Kinetic energy distribution of 2,′-bipyrimidine

Spectrochimica

Pergamon

Acta.

Vol. 50A. No. 7, pp. 1317-1328. Copyright @ IW4 Elscvicr

IVY4

Kinetic energy distribution of 2,2’-bipyrimidine

Department

N. NETO* and M. MUNIZ-MIRANDA of Chemistry, University of Firenze, Via G. Capponi 9, I-50121, Firenze, Italy

and G.

SBRANA

Centro Studi su Composti Eterociclici del CNR, Via G. Capponi 9, I-50121, Firenze, Italy (Received

9 August

1993;in$nal

form

20 September

1993; accepted 21 September

1993)

Abstract-A

method is proposed for the vibrational assignment of large molecules, based on the kinetic energy distribution among different groups of internal coordinates. It is related to the treatment of constrained motion and requires a coordinate change for a transformation of the whole vibrational space into two orthogonal subspaces. The normal mode description obtained by this procedure is compared with that suggested by the familiar potential energy distribution, using 2,2’-bipyrimidine as a test molecule. It is found that a criterion based on kinetic energy distribution gives a correct representation of the low frequency modes, associated with translation or rotation of the pyrimidine rings. It also attributes a proper weight to internal displacements associated with large diagonal force constants and is not influenced by the value of the vibrational frequencies.

INTRODUCTION THE

VIBRATIONAL analysis of complex molecules relies on normal mode calculations which, through comparison of experimental and calculated frequencies, allow determination of a force field and identification of the fundamental vibrations. The solution of the harmonic equation of motion provides a set of eigenvectors which contain all the information about the normal modes of vibrations. It is, however, not obvious how this information can be extracted from the eigenvectors in order to proceed to a vibrational assignment, i.e. to describe each mode in terms of bond stretchings, angle bendings, etc., or combinations of, involving specific chemical groups. Plotting Cartesian atomic displacements for each normal coordinate is useful for molecules with a small number of vibrational degrees of freedom but is of little help for large molecules. The only practical method to extract information from the eigenvectors is to analyse the potential energy distribution (PED) among diagonal force constants [l]. Through this procedure one obtains numbers, PEDik, representing how the contribution of the ith normal mode to the total potential energy is distributed among force constants. For a given subset of internal coordinates {R,} defining the kth diagonal force constant, a criterion for the vibrational assignment is thus based on the value of PEDik: the larger it is, the more the ith normal mode is attributed to internal coordinates of the kind R,. This procedure fails when, for instance, the members of {R,} are redundant coordinates. In this case a large value of PEDik has nothing to do with the description of the motion associated with Qi which actually depends on different coordinates, namely those forming the independent set. On the other hand, the PED dependence on both force constants and vibrational frequencies tends to emphasize contributions of lower frequency modes for a given force constant, or of larger force constants for a given normal mode, with possible misleading interpretation of the molecular motion. For these reasons we propose a different criterion based on the kinetic rather than the potential energy distribution. The convenience of considering the atomic motion was investigated in a lattice dynamics study of the DNA helix [2] in which emphasis was placed on the fraction of kinetic energy associated with a group of atoms. Here we consider how the kinetic energy of each normal mode is distributed in the space of linear

* Author to whom correspondence should be addressed. 1317

N. NETOet al.

1318

8

18

16

Fig. 1. Moleculargeometryof 2,2’-bipyrimidine (BPM).

internal displacements. This requires a factorization of the space of internal displacements in two orthogonal subspaces, so that contribution to the kinetic energy for each normal coordinate can be distributed between the two subspaces. A procedure by which the kinetic energy matrix can be exactly factorized in two blocks of variable dimensions is presented in this paper and applied to the vibrational analysis of 2,2’-bipyrimidine (BPM) which was examined in a previous paper [3]. This is a planar molecule formed by two pyrimidine molecules connected through a central CC bond, which could be treated as two interacting hetero-aromatic rings. Although such a model is actually not considered here, its main features are necessarily reflected by normal coordinates associated with low frequency vibrations, which turn out to be wrongly described on the basis of the PED.

NORMAL MODE CALCULATION

The planar molecular conformation of BPM, shown in Fig. 1, relies on distances and angles averaged from the X-ray structure [4], as reported in Ref. [3] to which the reader is referred for a description of both experiment and force field. It should be mentioned, however, that all calculated frequencies reported here are derived from a new set of force constants, obtained through a refinement procedure based on the experimental data of both BPM and bipyrazine [5]. Both initial and refined values are given in Table 1. The only change in the definition of the initial force field refers to the 24th and 25th force constants, associated with interactions among CH waggings and central CC waggings, respectively, grouped together under the unique force constant it. 24 in Ref. [3]. All local and cyclic redundancies have been eliminated to the first order. To do this, the internal coordinates which describe the force field were defined in terms of an independent set, which includes group coordinates listed in Table 2. The fundamentals identified earlier were confirmed by the new force constants, with the only exception of a weak Raman band at 401 cm-‘, previously considered a B,, fundamental and now attributed to the BzR species. This latter mode was earlier left without an experimental counterpart, a role now taken by the B,, mode. All observed and calculated in- and out-of-plane frequencies of the isolated molecule are listed in Table 3, together with an approximate vibrational assignment discussed in the last section.

KINETICENERGY DISTRIBUTION

The potential energy distribution is derived from the eigenvector matrix, L, and the diagonal matrix of the eigenvalues, A, obtained from the solution of the equation of

Kinetic energy distribution of 2,2’-bipyrimidine

1319

Table 1. Initial and refined values of forceconstantsfor BPM, numbered as in Ref. [3]. Diagonal terms for stretching in units of mdyne A-‘, for bending, torsion and out-ofplane wagging in units of mdyne A rad-*; stretchingbending interactions in units of mdyne rad-’ No.

Initial

Refined

No.

Initial

1

5.042 6.241 6.663 5.737 1.270 1.130 1.947 0.540 1.222 0.209 0.397 0.643 0.097 0.573

5.042 6.368 6.150 5.823 1.318 1.236 1.775 0.538 1.423 0.134 0.376 0.532 0.105 0.662

15 16 17 18 19 20 21 22 23 24 25 26 27

-0.161 0.189 0.103 0.849 0.559 0.285 0.464 0.213 0.204 -0.024 -0.024 -0.168 -0.079

2 3 4 5 6 7 8 9 10 11 12 13 14

Refined 0.097 0.226

-0.096 1.374 0.470 0.217 0.208 0.374 0.185 -0.004 -0.044 -0.121 -0.088

motion in internal coordinates. These quantities are related [6] to the potential, F, and kinetic, G, energy matrices by the following two equations: L+FL = A

W

LtG-‘L = E.

(lb) The first equation directly gives the following expression for contribution of the ith normal mode to the potential energy

where li is the ith eigenvalue and the sum is over a complete, independent linear set of N internal displacements {R,}. Since elements Fobare simple functions of the valence force field, Eqn (2) provides an expression for the potential energy distribution among force constants. Similar arguments apply to the kinetic energy distribution, using Eqn (lb) as a starting point. Both expressions for the kinetic energy matrix, G in terms of generalized Table 2. Group internal coordinates adopted for BPM as combinations of bendings and torsions (atom numbers as in Fig. 1) Atoms s”

7

3

42

~CH,=2-“2(a-/S)

L

8

4

5 3

/%ZH2=2-“‘(y-6)

& tl II w rl r, B

9 95 2 6 45 56 6 1 2 3 6 2

5

4 6 10 10 6 12 2 3 4 5. 10 10

BCH,=2-“*(E-q)

r4

Ts rfl 77

=m

SA(A) 50:7-H

Group internal coordinates

1 1

1 2 3 4 1 1

#Ii= 2- “2(Jr- 0) 1 3 4 5 6 15 11

r, = 6-“‘(r, - r2 + r3 - t4+ r, - q,) r2=o.5(q-T,+r~-qJ r, = 12- “?(2t, - r* - q + 27, - rs - m) I-,=2-“*(r,+r”)

N. NETOet M.

1320

Table 3. Observed and calculated frequencies of BPM and vibrational assignment Obs.

Calc.

As

3049 3016 1567 1451 1332 1075 998 782 339

3051 3050 1572 1449 1332 1083 1000 803 328

CH stretching CH stretching ring deformation interring stretching CH bending ring stretching ring stretching ring bending ring-ring translation

Blx

3077 1577 1432 1335 1291 1172 629

3049 1564 1446 1343 1275 1155 617 468

CH stretching ring deformation CH bending + ring stretching ring stretching CH bending ring deformation + CH bending ring bending ring-ring rotation

B2#

845 401

843 401

CH wagging ring torsion

B3s

990 809 794 550 222

983 823 796 536 211

interring wagging + ring torsion CH wagging CH wagging ring torsion ring torsion

89

843 413 91

CH wagging ring torsion + interring torsion interring torsion

116

902 802 779 397 96

CH wagging + ring torsion CH wagging CH wagging + ring torsion ring torsion ring-ring rotation

B2u

3049 3022 1568 1404 1188 1090 992 682

3051 3050 1562 1410 1181 1098 1006 669

CH stretching CH stretching ring deformation CH bending ring bending ring deformation ring breathing ring bending

B3u

3067 1557 1425 1380 1259 1144 648 167

3049 1565 1432 1370 1250 1132 651 182

CH stretching ring deformation CH bending + ring stretching ring stretching CH bending interring bending + ring stretching ring bending ring-ring rotation

Au

Blu

893 810 772 395

Description

momenta with elements G °b and G-~ in terms of velocities with elements Gob, will be considered. Hence, from Eqn (lb) the contribution to the kinetic energy due to the ith normal mode is such that

~_, ~_, GabLaiLbi----1. a

b

(3a)

Kinetic

energy distribution

The

same can be done in terms of momenta, result

C C a

1321

of 2,2’-bipyrimidine

using the inverse of Eqn (lb), with the

1.

G"b(L-')j~(L-')ib=

b

Equations (3a) and (3b) are the counterparts of Eqn (2) and will be used to show how the kinetic energy of each normal mode is distributed among internal coordinates. The problem is that the kinetic energy matrix contains a large number of off diagonal elements. Ignoring selected cross termS Gab in (3a) is not feasible since, in general, the same cannot be realized in the space of momenta. Only if a given group of internal coordinates, {R,}, were orthogonal to all remaining ones, {R,}, their contribution to the kinetic energy could be extracted from Eqns (3a) and (3b) and used to describe normal modes. Such a factorization is accomplished through a linear coordinate transformation of the kind r, =

r,=R,

(n=1,2,.

. . ,N--t),(a=1,2,.

. . ,z).

Here a, fi and y label internal displacements unaffected by the transformation, later associated with r constraints imposed to the vibrational motion, while 12,m identify all remaining coordinates. Whenever a distinction is not necessary, subscripts a, b will be used which thus run over N values. The transformation is such that the kinetic energy in terms of velocities

is transformed,

using the new coordinates,. into

n

m

a

B

with no cross terms between the (N- r) coordinates r, and the t coordinates r,. An explicit expression for the constant coefficients S,laR, of Eqn (4) is given, using the elements gnk of a (N - r) square matrix such that (7) k

This new matrix is thus obtained by: (1) dropping from G-’ all rows and columns labelled by coordinates R,; (2) inverting the resulting (N- t) square matrix. If t is small, as in the case in which just one coordinate R, is factored out, this procedure requires an inversion of a large matrix. This can be avoided using an expression [6] for the elements g”” given by

g”“=G”“-

cc

(8)

G”aglgnDG@m.

Here gaBare elements of a t X t matrix inverse to that obtained from G after dropping all rows and columns referring to the subset {R,}, such that

= c gayGYP

(9)

uaB.

Y

At this point, the new basis can be defined as r,,=R,+

xc g”“GaRa m

a

r, =

R

0’

(10)

N. NETO et al.

1322

Once each R, is introduced in (5) as a function of the rg, the desired result (6) is readily obtained after noting that elements gap, defined in (8), can also be written as

gap = G, - x c G&“‘G,,. n m

(11)

In fact this equation is just the same as Eqn (8), once the roles of the two subsets {R,,}and as well as of sub- and superscripts, are interchanged. Since the kinetic energy in Eqn (6) is in block diagonal form, the corresponding expression in terms of momenta is readily obtained as

{R,},

02) This means that the new momenta pa are given by pa = (a Tlaq

=

C gapis.

(13)

Using Eqn (11) for gaS and the fact that, from Eqn (lo), ra = R,, these new generalized momenta pa are readily obtained in terms of those, p,, and pa, in the original space of the R,. The same can be done for pn and we have the result Pn=Pn

pa=pa-

G,mgmnpn

zc n

(14)

m

which completes the transformation introduced in Eqn (10). Thus the new basis does not affect the coordinates R, while producing new momenta pa. Exactly the opposite applies to the other subset, for which new velocities are defined while the momenta remain unchanged. This also means that Wilson’s B -’ matrix for the subset {ra} is different from that obtained for the original coordinates R,. Since now the kinetic energy matrix is factorized in block diagonal form, Eqn (3a) can be applied giving, for each normal mode Qi, the contribution of the a coordinates to the kinetic energy expressed as

P

Y

where the subscript a refers to all coordinates of the subset {R,}. equivalent, expression can be obtained from (3b) as KED,, = 1 -c

c ”

g”“(L-‘),(L-I),.

An alternative,

Wb)

m

The latter form is less convenient than (15a) as it requires an inversion of the L matrix. Since KED;, gives the fractional {R,} character attributed to a given normal mode, it provides with a criterion, for the vibrational assignment, alternative to that based on the potential energy distribution. The same coordinate transformation discussed so far can also be applied to the potential energy which, in terms of the new coordinates r,, and r,, becomes 2V=c

2 n

F,,r,r,,,+

m (16)

Here constant coefficients ar,laR,, introduced in Eqn (4), are explicitly obtained from Eqn (10). The same vibrational frequencies are then obtained if the new coordinates r,

1323

Kinetic energy distribution of 2,2’-bipyrimidine

R”

‘n

”

Fig. 2. NCN bending coordinate for BPM, r, = R, + hv, when the inter-ring CC stretching, v, is kept rigid (see text).

rather than R,are used, only the eigenvectors change. Setting the secular equation in the new coordinates r, is not required to compute the KED but it is very appropriate when a set of rigid constraints are imposed to the vibrational motion:

R,=r,=O

(a=1,2,...,r).

(17)

In fact the kinetic energy is already in a suitable block form and is given by 2T= c

c n

G,,k,l?,=

c, 2 n

m

g""P,,P,,,

m

under constraints (17) while the potential energy (16) reduces to 2V= c

c n

F,,R,R,.

(19)

m

This means that an approximate, constrained, F' matrix is obtained from F by setting to zero all force constants F,,, and F,,.It is convenient to adopt the kinetic energy expression in terms of momenta as components Gub are directly evaluated from the definition of internal coordinates. Hence in the product G'F',for a constrained motion, G’ has elements g”” obtained from Eqn (8). This conclusion agrees with that proposed in Ref. [6], in which Eqn (8) was in fact introduced in connection with separation of high and low frequency. This does not imply, however, an approximation for the kinetic energy, which is simply transformed in the new basis of the r,, with constraints acting only on the potential energy. The share of the total kinetic energy associated with the t internal coordinates of a subset {R,}is obviously given by

c

KEDi, = t

with the sum extended to all normal modes. It provides us with a convenient check for KED calculation. If the sum is restricted to the normal coordinates belonging to a given irreducible representation, Eqn (20) gives the number of zero roots in the secular equation, for the given symmetry block, when constraints (17) are imposed. The coordinate transformation discussed so far will be now illustrated with the help of two examples, the first one given in Fig. 2 for the case of BPM when the inter-ring CC stretching, v, is chosen as the only member of the set {R,}.In a space factorized accordingly, a new NCN bending coordinate is given by r,= R,+hv, R, being the original bending with the factor h chosen so that r, is orthogonal to v, that is, g”‘= 0 while G,,, #O. Similar considerations apply to all other (planar, in the present case) coordinates R, having at least one atom in common with v. As far as v is concerned, Wilson’s B matrix elements with Cartesian components &lax, are not affected but the same is not true for the inverse coefficients, axJav, represented in Fig. 3(a). This is immediately

N. NETO et al.

1324

evident when these are compared with the corresponding coefficients for the original, not factorized, space shown in Fig. 3(b). PED data for BPM are listed in Table 4 while KED data, calculated by choosing, in turn, different groups of coordinates {R,}, are given in Table 5. For each irreducible representation, the last row of the KED table gives the number of “zeros” as obtained from Eqn (20). For the A, species and under the heading Vicorresponding to the central CC bond, the larger coefficient is found for QkM9,with smaller values for Q,572and Q,cxx,. The PED data suggest, instead, that the same V,character for Q,449and QZZH, due to the factor l/1 which favours the latter mode despite a much smaller value of (avlaQk)‘. Since the normal mode at 328 cm-’ is very similar to a translation of the two rings in opposite directions, all internal coordinates involving atoms of one ring must have nearly zero projections on it. Small components may arise only from inter-ring internal displacements. Such behaviour is evident from the KED data, which attribute nearly zero character to this mode for all internal coordinates. This means that the QsZRmode must be associated to a motion of two separate, interacting, pyrimidine rings whose frequency, but not dynamics, depends on the interaction forces. For this reason we assign the QjZRmode as a ring-ring translation on the basis of kinetic energy considerations. The PED data instead suggest an assignment as a central C-C stretching, mixed with ring bending and stretching. We conclude that a coordinate like QXzxrepresents the typical example of a normal mode whose description cannot rely on the PED while the KED is appropriate. In order to show how the KED description is related to a constrained motion, Fig. 4 shows the exact A, normal coordinates of BPM are shown and compared with those of the upper row obtained by keeping the inter-ring C-C stretching rigid. Such constraint affects only the A, symmetry block of the secular equation, producing a zero root visualized by a missing diagram in the upper row. Although constrained frequencies are consistently higher than the exact ones, as expected [6], the normal coordinates are surprisingly similar in the two cases. Excluding the central CC bond, a variation of atomic displacements can be appreciated from Fig. 4, only for the normal modes Qlooo and Qlos3, while Q32Ris the least affected by the constraint even if a consistent frequency shift occurs for this mode. Despite the imposed rigidity, it is possible to identify unambiguously all normal modes, hence the one which ‘goes to zero’ must correspond to Q 14497which is thus mainly due to an inter-ring CC stretching in agreement with KED description.

CONCLUSIONS

The approximate vibrational assignment of BPM, reported in Table 3, is based on the description provided by both kinetic and potential energy distributions. As discussed in

(a)

(b)

Fig. 3. Cartesian coefficients (dx,,l&) for Y= inter-ring CC stretching of BPM. (a) Y is factorized out; (b) all degrees of freedom are used (see text).

Kinetic energy distribution of 2,2’-bipyrimidine

1325

Table 4. Potential energy distribution (%) for BPM diagonal force constants and two out-of-plane interactions (force constants numbers as in Table 1)

2 vcc

F.C. No.

3 vCN

4

1572 1449 1332 1083 loo0 803 328

19 14 1 40 5 5 0

40 11 29 13 53 45 17

81,

1564 1446 1343 1275 1155 617 468

56 1 32 0 11 14 0

&”

1562 1410 1181 1098 1006 669

&I

1565 1432 1370 1250 1132 651 182

F.C. No.

9 6 3 4 7 0 10

4 7 9 6 19 33 8

18 42 98 14 23 13 14

5 4 0 1 0 42 0

8 0 0 0 5 33 1

23 11 3 24 20 3

51 23 4 17 57 35

4 0 18 11 0 5

7 1 26 12 0 38

54 1 31 0 13 12 1

18 40 92 18 33 9 9

4 4 0 2 1 35 5

8 0 0 0 4 32 2

10 r 4 79

18 41 3 1 22 1 36

Out-of-plane vibrations 11 12 13 26 wCH wi T---W ri 123 14

~3

44

823 797 536 211

9 0 46 70

21 140 101 11 24

A,

843 413 91

4 74 5

123 13 0

B,,

!I02 802 779 397 %

53 4 8 89 17

84 104 72 31 7

B,

In-plane vibrations 4 v, /5&C /?N6cC BCCC

57 10 1 43 3 0 6 94 14 3 10 57 56

vN:N

13 15 0 1 1 0 21

7 9 17 0 0 15

$H

ii

16 24 39 28 5 9 2

5 6 0 0 0 0 8

26 56 6 71 49 7 1

6 19 7 14 7 5 91

18 47 20 27 1 10

3 4 7 0 0 6

27 63 8 76 35 7 0

0 1 0 1 3 5 73

27 r--I

-30 -45

3 52

-84 -68 0 -48 -43

56 7 0 44 46

-30 -42 -3

3 49 3

-121 -13 2 -136 13

70 3 9 63 11

vCC, vCN: ring stretching; Vi: CC inter-ring stretching; /3CNC, BNCC, /?CCC, BNCN: ring bending; BCH, pi: CH, NCC inter-ring bending; r. ri: ring, inter-ring torsion; wCH, wi: CH, inter-ring CC wagging; r-w. r-r: torsion-wagging, torsion-torsion interaction.

the previous section, the major discrepancy among PED and ICED data is found for the low vibrational modes, either in-plane or out-of-plane. Their translational or rotational character is only indirectly suggested by the present KED data. It does, however, come

N. NETO et al.

1326

Table 5. Kinetic energy distribution (X 100) for BPM (zeros: number of zero roots when corresponding group of coordinates are kept rigid)

vcc

4

B2U

B %I

B, 4

A”

zeros

0

26 28 35 0 0 10 1

22 50 11 16 0 0 1

0

0

47 0 1 0 12 32 0 1

0

35 66 8 67 24 0 0 2

5 16 2 16 29 15 17 1

Out-of-plane vibrations wCH w, I-, l-?

f-3

L

1565 1432 1370 1250 1132 651 182

60 1 27 0 8 2 0 1

21 46 85 16 29 2 0 2

29 11 54 25 1 2

0

18 82 1

0

13 27 0 42 19 1

0

0

0

0

18 82 0 1

0 56 44 1

90 1 7 2 0 1

0 11 10 78 1 1

0

0

97 3 1

0

0

0

983 823 7% 536 211

18 80 98 3 1 2

74 9 0 17 0 1

84 3 0 13 0 1

843 413 91

97 3 0 1

0

59 83 54 4 0 2

40 6 20 32 2 1

zeros B,,

0

8 1 42 28 1 12 1

71 37 4 19 58 9 2

zeros

z 779 397 %

0 13 32 10 19 8 1 16 1

0

40 15 3 20 15 1 1

zeros

21 26 35 17 0 1 0 1

30 3 33 1 0 24 1

1562 1410 1181 1098 1006 669

E

48 46 0 1 1 0 3 1

0

9 0 1

23 54 84 15 20 2 2 2

zeros

j3,

33 61 5 65 35 0 0 2

62 1 26 0

zeros

/3CH

45 1 2 1 15 29 1 1

1564 1446 1343 1275 115.5 617 468

::

27 52 4 1 13 0 2 1

/3NCN

2 10 23 13 27 16 0 1

33 21 1 33 4 3 0 1

zeros

In-plane vibrations v, /3CCC j!?NCC 20 20 7 0 12 30 1 1

1572 1449 1332 1083 1000 803 328

zeros Bl,

vCN

-vCC, VCN: ring stretching; vi: inter-ring stretching; BCCC, BNCC, BNCN: ring bending; /?CH, pi: CH, NCC inter-ring bending (see Table 2); wCH: CH wagging; wi: inter-ring CC wagging; r,, r , r,: ring torsions (see Table 2); r,: inter-ring torsion (see Table 2).

out directly when a set of internal+ external coordinates is used for each of the two pyrimidine groups interacting, as in a molecular crystal, via inter-ring force constants. The three lowest planar modes ar assigned in Table 3 to ring-ring translation or rotations

Kinetic energy distribution of 2.2’-bipyrimidine

a +--+

43-e

1327

1328

N. NETO et al.

while, from the PED, they would be described as ring deformation or bending about the central bond, the latter coordinate giving the larger contribution to the potential energy when one ring rotates against the other. A similar difference arises for one of the two out-of-plane modes at about 90cm-‘, which can both be approximately viewed as out-of-plane rotations of each ring. The A, mode at 91 cm-’ is an out-of-phase rotation about the long molecular axis, with zero displacement for all atoms aligned with the CC central bond, hence this mode actually coincides with a torsion about the central bond and it is accordingly assigned in Table 3. The motion associated with the B,, vibration at 96 cm-’ is very similar to an out of phase ring rotation about an axis bisecting opposite N-CH bonds and, when projected on the wagging of the central carbon atoms, produces a very small contribution. When divided by 1, see Eqn (2), this contribution becomes a large one and causes a PED assignment of the B,, mode at 96 cm-’ as a CC inter-ring wagging. Such a description does not reflect the actual atomic motion, indicated as an out-of-plane ring-ring rotation in Table 3. The difference between PED and KED representations is also verified for the inplane, ring deformation modes in the 1500-16OOcm-’ region which are invariably described as ring stretchings according to the PED while large contributions from ring bendings are present in the KED. The much larger values of stretching force constants, as compared to bending, reduce the role of the ring bending contributions in the PED description. Such contributions are instead evident, for instance, for the 1572 cm-’ mode shown in Fig. 4. Thus an assignment of these normal modes as ring deformations is reported in Table 3, implying a substantial mixing of ring stretchings with bendings not evident from PED. Identical conclusions are obviously drawn from both distributions for normal modes nearly coincident with displacements of a specific group of internal coordinates. This is certainly the case for CH stretching modes, not listed in the PED or KED table, but also for ring stretching vibrations calculated at 1343 and 1370 cm-‘. The two distributions also suggest a unique assignment as ring breathing for the normal mode calculated at 1006 cm- ‘. This mode involves only CC and CN ring stretchings and is remarkably free of contributions from other internal coordinates. A unique identification of the CH bendings is also equally possible from either distributions even if, in this case, a considerable mixing with other internal coordinates is observed. Finally, it should be mentioned that in Table 4 only the percent distribution among diagonal, in-plane force constants is reported, with data in each row summing up approximately to 100% since the contribution of interactions is negligible. The same is not true for the out-of-plane valence field, as in this case the potential energy evenly distributes among all force constants and implies a considerable torsion-wagging mixing. A similar mixing arises from corresponding KED data, without the need of interpreting a distribution among off diagonal terms. Acknowledgement-Financial support from the Italian Consiglio Nazionale delle Ricerche and Minister0 della Ricerca Scientifica e Tecnologica is acknowledged.

REFERENCES [l] J. Overend and J. R. Scherer, .I. Chem. Phys. 24, 1119 (1960). [2] X. M. Hua and E. W. Prohofsky, Biopolymers 27, 645 (1988). [3] N. Neto, G. Sbrana and M. Muniz-Miranda, Specrrochim. Acfa 46A, 705 (1990). [4] J. Fernholt, C. Romming and S. Samdal, Acfa Chem. &and. A353, 706 (1981). (51 N. Neto, G. Sbrana and M. Muniz-Miranda, Specrrochim. Acra SOA, 357 (1994). [6] E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations. McGraw-Hill, London (1955).