A normal coordinate analysis of the planar vibrations of s-trifluorobenzene

SpectrochiraicaActa, Vol. 51A, No. 6, pp. 961-968, 1995

Pergamon 0584-$53904)00230-4

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A normal coordinate analysis of the planar vibrations of s-trifluorobenzene K. TAVLADORAKIStand J. E. PARKIN Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, U.K. (Received 27 September 1994; in final form 1 November 1994; accepted 3 November 1994)

Abstract--A 19 parameter harmonic in-plane force field for the s-trifluorobenzene-h3molecule is constructed. Internal force constants and symmetry force constants are derived and normal coordinates and Cartesian displacements are calculated for the first time. The observed data consist of fundamental frequencies, firstorder Coriolis constants and centrifugal distortion constants for both s-trifluorobenzene-h3and -d3. INTRODUCTION THERE are several reviews of calculations which seek to establish the force fields controlling the normal vibrations of polyatomic molecules in the literature [1-4]. In these reviews the types of observed data that are needed to determine the force constants as well as the mathematical and computational methods used to simplify calculations are included. The greatest difficulty which is usually encountered in this type of calculation is that the n u m b e r of force constants is larger than the number of observed data available for their determination. Consequently, the force constants are not well determined by the data. Obviously, the problem becomes worse with increasing size and decreasing symmetry of a particular molecule. To overcome this problem a n u m b e r of approximate force fields, containing fewer force constants than the General Force Field (G.F.F.) have been proposed (e.g. the Valence Force Field (V.F.F.) and the U r e y - B r a d l e y Force Field (U.B.F.F.)). THE CALCULATION AND REFINEMENT OF FORCE CONSTANTS The vibrational secular equation is solved using the so-called G F method [5]. There are several textbooks (e.g. Ref. [6]) which describe the detailed method involved in the force constant calculations. The process of the refinement of the force constants can be divided into the following steps [7]: (1) A trial set of force constants is guessed. These are, then, used to calculate the vibrational frequencies the Coriolis constants and the centrifugal distortion constants. Then, the Jacobian matrix elements of the frequencies, Coriolis constants and distortion constants, defined as the first term in t h e Taylor expansion of the calculated data with respect to each force constant, are calculated. (2) The error vector e is formed. Thus, the Jacobian matrix J is used to set up linear equations relating the first-order changes in the force constants ( 6 F ) to the consequent small changes (i.e. the error vector e) in all the calculated data. (3) A residual vector is formed. Then, the solution 6 F w h i c h minimizes the sum of the weighted squares of residuals is determined by forming the normal equations. These are, then, related to the estimated probable error of the observed values. (4) The values of 6 F which are determined by the mentioned procedure are used to form a new set of force constants and the whole cycle is repeated a n u m b e r of times until th calculation has converged. There are three main mathematical difficulties which arise in force field calculations [7]. These are the following: ? Author to whom correspondence should be addressed: Tsakalof 8, Ilioupolis, 16346 Athens, Greece. 961

962

K. TAVLADORAKIStand J. E. PARKIN

(a) Non-linearity. If the initial F matrix is not a good guess, the errors e and the corrections 6F will contain some relatively large elements and the relationship between the Jacobian matrix and the error vector will not hold. This will also happen if some elements of J are very sensitive to changes in the force constants which is usually observed as oscillations in the calculation. This can be avoided by adding only (1/2)6F for the first few cycles of the calculation increasing it until it reaches 1 in the final cycle. (b) Singularity. If the matrix [J' WJ] is almost singular, large rounding errors may occur in taking its inverse causing corresponding errors in the calculated corrections 6F. This, generally, means that the original data do not suffice to fix the force field. An easy way to detect this is by observing the diagonal elements of the inverse of the [J'WJ] matrix. If at least one diagonal element of the matrix becomes unusually large, it will lead to a corresponding large uncertainty in at least one of the force constants. If this is indeed the case then either further data must be added or the force field must be constrained in some way. (c) Multiple solutions. There may be several distinct solutions to the force field which fit all the available data satisfactorily. The likelihood of this occurrence generally decreases with increasing variety in the type of the experimental data used. If two or more solutions are found, it may be necessary to choose between the possible solutions on physical and chemical grounds.

THE PLANAR NORMAL COORDINATES OF s-TRIFLUOROBENZENE

The planar normal vibrations of s-trifluorobenzene are distributed among the irreducible representation as follows: F(Qv, planar) = 4a'1 + 3a~ + 7e'. This problem is more complicated than the corresponding one of benzene [8-10] due to the fact that the blocks into which the planar vibrational secular determinant factorizes are larger than the correspnding ones for benzene. There are 11 (4a'~ + 7 e ' ) assigned fundamental frequencies and three uncertain ones (3a~) for each isotopic species indicating that the G.F.F. is seriously underdetermined due to the fact that there are 44 independent symmetry force constants controlling the planar vibrations in it. So, it is not surprising that there are only two published planar force field calculations. The first, published in 1962, was a modified U.B.F.F. [11], the other, published in 1973, was a general study of fluoro derivatives of benzene [12] which included the s-trifluorobenzene, using the overlay technique. In the mentioned calculations, the authors did not evaluate normal coordinates or centrifugal distortion constants from their force fields. However, in more recent papers [13, 14] calculated Coriolis constants have been quoted from the second force field. In this study, a 19-parameter force field was constructed and also the symmetry coordinates, the symmetry force constants, the normal coordinates and the Cartesian displacements were derived for the planar vibrations of s-trifluorobenzene which have not been reported before, utilizing data from a high resolution mid-IR study of the molecule [15].

THE SECULAR EQUATION AND THE CONSTRUCTION OF SYMMETRY FORCE CONSTANTS

The equilibrium geometry of s-trifluorobenzene is [12] R0(C-C) = 1.397 A,

R0(C-H) = 1.084 A,

R0(C-F) = 1.327 A.

It is also assumed that s-trifluorobenzene is a regular hexagonal ring. The definitions of the internal displacement coordinates are given in Table 1. In constructing the planar symmetry coordinates from the given set of internal coordinates redundancies are introduced into the a'l and e' blocks. A detailed description of the procedure for

Normal

coordinate

analysis of the planar vibrations of s-trifluorobenzene

Table I. Definition

of internal

displacement

963

coordi-

nates for s-trifluorohenzene r~

i n c r e a s e in l e n g t h o f C,--H~ b o n d

l~

increase in length of C,-E bond

Ri

i n c r e a s e in l e n g t h o f C,-C~+I b o n d H

ai

I

i n c r e a s e in C ~ _ : C , - C i + I a n g l e F

I Yi

i n c r e a s e in C i - t - C + l

angle

fli C , - H ~ b e n d , i n c r e a s e in a n g l e b e t w e e n

a n d e x t e r n a l b i s e c t o r o f C~_ : C : - C ~ .

C,-H,

~;

p o s i t i v e w h e n h, m o v e s t o w a r d s C,_ ~ ~o~ C,-F~ b e n d . i n c r e a s e in a n g l e b e t w e e n

C,--F,

a n d e x t e r n a l b i s e c t o r o f C~_ j - C ~ C . , ; p o s i t i v e w h e n F, m o v e s t o w a r d s C~ t

removing the redundancies (both for the non-degenerate and the degenerate species) is given in Refs [5, 15]. The symmetry coordinates which were constructed specifically for this study but can also be used for related molecules (e.g. s-trichlorobenzene, etc.) are listed in Table 2. The symmetry force constants were, then, constructed using the method developed by WILSON et al. [5]. T a b l e 2. S y m m e t r y F

Sr

A~

1

2

E'

3

for s-trifluorobenzene

4

5

6

N

Sr

St

1

1

1

--

--

--

3 1/2

rr

$2

1

1

1

--

--

--

3 1/2

li

S3

1

1

1

S4

1

1

1

-1 A '2

coordinates

-1

$5

1

S6 $7

1 1

S~

- 1

$9~

2

-1

-1

1 --

--

--

1

-1

1 1

1 1

---

2

- 1 -1

-1

1

--

1/2

Ri Royr Roa r

-1

6 -1:2

Rr

---

---

3 - I/2 3 I/2

/(go r

--

--

--

6 - I/2

ri

--

--

--

6

1,

-1

2

-1

-1

Sil Si2a

-1

0 1

1 - 1

---

---

-1

--

-1

--

2

1/2

rofli

-1

12 -I/2

---

2 /2 2-1/2

Rr

--

--

12 1/2

Royr

--

-- 1

rofl, lt¢o i

Si3 a

2

$14a

-1 - 1

2 0

2 -3:2

Roar Rr

2

- 1

- 1

--

--

--

2 4 - i/2

RoTr

1 1

-2 0

1 -- 1

---

---

---

2 - I/2

Roar r,

S9b

0

-- 1

1

--

--

2 1/2

1,

S.~b

-- 1

0

1

-- 1

2- l

Rr

SHb

--1

2

--1

--

--

--

6 1/2

roflr

Sj2b SL~b

2 0

-- 1 -- 1

-- 1 1

---

---

---

6-1/2 2- t

loto i

-

1

--

--

--

Ssb

1

Si4b

-1

6

6 - J/2

-1

St0a

0

1 --

0

1

1

0

-0

1

1 0

2 - 1

1 1

--1 --

--2 --

--1 --

- 1

0

1

--

--

--

Royr R(~

24 -la 2 3r2

i

Ri Royi

R~ti

N o t e : R . = e q m C - C b o n d l e n g t h , r , = e q m C - H b o n d l e n g t h , l0 = e q m C-F bond length.

K. TAVLADORAKIStand J. E. PARKIN

964

Table 3. Internal force constant notation for s-trifluorobenzene

R,

Ri

ri

li

Roai

Ro)'i

rofli

lq~oi

D d,, d,,

ho hm hp

hl hF,, hVp

i,, i,. ip

iF,, iF "F lp

],, j,. jp

j~ .i F ]p"F

E e,,

eVo epv E~

k

iv, '

kFo'

kVo kpr kF

l,,

k,,

IF lpv lF

e~

g~

k~

t~

I~

F

fFo

nm

nVo

Fv

nVo'

nv

G

gF

gm

g~

dp ri l, R,,ai Ro)~, rofli ll¢oi

fro'

Gr

Note: capital letters indicate diagonall force constants. Also, o = ortho, rn =- meta, p = para.

The internal force constants are given in Table 3 while the symmetry force constants derived from them are listed in Table 4. These, again, were constructed specifically for the present study but can also be used for related molecules.

OBSERVED DATA

The observed data are summarized in Tables 5 and 6. The observed fundamentals the Coriolis coupling constants ~8-~12 and the centrifugal distortion constants Dj, D~K and D r for the s-trifluorobenzene-h3 were taken from the high resolution mid-IR study of the molecule [15] while the rest of the observed data for s-trifluorobenzene-h3 and all of the observed data for s-trifluorobenzene-d3 were taken from Ref. [14]. It should also be noted that none of the data were corrected for any anharmonic effects. It was expected that because of the great variety of data used that there would be an improvement in the possibility of determining an acceptable force field. However, the main problem still remains, namely that the number of observables is not sufficient to determine a unique solution for the G.F.F. The situation is summarized as follows: for symmetry species A'I, A~ and E ' , the number of independent parameters in G.F.F. (FH to F1414) is 44 while the number of observables (that is 22 fundamental frequencies, 11 Coriolis constants and 3 distortion constants) is 36. Vs-Vj2,

RESULTS AND DISCUSSION

A model force field of 19 parameters was constructed for the case of the planar vibrations of s-trifluorobenzene to see how well the 5 e' fundamental modes, the 4 Coriolis coupling constants and 3 centrifugal distortion constants measured in Ref. [15] were reproduced. The calculated values for the observed data, the internal force constants and the symmetry force constants are given in Tables 5-8. The normal coordinates and the pictorial representation of the Cartesian displacements for each normal mode for s-trifluorobenzene-h3 and -d3 are available on request. The program ASYM20 [16] was used in all our calculations.

N o r m a l c o o r d i n a t e analysis o f the planar vibrations o f s - t r i f l u o r o b e n z e n e

965

T a b l e 4. S y m m e t r y force constants for s-trifluorbenzene

AI:

F . = E + 2e,,, F21 = 2eoF + epF F22 = E F + 2e,V.

E~t = 2U2(ho + h,,, + hp) E~2 = 21'2(hoF + hVm+ hpv)

F.~3= D + 2do + 2d,,, + dp

F,,= 2 "~(-k + 2k~- 2km + k~) F 4 2 = 2 - 1 / 2 (k F - 2 k o F ' + 2 k F - k • • " 'F 'F 10 - - I m - - lp .st. to + l p

F')

/743 -_-_

A~:

V44= Z - ' ( F + 2f., + FV--4fVo + 2 f F - - 2fF ) Fss = D - Zdo + Z d . , - dp F.s = 2 " 2 ( - J o + J., -Jp ) F~6 = G + 2g., __ 1/2 "F "F "F F75-2 (lo-lm+lp)

F~ = 2go~+ g.~ E':

F77 = G F + 2 g ~ F ~ = E - e,. /798 = - e o +Fe p F F99 = E r - e,V,, F.~=2 I n ( - h o + 2h,,,-hp) E l ( ~ = 2 - " 2 ( - h o F + 2 h F _ h F) Fi0,0 = = D - do - d., + d e

F , s = _ 31/2l,,, FI J9 = 31nlVo' F n .) = (3/2)l/2(_jo + lp)

Fml=G-g,,, Fl2 8 = 31:21V °

Fl29= Y/2F.. F,2,0 = ( 3 / 2 ) ~ ' 2 ( - j o F +jpv ) F~2H -- - - g ov+ g p F FI212 = G F __ gV F,~ = 2 "~(k - I,o~ - k.. + G% F I 3 9_- 2 ll2 (k r - kor - k.,F + k ~ ' ) F B I 0 = 2 l(--io+2i,.--ip--'Fto+2t,.--tp)'F F,3,, = (3/2)'/2(noF ' - n,,,) FB,2 = ( 3 / 2 ) " 2 ( n o r -- n F)

F13,3 = 2 - ' ( V - f ~ + F F r~4s = 2 '[3~/2(ho - hp) Fi49 = 2 i 13 1 / 2 ( - h oF+ h p )Fk Ft40j = 2 -3/2(i o - 2i,,, + ip FI411 2-1[31/2(noV' + n,.) :

_

.F

2 f v _ f v + 2fV) k - kVo+ km+ k F] F + k oF-' k , . , -Fk p ] F ' lo+2lm__lp "F "F - Jo - 2j,. - j.]

The starting values for the internal force constants were taken from Ref. [12]. It is obvious, from Tables 5 and 6 that the observed frequencies, Coriolis coupling constants and centrifugal distortion constants are reproduced reasonably well within the limits of the assumed experimental uncertainties, in almost all of the cases. It should be stressed that the "Kekule assumption" [16, 17] was used (i.e. d = d o = - d m = d p ) . A more detailed analysis of the individual force constants used in the construction of the model force field can be found in Ref. [15]. The internal force constants that are not included in Table 7 were constrained to zero. The internal constants that are included in Table 7 were chosen because of the relatively large values associated with them in Ref. [12]. Initially, some of them were constrained to certain plausible values until a satisfactory fit was achieved. In the next stage, all the internal force constants associated with s-trifluorobenzene in 39 parameter force field [12] were included. This led to the construction of a 27 parameter force field where the number of parameters was obviously too large compared with the number of observed data. Therefore, some of the

K. TAVLADORAKISt and J. E. PARKIN

966

Table 5. Observed and calculated data for planar vibrations of s-trifluorobenzene-h3

Ai

A~

E'

vl v2 v3 v4 v5 v6 v7 ~8 ~ vl0 vtl vl2 vl3 v14 ~ ~ ~ll~ Ell ~12 ~13 ~14 Dj DjK DK

Observed

Uncertainty

Calculated

Error

3076.0 1362.6 1012.4 589.9 ---3113.0 1629.0 1475.4 1127.6 996.3 502.4 324.2

31 14 10 6 ---31 16 15 11 10 5 3

3093.5 1363.6 1018.0 577.1 1304.6 1173.8 533.7 3086.2 1622.2 1472.9 1139.6 1003.0 497.6 323.3

- 17.5 - 1.0 -5.6 2.8 ---26.7 6.8 2.5 -12.0 -6.7 4.8 1.0

- 0 . 0 0 (2)

0.00 (2)

- 0 . 0 0 (2)

0.00

- -

-0.4 -0.05 -0.35 -0.30 -0.20 0.1573 -0.2949 0.1425

- -

0.20 0.20 0.05 0.02 0.15 0.02 0.10 0.10

0 . 1 9

-0.19 -0.02 -0.38 -0.30 -0.34 0.0932 -0.1540 0.0689

- -

-0.21 0.03 0.03 -0.00 0.14 0.0641 -0.1409 0.0736

Note: wave numbers are in cm ~, Coriolis coupling constants are dimensionless and centrifugal distortion constants are in kHz.

Table 6. Observed and calculated data for planar vibrations of s-trifluorobenzene-d3

A '~

~1 v2 v3

A~

vs v6 v7 ~ ~ vl0 vii vJ2 vl3 vl4

"~4

E'

~8 ~9 ~10 ~, ~12 ~t3 ~14 Dj

Observed

Uncertainty

Calculated

Error

2319.2 1359.7 969.2 577.3 ---2314.0 1617.0 1425.0 1054.0 792.0 487.0 322.4

23 14 10 6 ---23 16 14 11 8 5 3

2327.4 1363.5 957.4 577.1 1261.3 947.7 509.8 2312.4 1621.5 1431.2 1044.3 785.1 487.7 319.8

-8.2 -3.7 11.8 0.2 ---1.6 -4.5 -6.2 9.7 6.9 -0.7 2.6

-0.05 --0.40 0.02 -0.40 -0.25 -0.25 --

0.04 -0.4 0.02 0.1 0.1 0.05 --

0.00 (4) 0.13 -0.24 0.02 -0.32 -0.30 -0.29 0.0829 -0.1367 0.0611

0.04 (5) --0.16 0.00 -0.08 0.05 0.04 ----

D:x

--

--

Dx

--

--

Note: wave numbers are in cm- ~, Coriolis coupling constants are dimensionless and centrifugal distortion constants are in kHz.

Normal coordinate analysis of the planar vibrations of s-trifluorobenzene

967

Table 7. Model force field for planar vibrations of s-trifluorobenzene Force constant

Typical coefficient

D E EF

R~ r,.z l~

F F~ G GF d hV,, io j,, jF jF kr nFo k F' k gF g~

(Roai) 2 (Roy;) 2 (rofli) 2

(lotoi)2 (R~R/) Rill RoRiai r,,R,fl; l,,R;w i loRitoi+l Rol;y; RoloaiW; Roliai Roria, r,,loflitoi rolofl,wi+l

Best value

Dispersion

7.460 5.081 6.238 0.206 1.249 0.758 1.749 0.468 0.346 0.528 0.536 1.038 -0.457 0.856 0.191 0.765 0.319 -0.015 -0.088

0.455 0.115 0.315 0.120 0.101 0.023 0.208 0.093 0.201 0.172 0.061 0.207 0.109 0.135 0.098 0.093 0.166 0.023 0.039

Note: all constants are in units of mdyn/~-

parameters were constrained to zero beginning with ho, h F nd hpe which were chosen because it had been predicted in the 39 parameter field that their values would be almost zero [12]. This was confirmed by the present analysis. The removal of the three mentioned parameters led to the construction of a 24 parameter force field which was found to be ill-conditioned, when all of the parameters Table 8. Symmetry force constants for the planar vibrations of s-trifluorobenzene Symmetry force constants

F AI

Fil F21

F22 F31

F~2

A~

E'

F33 F4j F42 F43 F44 F55 F65 F66 F75 F76 F77

F~ F~s F~ Fu~ Fn~ Fi010

Best value 5.081 0.000 6.238 0.000 0.488 7.928 -0.227 -0.471 0.000 0.727 5.119 -0.755 0.758 2.108 -0.117 1.749 5.081 0.000 6.238 0.000 -0.246 7.928

F E'

Symmetry force constants Fits F.9

Fill0 Fire Fl2s F129 Fm{i FI211 FI2L2 Fl3s Fi39 F1310

El311 F1311

F1313 F148 Fl4y

Fi41o Fj411

FI412 Ft4t3 FI414

Note: all force constants are in units of mdyn ,~-~.

Best value 0.000 0.000 -0.653 0.758 0.000 0.000 - 1.266 -0.073 1.749 0.226 0.065 -0.528 0.000 0.233 0.727 -0.159 0.510 0.000 --0.268 -0.105 0.365 2.837

968

K. TAVLADORAKIStand J. E. PARKIN

were allowed to vary. After constraining several parameters to zero and allowing the rest to vary, the present 19 parameter force field was constructed. It should be noted that further force constants adjustment calculations were tried applying different constraints to individual parameters in an attempt to find best values for the internal force constants, but a more satisfactory solution could not be found. However, due to the essential complexity and underdeterminacy of this problem, this solution is regarded as a possible solution rather than the best one. The British Library Document Supply Centre Supplementary Publication No. Sup 13096 contains 6 pages of tables and figures. Retrieval information is given in the Notes for Contributors of each issue of Spectrochimica Acta Part A.

REFERENCES [1] T. Shimanouchi and I. Nagakawa, Ann. Rev. Phys. Chem. 23, 217 (1972). [2] P. Gans, Advances in IR and Raman Spectroscopy (Edited by R. J. H. Clark and R. E. Hester), Vol. 3 (1977). [3] T. Shimanouchi, Physical Chemistry: An Advanced Treatise, Vol. 4, p. 233 (1970). [4] J. Overend, Ann. Rev. Phys. Chem. 21,265 (1970). [5] E. Wilson, J. C. Decious and P. Cross, Molecular Vibrations. McGraw-Hill, New York (1955). [6] S. R Califano, Vibrational States. Wiley, New York (1976). [7] J. Aldous and I. M. Mills, Spectrochim. Acta 18, 1073 (1962). [8] A. G. Ozkabak, L. Goodman, S. N. Thakur and K. K. Jespersen, J. Chem. Phys. 83, 6047 (1985). [9] S. N. Thakur, L. Goodman and A. G. Ozkabak, J. Chem. Phys. 84, 6642 (1986). [10] A. G. Ozkabac and L. Goodman, J. Chem. Phys. 87, 2564 (1987). [11] J. R. Scherer, C. Evans and M. M. Muelder, Spectrochim. Acta 18, 1579 (1962). [12] V. J. Eaton and D. Steele, J. Molec. Spectrosc. 48, 446 (1973). [13] H. F. Shurvell, T. E. Cameron, D. B. Baker and S. J. Daunt, Spectrochim. Acta 35A, 757 (1979). [14] J. Korppi-Tommola, H. F. Shurvell, S. J. Daunt and D. Steele, J. Molec. Spectrosc. 87, 382 (1981). [15] K. Tavladorakis, Ph.D. Thesis, University College London (1990). [16] J. C. Duinker and I. M. Mills, Spectrochim. Acta 24A, 417 (1968). [17] J. R. Scherer and J. Overend, Spectrochim. Acta 17,719 (1961).