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Point Group Symmetry E
It is assumed that the reader has previously learned, in undergraduate inorganic or physical chemistry classes, how symmetry arises in molecular shapes and structures and what symmetry elements are (e.g., planes, axes of rotation, centers of inversion, etc.). For the reader who feels, after reading this appendix, that additional background is needed, the texts by Cotton and EWK, as well as most physical chemistry texts can be consulted. We review and teach here only that material that is of direct application to symmetry analysis of molecular orbitals and vibrations and rotations of molecules. We use a specific example, the ammonia molecule, to introduce and illustrate the important aspects of point group symmetry.
I. The C3v Symmetry Group of Ammonia - An Example
The ammonia molecule NH 3 belongs, in its ground-state equilibrium geometry, to
the C3v point group. Its symmetry operations consist of two C 3 rotations, C 3 , C 32
(rotations by 120° and 240°, respectively about an axis passing through the nitrogen atom and lying perpendicular to the plane formed by the three hydrogen atoms), three vertical
reflections, σv, σv' (^) , σv", and the identity operation. Corresponding to these six operations
are symmetry elements: the three-fold rotation axis, C 3 and the three symmetry planes σv,
σv' and σv" that contain the three NH bonds and the z-axis (see figure below).
N
σv'
σv
σxz is σv'' H 3
H 2
H 1
C 3 axis (z)
x- axis
y- axis
These six symmetry operations form a mathematical group. A group is defined as a set of objects satisfying four properties.
- A combination rule is defined through which two group elements are combined to give a result which we call the product. The product of two elements in the group must also be a member of the group (i.e., the group is closed under the combination rule).
- One special member of the group, when combined with any other member of the group, must leave the group member unchanged (i.e., the group contains an identity element).
- Every group member must have a reciprocal in the group. When any group member is combined with its reciprocal, the product is the identity element.
- The associative law must hold when combining three group members (i.e., (AB)C must equal A(BC)).
The members of symmetry groups are symmetry operations; the combination rule is successive operation. The identity element is the operation of doing nothing at all. The group properties can be demonstrated by forming a multiplication table. Let us label the rows of the table by the first operation and the columns by the second operation. Note that this order is important because most groups are not commutative. The C3v group multiplication table is as follows:
E C (^3) C 32 σv σv'^ σv"^ second operation
E E C (^3) C 32 σv σv'^ σv" C 3 C (^3) C 32 E (^) σ v'^ σv"^ σv
C 32 C 32 E^ C^3 σv" σv σv'
σv σv σv" σv' E^ C 32 C^3
σv' σv' σv σv" C^3 E^ C 32
σv" σv" σv' σv C 32 C^3 E First operation
Note the reflection plane labels do not move. That is, although we start with H 1 in the σv
plane, H 2 in σv'', and H 3 in σv", if H 1 moves due to the first symmetry operation, σv
remains fixed and a different H atom lies in the σv plane.
II. Matrices as Group Representations
In using symmetry to help simplify molecular orbital or vibration/rotation energy level calculations, the following strategy is followed:
- A set of M objects belonging to the constituent atoms (or molecular fragments, in a more general case) is introduced. These objects are the orbitals of the individual atoms (or of the fragments) in the m.o. case; they are unit vectors along the x, y, and z directions located on each of the atoms, and representing displacements along each of these directions, in the vibration/rotation case.
- Symmetry tools are used to combine these M objects into M new objects each of which belongs to a specific symmetry of the point group. Because the hamiltonian (electronic in the m.o. case and vibration/rotation in the latter case) commutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "block diagonal". That is, objects of different symmetry will not interact; only interactions among those of the same symmetry need be considered.
To illustrate such symmetry adaptation, consider symmetry adapting the 2s orbital of N and the three 1s orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3v point group. The act of each of the six symmetry operations on the four atomic orbitals can be denoted as follows:
D(4)(σv") =
It is easy to verify that a C 3 rotation followed by a σv reflection is equivalent to a σv' reflection alone. In other words
σv C 3 = σv' , or,
S 1
S 2 S 3
C 3
S 3
S 1 S 2
σv
S 3
S 2 S 1
Note that this same relationship is carried by the matrices:
D(4)(σv) D(4)(C 3 ) =
0 0 1 0
= D(4)(σv')
Likewise we can verify that C 3 σv = σv" directly and we can notice that the matrices also show the same identity:
D(4)(C 3 ) D(4)(σv) =
0 0 1 0
= D(4)(σv").
In fact, one finds that the six matrices, D(4)(R), when multiplied together in all 36 possible ways obey the same multiplication table as did the six symmetry operations. We say the matrices form a representation of the group because the matrices have all the properties of the group.
A. Characters of Representations
One important property of a matrix is the sum of its diagonal elements Tr(D) = (^) ∑
i
Dii = χ.
χ (^) is called the trace or character of the matrix. In the above example
χ(E) = 4 χ(C 3 ) = χ(C 32 ) = 1
χ(σv) = χ(σv') = χ(σv") = 2.
The importance of the characters of the symmetry operations lies in the fact that they do not depend on the specific basis used to form them. That is, they are invariant to a unitary or orthorgonal transformation of the objects used to define the matrices. As a result, they contain information about the symmetry operation itself and about the space spanned by the set of objects. The significance of this observation for our symmetry adaptation process will become clear later. Note that the characters of both rotations are the same as are those of all three reflections. Collections of operations having identical characters are called classes. Each operation in a class of operations has the same character as other members of the class. The character of a class depends on the space spanned by the basis of functions on which the symmetry operations act. Above we used (SN,S 1 ,S 2 ,S 3 ) as a basis.
B. Another Basis and Another Representation
If, alternatively, we use the one-dimensional basis consisting of the 1s orbital on the N-atom, we obtain different characters, as we now demonstrate. The act of the six symmetry operations on this SN can be represented as follows:
SN →
E
SN; SN →
C 3
SN; SN →
C 32
SN;
SN →
σv SN; SN →
σv' SN; SN →
σv" SN. We can represent this group of operations in this basis by the one-dimensional set of matrices:
D(1)^ (E) = 1; D(1)^ (C 3 ) = 1; D(1)^ (C 32 ) = 1, D(1)^ (σv) = 1; D(1)(σv") = 1; D(1)^ (σv') = 1. Again we have
D(1)^ (σv) D(1)^ (C 3 ) = 1 ⋅ 1 = D(1)^ (σv"), and
D(1)^ (C 3 ) D(1)^ (σv) = 1 ⋅ 1 = D(1)^ (σv').
These six matrices form another representation of the group. In this basis, each character is equal to unity. The representation formed by allowing the six symmetry operations to act on the 1s N-atom orbital is clearly not the same as that formed when the same six operations acted on the (SN,S 1 ,S 2 ,S 3 ) basis. We now need to learn how to further analyze the information content of a specific representation of the group formed when the symmetry operations act on any specific set of objects.
III. Reducible and Irreducible Representations
A. A Reducible Representation
Note that every matrix in the four dimensional group representation labeled D(4)^ has the so-called block diagonal form
0 3 x 3 matrix 0
D(3)(E) =
;D(3)(C 3 ) =
1 2
3 2
1 2
1 2
D(3)(C 32 ) =
1 2
3 2
1 2
1 2
;D(3)(σv) =
D(3)(σv') =
1 2
3 2
1 2
1 2
;D(3)(σv") =
1 2
3 2
1 2
1 2
C. Reduction of the Reducible Representation
These six matrices can be verified to multiply just as the symmetry operations do; thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Ti functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. The one-dimensional part of the above reducible three-dimensional representation is seen to be the same as the totally symmetric representation we arrived at
before, D(1). The two-dimensional representation that is left can be shown to be irreducible ; it has the following matrix representations:
D(2)(E) =
; D(2)(C 3 ) =
3 2
1 2
1 2
; D(2)(C 32 ) =
3 2
1 2
1 2
D(2)(σv) =
; D(2)(σv') =
3 2
1 2
1 2
; D(2)(σv") =
3 2
1 2
1 2
The characters can be obtained by summing diagonal elements:
χ(E) = 2, χ(2C 3 ) = -1, χ(3σv) = 0.
D. Rotations as a Basis
Another one-dimensional representation of the group can be obtained by taking rotation about the Z-axis (the C 3 axis) as the object on which the symmetry operations act:
Rz →
E
Rz; Rz →
C 3
Rz; Rz →
C 32
Rz';
Rz →
σv -Rz; Rz →
σv" -Rz; Rz →
σv' -Rz.
In writing these relations, we use the fact that reflection reverses the sense of a rotation. The matrix representations corresponding to this one-dimensional basis are:
D(1)(E) = 1; D(1)(C 3 ) = 1; D(1)(C 32 ) = 1; D(1)(σv) = -1;D(1)(σv") = -1; D(1)^ (σv') = -1. These one-dimensional matrices can be shown to multiply together just like the symmetry operations of the C3v group. They form an irreducible representation of the group (because it is one-dimensional, it can not be further reduced). Note that this one-dimensional representation is not identical to that found above for the 1s N-atom orbital, or the T 1 function.
E. Overview
We have found three distinct irreducible representations for the C3v symmetry group; two different one-dimensional and one two dimensional representations. Are there any more? An important theorem of group theory shows that the number of irreducible representations of a group is equal to the number of classes. Since there are three classes of operation, we have found all the irreducible representations of the C3v point group. There are no more.
The irreducible representations have standard names the first D(1)^ (that arising
from the T 1 and 1sN orbitals) is called A 1 , the D(1)^ arising from Rz is called A 2 and D(2)^ is called E (not to be confused with the identity operation E).
Thus, our original D(4)^ representation was a combination of two A 1
representations and one E representation. We say that D(4)^ is a direct sum representation:
D(4)^ = 2A 1 ⊕ E. A consequence is that the characters of the combination representation
D(4)^ can be obtained by adding the characters of its constituent irreducible representations.
E 2C (^3 3) σ v A 1 1 1 1 A 1 1 1 1 E 2 -1 0 2A 1 ⊕ E 4 1 2
F. How to Decompose Reducible Representations in General
Suppose you were given only the characters (4,1,2). How can you find out
how many times A 1 , E, and A 2 appear when you reduce D(4)^ to its irreducible parts? You want to find a linear combination of the characters of A 1 , A 2 and E that add up (4,1,2).
You can treat the characters of matrices as vectors and take the dot product of A 1 with D(4)
since the characters (the numbers listed opposite A 1 and below E, 2C 3 , and 3σv in the C3v character table) of all six symmetry operations are 1 for the A 1 irreducible representation. The 2px and 2py orbitals on the nitrogen atom transform as the E representation
since C 3 , C 32 , σv, σv', σv" and the identity operation map 2px and 2py among one another. Specifically,
C 3
2px
2py
Cos120° -Sin120°
Sin120° Cos120°
2px
2py
C 32
2px
2py
Cos240° -Sin240°
Sin240° Cos240°
2px
2py
E
2px
2py
2px
2py
σv
2px
2py
2px
2py
σv'
2px
2py
+^1 2
3 2
3 2
1 2
2px
2py
σv"
2px
2py
+^1 2
3 2
3 2
1 2
2px
2py
The 2 x 2 matrices, which indicate how each symmetry operation maps 2px and 2py into
some combinations of 2px and 2py, are the representation matrices ( D(IR)) for that particular operation and for this particular irreducible representation (IR). For example,
+^1 2
3 2
3 2
1 2
= D(E)(σv')
This set of matrices have the same characters as D(2)^ above, but the individual matrix elements are different because we used a different basis set (here 2px and 2py ; above it was T 2 and T 3 ). This illustrates the invariance of the trace to the specific representation; the trace only depends on the space spanned, not on the specific manner in which it is spanned.
B. A Short-Cut
A short-cut device exists for evaluating the trace of such representation matrices (that is, for computing the characters). The diagonal elements of the representation matrices
are the projections along each orbital of the effect of the symmetry operation acting on that orbital. For example, a diagonal element of the C 3 matrix is the component of C 3 2py along the 2py direction. More rigorously, it
is ⌡⌠2pyC 3 2pydτ. Thus, the character of the C 3 matrix is the sum of ⌡⌠2pxC 3 2pydτ and
⌡⌠2px*C 3 2pxdτ. In general, the character χ^ of any symmetry operation S can be computed
by allowing S to operate on each orbital φi, then projecting Sφi along φi (i.e., forming
⌡⌠φi*Sφidτ ), and summing these terms,
i
⌡⌠φi*Sφidτ = χ(S).
If these rules are applied to the 2px and 2py orbitals of nitrogen within the C3v point group, one obtains
χ(E) = 2, χ(C 3 ) = χ(C 32 ) = -1, χ(σv) = χ(σv") = χ(σv') = 0.
This set of characters is the same as D(2)^ above and agrees with those of the E representation for the C3v point group. Hence, 2px and 2py belong to or transform as the E representation. This is why (x,y) is to the right of the row of characters for the E representation in the C3v character table. In similar fashion, the C3v character table states
that dx^2 −y^2 and dxy orbitals on nitrogen transform as E, as do dxy and dyz, but dz^2 transforms as A 1. Earlier, we considered in some detail how the three 1sH orbitals on the hydrogen atoms transform. Repeating this analysis using the short-cut rule just described, the traces (characters) of the 3 x 3 representation matrices are computed by allowing E, 2C 3 , and 3 σv to operate on 1sH 1 , 1sH 2 , and 1sH 3 and then computing the component of the
resulting function along the original function. The resulting characters are χ(E) = 3, χ(C 3 )
= χ(C 32 ) = 0, and χ(σv) = χ(σv') = χ(σv") = 1, in agreement with what we calculated before. Using the orthogonality of characters taken as vectors we can reduce the above set of characters to A 1 + E. Hence, we say that our orbital set of three 1sH orbitals forms a reducible representation consisting of the sum of A 1 and E IR's. This means that the three 1sH orbitals can be combined to yield one orbital of A 1 symmetry and a pair that transform according to the E representation.
IV. Projector Operators: Symmetry Adapted Linear Combinations of Atomic Orbitals
To generate the above A 1 and E symmetry-adapted orbitals, we make use of so- called symmetry projection operators PE and PA 1. These operators are given in terms of linear combinations of products of characters times elementary symmetry operations as follows:
PA 1 = ∑
S
χ A(S) S
χ(S) = (^) ∑
i
ni χi(S) ,
it is necessary to determine how many times, ni, the i-th irreducible representation occurs in the reducible representation. The expression for ni is (see the text by Cotton)
ni =
g ∑ S
χ(S) χi(S)
in which g is the order of the point group; the total number of symmetry operations in the group (e.g., g = 6 for C3v).
For example, the reducible representation χ(E) = 3, χ(C 3 ) = 0, and χ(σv) = 1 formed by the three 1sH orbitals discussed above can be decomposed as follows:
nA 1 =
(3 ⋅ 1 + 2 ⋅ 0 ⋅ 1 + 3 ⋅ 1 ⋅ 1) = 1,
nA 2 =
(3 ⋅ 1 + 2 ⋅ 0 ⋅ 1 + 3 ⋅ 1 ⋅ (-1)) = 0,
nE =
(3 ⋅ 2 + 2 ⋅ 0 ⋅ (-1) + 3 ⋅ 1 ⋅ 0) = 1.
These equations state that the three 1sH orbitals can be combined to give one A 1 orbital and, since E is degenerate, one pair of E orbitals, as established above. With knowledge of the ni, the symmetry-adapted orbitals can be formed by allowing the projectors
Pi = (^) ∑ i
χi(S) S
to operate on each of the primitive atomic orbitals. How this is carried out was illustrated for the 1sH orbitals in our earlier discussion. These tools allow a symmetry decomposition of any set of atomic orbitals into appropriate symmetry-adapted orbitals. Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH 3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations {S} of the point group. In the case of NH 3 , the characters of the resultant 12 x 12 representation matrices form a reducible representation
in the C2v point group: χ(E) = 12, χ(C 3 ) = χ(C 32 ) = 0, χ(σv) = χ(σv') = χ^ (σv") = 2.
(You should try to prove this. For example under σv, the H 2 and H 3 atoms are interchanged, so unit vectors on either one will not contribute to the trace. Unit z-vectors on N and H 1 remain unchanged as well as the corresponding y-vectors. However, the x-
vectors on N and H 1 are reversed in sign. The total character for σv' the H 2 and H 3 atoms are interchanged, so unit vectors on either one will not contribute to the trace. Unit z-vectors on N and H 1 remain unchanged as well as the corresponding y-vectors.
However, the x-vectors on N and H 1 are reversed in sign. The total character for σv is thus 4 - 2 = 2. This representation can be decomposed as follows:
nA 1 =
[1⋅ 1 ⋅ 12 + 2⋅ 1 ⋅ 0 + 3⋅ 1 ⋅ 2] = 3,
nA 2 =
[1⋅ 1 ⋅ 12 + 2⋅ 1 ⋅ 0 + 3⋅ (-1)⋅ 2] = 1,
nE =
[1⋅ 2 ⋅ 12 + 2⋅ (-1)⋅ 0 + 3⋅ 0 ⋅ 2] = 4.
From the information on the right side of the C3v character table, translations of all four atoms in the z, x and y directions transform as A 1 (z) and E(x,y), respectively, whereas rotations about the z(Rz), x(Rx), and y(Ry) axes transform as A 2 and E. Hence, of the twelve motions, three translations have A 1 and E symmetry and three rotations have A 2 and E symmetry. This leaves six vibrations, of which two have A 1 symmetry, none have A 2 symmetry, and two (pairs) have E symmetry. We could obtain symmetry-adapted vibrational and rotational bases by allowing symmetry projection operators of the irreducible representation symmetries to operate on various elementary cartesian (x,y,z) atomic displacement vectors. Both Cotton and Wilson, Decius and Cross show in detail how this is accomplished.
VI. Direct Product Representations
A. Direct Products in N-Electron Wavefunctions
We now return to the symmetry analysis of orbital products. Such knowledge is important because one is routinely faced with constructing symmetry-adapted N- electron configurations that consist of products of N individual orbitals. A point-group symmetry operator S, when acting on such a product of orbitals, gives the product of S acting on each of the individual orbitals
S(φ 1 φ 2 φ 3 ...φN) = (Sφ 1 ) (Sφ 2 ) (Sφ 3 ) ... (SφN).
For example, reflection of an N-orbital product through the σv plane in NH 3 applies the reflection operation to all N electrons. Just as the individual orbitals formed a basis for action of the point-group operators, the configurations (N-orbital products) form a basis for the action of these same point-group operators. Hence, the various electronic configurations can be treated as functions on which S operates, and the machinery illustrated earlier for decomposing orbital symmetry can then be used to carry out a symmetry analysis of configurations. Another shortcut makes this task easier. Since the symmetry adapted individual
orbitals {φi, i = 1, ..., M} transform according to irreducible representations, the representation matrices for the N-term products shown above consist of products of the
matrices belonging to each φi. This matrix product is not a simple product but a direct product. To compute the characters of the direct product matrices, one multiplies the characters of the individual matrices of the irreducible representations of the N orbitals that appear in the electron configuration. The direct-product representation formed by the orbital products can therefore be symmetry-analyzed (reduced) using the same tools as we used earlier. For example, if one is interested in knowing the symmetry of an orbital
product of the form a 12 a 22 e^2 (note: lower case letters are used to denote the symmetry of electronic orbitals) in C3v symmetry, the following procedure is used. For each of the six symmetry operations in the C2v point group, the product of the characters associated with
each of the six spin orbitals (orbital multiplied by α or β spin) is formed
χ(S) = (^) ∏
i
χi(S) = (χA 1 (S))
(^2) (χA 2 (S))
(^2) (χE(S)) (^2).
symmetric representation of the group). In terms of the projectors introduced above in Sec. IV, of this Appendix we must have
∑ S
χA(S) S ψa*^ V ψb
not vanish. Here the subscript A denotes the totally symmetric representation of the
group. The symmetry of the product ψa*^ V ψb is, according to what was covered earlier
in this Section, given by the direct product of the symmetries of ψa*^ of V and of ψb. So, the conclusion is that the integral will vanish unless this triple direct product contains, when it is reduced to its irreducible components, a component of the totally symmetric representation. To see how this result is used, consider the integral that arises in formulating the interaction of electromagnetic radiation with a molecule within the electric-dipole approximation:
⌡⌠ψa*^ r ψb dτ.
Here r is the vector giving, together with e, the unit charge, the quantum mechanical dipole moment operator
r = e∑
n
Z (^) n R n - e∑
j
r j ,
where Zn and R n are the charge and position of the nth^ nucleus and r j is the position of
the jth^ electron. Now, consider evaluating this integral for the singlet n→π*^ transition in
formaldehyde. Here, the closed-shell ground state is of 1 A 1 symmetry and the excited state, which involves promoting an electron from the non-bonding b 2 lone pair orbital on
the Oxygen into the π*^ b 1 orbital on the CO moiety, is of 1 A 2 symmetry (b 1 x b 2 = a 2 ). The direct product of the two wavefunction symmetries thus contains only a 2 symmetry. The three components (x, y, and z) of the dipole operator have, respectively, b 1 , b 2 , and a 1 symmetry. Thus, the triple direct products give rise to the following possibilities: a 2 x b 1 = b 2 , a 2 x b 2 = b 1 , a 2 x a 1 = a 2. There is no component of a 1 symmetry in the triple direct product, so the integral
vanishes. This allows us to conclude that the n→π*^ excitation in formaldehyde is electric dipole forbidden.
VII. Overview
This appendix has reviewed how to make a symmetry decomposition of a basis of atomic orbitals (or cartesian displacements or orbital products) into irreducible representation components. This tool is most helpful when constructing the orbital correlation diagrams that form the basis of the Woodward-Hoffmann rules. We also learned how to form the direct-product symmetries that arise when considering configurations consisting of products of symmetry-adapted spin orbitals. This step is essential for the construction of configuration and state correlation diagrams upon which one ultimately bases a prediction about whether a reaction is allowed or forbidden. Finally, we learned how the direct product analysis allows one to determine whether or not integrals of products of wave functions with operators between them vanish. This tool is of utmost importance in determining selection rules in spectroscopy and for
determining the effects of external perturbations on the states of the species under investigation.
D 4 E 2C 4 C 2
(=C 42 )
2C 2 ' 2C 2 "
A 1 1 1 1 1 1 x^2 +y^2 ,z^2 A 2 1 1 1 -1 -1 z,Rz B 1 1 -1 1 1 -1 x^2 -y^2 B 2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (x,y)(Rx,Ry) (xz,yz)
C2v E C 2 σv(xz) σv'(yz) A 1 1 1 1 1 z x^2 ,y^2 ,z^2 A 2 1 1 -1 -1 Rz xy B 1 1 -1 1 -1 x,Ry xz B 2 1 -1 -1 1 y,Rx yz
C3v E 2C 3 3 σv A 1 1 1 1 z x^2 +y^2 ,z^2 A 2 1 1 -1 Rz E 2 -1 0 (x,y)(Rx,Ry) (x^2 -y^2 ,xy)(xz,yz)
C4v E 2C 4 C 2 2 σv 2 σd A 1 1 1 1 1 1 z x^2 +y^2 ,z^2 A 2 1 1 1 -1 -1 Rz B 1 1 -1 1 1 -1 x^2 -y^2 B 2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (x,y)(Rx,Ry) (xz,yz)
C2h E C 2 i σh
Ag 1 1 1 1 Rz x^2 ,y^2 ,z^2 ,xy Bg 1 -1 1 -1 Rx,Ry xz,yz Au 1 1 -1 -1 z Bu 1 -1 -1 1 x,y
D2h E C 2 (z) C 2 (y) C 2 (x) i σ(xy) σ(xz) σ(yz) Ag 1 1 1 1 1 1 1 1 x^2 ,y^2 ,z^2 B1g 1 1 -1 -1 1 1 -1 -1 Rz xy B2g 1 -1 1 -1 1 -1 1 -1 Ry xz B3g 1 -1 -1 1 1 -1 -1 1 Rx yz Au 1 1 1 1 -1 -1 -1 - B1u 1 1 -1 -1 -1 -1 1 1 z B2u 1 -1 1 -1 -1 1 -1 1 y B3u 1 -1 -1 1 -1 1 1 -1 x
D3h E 2C 3 3C 2 σh 2S 3 3 σv A 1 ' 1 1 1 1 1 1 x^2 +y^2 ,z^2 A 2 ' 1 1 -1 1 1 -1 Rz E' 2 -1 0 2 -1 0 (x,y) (x^2 -y^2 ,xy) A 1 " 1 1 1 -1 -1 - A 2 " 1 1 -1 -1 -1 1 z E" 2 -1 0 -2 1 0 (Rx,Ry) (xz,yz)
D4h E 2C 4 C 2 2C 2 ' 2C 2 " i 2S 4 σh 2 σv 2 σd
A1g 1 1 1 1 1 1 1 1 1 1 x^2 +y^2 ,z^2
A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz
B1g 1 -1 1 1 -1 1 -1 1 1 -1 x^2 -y^2 B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx,Ry) (xz,yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -
A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z
B1u 1 -1 1 1 -1 -1 1 -1 -1 1 B2u 1 -1 1 -1 1 -1 1 -1 1 - Eu 2 0 -2 0 0 -2 0 2 0 0 (x,y)
