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Revised mean amplitudes of vibration for naphthalene
Volume 2, number 5
REVISED
CHEMICAL
MEAN
AMPLITUDES
PHYSICS LETTERS
OF
VIBRATION
September 1968
FOR
NAPHTHALENE
S. 3. CYVIN, B. N. CYVIN and G. HAGEN In&%&e of Physical Chemistry, Technical
University
of Nm-zuay, Trondheim,
Received
11July
Norway
1968
The method of transferable compliants uer.stlS transferable force constants in vibrational analyses of large molecules is discussed. Revised calcuIatious of mean amplitudes for naphthalene indicate that the abnormally large value previously reported for one of the bonded CC distances is not correct, but is a consequence of the previously applied method.
In the previous spectroscopic calcuIations of mean amplitudes of vibration for naphthalene Hagen and Cyvin [l] produced an initial force field from valence compliance mainly transferred from benzene. The spectroscopic analysis of inplane vibrations for benzene, naphthalene and anthracene by Neto et al. [2] on the other hand employs transferable valence force constants. In the presence of redundants among the valence coordinates the corresponding compliance matrix, 17, is known to i_? s.in~qlzr [3], and can therefore not be established tentatively in numerical form. The corresponding force constant matrix, f, on the other hand can be set up numerically irrespectively of the presence of redundank. But in those cases certain combinations of the f matrix elements are indeterminate [4], and different f matrices may be physically equivalent. In the present communication we wish to point out some advantages and disadvantages of the two approaches [l, 21. Next we wish to report our revised calculations for mean amplitudes of CC distances in naphthalene; some of the previous results [I, 51 seem to be in error as a consequence of the method used without proper precaution. Force constants for given bond stretchings and other coordinates have a meaning only in an approximate sense, since all force constants of a force field should be regarded as a whole [6 J. A force constant fqr a given coordinate pair may change its value as a consequence of the choice of the other coordinates of the set, i.e. on account of a definition with no physical implications. This feature is referred to as the lack of invariance [3,7,8 3. Since compliants do have the property of invariance in this sense thejr may seem
much better suited for transferring from one molecule to the other [7]. In practice these considerations are often not so important as they may seem because the whole sets of coordinates for two related molecules usually are set up in a similar way almost automatically before force constants are transferred. Complications occur when redundants among the vibrational coordinates are involved. They are inevitable among valence coordinates for ring structures of high symmetries Like benzene, which is a good ewmple to illustrate one of &he points of the first paragraph. Hagen in an unpub; lished thesis (1967) reported fd = 4.331 mdyne/A for the CC stretching force constant in benzene; the compliants from the same work were those used in the previous naphthalene calculations [I, 51. This value is significantly different from 6.433 mdyne/A repxted by Neto et al. [ZI. Erevertheless hvo force constant sets with these two values need not necessarily be physically different, since the meaning of & is obscured due to redundancy. ti order to clarify this matter we computed the valence cympliant ‘Ed from a set with & = 6.433 mdyne/A according to Neto et aL The result was 0.162 urndyne in quite striking agreement with Hagen’s 0.173 i/mdyne. In connection with the analysis cf naphthalene we wish to give
a warning
against
uncritical
use
of the method of transferring compLiar,ts, aIthough this method may seem very plausibIe from the above considerations. It should be cLear t&at the employmer& of a coordinate set without redundants is iif essential requirement for this method; such a-set for naphthalene type molecules is given elsewhere [9]. In this particular set one type of outer CC stretchings VJas Left out~(in ad341
Volume
2, number
5
CHEMICAL
PHYS:cZS LETTERS
&ion to some ring bendings) 3.s unnecessary coordinates in order to avgid redundants. Exactly for the same type of CC distances the calculated mean ?mplitude [l, 51 seemingly came out with an abnorlliarly high value (0.061 b) when compared to the ocner mean amplitudes for bonded CC in naphthalene and benzene (about 0.048 A). AH values pertain to 298OK. Although r;he employed analysis [II] is mathematicaliy correct, as also are the numerical computations usixz this method, we became suspicious that the results might not be physically real, but depend on the special circumstances in the choice of coordinates. We had namely lost the control of the magnitude of the compliant for the CC stretchings in question. In the present w.ork we base4 the calculations on transferable valence force constants rather than compliants. In the initial set of coordinates we included all ring stretchings and planar bendings, thus introducing redunciants. We performed two sets of cakulations based on (a) force constznts from the afore-mentioned thesis of Hagen and (b) those from Neto et at 12). Actually we were not able to make full use of the tables from the latter work [Z] because of unsufficient definition of CH bending coordinate*; the values for the corresponding force constants were consequently taken f ram Bagen in both sets of calculations. The problem of the mean amplitude for the outer CC distances was resolved, a5 it became clear that a revision of the previous result [l, 51 really was needed. In tab1 e 1 the calculated mean amplitudes in the columns under (a) and (b) pertain to the calculations explained above. in both cases we have listed the results from (i) the initial force field using calculated frequencies throughout the computztions and (ii) a refined force field adjusted to observed frequencies. As observed .frequencies we have taken the sets quoted by Hagen and Cyvin [l] and by Neto et al. [2] in the two cases (a) and (b), respectively. The differences between frequencies of these two sets are not substantial, however, and are not supposed to have significant effect on the mean amplitudes. On comparing the figures in table 1 it is seen that all values for the bonded CC distances according td the revised calculations are similar, as was expected from physical reasons. We conclude that the value of 0.048 A as in benzene is as good as any for all the bonded CC mean amplitudes in naphthalene, in spite of a slight tendency to larger values, amounting to 0.052 A as a maxiI&L lt is also seen that most of the non-bonded CC_@ean ain$itudes have not changed substan-
,
:
,
September
1968
Tabk 1 Mean amplitudes of vibration (A units) at 298% for CC distances in naphthaleneaccording to different calculations (see the text). The numbering of atoms is consistent with the usage
in ref. [l]
Revised calculations @I (0
Distance
Ref. Cl*51
c2-c3
0.061
0.050
0.051
0.048
0.047
Cl-c2
0.046
0.052
0.051
0.046
0.045
Cl-C9
0.047
0.049
0.049
0.048
0.047
Cs-Cl0 Cl_. . c3 c2._.c9 Cl.. . cl() Cl...C6
0.046
0.045
0.047
0.047
0.046
0.062 0.052 0.054 0.060
0.059 0.058 0.05Q
0.055 -0.055 0.051 0.061
0.058
0.054
0.058 0.058 0.075
0.054
0.071
c2.. . CIQ
0.057
0.055
0.051
Cl...C4
0.056
0.060
0.058
Cl.. .CI Cl.._C5
0.060 0.058 0.057 0.063 0.061
c2...c, C1...C6 C2.._C6
0.065 0.065
0.075
0.065
0.060
0.054
0.074 0.061
0.073
0.064
0.072
0.064
0.056
0.071
0.065
0.058
0.071
W
0.054 0.061 0.060 0.060 0.062 0.057 0.062 0.064 0.065
tially during the revision. This is still more striking for the other non-bonded distances along with bonded CH; the mean amplitudes for all these distances hardly need revision, and they are not included in table 1 for the sake of brevity.
REFERENCES [l] G. Hagen and S. J. Cyvin, J. Phys. Chem. 72 (1968) 1446. [Z] N. Neto, M. Scrocco and S. Califano, Spectrochim. Acta 22 (1966) 1981.
[3] S. J. Q&n, Molecular Vibrations and Mean Square Amnlitudes Wniversitetsforlawt. Oslo. and Elsevie;,
Amst&ciaxn,
1968).
v
s
141 _ _ B.Crawford Jr. and J. Overend. J.Mol.Suectrv. _ 12
(1964) 307. 151 B.N. Cyvin. S. J. Cyvin and G. Hagen. Chem. Phys. Letters 1 (1967) 211. 161 E. B. Wilson Jr., J. C. Decius and P. C. Cross, Molecular Vibrations mcGraw-Hill. New York, 1955). [71 S. J. Cyvin and N. B. Slater, Nature 188 (1960) 485. [Sl 3. C. Decius, J. Chem.Phys. 39 (1963) 1130. [!?I y&Ilagen and S. J_Cyvin, J. Phys. Chem. 72 (1968)
.
REVISED
CHEMICAL
MEAN
AMPLITUDES
PHYSICS LETTERS
OF
VIBRATION
September 1968
FOR
NAPHTHALENE
S. 3. CYVIN, B. N. CYVIN and G. HAGEN In&%&e of Physical Chemistry, Technical
University
of Nm-zuay, Trondheim,
Received
11July
Norway
1968
The method of transferable compliants uer.stlS transferable force constants in vibrational analyses of large molecules is discussed. Revised calcuIatious of mean amplitudes for naphthalene indicate that the abnormally large value previously reported for one of the bonded CC distances is not correct, but is a consequence of the previously applied method.
In the previous spectroscopic calcuIations of mean amplitudes of vibration for naphthalene Hagen and Cyvin [l] produced an initial force field from valence compliance mainly transferred from benzene. The spectroscopic analysis of inplane vibrations for benzene, naphthalene and anthracene by Neto et al. [2] on the other hand employs transferable valence force constants. In the presence of redundants among the valence coordinates the corresponding compliance matrix, 17, is known to i_? s.in~qlzr [3], and can therefore not be established tentatively in numerical form. The corresponding force constant matrix, f, on the other hand can be set up numerically irrespectively of the presence of redundank. But in those cases certain combinations of the f matrix elements are indeterminate [4], and different f matrices may be physically equivalent. In the present communication we wish to point out some advantages and disadvantages of the two approaches [l, 21. Next we wish to report our revised calculations for mean amplitudes of CC distances in naphthalene; some of the previous results [I, 51 seem to be in error as a consequence of the method used without proper precaution. Force constants for given bond stretchings and other coordinates have a meaning only in an approximate sense, since all force constants of a force field should be regarded as a whole [6 J. A force constant fqr a given coordinate pair may change its value as a consequence of the choice of the other coordinates of the set, i.e. on account of a definition with no physical implications. This feature is referred to as the lack of invariance [3,7,8 3. Since compliants do have the property of invariance in this sense thejr may seem
much better suited for transferring from one molecule to the other [7]. In practice these considerations are often not so important as they may seem because the whole sets of coordinates for two related molecules usually are set up in a similar way almost automatically before force constants are transferred. Complications occur when redundants among the vibrational coordinates are involved. They are inevitable among valence coordinates for ring structures of high symmetries Like benzene, which is a good ewmple to illustrate one of &he points of the first paragraph. Hagen in an unpub; lished thesis (1967) reported fd = 4.331 mdyne/A for the CC stretching force constant in benzene; the compliants from the same work were those used in the previous naphthalene calculations [I, 51. This value is significantly different from 6.433 mdyne/A repxted by Neto et al. [ZI. Erevertheless hvo force constant sets with these two values need not necessarily be physically different, since the meaning of & is obscured due to redundancy. ti order to clarify this matter we computed the valence cympliant ‘Ed from a set with & = 6.433 mdyne/A according to Neto et aL The result was 0.162 urndyne in quite striking agreement with Hagen’s 0.173 i/mdyne. In connection with the analysis cf naphthalene we wish to give
a warning
against
uncritical
use
of the method of transferring compLiar,ts, aIthough this method may seem very plausibIe from the above considerations. It should be cLear t&at the employmer& of a coordinate set without redundants is iif essential requirement for this method; such a-set for naphthalene type molecules is given elsewhere [9]. In this particular set one type of outer CC stretchings VJas Left out~(in ad341
Volume
2, number
5
CHEMICAL
PHYS:cZS LETTERS
&ion to some ring bendings) 3.s unnecessary coordinates in order to avgid redundants. Exactly for the same type of CC distances the calculated mean ?mplitude [l, 51 seemingly came out with an abnorlliarly high value (0.061 b) when compared to the ocner mean amplitudes for bonded CC in naphthalene and benzene (about 0.048 A). AH values pertain to 298OK. Although r;he employed analysis [II] is mathematicaliy correct, as also are the numerical computations usixz this method, we became suspicious that the results might not be physically real, but depend on the special circumstances in the choice of coordinates. We had namely lost the control of the magnitude of the compliant for the CC stretchings in question. In the present w.ork we base4 the calculations on transferable valence force constants rather than compliants. In the initial set of coordinates we included all ring stretchings and planar bendings, thus introducing redunciants. We performed two sets of cakulations based on (a) force constznts from the afore-mentioned thesis of Hagen and (b) those from Neto et at 12). Actually we were not able to make full use of the tables from the latter work [Z] because of unsufficient definition of CH bending coordinate*; the values for the corresponding force constants were consequently taken f ram Bagen in both sets of calculations. The problem of the mean amplitude for the outer CC distances was resolved, a5 it became clear that a revision of the previous result [l, 51 really was needed. In tab1 e 1 the calculated mean amplitudes in the columns under (a) and (b) pertain to the calculations explained above. in both cases we have listed the results from (i) the initial force field using calculated frequencies throughout the computztions and (ii) a refined force field adjusted to observed frequencies. As observed .frequencies we have taken the sets quoted by Hagen and Cyvin [l] and by Neto et al. [2] in the two cases (a) and (b), respectively. The differences between frequencies of these two sets are not substantial, however, and are not supposed to have significant effect on the mean amplitudes. On comparing the figures in table 1 it is seen that all values for the bonded CC distances according td the revised calculations are similar, as was expected from physical reasons. We conclude that the value of 0.048 A as in benzene is as good as any for all the bonded CC mean amplitudes in naphthalene, in spite of a slight tendency to larger values, amounting to 0.052 A as a maxiI&L lt is also seen that most of the non-bonded CC_@ean ain$itudes have not changed substan-
,
:
,
September
1968
Tabk 1 Mean amplitudes of vibration (A units) at 298% for CC distances in naphthaleneaccording to different calculations (see the text). The numbering of atoms is consistent with the usage
in ref. [l]
Revised calculations @I (0
Distance
Ref. Cl*51
c2-c3
0.061
0.050
0.051
0.048
0.047
Cl-c2
0.046
0.052
0.051
0.046
0.045
Cl-C9
0.047
0.049
0.049
0.048
0.047
Cs-Cl0 Cl_. . c3 c2._.c9 Cl.. . cl() Cl...C6
0.046
0.045
0.047
0.047
0.046
0.062 0.052 0.054 0.060
0.059 0.058 0.05Q
0.055 -0.055 0.051 0.061
0.058
0.054
0.058 0.058 0.075
0.054
0.071
c2.. . CIQ
0.057
0.055
0.051
Cl...C4
0.056
0.060
0.058
Cl.. .CI Cl.._C5
0.060 0.058 0.057 0.063 0.061
c2...c, C1...C6 C2.._C6
0.065 0.065
0.075
0.065
0.060
0.054
0.074 0.061
0.073
0.064
0.072
0.064
0.056
0.071
0.065
0.058
0.071
W
0.054 0.061 0.060 0.060 0.062 0.057 0.062 0.064 0.065
tially during the revision. This is still more striking for the other non-bonded distances along with bonded CH; the mean amplitudes for all these distances hardly need revision, and they are not included in table 1 for the sake of brevity.
REFERENCES [l] G. Hagen and S. J. Cyvin, J. Phys. Chem. 72 (1968) 1446. [Z] N. Neto, M. Scrocco and S. Califano, Spectrochim. Acta 22 (1966) 1981.
[3] S. J. Q&n, Molecular Vibrations and Mean Square Amnlitudes Wniversitetsforlawt. Oslo. and Elsevie;,
Amst&ciaxn,
1968).
v
s
141 _ _ B.Crawford Jr. and J. Overend. J.Mol.Suectrv. _ 12
(1964) 307. 151 B.N. Cyvin. S. J. Cyvin and G. Hagen. Chem. Phys. Letters 1 (1967) 211. 161 E. B. Wilson Jr., J. C. Decius and P. C. Cross, Molecular Vibrations mcGraw-Hill. New York, 1955). [71 S. J. Cyvin and N. B. Slater, Nature 188 (1960) 485. [Sl 3. C. Decius, J. Chem.Phys. 39 (1963) 1130. [!?I y&Ilagen and S. J_Cyvin, J. Phys. Chem. 72 (1968)
.