Research: Science and Education
Comparison of Stretching Force Constants in Symmetry Coordinates between Td and C3v Point Groups Maureen M. Julian Department of Geological Sciences, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0420;
[email protected] Just as numbers measure quantity, groups measure symmetry. M. A. Armstrong
In a previous paper (1) the force constants for structures with tetrahedral symmetry were calculated. This paper is addressed to students and researchers in applied group theory who wish to compare force constants between two similar molecules. We assume the reader has some familiarity with the group theoretical methods presented by Wilson, Decius, and Cross (2). Specifically we consider what happens to the force constants of a silicate moiety (SiO4) when the length of one of its bonds is changed. This situation exists in the molecule O 3SiO brSiO 3, where Obr is the bridging oxygen atom connecting the two SiO3 moieties. The problem is to present a set of force constants such that when the structure of a molecule is perturbed, the relevant force constants are also perturbed. As we shall show, we cannot directly apply Wilson’s method for obtaining symmetry coordinates from internal coordinates. We start with the irreducible representations of the symmetries of the moiety with the higher symmetry and reduce them to the representations of the symmetries of the moiety with the lower symmetry. The same concept is used by Merzbacher in describing the Stark effect for the 2s and 2p orbitals of the hydrogen atom (3). More generally, when molecule A has a symmetry of a subgroup of molecule B, then as molecule A approaches molecule B, the numerical values of the force constants of A should approach those of B. In order to demonstrate this, it is imperative that the appropriate symmetry coordinates be used. For simplicity in this introduction, only the stretching coordinates will be considered. The bending coordinates are complicated by redundancy and will be treated in a later paper. This paper compares the stretching force constants of SiO4 (tetrahedral point group Td) and ObrSiO3 (point group C3v) in symmetry coordinates. The C3v point group is a subgroup of the Td point group. The SiObr bond is longer than the SiO bond. After the point group has been assigned and the character table exhibited, we compute the number of occurrences expected for each symmetry species. The symmetry coordinate(s) are calculated for each species in order to factor the secular equation. The matrix representations of the generators of these point groups are given. These matrices are a function of the specific symmetry coordinates. Finally, the symmetry coordinates are applied to the force constant matrix and the results are examined. A major pedagogical point is that the most familiar symmetry coordinate set may not be the desired one. The Td group has nine proper subgroups (T, D2d , C3v, S4, D2, C2v, C3, C2, and Cs), each with its own set of force constants. Using the symmetry coordinates constructed directly from a projection operator for both Td and C3v point groups, two force constant matrices will be constructed which are not limiting cases of one another. The matrix representations of the generators in common of the group and subgroup are different. Thus it
Table 1. Character Tables of Related Point Groups Td and C3v 8 C3 6 σd
E
C3v
E
A1
1
1
1
1
1
A1
1
1
1
A2
1
1
{1
{1
1
A2
1
1
{1
E
2
{1
0
0
2
E
2
{1
0
1
{1
{1
{1
F1
3
0
{1
F2
3
0
1
6 S4 3 C2
2 C3 3 σv
Td
Table 2. Characters of Transformation Matrices For R, Applied to SiO4 Moiety
Td
E
8 C3 6 σd 6 S4
R
4
1
2
0
For r and R, Applied to Obr SiO3 Moiety 2 C3
3 σv
3C2
C3v
E
0
r
1
1
1
R
3
0
1
N OTE: R and r are stretching coordinates. For the SiO4 moiety, R is R1 to R4 . For the ObrSiO3 moiety, r is the single r1 stretching coordinate and R is R2 to R4 .
is necessary to design a single set of symmetry coordinates that in the limit satisfy both point groups. These coordinates will be applied to transform both stretching force constant matrices. In this case, as the bond length SiObr approaches the bond length of SiO, the force constant matrix representing the O brSiO3 moiety approaches the force constant matrix representing SiO4. The matrix representations of the generators in common (see Tables 8 and 9) are given and shown to be identical. Thus the matrix representations of the generators confirm the relationship between the subgroup and the group (see eqs 11 and 12). Character Tables and Number of Symmetry Species The character tables of the two point groups are given in Table 1. These tables are found in many physical chemistry texts (4). The Td point group contains the C3v group. The mirror planes σv and σd can be relabeled. The following discussion is limited to the SiO4 and the ObrSiO3 moieties. Since both these moieties have the same number of oxygen atoms about a central Si atom, the number of stretching coordinates is the same. The SiO 4 group with its tetrahedral symmetry has four stretching coordinates, R1, R2, R3, and R4. In this paper R refers to the four stretching coordinates R1, …, R4 when the SiO4 group is considered. By contrast, the ObrSiO3 group has two kinds of stretching coordinates: r1 is the elongated SiObr bond; and R2, R3, and R4 are the three remaining stretching coordinates, related through a 3-fold axis. In this paper r refers to the single r1 stretching coordinate and R refers to R2, R3, and R4 when the O brSiO 3 group is considered. Thus r1 approaches R1 as the length of the SiObr bond (C3v point group) approaches that of the SiO bond (Td point group). The traces (or characters) of the transformation matrices for the stretching coordinates are given in Table 2.
JChemEd.chem.wisc.edu • Vol. 76 No. 5 May 1999 • Journal of Chemical Education
679
Research: Science and Education
Table 4 shows the correlation between the Td point group and its subgroup C 3v for stretching only. This table is generated by applying eq 1 and obtaining the first χ j(γ) from the C3v table and the χ j from the Td table. Note that the dimensionality is preserved across the row.
Table 3. Number of Symmetry Species a
a
For SiO4 Moiety
For O br SiO3 Moiety
Td
R
C3v
r
R
A1
1
A1
1
1
A2
0
A2
0
0
E
0
E
0
1
F1
0
F2
1
a Calculated
Td
C3v
A1
A1
F2
A1 + E
Symmetry Coordinates from eq 1.
For a given representation, the number, n(γ), of occurrences of the symmetry species γ can be calculated by the formula (5):
n (γ) = 1g Σ g j χ j(γ) *χ j j
(1)
where g is the order of the group (total number of operations in the symmetry group), g j is number of operations for each class, χ j(γ) is character for a given symmetry in the character table, and χ j is the character for the transformation representing a symmetry operation. Applying eq 1 to symmetry species A1 for SiO4, we get the number of stretching species, which is n A1 = 1/24 (1?1?4 + 8?1?1 + 6?1?2 + 6?1?0 + 3?1?0) = 1 The rest of the calculations in Table 3 are done in the same way. By combining this information, the structure of the representation formed by the stretching coordinates is
Now that we know which of the species is occupied for each of the point groups, the symmetry coordinates for stretching can be calculated. The equation (6 ) for calculating the symmetry coordinates is S(γ) = n ∑ χ(γ)P(S1) (2) where S(γ) is the symmetry coordinate to be calculated, P(S1) is the coordinate to which S1 is transformed by the operation P, χ(γ) is character for a given symmetry in the character table in Table 1 (see eq 1), and n is the normalizing factor. The summation is over the operations, P, of the point group. Table 5 gives the information to calculate the symmetry coordinates for the stretching coordinates in C3v. Applying the information in Table 5 to eq 2, we get for A1 symmetry species:
S 1 3v r 1 = 1 1⋅r 1 + 1⋅r 1 + 1⋅r 1 + 1⋅r 1 + 1⋅r 1 + 1⋅r 1 = r 1 6 C
This notation means that there is one totally symmetric coordinate with symmetry species A1, none with symmetry species A2, E, or F1, and one triple of coordinates with symmetry species F2. Thus for the SiO4 moiety there are only two distinct stretching symmetry species present. The force constants relating to stretching only of SiO 4 can be represented as a 4 × 4 symmetric matrix that contains a 1 × 1 and a 3 × 3 submatrix along the main diagonal and zeros elsewhere. In other words, there exists a set of transformation coordinates or symmetry coordinates that reduces the stretching force constant matrix to the above form. Similarly for the C3v point group, the structure of the representation formed by the stretching coordinates r and R for the ObrSiO3 moiety is: Γ(r)= A1(r) Γ(R)= A1(R) + E(R) This information can be combined to get Γ(r,R) = 2A1(r,R) + E(R) In the C3v point group, there are two totally symmetric coordinates that are functions of r and R and one pair of coordinates with the symmetry species E. Thus the force constants can be represented by a 4 × 4 symmetric matrix that contains two 2 × 2 submatrices along the diagonals and zeros elsewhere. The object of this paper is to find a formulation where the C3v force matrix in symmetry coordinates will converge to the Td force matrix as the SiObr bond converges to the SiO bond.
(3)
for A1 symmetry species: C S 2 3v R 2 = 1 R 2 + R 3 + R 4 3
Γ(R) = A1(R) + F2(R)
680
Table 4. Correlation of Symmetr y Species between Point Groups for Stretching Only
(4)
for E symmetry species: C S 3 3v R 2 = 1 2R 2 – R 3 – R 4 6
(5)
Another symmetry coordinate in the E symmetry species is needed (because E is two dimensional). Candidates can be obtained by applying eq 2 to R3 or R 4. Since we wish the set of symmetry coordinates to form an orthonormal set, we choose for E symmetry species C S 4 3v R = 1 {S 3 R 3 + S 3 R 4 = 1 {R 3 + R 4 3 2
(6)
Table 5. Information for Calculating Stretching Symmetr y Coordinate for C3v P(r1)
P(R2)
χ(A 1)
E
r1
R2
1
2
C31
r1
R3
1
{1
C32
r1
R4
1
{1
σ34
r1
R2
1
0
σ23
r1
R3
1
0
σ24
r1
R4
1
0
P
χ(E )
NOTE : P is an operation of the point group; σij is the mirror plane that reflects Ri into Rj (this is not standard notation; it is used in this paper for clarity in this particular application). In the second column S 1 = r1 and in the third column S1 = R 2. The last two columns are the characters found in Table 1.
Journal of Chemical Education • Vol. 76 No. 5 May 1999 • JChemEd.chem.wisc.edu
Research: Science and Education
Note that r appears only in the first symmetry coordinates S1 (A1 species), while R appears only in the last three symmetry coordinates S2 (A1 species) and S3 and S4 (E species). For the Td point group a table similar to Table 5 is constructed and applied to eq 2, and we get for A1 symmetry species:
S 1 d R1 = 1 R1 + R2 + R3 + R4 2 T
Table 6. Matrix Representations for Generators of Stretching Symmetr y Coordinates in Td Symmetr y Group
T
C2Z
1
1
1
F2
010 001 100
010 100 001
100 010 001
Table 7. Matrix Representations for Generators of Stretching Symmetr y Coordinates in C3v Symmetr y Group
(8)
Two more symmetry coordinates in the F2 symmetry species are needed (because F2 is three dimensional). Since we wish the set of symmetry coordinates to form an orthonormal set, we choose for F2 symmetry species: T S 3 d R = 1 R1 + R2 – R3 – R4 (9) 2
S 4 d R = 1 R1 – R2 + R3 – R4 (10) 2 Note that all four stretching coordinates R appear in all the symmetry coordinates (see Gans [7 ]).
R1 R3 R4 R2
or
111 3+
1000 = 0010 0001 0100
3+111
σ34
A1
1
1
A1
1
1
{1 { 3 2 2 3 {1 2 2
10 01
Note: From eqs 3–6.
Consider the transformation for the Td symmetry given in eqs 7–10.
Matrix Representations of Groups If to every member A1, A2, … of a group G we can associate a square, nonsingular matrix D(A1), D(A2), … in such a way that if Ai?Aj = Ak and D(Ai)?D(Aj) = D(Ak), then the matrices themselves form a group homomorphic with group G. Such matrices are a representation of the group; their order is the degree or dimension of the representation. The character of a matrix is its trace, which is invariant under changes of coordinates. This invariance makes the trace a valuable indicator in these symmetry relationships. We do not have to consider all the members of a group, just the generators. Only two generators, a threefold rotation and an improper fourfold rotation, are needed for the Td point group (8). However, for clarity, in this exercise we choose the generators of the group Td to be 3+111, σ34, C2z and the generators of the subgroup C3v to be 3+111, σ34. The operation 3+111 represents a positive threefold rotation about R1 (or [111]) where R2 goes into R3, R3 goes into R4, and R4 goes into R2. C2z is a twofold rotation about the z axis and σ34 is a mirror plane that reflects R3 into R4. Thus
C3v
E
T
=
σ34
A1
Note: From eqs 7–10.
S 2 d R = 1 R1 – R2 – R3 + R4 2
111 3+
3+111
(7)
for F2 symmetry species:
R1 R2 R3 R4
Td
T 111 S 3+ S
1111 1111 1000 1000 1 1 1 1 1 1 1 1 1 0 0 1 0 = ⋅ ⋅ = 0010 0001 0001 1111 4 1111 0100 0100 1111 1111
The last matrix can be partitioned
1000 0010 = 0001 0100
A1 F2
Table 6 can likewise be constructed for the generators of the stretching symmetry coordinates in the Td symmetry group. Using eq 3, we can construct Table 7 for the C3v symmetry group. Stretching Force Constants Now we have a transformation we can use on the force constant matrices. Consider first the Td group SiO 4. This moiety has only two stretching force constants: fr, which is the force constant associated with each SiO bond, and frr, which is the force constant associated with each pair of SiO bonds. Since there are four bonds, the 4 × 4 symmetric stretching force constant matrix is
Likewise,
1000 σ34 = 0 1 0 0 ; C 2z = 0001 0010
0100 1000 0001 0010
FT = d
fr f rr f r f rr f rr f r f rr f rr f rr f r
JChemEd.chem.wisc.edu • Vol. 76 No. 5 May 1999 • Journal of Chemical Education
681
Research: Science and Education
As before, the transformation matrix from the symmetry coordinates from eqs 7–10 is
1 1 1 1
ST = 1 d 2
1 1 1 {1 1 {1 {1 {1 1 1 {1 {1
SC
FC = 3v
When the transformation ST F S is performed to transform the force constant matrix to symmetry coordinates, we get the symmetric matrix
SC FT = d
f r + 3f rr 0 f r – f rr 0 0 f r – f rr 0 0 0 f r – f rr
The superscript SC is used to indicate the use of symmetry coordinates. The symmetry coordinates have transformed the stretching force constant matrix into a diagonal matrix. From the symmetries we would expect a 1 × 1 matrix associated with the A 1 species and a 3 × 3 matrix associated with the F2 symmetry species. Indeed, F11 = fr + 3frr is associated with the A1 symmetry species and F22 = F33 = F44 = fr – frr are associated with the F2 symmetry species. Note that the cross terms in the F2 species are all zero. Now let us repeat the calculations for the C3v point group and the moiety ObrSiO3. This group has four stretching force constants: f r, which is the force constant associated with the SiObr bond; fR, which is the force constant associated with each SiO bond; frR , which is the force constant associated with each pair of SiObr and SiO bonds; and fRR, which is the force constant associated with each pair of SiO bonds. Since again there are four bonds the 4 × 4 symmetric stretching force constant matrix is
FC = 3v
fr f rR f R f rR f RR f R f rR f RR f RR f R
The transformation matrix from the symmetry coordinates (eqs 3–5) is
1
0
0
1 2 0 3 6 1 { 1 { 1 3 6 2 1 { 1 1 3 6 2
SC = 3v
0 0
0
0
When the transformation ST F S is performed on the force constant matrix we get 682
fr 3f rR f R + 2f RR 0 0 f R – f RR 0 0 0 f R – f RR
This matrix is decomposed into two 2 × 2 submatrices. The first is associated with two A1 symmetry species and shows an interdependence between – the r and the R internal coordinates. Here F11 = f r, F12 = √3 frR, and F22 = fR + 2fRR. Notice that all four force constants are present. The second 2 × 2 matrix is associated with the E symmetry species. Note here that only the R coordinates are present. Here F33 = F44 = fR – fRR. Again, the cross terms in the E species are all zero. Note, however, in the limit as the moiety with C3v symmetry approaches Td symmetry that the corresponding force constant matrices in symmetry coordinates do not approach one another. Symmetry Coordinates, Force Constants, and Matrix Representations for Related Point Groups So far we have just directly applied eq 2 to calculate the symmetry coordinates. We are encouraged by the fact that there is a striking similarity in these two force constant matrices, namely, that the lower right 2 × 2 matrices look alike. So we take linear combinations of the symmetry coordinates from eqs 3–6 and eqs 7–10 to form a new set of symmetry coordinates that apply to both point groups. This is possible because C3v is a subgroup of Td . In the representation for the Td point group, there is only one coordinate in the A1 species with symmetry coordinate S1 = 1/2(R1 + R2 + R3 + R4). From the correlation table in C3v, the corresponding A1 coordinate must be identical (see Table 4). The linear combination C C S 1 = 1 S 1 3v + 3 S 2 3v = 1 r 1 + R 2 + R 3 + R 4 2 2
fulfills this condition. Since the coordinates of the E species of the C3v group cannot contain any r stretching coordinates, let S 3 and S 4 be identical to S C3 3v and S C4 3v. Appropriate Td group linear combinations can be found. In a similar manner S 2 is constructed. For Td point group, the symmetry coordinates are A1 symmetry species: T S 1 = S 1 d = 1 R1 + R2 + R3 + R4 2 F2 symmetry species: T T T S 2 = 2 S 2 d + S 3 d + S 4 d = 1 3R 1 – R 2 + R 3 + R 4 12 12 T T T S 3 = 1 {S 2 d + 2S 3 d – S 4 d = 1 2R 2 – R 3 – R 4 6 6 T T S 4 = 1 S 2 d – S 4 d = 1 {R 3 + R 4 6 2
Similarly, for the C3v point group the symmetry coordinates are
Journal of Chemical Education • Vol. 76 No. 5 May 1999 • JChemEd.chem.wisc.edu
Research: Science and Education
A1 symmetry species: C C S 1 = 1 S 1 3v + 3 S 2 3v = 1 r 1 + R 2 + R 3 + R 4 2 2 C C S 2 = 1 3S 1 3v – 3 S 2 3v = 1 3r 1 – R 2 + R 3 + R 4 12 12
Table 8. Matrix Representations for Generators of Stretching Symmetry Coordinates in Td Symmetry Group for Symmetry Coordinates Applying in the Limit to both Td and C3v Point Groups
Td
3+111
σ34
C2Z
A1
1
1
1
F2
1
E symmetry species:
S3 =
C S 3 3v
= 1 2R 2 – R 3 – R 4 6
C S 4 = S 4 3v = 1 {R 3 + R 4 2 Note that if we allow r1 to equal R1, then the two sets of symmetry coordinates are identical. This is the relationship we need to compare the two moieties SiO4 and ObrSiO3 as the length of the SiO bond approaches that of the SiObr bond. Now let us apply this new set of symmetry coordinates to the force constant matrices. For C3v point group, when the transformation ST F S is performed to transform the force matrix we get
SC
FC = 3v
1f + 3f + 3f + 3f 4 r 4 R 2 rR 2 RR 1 3 f – 3 f + 3f – 3f r rR RR 2 R 12 2 0 0
3f + 1f – 3f + 1f 4 r 4 R 2 rR 2 RR 0 f R – f RR 0 0 f R – f RR
When we relax to Td symmetry, that is, we let fr equal fR and frR equal fRR, then we get
F SC =
f r + 3f rr 0 f r – f rr 0 0 f r – f rr 0 0 0 f r – f rr
which is what we got the first time we transformed the force constant matrix in the Td point group. The same force constant matrix is obtained using the above symmetry coordinates for the Td point group. As before, Table 8 can be constructed for the matrices of the generators of the stretching symmetry coordinates in the Td symmetry group. We can likewise construct Table 9 for the C3v symmetry group. A comparison of the two representations shows the intimate relationship between Td and C3v when the appropriate symmetry coordinates are designed and applied to each group. Conclusions We have shown that if the symmetry coordinates are limits of each other for two point groups, then the corresponding force constants are also limits of each other. Also, different force constants can be obtained with different selections of the symmetry coordinates. Thus when one is comparing results of structure in related point groups using symmetry coordinates, it is imperative to use compatible symmetry coordinates. The correlation tables show the relationship between
0 0
0
0
{1 { 3 2 2 3 {1 2 2
1 2 2 3 3 2 2 1 3 3
100 010 001
0
0
0
0 1
Table 9. Matrix Representations for Generators of Stretching Symmetry Coordinates in C3v Symmetry Group for Symmetry Coordinates Applying in the Limit to both Td and C3v Point Groups
C3v
3+111
σ34
A1
1
1
A1
1
1
E
{1 { 3 2 2 3 {1 2 2
10 01
symmetry species of a given group and those of its subgroup. Note that the force constant matrix for the C3v point group is more complicated for the second set of symmetry coordinates. This is because S C1 3v and S C2 3v now contain mixes of both r and R. This paper emphasizes the importance of constructing the underlying matrix representations to understand the relationships between groups and their subgroups. For students: as an exercise, show what happens to the force constants when a tetrahedral sulfate ion attaches to a metal atom through two of its oxygen atoms. Hint: the symmetry of the ion is lowered to C2v point group in the complex and the appropriate symmetry coordinates for the sulfate are related to eqs 7–10. CH 2D2 is another example of a molecule amenable to this approach. Another application would be a comparison between methane, CH4, and its deuterated form, CH3D. Acknowledgments I would like to thank the three referees for comments. Their assistance helped improve the clarity of this paper. Literature Cited 1. Julian, M. M. J. Chem. Educ. 1998, 75, 497–502. 2. Wilson, E. B. Jr.; Decius, J. C.; Cross, P. C. Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra; Dover: New York, 1980. 3. Merzbacher, E. Quantum Mechanics; Wiley: New York, 1961; p 388. 4. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.; Wiley: New York, 1990. Atkins, P. W. Physical Chemistry, 5th ed.; Freeman: New York, 1994. Baraldi, I.; Vanossi, D. J. Chem. Educ. 1997, 74, 806–809. 5. Wilson, E. B. Jr.; Decius, J. C.; Cross, P. C. Op. cit.; p 107. 6. Wilson, E. B. Jr.; Decius, J. C.; Cross, P. C. Op. cit.; p 119. 7. Gans, P. Vibrating Molecules; Chapman and Hall: London, 1971; p 110. 8. Baraldi, I.; Carnevali, A. J. Chem. Educ. 1993, 70, 964–996.
JChemEd.chem.wisc.edu • Vol. 76 No. 5 May 1999 • Journal of Chemical Education
683
