A Simple Reduction Process for the Normal Vibrational Modes

Research: Science and Education

A Simple Reduction Process for the Normal Vibrational Modes Occurring in Linear Molecules William McInerny Department of Chemistry, City College of San Francisco, San Francisco, CA 94112; [email protected]

General Theory In his molecular spectroscopy book, Daniel Harris comments that the reduction formula that works well for finite groups fails for the continuous groups C∞v and D∞h, and the reduction might as well be done by the “eyeball technique” (1). The failure occurs because the number of group elements (h) is infinite, and this value leads to a zero result because of the necessary division by ∞ in the finite character reduction formula. Thus one is faced with a somewhat more sophisticated and difficult problem than occurs in the finite group case. The formula for the finite group case is found in Cotton’s book (2), where the derivation is shown to be a relatively straightforward consequence of the grand orthogonality theorem, and the derivation of that theorem can be found in either Tinkham’s book (3) or in Wigner’s classic work (4). The pertinent reduction formula is number of 1 representations = ai = ∑ χ ( R ) χi ( R ) h R of type i where h is the number of group elements; χ is the character of the given reducible representation; χi is the character of the ith irreducible representation; and the summation is taken over all group elements R. The complete determination of the normal modes of motion has its powerful formulation in the method of Lagrange–Euler in the general context of the calculus of variations and the Lagrangian form of classical dynamics. The explicit formulation of Wilson (5) is the traditional method employed by physical chemists to determine the normal modes, and the symmetries of those modes appear naturally and automatically as a consequence of the solutions themselves. Thus, a complete solution of the dynamical problem produces the group theoretical results without any recourse to the group theoretical approach being necessary. Much can be said for the simpler group theoretical approach to the characterization of the motions through algebraic and arithmetic methods. This is certainly true for the linear molecular problem that falls outside the scope of the usual resolution formula applicable to molecules associated with finite group representations. The reduction problem for C∞v and D∞h can be addressed by an integration over group elements, and a good example of that general technique is to be found in the work by Ferraro and Ziomek (6), as well as in the article by Alvarino and Chamorro (7). The article by Jaffe and David (8) addresses a similar problem in the context of projection operators for these infinite order groups. But there is also available a very simple algebraic equation approach that encompasses both the finite and continuous problems, and it is well illustrated by McNaught (9) and by Shäfer and Cyvin (10). Thus it is possible to solve all reduction problems by one approach that involves only simple linear equations with integral coefficients 140

Journal of Chemical Education

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and their solution methods are generally available to all undergraduates. The method employs algebraic equations that are known to have solutions in integers, and the equations can be manipulated easily by hand, even in the general cases considered below. Although this algebraic method can be applied to finite groups, one would ordinarily prefer the finite reduction formula indicated above for working with such cases. The finite order cases that are subgroups of C∞v and D∞h can be used to generate the solutions to the continuous problems by a suitable limiting procedure as shown by Strommen and Lippincott (11). Furthermore, Strommen has shown how to sort out correlations of their results with the continuous results based on coordinate bases analysis (12). The basic algebraic method, however, is based only on the linear independence of the character components themselves, and it does not require any complicated or sophisticated formalism for full implementation. A similar reduction procedure is shown by Flurry, but it requires that one proceed from the vantage of the three-dimensional rotation group R(3) (13). The results for the group C∞v for linear molecules are indicated below, where the character table is taken from the classic book by Hamermesh (14): C ∞ v

E

C ( ϕ)

σv

A1 A2

1 1

1 1

1 −1

E1 E2

2 2 cos ( ϕ ) 2 2 cos ( 2ϕ )

⯗

0 0

⯗

The cosine functions make it possible to set up separate relations based on the linear independence of these quantities. Recall that linear independence requires that if acos(ϕ) + bcos(2ϕ) = 0, then both a and b must be 0 themselves. This fact is easily established by choosing appropriate values for ϕ, such as ϕ = 0 and ϕ = π. Therefore, one can easily reduce the reducible representation Γ shown below. The astute student will recognize Γ as the sum A1 + A2 + E1 + E2. This result is essentially obvious by inspection since the cos(ϕ) and cos(2ϕ) terms can arise only from the E1 and E2 representations respectively, but the validity of the algebraic approach is revealed by the reduction of an especially transparent example:

Γ

E

C (ϕ)

σv

6

[2 + 2 cos ( ϕ ) + 2 cos ( 2ϕ )]

0

One assumes that Γ = aA1 + bA2 + cE1 + dE2 , where no higher-order cosine functions are needed because of the linear independence of the several cosine functions. Here the

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summation is over the group representations rather than over the group elements. This approach is similar to that of Alvarino, where he restricts his summation by a logical elimination of higher-multiple cosine functions (15). Thus the linear system of equations is

a (1) + b (1) + c ( 2 ) + d ( 2 ) = 6

E

a (1) + b (1) + c ( 2 cos ϕ ) + d ( 2 cos 2ϕ ) C (ϕ) = 2 + 2 cos ϕ + 2 cos 2ϕ a (1) + b ( −1) + c ( 0 ) + d (0 ) = 0 σv The equations above appear to be too few in number to determine all four unknowns, but the middle equation is decomposable into three equations because of the linear independence of the two cosine functions: a (1) + b (1) = 2 c ( 2 cos ϕ ) = 2 cos ϕ d ( 2 cos 2ϕ ) = 2 cos 2ϕ

Thus the values of c and d are simply determined by inspection: c = 1, d = 1. Then using the relation in the σv equation that shows a = b, one gets a(1) + a(1) + 1(2) + 1(2) = 6; thus a(2) = 6 − 4, or a = 1 and b = 1. Therefore one concludes that a = b = c = d = 1, as was simply and intuitively obvious, but now the conclusion is algebraically rigorous. With this simple algebraic approach one can reduce any given representation for C∞v into its irreducible components. The corresponding reduction for D∞h proceeds in a similar manner, but the equation systems are correspondingly larger. The character table from Hamermesh (14) for D∞h is shown below: D ∞ h

E

C (ϕ)

σv

i

i C ( ϕ)

i σv

Σg +

1

1

1

1

1

1

Σu +

1

1

1

−1

−11

−1

Σg −

1

1

−1

1

1

−1

−

1

1

−1

−1

−1

1

Πg Πu

2 2

2 cos ( ϕ ) 2 cos ( ϕ )

0 0

2 −2

2 cos ( ϕ ) −2 cos ( ϕ )

0 0

∆g ∆u

2 2 cos ( 2ϕ ) 2 2 cos ( 2ϕ )

0 0

2 2 cos ( 2ϕ ) −2 −2 cos ( 2ϕ )

Σu

⯗

0 0

Example of CO2 A simple example—the inspiration for this article—is the reducible representation actually used for CO2 by Harris (16). The total character based on linear displacements for the normal modes is described as Γtot, and it is seen correctly to be

Γ tot 9

C ( ϕ)

[3 +

σv

iC ( ϕ )

i

6 cos ( ϕ )] 3 −3

[−1 −

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i σv

2 cos (ϕ)] −1

•

a (1) + b (1) + c (1) + d (1) + e ( 2 ) + f ( 2 ) = 9

E

a (1) + b (1) + c (1) + d (1) + e ( 2 cos ϕ) + f ( 2 cos ϕ ) = 3 + 6 cos ϕ a (1) + b (1) + c ( −1) + d ( −1) + e ( 0 ) + f ( 0 ) = 3

C ( ϕ) σv

a (1) + b ( −1) + c (1) + d ( −1) + e ( 2 ) + f ( −2 ) = −3

i

a (1) + b ( −1) + c (1) + d ( −1) + e ( 2 cos ϕ ) + f ( −2 cos ϕ ) = −1 − 2 cos ϕ a (1) + b ( −1) + c (1) + d ( −1) + e ( 0 ) + f ( 0 ) = −1

iC ( ϕ) i σv

These equations have a relatively straightforward solution that can be effected by the same considerations that were used for C∞v. Although these equations may look daunting at first sight, their solution is easier than one might expect. Utilizing the same considerations for the cosine terms as was done for C∞v, there result eight algebraic equations that can be arranged as follows,

(1) a + b + c + d + 2e + 2 f =

9

(2 ) a + b + c + d (3)

3 3

e + f

= =

( 4) a + b − c − d = 3 (5) a − b + c − d + 2e − 2 f = −3 (6 ) a − b + c − d (7 ) (8) a − b − c + d

⯗

E

One finds that the sum over irreducible representations only needs to proceed through Πu since the cos(2ϕ) and cosines of higher multiples of ϕ are not present in the character of Γ tot . Therefore one has the character equation Γtot = a∑g+ + b∑u+ + c∑g− + d ∑u− + eΠg + f Πu, which implies the following six algebraic equations:

e − f

= −1 = −1 = −1

and a solution can be found easily. Thus these eight equations must be satisfied for the six unknowns. This overdetermined system can be solved first for the decoupled unknowns e and f, in eqs 3 and 7, and then for a and b, after first eliminating c and d from eqs 2, 4, 6, and 8, and lastly for c and d themselves. One gets the solution as follows: a = 1, b = 2, c = 0, d = 0, e = 1, f = 2. Therefore, one finds the same result that Harris reports (17), namely, Γtot = ∑g+ + 2∑u+ + Πg + 2Πu. If one next removes the translational character Γtrans and the rotational character Γrot from Γtot, one obtains the vibrational result for the molecule CO2. One sees that Γtrans = ∑u+ + Πu, and Γrot = Πg, and thus the vibrations are given by Γvib = ∑g+ + ∑u+ + Πu. These representations are reported by Harris as the correct symmetries of the normal vibrations of the CO2 molecule. It is quite feasible to find the solution to the equation system above by any standard method or even by inspection. General Formula for C∞v The general problem of the normal vibrations of a molecule with C∞v symmetry can be cast in Cartesian coordinates for the n atoms and resolved by the application of the same considerations given to the CO2 problem. The repre-


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sentations shown below are again those from Hamermesh for C∞v symmetry (14): E

C ( ϕ)

σv

Σ

+

1

1

1

Σ

−

1

1

−1

Π ∆

2 2

C ∞ v

2 cos ( ϕ ) 2 cos (2ϕ)

0 0

⯗ Γ tot

3n n [1 + 2 cos ( ϕ )]

⯗ n

The determination of the character of the Cartesian coordinate representation for the molecule is found simply by noting which atoms are unmoved by the operations considered. Thus, for E all the atoms are fixed in space, and all three coordinates for each atom are unchanged. One has therefore the value 3n for the trace of the matrix of the coordinates. When one performs a rotation C(ϕ) about the common z axis, the x and y coordinates are appropriately transformed as in a two-dimensional rotation, and the trace for their transformations is 2ncos(ϕ), while the z coordinates are left unchanged, and they contribute n to the total trace of the matrix of the coordinates. Lastly, the operation σv leaves all the z and x coordinates unchanged, while the y coordinates all change sign. Thus, the x and y contributions to the trace simply cancel, and the z values are all unity, and they give a total contribution of n. These results are perfectly general, and they can be simply seen if one will consider the appropriate matrix for the two atom case. One can now assume that Γtot = a∑+ + b∑− + cΠ, where again no higher-order cosine functions are needed because of the linear independence of the cosine function. Thus the appropriate algebraic equations are

a (1) + b (1) + c ( 2 ) = 3n

E

a (1) + b (1) + c ( 2 cos ϕ ) = n + 2n cos ϕ C ( ϕ ) σv a (1) + b ( −1) + c ( 0 ) = n These equations reduce to

= n c = n

( 4) a − b

= n


There appear to be two general cases for this reduction process: one for molecules with an even number of atoms, and one for molecules with an odd number of atoms. We consider the odd numbered case first. Let n be the odd number of atoms in a linear molecule possessing D∞h symmetry. Using the same kind of deductions for the D∞h case as were used above, one finds the reducible character Γtot as follows: D ∞ h

E

C (ϕ)

σv

i

Σg

+

iC ( ϕ )

1

1

1

1

1

1

Σu

+

1

1

1

−1

−1

−1

Σg

−

1

1

−11

1

1

−1

Σu

−

1

1

−1

−1

−1

1

Πg Πu

2 2

2 cos ( ϕ ) 2 cos ( ϕ )

0 0

2 −2

2 cos ( ϕ ) −2 cos ( ϕ )

0 0

∆g ∆u

2 2

2 cos ( 2ϕ ) 2 cos ( 2ϕ )

0 0

2 −2

2 cos ( 2ϕ ) −2 cos ( 2ϕ )

0 0

⯗ Γ tot

n −3

[−1 − 2 cos ( ϕ)] −1

Again, for E all the atoms are fixed in space, and the Cartesian coordinate representation matrix trace is 3n. The C(ϕ) and the σv cases go exactly as in the C∞v results. Thus, one gets n(1 + 2cosϕ) and n, respectively. The inversion i interchanges all atoms except the central one, and therefore it yields ᎑3. The rotation followed by the inversion also interchanges all atoms except the cental one, and produces a term (᎑1 − 2cosϕ) as its result. Finally, iσv yields the result ᎑1 because only the trace on the central atom contributes to the total character, and x and z both add ᎑1, while the y coordinate adds +1 to the total trace. The corresponding algebraic equations for the general reduction in this case when Γtot = a∑g+ + b∑u+ c∑g− + d ∑u− + eΠg + f Πu are

= n (1 + 2 cos ϕ )

•

i σv

⯗ 3n n   1 + 2 cos ( ϕ )  

(2) a + b + c + d + 2e cos ϕ + 2 f cos ϕ

Here the value of c is trivially equal to n, and the values of a and b are easily found from the remaining pair of equations: a = n and b = 0. One has, therefore, the basic conclusion that Γtot = n∑+ + nΠ, an intuitively satisfying result. In order to obtain the numbers and types of normal vibrations, one must subtract the characters for translation and rotation from Γtot. It is observed that Γtrans = ∑+ + Π, and Γrot = Π. Therefore, the vibrational result is Γvib = (n − 1)∑ + + (n − 2)Π. This result is seen to be correct for HCN, N2O, 142

General Formula for D∞h

(1) a + b + c + d + 2e + 2 f = 3n

(1) a + b + 2c = 3n (2 ) a + b (3 )

and HCCCl, which have 2∑+ and Π modes, and 3∑+ and 2Π modes, respectively. These results are in agreement with the formula reported by Herzberg (18).

E C (ϕ)

(3) a + b − c − d = n

σv

( 4 ) a − b + c − d + 2 e − 2 f = −3

i

(5) a − b + c − d + 2e cos ϕ − 2 f cos ϕ = −1 − 2 cos ϕ ( 6 ) a − b − c + d = −1

iC ( ϕ ) i σv

These equations can be sorted into eight equations using the linear independence of the cosϕ. One obtains the overcomplete system shown below as a result of the separa-


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tion. The solution is just as straightforward as in the previous examples:

(1) a + b + c + d + 2e + 2 f = 3n (2 ) a + b + c + d (3 )

e + f

= −1 = −1

e − f

(8) a − b − c + d

(1) a + b + c + d + 2e + 2 f = 3n (2 ) a + b + c + d = n e + f

= n = n

(5) a − b + c − d + 2e − 2 f = 0 (6 ) a − b + c − d = 0 (7 ) (8) a − b − c + d

Γ vib = ( n − 1) Σ + + ( n − 2 ) Π Γ vib =

D∞ h : Γ vib =

= −1

The values are a = (n − 1)兾2, b = (n + 1)兾2, c = 0, d = 0, e = (n − 1)兾2, and f = (n + 1)兾2. Thus, one has the result: Γtot = [(n − 1)兾2]∑g+ + [(n + 1)兾2]∑u+ + [(n − 1)兾2]Πg + [(n + 1)兾2]Πu. Finally, subtracting the coefficients for the translations Γtrans = ∑u+ + Πu and the rotations Γrot = Πg, one obtains the result for the number and kind of the normal modes Γ vib = [(n − 1)兾2]∑ g+ + [(n − 1)兾2]∑ u+ + [(n − 3)兾2] Πg + [(n − 1)兾2]Πu. We turn ultimately to the even atom number case to conclude this discussion. Now, let n be the even number of atoms in a linear molecule possessing D∞h symmetry. The equations of interest are

(3 ) ( 4) a + b − c − d

C ∞ v :

= n = n

( 4) a + b − c − d = n (5) a − b + c − d + 2e − 2 f = −3 (6 ) a − b + c − d (7 )

can be summarized as

e − f

= 0 = 0

One finds the general solution to be a = n兾2, b = n兾2, c = 0, d = 0, e = n兾2, and f = n兾2. Thus, one has, again after the subtraction of the translational and rotational contributions, the final result: Γ vib = (n兾2)∑ g + + [(n − 2)兾2]∑ u + + [(n − 2)兾2]Πg + [(n − 2)兾2] Πu. One can see clearly that the problem of resolving a reducible representation into irreducible components for the D∞h linear molecular vibration problem is simply a matter of inserting values of n in the three formulas rederived here. Thus all normal vibrational possibilities for linear molecules are enumerated by these three simple formulas. These results are stated by Herzberg in an alternative form in his classic work (19). Summary Therefore, one has available a method that encompasses the finite reduction formula and the reduction formulas for the two continuous groups in a unified approach. The formulas for the normal vibrational modes in linear molecules www.JCE.DivCHED.org

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n −1 + n −1 + Σg + Σu 2 2 n − 3 n −1 + Πg + Π u for n odd 2 2 n + n − 2 + Σg + Σu 2 2 n − 2 n − 2 + Πg + Π u for n even 2 2

where n = number of atoms in the linear molecule. While the finite groups can be dealt with by this technique, it is generally easier to use the well-known finite reduction formula for those cases. The author has successfully applied the method to the well-known resolution problem for the normal vibrations permitted for the water molecule with its C2v symmetry when using a Cartesian coordinate basis. This basic algebraic approach has a long history, going back to the seminal work of Cabannes (20). While not completely establishing the full character of the normal vibrations [a normal coordinate analysis using the full Lagrange–Euler method is needed (21, 22)], the enumeration of the symmetry species permitted will be useful in directing the constructive approaches to be used in combining the various symmetry adapted coordinate basis sets available in characterizing the full three dimensional motion (23). The approach indicated here can readily be extended to the derivation of the dipole selection rules for vibrational transitions in linear molecules through the resolution of the needed direct products. Literature Cited 1. Harris, D. Symmetry and Spectroscopy; Dover: New York, 1989; p 142. 2. Cotton, F. A. Chemical Applications of Group Theory; Wiley Interscience: New York, 1971; p 84. 3. Tinkham, M. Group Theory and Quantum Mechanics; McGraw-Hill: New York, 1964; p 23. 4. Wigner, E. Group Theory; Academic Press: New York, 1959; p 83. 5. Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955; p 11–76. 6. Ferraro, J. R.; Ziomek, J. S. Introductory Group Theory; Plenum Press: New York, 1975; p 51. 7. Alvarino, J. M.; Chamorro, A. J. Chem. Educ. 1980, 57, 785– 786. 8. Jaffe, H. H.; David, S. J. J. Chem. Educ. 1984, 61, 503. 9. McNaught, I. J. Chem. Educ. 1997, 74, 809–810. 10. Shäfer, L; Cyvin, S. J. Chem. Educ. 1971, 48, 295–296. 11. Strommen, D. P.; Lippincott, E. R. J. Chem. Educ. 1972, 49, 341–342. 12. Strommen, D. P. J. Chem. Educ. 1979, 56, 640. 13. Flurry, R. L. J. Chem. Educ. 1979, 56, 638. 14. Hamermesh, M. Group Theory; Addison-Wesley: Reading, MA, 1962; p 324.


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Research: Science and Education 15. Alvarino, J. M. J. Chem. Educ. 1978, 55, 307–308 16. Harris, D. Symmetry and Spectroscopy; Dover: New York, 1989; p 143. 17. Harris, D. Symmetry and Spectroscopy; Dover: New York, 1989; p 144. 18. Herzberg, G. Molecular Spectra and Molecular Structure; Van Nostrand Reinhold: New York, 1945; Vol. 2, p 137. 19. Herzberg, G. Molecular Spectra and Molecular Structure; Van

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Nostrand Reinhold: New York, 1945; Vol. 2, p 139. 20. Cebannes, J. Ann. Phys. 1932, 18, 285. 21. Goldstein, H. S.; Poole, C.; Safko, J. Classical Mechanics; Addison-Wesley: New York, 2002; p 250. 22. Barrow, G. M. Molecular Spectroscopy; McGraw-Hill: New York, 1962; p 207–229. 23. Hargittai, I.; Hargittai, M. Symmetry Through The Eyes Of A Chemist; VCH Publishers: New York, 1986; p 207.


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A Simple Reduction Process for the Normal Vibrational Modes

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