmetry, e.g., groups having an unique order n finite main rotation axis, namely axial or dihedral groups (Cn, Cnh, Cnv, Dn, Dnh, Dnd, Sn). S y m m e t r y - a d a p t e d functions for these last groups appear tabulated in the literature [4-6]. We will first study the case of the Dn group to any n value in a simple way and then the obtained results can be applied to the remaining finite axial point groups by using the mathematical relationships among them. Since Dn is a subgroup of K, we can obtain s y m m e t r y - a d a p t e d functions to the first taking suitable linear combinations of surface spherical harmonics (SALC's) [7].
2. S A L C ' S F O R G R O U P S
Dn
In order to apply the topological geometric method, let's build a set, C(n), of equivalent particles t h a t generates a representation of Dn in which all its irreducible representations are contained. This can be accomplished by applying the s y m m e t r y operations of Dn to a particle in Typeset by ~4A/~S-TEX 41
E. MARTINEZ-TOItRESet al.
42
an a r b i t r a r y spatial position. We thus obtain a set of 2n particles whose Cartesian coordinates are:
xg=r
cos(~r(2~ -1)
yg = r sin ( 7 r ( 2 ~ - 1 ) Z~ = (--1)'Z,
+ (-1)"¢) , (i)
+ (-i)"¢),
# = 1, 2, 3 , . . . , 2n,
where ¢ and z are arbitrary constants. T h e set C(n) generates the representation regular A n of the Dn group, whose reduction is the following: A n = A i q~ A2 ~ B1 @ B2 (~ 2E1 ~ 2E2 E~... ~ 2E(n/2)_1,
for even n,
A n = A1 (B A2 E~ 2Ei ~ 2 E - . . (B 2E(n-i)/2,
for odd n.
(2)
From equation (29) of [1], the elements of the adjacency matrix of the set C(n) are:
M(n)ij
[" 1,
if [ i - j l ( m o d 2 n -
I
otherwise.
0,
3) = 1,2, (3)
T h e eigenvalues and the eigenvectors' coefficients of M(n) are shown in Table 1. T h e general expressions of the S A L C ' s for groups Dn, which are shown in Table 2, have been obtained by substituting the spatial coordinates of the particles of the set C(n) from equation (1) and the coefficients S(F)~i from Table 1 in equation (4) of [3]. Table 1. Eigenvalue and coefficients for the eigenvectors of M(n). F eigenvalue i
S(F)~i
A1 A2
4 0
B1
-2
1 (-1)~/2 (for # even) (-1) (~-U/2 (for # odd)
B2
-2
1 (-1) (/~-2)/2 (for # even) (-1) (~-1)/2 (for # odd)
Ek
0
1 1 1 (-1) (~-1)
Table 2. SeAL's for groups D,~. (x axis coincident with C~ axis.) F
n
_n 2
1
i
¢~l)r
A1
even even odd odd
even/odd even/odd -
even odd even odd
1 1 1 1
~ cvYlnp/2'c ~ cpY~r'p/2'8 ~%Yz np'c
~cpYl np's
A2
even even odd odd
even/odd even/odd
even odd even odd
1
V nl p / 2 , /_..cp.
1
~'~CpYlnp/2'c
1 1
~ cpYlnp's ~-~cpYlnp'c
even even even even
even even even even
even odd even odd
1
~ opYln(2p+l)/4'c
1
~ CpYln(2p+l)/4'8
1
Y~C~pYln(2p+l)/4's ~ CpYln(2p+l)/4"c
1 ei~rk(~-l)/n 2 e -i~rk(~-l)/n B1 B2
1
s
Ek
odd even/odd 1 ~ [cpYta + (cpyta) *] odd even/odd 2 ~ [cpylb + (c~ylb)*] where p=0,1,2,... ; a= n(2p+l)+k]/2, b--[n(2p+l)-k]/2 for k odd; and a----(2pn+k)/2, b=[2n(p+l)-k]/2 for k even.
3. S A L C ' S
FOR THE
OTHERS
DIHEDRAL
GROUPS
According to the s y m m e t r y descent from Dn to Cn displayed in Table 3 [8], the S A L C ' s for Cn groups are the same as for Dn. O n the other hand, for the Dnh and Sn groups, we m u s t take into account the direct products shown in Table 4 [8]; thus, bearing in mind the s y m m e t r y properties of the spherical harmonics with respect to inversion, i, and the reflection a,~ [6]: i[Yzm(0, ¢)] = Ylm(Tr - 0, ¢ + 7r) = (-1)tYtm(0, ¢), ah[ylm(0, ¢)] = Ylm(Tr - 0, ¢) = (--1)l+mylm(0, ¢),
(4)
The Dihedral Groups
43
the function ~/~(0r ~i of Dn must be the ith basis function of the irreducible representation Fg or r~ (F ~ or r " ) of the group Ci®D,~ (Cs®Dn) if 1 (l+m) is even or odd, respectively. Finally, as shown in Table 5 for groups Dnd (with even values of n) and C~v, we can make the symmetry descent from D2nh and Dnh, respectively. Table 3. Symmetry descent from Dn to Cn. Dn A1 A2 B~ B2
Cn A A B B
Ek
Eka~]~ Ekb
Table 5. Symmetry descents from Dnh and D2nh. Dnh (odd n) Cnv A~ AI
A~ E~ A~ A~ Eg
D2nh Dnd Alg A2g Big B2g
A2 E~ A2 A1 Ek
Ekg Alu A2u Blu B2u Eku
Table 4. Direct products. even n odd n
Cnh = Ci® Cn Dnh = Ci® Dn $2, = Ci® C, C,h = Cs® C, Dnh = C8® Dn Dnd =- Ci® Dn
(even n) C2nv (even n) C2nv (odd n) AI AI A1
A2 B1 B2 E~ ) B1 B2 A1 A2
A2 B1 B2 Ek A2 AI B2 B1 Ek
A2 B1 B2 Ek A2 AI B1 B2 Ek
(*)see the Appendix
APPENDIX
In making the symmetry descent from D2nh to Dnh, the irreps Ekg and Eku are transformed into Er and Et, respectively. In this Appendix, we will obtain the relationship between the parameters k, r and t. The irreducible representations E8 of the Dnd group is that where the character of the symmetry operation S~n (=i c~n+l~ "~2, J is 2cos(2rs/2n). On the other hand, the character of this transformation is 2cos[27r(n+l)k/2n] and - 2 cos[21r(n+l)k/2n] in the irreducible representations Ek9 and Eku of the D2nh group, respectively [8]. Therefore, the representations Ekg and Ek~ of the D2nh reduce to Er and Et of the Dnd group respectively, where r and t are such that: cos
~(n + 1)k ~r = cos-n n
and
-cos
~(n + 1)k ~t - cos--. n n
(5)
Since Dnd has n - 1 twofold irreducible representations, we conclude from equation (5) that v, r
=
ifv
(w, t = ~
-
2n-v,
ifv>n-1,
2n-w,
ifw
ifw>n-1,
(6)
where v = k(n + 1)(mod 2n)
and
w = [k(n + 1) + n ] ( m o d 2 n ) .
(7)
REFERENCES
1. E. Martinez Torres, J.J. L6pez Gonz~lez and M. Fernandez G6mez, A topological geometric method for the obtention of symmetry-adapted functions for point groups--I. General theory, Computers Math. Applic. 26 (2), 79-85, (1993). 2. E. Martfnez Torres, J.J. L6pez Gonz~lez and M. Fern~.ndez G6mez, A topological geometric method for the obtention of symmetry-adapted functions for point groups--II. The icosahedral group, Computers Math. Applic. 26 (7), 67-77, (1993). 3. E. Martlnez Torres, J.J. L6pez Gonz~lez and M. Fernandez G6mez, A topological geometric method for the obtention of symmetry-adapted functions for point groups--III. Cubic group, Computers Math. Applic. (to appear).
44
E. MARTINEZ-TORRESet al.
4. S.L. Altmann, On the symmetries of spherical harmonics, Proc. Cam. Phil. Soc. 53, 343-367, (1957). 5. S.L. Altmann and C.J. Bradley, On the symmetrics of spherical harmonics, Phil. Trans. Royal Soc. A255, 199-215, (1963). 6. C.J. Bradley and A.P. Cracknell, The Mathematical Theory of Symmetry in Solids, Clarendon Press, London, (1972). 7. A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, (1957). 8. J.A. Salthouse and M.J. Ware, Point Group Character Tables, Cambridge University Press, Cambridge, (1972).