Calculation of vibrational cyclic redundancies in planar rings

26 November 1999

Chemical Physics Letters 314 Ž1999. 189–193 www.elsevier.nlrlocatercplett

Calculation of vibrational cyclic redundancies in planar rings C. Mapelli, C. Castiglioni ) , G. Zerbi Dipartimento di Chimica industriale e Ingegneria Chimica, Politecnico di Milano, P. Leonardo da Vinci 32, 20133 Milano, Italy Received 19 May 1999; in final form 29 September 1999

Abstract A simple method is presented which allows us to compute in-plane cyclic redundancies. The case of a planar six-membered ring Žsuch as the benzene molecule. is solved in detail, thus giving expressions of redundancies that may be used for any condensed aromatic hydrocarbon and for graphite. A short example of the application to graphite is shown. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction At present polycyclic aromatic hydrocarbons ŽPAH’s. are the center of strong scientific interest because of their potential application as liquid crystals and because they can be considered as molecular models Žoligomers. of a graphite layer. In fact, if graphite is considered as a two-dimensional polymer, PAH’s play the same role as short chains with respect to polymers. The vibrational dynamics of PAH systems is under investigation w1–3x, but it is made difficult by the presence of many cyclic redundancies. When studying molecular vibrations, the problem of redundancies arises whenever the geometry of the molecule is such that natural internal coordinates Ži.e. bond stretching and valence angle bending as defined by Wilson w4x. are not linearly independent

) Corresponding author. Tel.: q39-2399-3235; fax: q39-23993231; e-mail: [email protected]

but are linked by relationships called redundancies w5x. It is well known that, if redundant valence coordinates are used in order to study molecular vibrations according to Wilson GF method w4x, the G matrix turns out to be singular. The inconvenience of using redundant internal coordinates becomes apparent when valence coordinates are used to build symmetry coordinates. The symmetry coordinates treatment is very useful in the analysis of vibrational normal modes of a molecule: sometimes w1x it permits determination of the species and the approximate shape of each normal mode, without explicit diagonalization of the dynamical matrix. Nevertheless, if redundant internal coordinates are used, symmetry coordinates will be overabundant, thus hindering a unique identification of the normal modes. Removal of redundancies becomes vital in a refinement of valence force constants. In fact when the FR matrix Ži.e. the matrix whose elements are the valence force constants f i j . is based on redundant internal coordinates, the refinement procedure may

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 1 3 9 - 2

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C. Mapelli et al.r Chemical Physics Letters 314 (1999) 189–193

not converge. This has been the difficulty encountered by many authors of ‘grand least squares’ refinement of valence force constants. Let P be a redundancy Ž P s Ý i c i R i , where the R i are the internal coordinates, and c i are the coefficients of the combination., FP the collective diagonal force constant for the coordinate P , and fP j the interaction constants between P and the other coordinates. FP and the fP j cannot be determined by a refinement procedure; however, the vibrational frequencies are independent of the values taken by FP and fP j , because the rows of the G matrix that compete to P are linearly dependent, i.e. Ý k c k g k i s 0, where the c k are the coefficients of the combination of internal coordinates that defines P , and the g k i are elements of G. In order to solve this problem, two different methods may be used, each of them requiring the knowledge of the redundancy relations. The first option is to build a new set  R 4 of internal or symmetry coordinates orthogonal to the redundancies and to refine the FR matrix based on these new coordinates, which are linearly independent. The second possibility is to refine FR Žbased on the set  R4 of redundant internal coordinates., with the constraint that FP and fP j Žobtained combining the f i j in a suitable way. should be fixed to a constant value. Since along P the nuclei do not move, the most natural choice is to set FP and fP j equal to zero w7x. The knowledge of redundancies is therefore crucial in vibrational dynamics. At present, while local redundancies are usually removed using a procedure that is well established w8x and easy to apply, cyclic redundancies are harder to find and in most cases they mix bending and stretching coordinates. A general treatment of cyclic redundancies and the description of a method for their removal can be found in Refs. w5x and w6x. The redundancies of benzene have also been treated several times in literature by authors who calculated valence force fields and normal modes of vibration of benzene w9,10x. The method presented in the present paper is a simpler alternative to obtain in-plane redundancies in any planar cyclic molecule. Here the case of a hexagonal planar ring Žbenzene. is solved first and the final expressions of the redundancies can be applied to any system that contains six-membered rings.

In a recent paper w1x we have treated the vibrational dynamics of a 2-D graphite sheet and the removal of cyclic redundancies in that system has been carried out using the method here reported. In that work we show that the method is very successful in obtaining a completely analytic expression of phonon eigenvectors in internal valence coordinates at the high symmetry points of the first Brillouin Zone of graphite.

2. Calculation In a planar N-atomic closed ring there are always 2 N y 3 in-plane normal modes of vibration, while 2 N in-plane internal coordinates Žbond stretching and bond angle variations. can be defined. This means that in a planar ring always 3 redundancy relations are to be found between in-plane internal coordinates. In the following we will lay out a general method to obtain in-plane redundancies in any planar ring and will solve the case of a hexagonal ring as an example. The bonds and angles of a hexagonal planar molecule are defined and labeled in Fig. 1. Cyclic redundancies in a planar ring arise from the constraint that during any in-plane vibration the ring must remain closed. This is translated in analytical form by writing the condition: N

R s Ý rn s 0

Ž 1.

n

where N is the number of bonds in the ring, and rn is a vector that represents the nth bond. We point out that rn and a n indicate bond lengths and angles and not their variations.

Fig. 1. Definition of the labels of bonds and angles of the hexagonal molecule.

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These equations become simpler considering the periodicity of trigonometric functions: Ø r6 y r 1 cos Ž a 6 . q r 2 cos Ž a 6 q a 1 . yr 3 cos Ž a 6 q a 1 q a 2 . qr4 cos Ž a 6 q a 1 q a 2 q a 3 .

Fig. 2. Definition of phases V i .

yr5 cos Ž a 6 q a 1 q a 2 q a 3 q a 4 . s0 The vectors rn may be written using the exponential notation: rn s rn e

iVn

V n s V ny1 q Ž p y a ny1 .

Ž 3.

as can be inferred from Fig. 2. Using Ž2. and Ž3. recursively, Ž1. can be written in terms of ri and a i only. For the hexagonal ring Ž1. becomes: r6 q r 1e iŽ py a 6 . q r 2 e iŽ2 py a 6ya 1 . q r 3 e iŽ3 py a 6ya 1ya 2 . q r4 e iŽ4 py a 6ya 1ya 2ya 3 . q r5 e iŽ5 py a 6ya 1ya 2ya 3ya 4 .s0

Ø r 1 sin Ž a 6 . y r 2 sin Ž a 6 q a 1 . qr 3 sin Ž a 6 q a 1 q a 2 .

Ž 2.

If the origin of the phases is chosen so that V N s 0, and a 1 is the angle between rN and r 1 , the phases of all other vectors are given by:

Ž 4.

By writing each exponential e i V n in the extended form cosŽ V n . q i sinŽ V n ., this equation can be divided in two equations which apply, respectively, to the real and to the imaginary parts of the first member. In the case N s 6, Eq. Ž4. becomes:

yr4 sin Ž a 6 q a 1 q a 2 q a 3 . qr5 sin Ž a 6 q a 1 q a 2 q a 3 q a 4 . s 0

Ž 8.

The above equations are relationships between bond lengths and valence angles. Since internal coordinates are defined by Wilson as variations of bond lengths and of angles, Ž7. and Ž8. must be differentiated in order to find the expressions for the redundancies between internal coordinates. We have thus found two of the three redundancies we are looking for. The third condition comes from the equation: N

Ý a n s p Ž N y 2.

Ž 9.

ns1

In fact for any polygon the sum of internal angles equals p Ž N y 2., where N is the number of sides. By differentiation, Ž9. becomes: N

Ý d an s 0

Ø r6 q r 1 cos Ž p y a 6 . q r 2 cos Ž 2 p y a 6 y a 1 .

Ž 7.

Ž 10 .

ns1

qr 3 cos Ž 3p y a 6 y a 1 y a 2 . qr4 cos Ž 4 p y a 6 y a 1 y a 2 y a 3 . qr5 cos Ž 5p y a 6 y a 1 y a 2 y a 3 y a 4 . s 0 Ž 5. Ø r 1 sin Ž p y a 6 . q r 2 sin Ž 2 p y a 6 y a 1 . qr 3 sin Ž 3p y a 6 y a 1 y a 2 . qr4 sin Ž 4 p y a 6 y a 1 y a 2 y a 3 . qr5 sin Ž 5p y a 6 y a 1 y a 2 y a 3 y a 4 . s 0

Ž 6.

Expressions Ž7., Ž8. and Ž10., written with N s 6, constitute the complete set of redundancy relations in any six-membered planar ring. By comparison with Ref. w6x one can easily notice that the three redundancy relations here obtained correspond, respectively, to the three conditions for a planar ring imposed by Califano, i.e. that the two projections of R Ždefined in Eq. Ž1.. along x and y axes and the modulus of R be zero. Since molecular vibrations are small vibrations around the equilibrium position, all rn and all a n can be taken equal to their equilibrium values.

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In a benzene ring all bonds have the same length, which we indicate as r 0 , and all valence angles are equal; this means that the hexagon is regular. Therefore: rn s r 0

for n s 1, . . . , 6

Ž 11. Ž 12.

a n s a 0 s 1208 for n s 1, . . . , 6 Using these values, Ž7. and Ž8. become: 1 2

Ž d r1 y d r 2 y 2d r 3 y d r4 q d r5 q 2d r6 .

'3 y 2

'3 2

r 0 Ž d a 1 q 2d a 2 q 2d a 3 q d a 4 . s0

Ž 13 .

Ž d r 1 q d r 2 y d r 4 y d r5 .

1 q r 0 Ž 3d a 1 q 2d a 2 y d a 4 q 2d a 6 . s0 Ž 14 . 2 A more symmetric form of Eqs. Ž13. and Ž14. can be obtained, by summation or subtraction of Ž12., thus yielding: 1 2

Ž d r1 y d r 2 y 2d r 3 y d r4 q d r5 q 2d r6 .

'3 y 2

'3 2

r 0 Ž d a 2 q d a 3 y d a 5 y d a 6 . s0

Ž 13bis.

1

Ž d r1 q d r 2 y d r4 y d r5 . q r 0 Ž 2d a 1 2

qd a 2 y d a 3 y 2d a 4 y d a 5 q d a 6 . s0 Ž 14bis . These relationships link the variations of bond lengths and of valence angles for any ring that in the equilib-

rium position is a planar regular hexagon with side length r 0 . A single hexagonal ring Že.g. benzene. belongs to the symmetry point group D6h , accordingly the redundancies relations can be assigned to symmetry species of the D6h group: Ž10. spans A 1g , Ž13bis. and Ž14bis. span E 1u . For a polycyclic aromatic hydrocarbon conditions Ž1. and Ž10. must be imposed to each of the rings forming the molecule. If it is reasonable to assume equal lengths for all bonds, expressions Ž13. and Ž14. still hold. On the contrary, if bonds have different lengths and angles are not equal to 1208, more general expressions Ž7. and Ž8. are to be used, substituting to all rn and a n the actual values in the equilibrium position. The same method applies to any planar ring.

3. Redundancies in graphite The redundancy relations found above can be used successfully to remove cyclic redundancies in graphite. In Fig. 3 we define the internal coordinates in the elementary cell of the graphite lattice. Applying Bloch’s theorem, the generic internal coordinate qi of the cell Ž n1 , n 2 . can be written as: qiŽ n1 , n 2 . s qiŽ0 ,0. e iŽ n1 u 1qn 2 u 2 . where u 1 and u 2 are the components of the phonon wavevector along the two principal directions of the reciprocal lattice.

Fig. 3. Definition of internal coordinates of graphite.

C. Mapelli et al.r Chemical Physics Letters 314 (1999) 189–193

Consequently relations Ž10., Ž13bis. and Ž14bis. become:

P 1 s v 1 q v 2 e i u 1 q v 3 e i u 2 q v X1e iŽ u 2qu 2 . q v X2 e i u 2 q v X3 e i u 1s0

P 2 s 2 R1Ž e i u 1 y e i u 2 . q R 2 Ž 1 y e i u 2 . y R3 Ž1 y e

iu1

. q '3 R cc Ž v 1 q 2 v 3 e

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aromatic hydrocarbon w1,3x. In particular these expressions have been used in recent work w1x to remove analytically the vibrational redundancies of a two-dimensional layer of graphite. Acknowledgements

iu 2

qv X1e iŽ u 1qu 2 . q 2 v X2 e i u 2 . s0

We would like to thank the referee for the helpful comments that fostered improving the quality of this manuscript.

P 3 s '3 Ž R 2 Ž 1 y e i u 2 . q R 3 Ž 1 y e i u 1 . . q R cc Ž 3 v 1 q v X1e iŽ u 1qu 2 . q 2 v X2 e i u 2

References

q2 v X3 e i u 1

w1x C. Mapelli, C. Castiglioni, G. Zerbi, Phys. Rev. B, in press. w2x C. Mapelli, C. Castiglioni, E. Meroni, G. Zerbi, J. Mol. Struct. 480r481 Ž1999. 615. w3x C. Mapelli, Thesis in Materials Engineering, Politecnico di Milano, Milan, Italy, 1998. w4x E.B. Wilson, Jr., J.C. Decius, P.C. Cross, Molecular Vibrations, McGraw-Hill, 1955. w5x S. Califano, Vibrational States, Wiley, 1976. w6x S. Califano, B. Crawford, Z. Elektrochem. 64 Ž1960. 571. w7x M. Gussoni, G. Dellepiane, S. Abbate, J. Mol. Spectros. 57 Ž1975. 323. w8x P. Pulay, G. Fogarasi, F. Pang, J.E. Boggs, J. Am. Chem. Soc. 101 Ž1979. 2550. w9x D.H. Whiffen, Philos. Trans. R. Soc. A 248 Ž1955. 131. w10x J.C. Duinker, I.M. Mills, Spectrochim. Acta 24A Ž1968. 417.

. s0

The treatment of in plane vibrations of graphite and a more extensive descriptions of the elimination of redundancies can be found in Ref. w1x.

4. Summary A simple method has been proposed in order to write down in-plane cyclic redundancies in any planar cyclic molecule. The expressions obtained for benzene turn out to be useful for any polycyclic