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The evaluation of Urey-Bradley force constants in planar XY3 type molecules
,,OL-RXAL
OF
MOLE(‘lTLAR
The
SI’E(‘TROsc%I’Y
Evaluation
of Chemistry,
92-100
(1!)6())
of Urey-Bradley
in Planar
Department
1,
XY3 Type
Rensselaer
Force Constants
Molecules
Polytechnic
Institute,
Troy, New 1’ork
The force const,:tnts for the planar SY, species, BCI, , SO3, BOb, CO:-, and XOsm have been calculated on the basis of the Urey-Bradley type potent,ial force field. The existence of a relation between t,he repulsion force constant and the separation distance for nonbonded at,oms is shoxn for the case of chlorine compounds and oxygen compounds using the results for the above together with a survey of published data calculated by this force field for other species. The species BQ and X0deviate noticeably from the empirical correlation thus found. The structure of the nitrate ion is considered in relation to the position of NO,- in t,his correlation. I. INTRODUCTION
In the normal coordinate t)reatment of polyatomic molecules, it has been shown that the Urey-Bradley field type of poter1t)ia.l is a good approximation for the force field in molecules or ions. Shimanouchi (1) applied the force field for various methane derivatives and other molecular systems and found that the force constant,s in this type force field were t’ransferahle from molecule to molecule. It was noted (1) also that the forces between nonhonded at)oms are repulsive and their magnitude was reasonable compared with the interatomic forces of argon and krypton. Heath and Linnett (2) have also introduced t,he Urey-Bradley type potential in the orbital valency force field and showed (3) that the repulsion hetween nonbonded atoms accounted for the major part’ of the deviation from the simple valence force field. The calculation of normal vibration of the planar symmetric XY, type molecule 1 5). Venkatesmarlu et al. (,G) calcuhas been made by several investigators (2, +, lated the force constants of nine molecules or ions, i.e., BI?, , BCl, , BBrB , AlCl:j , and Wilson (5) investigated the SO, , NOa-, CO;-, BOi-, and PO:-. Lindeman vibrational spectraof the mixed halidesof boron and calculated the forceconstants. Both calculat,ions above were made on the basis of the general valence force potential. The latter authors emphasized that the off-diagonal elements of F-matrices of the three boron halides could be transferred to t,he mixed boron halides. The orbital valence force field invest,igations by Heath and Limiett ( 2) included the calculation of force constants of the planar SY, molecules. Application
UREY-BRADLEY
FORCE
CONSTANTS
IN MOLECULES
93
of this approach to more complicated molecules t’han those treat,ed (5’) is more difficult.. The present, communication reports calculations of force constants using the Urey-Bradley type potential using revised data for the frequencies of BF, , BCl, , BBr3 , SO, , NOa-, CO:-, and BO:-. These systems are known to have the planar configurations, belonging to t)he symmetry group D,, . The repulsion force constants so obtained were considered relative to interatomic distances for nonbonded atoms and also to the electrical charge of the polyatomic system concerned. 2. CALCULATIONS
Calculations of the normal vibrations of the planar symmetric XU, type molecule were made by the method of Wilson (6) on t’he basis of the Urey-Bradley type potent,ial. The G-matrix follows directly from the original methods of Wilson (6) and the expressions may be checked by the equations published by Shimanouchi (7). The internal coordinates and t’he geometry of the system are shown in Fig. 1. Such systems give rise to one A, vibration and two doubly degenerat.e E vibrations. The symmetry coordinates are constructed from internal coordinat,es by the equations
r(E,)
=
2 d6
Ar,
-
A
46
Ar,
-
1
Ara
,
46
(1)
By t,he transformation to the symmetry coordinat,es into t,wo matrices, G( A,) and G(E), as shown below: G(&)
:(P),
the G-matrix
is reduced
JAKZ
94
FIG.
1. Geometry
AND
and internal
MIIiAWA
coordinates
of the XY3 species
where px and pLyare the reciprocal mass of the atom X and Y, respecbively, and r denotes the equilibrium distance of X-Y. The Urey-Bradley field t’ype of potent,ial funct’ion is expressed, in general, as: V = 5
i=l
[KJriAri
+ f/dric(Ari)‘] + $
[HI,jiijAa,j + &
+ $iHij(r
[FijQ;jAq;j + :‘$F;j(
Ap;j)‘Ip
(3)
where r’s are bond lengths, (Y’Sthe bond angles, q’s the distances between atoms not bonded directly, and rij represents (rirj)“‘. K’, K, H, H’, F, and F’ are the force constants, the last two of which are the repulsion constants between nonbonded atoms. Through the relation qij
=
Ti2 +
rf -
2rJj
(4)
COS Cyjj,
AQij can be expressed by Ari , Arj , and AcY,~. Consequently, the potential energy is obtained as a function of valence-force coordinates. Similarly the F matrix can be reduced into two matrices F(A1) and F(E) :
F(A,): F(E):
(K + 3F) K + BiF’ + 3iF ( - 2/3/4r(F’
+ F)
-2/3/4r(F’ rZH -
!ir’F’
+ F)
(5)
+ >ir’F )-
The normal frequencies are calculated by the secular equation 1GF - XI 1 = 0, where G and F are expressed by (2) and (5)) respectively, the observed value o in cm-r by the relation x = 4~r%~/N.
(6) and X is related with (7)
UREY-BRADLEY From (2)) expressed by to determine An additional showed that,
FORCE CONSTANTS
IN MOLECULES
95
(5)) and (6) it is apparent that the three normal frequencies are the four force constants K, H, F, and F’. Therefore it is impossible the four force constants uniquely from the three observed values. relation between t’he force constant’s is required. Shimanouchi F’ can be approximated by t#he relation: F
,l,ioF.
=
(8)
Using t#his, the three force constants can be readily calculated. The final expression for the normal ferquencies t.hus is XI = ~ly(K + 3F), x3 + x4 = (K + 3iF’ + (H x3x4 = ((K
+ :SF’
(3/d
+ .3iF)(%x ?;F’
+ PY) -
+ ?P~(%clx
+ 3,iF)(H
-
%(F
+ F)PX
+ 3pcly),
YiF’ + f,iF)
(9) -
9fs(F’
+ F)‘}
+ %xPv),
where F’ = -f,foF. From t,he observed values w1 , w3 , and wq , K, H, and F have been calculated with an IBM 650 Electronic Computor. The values used for c (velocity of light) and N (Avogadro’s number) were 2.997902 X 10” cm set? and 6.02544 X 10z3 Since this value of N is based on the physical scale of g mole-‘, respectively. atomic weights, the following values for the atomic weights were used; B1l, 11.012811; F1’, 19.004444; Cl’, 12.003844; S3’, 31.982274; N14, 14.007550; C135, of t’he spect#roscopic 34.98006; 016, 16.000000; and Br”, 78.94366. A summary data used is given in Table I. Although PO!- and A1C13 also are listed as planar XY, type molecules in Landolt-Bbrnstein Tables (9), these two systems have not been included since their existence as planar XY, species still remains doubtful. 3. RESULTS The values of force constants and repulsion constants thus calculated are summarized in Table II, where K, H, and F are the st,retching and bending force constant, and repulsion constant, respectively. The corresponding values obtained by Heat,h and Linnett (9) are listed in the table for comparison. The relationship between the t)wo is discussed lat’er. 4. DISCUSSION Some comparison of the present results with the calculations of Venkat,eswarlu and Sunderam (4) and Lindeman and Wilson (5) is possible. Although the force fields used were different, t,he G-matrix should have t#he same expression in each
JANZ
96
AND
RIIK4WA
TABLE
I
NORMAL FREQUENCIESOF PLANAR slv,
(CL%) B”Fa B”Cl3 B”Br3 SOS NO,co;BO;-
888 471 279 1068 1050 1060 910
~IOLE~ULES AND IONR
w:,
WP
(cm-‘)
icm-‘)
Ref.
480.4 243.0 151 560 720 680 700
1453.5 954.2 819.2 1332 1390 1415 1445
6 6 5 8 9 9 9
TABLE
II
FORCECONSTANTSAND REPULSIONCONSTANTSOF PLANARXI’,
TYPE MOLECULESAND IONS
Cmdyne/Al Stretching force constant k? K B”F, B”CI, B”Bra SOS X03co:BO;-
6.04 3.01 2.39 9.08 5.62 5.43 4.64
Bending force constant H k/3/3 @)
5.98 3.10 2.39 9.2 5.4 5.4 -
0.212 0.058 0.060 0.521 0.541 0.341 0.744
0.215 0.102 0.063 0.489 0.370 0.352 -
Repulsion constant F 2A@) 0.927 0.519 0.408 0.555 1.59 1.72 1.05
0.95 0.430 0.426 0.50 1.85 1.75 -
case. Inspectjion of the results shows that these expressions differ in each case. The discrepancy with t)he work of Lindeman and Wilson (5) may be att’ributed to misprints in the publication since t,heir values, by recalculation, are those gained from the use of t,he G-matrix in its correct form. The errata in this publicat,ion are listed be1ow.l The disagreement with t,he expression of Venkat’eswarlu and Sunderam (4) could not be resolved similarly owing to insufficient’ detail in t,he published work. In general t,he values for force const’ants depend on the force field assumed for the vibrational analysis and comparison of t’he values is not meaningful unless the same treatment has been used for the systems. Inspection of t,he Urey-Bradley potent,ial and the orbital valence force field shows that for t,he in-plane vibrations of the XI’, system, the force const’ants K, H, F, and F’ in the former are equiva1Errata in Ref. 5: (a) p. 246, (b) p. 246, (c) p. 246, (d) p. 244,
RS = (l/fi)(Aa
-
Aw)
Gss = W_X/~~)+ (MY/W + [(2/~2) + (Wd)
+ (~/~@)IPB
633 = GSS = (1/@)[3~x + (9/q/2)1 Table II: All of Blo and B” in the first line should be exchanged.
UREY-BRADLEY
FORCE
CONSTANTS
IN MOLECULES
97
lent, t’o k, , 443, 2A, and -B/R, , respectively, in the lat’ter. The values for the force constants gained in the present work and those of Heath and Linnett (2) accordingly are summarized in Table II for comparison. The agreement leaves lit’tle t’o be desired. The small difference may possibly be attribut,ed to the fact that F’ was estimated as equal to -jisF in t’he notation of the present, calculations, rat,her than -$foF as assumed in (8). The former values follow from the LennardJones equation :
Jr+”
(101
a”
”
for the repulsion energy, whereas t,he lat’ter has been used wit#hgood results by various investigators for a variety of molecular systems (10). The close agreement of the results (Table II) indicates that this difference is of a minor mat#ter in vibrational analyses. The force constant)s for a variety of polyatomic systems have been calculated by t,he Urey-Bradley pot’ential field (10-17). Since t’his force field is basically simpler and less laborious than the orbital valence force field, it seemed of interest to investigate the results for a possible correlation between the repulsion force TABLE THE Cl-Cl
III
REPULSION CONSTANT AND THE DISTANCE OF THE Two Molecule
Distance
CHLORINE ATOMS
Repulsion Constant
Ref.
CCl,-ccl, ccl, ccI~-ccl~ CHClz-CHCl, CHClt-COCl BCl,
2.898 2.87 2.87 2.87 2.87 3.00
0.60 0.65 0.65 0.64 0.65 0.52
11 1 1 is 12 This work
BClz-BC12 SiCl, SiCl:,-Sic13
3.00 3.30 3.30
0.34 0.30 0.30
1Q 1 15
TABLE THE O-O
IV
REPULSION CONSTANT AND THE DISTANCE OF THE Two
Molecule
Distance
SOI
2.48A
H,N+-SOaBO:NOaco;HCO?-
2.43 2.36 2.11 2.27 2.24
OXYGEN ATOMS
Repulsion Constant
Ref.
0.555 0.76 1.05 1.59 1.72 3.00
This woik 16 This work This work This work 17
JANZ
98
AND MKAWA
constants and the separat)ion of nonbonded atoms much in the manner of Linnett, and Heath (3) based on the orbit#al valency force field results. The collected results for the Cl.. .Cl and 0.. .O repulsion force const,ants and separation dist,ances are listed in Tables III and IV, respectively, and are illustrated graphically in Fig. 2. Inspection of the graph shows that. wit,h the exception of B&I, and NOa-, the repulsion force constant between either the chlorine at,oms or the oxygen at,oms can be correlated directly with the separation distance of t,he nonbonded atoms. Relative to the nitrate ion, it is known t,hat the vibrational frequencies vary with the stat’e of aggregation, and some experimental error in the N-O bond distance is conceivable. However, even with consideration of this uncertainty, the position of NOa- relat#ive to t’he oxygen correlat,ion in Fig. 2 remains excepGonal. The reason for t’his is not obvious, but it may be noted that a distinctive feature of the nitrate ion is the existence in close proximity of positive and negative centers of charge. The possible canonical st’ructures of the molecules and ions are shown in Fig. 3. The lowest energy configuration, indicated by stars (*) in Fig. 3, has no formal positive charge on the central atom in each case except for the nitrate ion. For the latter, the configuration of lowest energy has a positive charge in close proximit,y of the negative charge in t,he ion. Thus the most
0' 2.0
1
I
2.5
3.0
Distance FIG. 2. Empirical nonbonded atoms.
correlation
of repulsion
force
(A) constant
and separation
distance
of
UREY-BRADLEY
FORCE CONSTANTS
IN MOLECULES
99
FIG. 3. Canonical structures of SOa , SOa-, BO:-, NOs-9 CO’,-, and HCOz-
stable state can be reasonably accounted for in terms of resonance between three equivalent structures of type (a) with the small additional contribution of type 0 ?-
&+ -0
d
ko-
-0
(4
(b) . It follows tive charge in ment,‘* 1.39D, of the positive tures
zo_ @)
that there is a pronounced localization of the positive and the negathe nitrate ion. Support for this is seen in the marked dipole moof NzOs . The latter is understood by the pronounced localization and negative charge densities in the resonance hydrid of the st,rucO\+/O\+/O)J
0 a
\j/o\#o 1 o-
1 o-
much as in the nitrat’e ion. The great,ly shortened length of the N-O bond, and t’he deviation of the nitrate ion from the correlation of the repulsion force constant-nonbonded atom separation, illustrated in Fig. 2, may possibly be attributed in large part to this st’ructural feature which distinguishes the nitrate ion from the other species examined. No explanation of the low value for B&14 in t’he chlorine atom correlation is apparent at present.
JAN% ANI)
100
MIKAWA
This work was made possible in large part by grttn-ill-aid support from the National Science Foundation, Witshington, I1.C. C;rctteful acknowledgment is made also to Rensselaer Polytechnic Instit,ute, Computer Laboratory, for use of the IBM 650 Computer in these calculations. RECEIVEI)
: January
28,
l!XiO REFERENCES
1. 2. S. 4. 5. 6. 7. 8.
9. 10. if. 12. 1s. 14. 16.
T. SHIMANOUCHI,J. Chenl. Phys. 17, 245, 734, 848 (1949). D. F. HEATH AND J. W. LINNETT, Trans. Faraday Sm. 44, 873, 884 (1948). J. W. LINNETT AND D. F. HEATH, Trans. Faraday Sot. 48,592-601 (1952). K. VENKATESWARLC ANI) S. STNDERAM, J. Chem. Phys. 23, 2368 (1955). I’. LINDEMAN AND M. KENT WILSON, J. Chem. Phys. 24, 242 (1956). E. B. WILSON, J. Chem. Phys. 7, 1047 (1939); 9, 76 (1941). T. SHIMANOUCHI,J. Chem. Phys. 26, 660 (1956). W. H. STOCKMAYER, C;. M. KAVANACTH,AND H. C. MICKLEY, J. Chem. Phys. 12, 408 (1944). LANDOLT-BARNSTEIN, “Zahlenwerre und Funktionen,” 6th ed., Vol. I, Par + 2. SpringerVerlag, Berlin, 1951. 8. MIZUSHIMA, “Structure of Molecules and Internal Rotation.” Academic Press, New York, 1954. D. E. MANX, L. FANO, J. H. MEAL, AND T. SHIMANOUCHI,J. (‘hem. Phys. 27, 51 (1957). A. MIYAKE, I. NAKAGA~A, T. M~Azaw~a, I. I~HISHIMA, T. SHIMANOU~HI AXD S. M~zuSHIMA, Spectrochim. Acta 13, 161 (1958). K. NAITO, I. NAKAGA~A, Ii. KURATANI, I. ICHISHIMA, .%ND s. MIZUSHIMA, J. Chem. Phys. 23, 1907 (1955). D. E. MANX AND L. FANG, J. Chew. Phys. 26, 1665 (1957). M. KATAYAMA, T. SHIMANO~CHI, Y. MORINO, AND 8. MIZUSHIMA, J. Chew{. Phys. 18,
506 (1950).
16. M. J. SCHMELZ, T. MIYAZA~A, S. MIZUSHIMA, T. J. LANE, AND J. V. Q~AGLIAXO, Spectrochirn. Acta 9, 51 (1957). 17. T. MIYAZAWA, J. Chem. Sot. Japan 77, 381 (1956). 18. Y. K. SYRKIN AND M. E. DYATIUNA, “Structure of Molecules Butterwort.hs, London, 1950.
and the Chemical
Bond.”
OF
MOLE(‘lTLAR
The
SI’E(‘TROsc%I’Y
Evaluation
of Chemistry,
92-100
(1!)6())
of Urey-Bradley
in Planar
Department
1,
XY3 Type
Rensselaer
Force Constants
Molecules
Polytechnic
Institute,
Troy, New 1’ork
The force const,:tnts for the planar SY, species, BCI, , SO3, BOb, CO:-, and XOsm have been calculated on the basis of the Urey-Bradley type potent,ial force field. The existence of a relation between t,he repulsion force constant and the separation distance for nonbonded at,oms is shoxn for the case of chlorine compounds and oxygen compounds using the results for the above together with a survey of published data calculated by this force field for other species. The species BQ and X0deviate noticeably from the empirical correlation thus found. The structure of the nitrate ion is considered in relation to the position of NO,- in t,his correlation. I. INTRODUCTION
In the normal coordinate t)reatment of polyatomic molecules, it has been shown that the Urey-Bradley field type of poter1t)ia.l is a good approximation for the force field in molecules or ions. Shimanouchi (1) applied the force field for various methane derivatives and other molecular systems and found that the force constant,s in this type force field were t’ransferahle from molecule to molecule. It was noted (1) also that the forces between nonhonded at)oms are repulsive and their magnitude was reasonable compared with the interatomic forces of argon and krypton. Heath and Linnett (2) have also introduced t,he Urey-Bradley type potential in the orbital valency force field and showed (3) that the repulsion hetween nonbonded atoms accounted for the major part’ of the deviation from the simple valence force field. The calculation of normal vibration of the planar symmetric XY, type molecule 1 5). Venkatesmarlu et al. (,G) calcuhas been made by several investigators (2, +, lated the force constants of nine molecules or ions, i.e., BI?, , BCl, , BBrB , AlCl:j , and Wilson (5) investigated the SO, , NOa-, CO;-, BOi-, and PO:-. Lindeman vibrational spectraof the mixed halidesof boron and calculated the forceconstants. Both calculat,ions above were made on the basis of the general valence force potential. The latter authors emphasized that the off-diagonal elements of F-matrices of the three boron halides could be transferred to t,he mixed boron halides. The orbital valence force field invest,igations by Heath and Limiett ( 2) included the calculation of force constants of the planar SY, molecules. Application
UREY-BRADLEY
FORCE
CONSTANTS
IN MOLECULES
93
of this approach to more complicated molecules t’han those treat,ed (5’) is more difficult.. The present, communication reports calculations of force constants using the Urey-Bradley type potential using revised data for the frequencies of BF, , BCl, , BBr3 , SO, , NOa-, CO:-, and BO:-. These systems are known to have the planar configurations, belonging to t)he symmetry group D,, . The repulsion force constants so obtained were considered relative to interatomic distances for nonbonded atoms and also to the electrical charge of the polyatomic system concerned. 2. CALCULATIONS
Calculations of the normal vibrations of the planar symmetric XU, type molecule were made by the method of Wilson (6) on t’he basis of the Urey-Bradley type potent,ial. The G-matrix follows directly from the original methods of Wilson (6) and the expressions may be checked by the equations published by Shimanouchi (7). The internal coordinates and t’he geometry of the system are shown in Fig. 1. Such systems give rise to one A, vibration and two doubly degenerat.e E vibrations. The symmetry coordinates are constructed from internal coordinat,es by the equations
r(E,)
=
2 d6
Ar,
-
A
46
Ar,
-
1
Ara
,
46
(1)
By t,he transformation to the symmetry coordinat,es into t,wo matrices, G( A,) and G(E), as shown below: G(&)
:(P),
the G-matrix
is reduced
JAKZ
94
FIG.
1. Geometry
AND
and internal
MIIiAWA
coordinates
of the XY3 species
where px and pLyare the reciprocal mass of the atom X and Y, respecbively, and r denotes the equilibrium distance of X-Y. The Urey-Bradley field t’ype of potent,ial funct’ion is expressed, in general, as: V = 5
i=l
[KJriAri
+ f/dric(Ari)‘] + $
[HI,jiijAa,j + &
+ $iHij(r
[FijQ;jAq;j + :‘$F;j(
Ap;j)‘Ip
(3)
where r’s are bond lengths, (Y’Sthe bond angles, q’s the distances between atoms not bonded directly, and rij represents (rirj)“‘. K’, K, H, H’, F, and F’ are the force constants, the last two of which are the repulsion constants between nonbonded atoms. Through the relation qij
=
Ti2 +
rf -
2rJj
(4)
COS Cyjj,
AQij can be expressed by Ari , Arj , and AcY,~. Consequently, the potential energy is obtained as a function of valence-force coordinates. Similarly the F matrix can be reduced into two matrices F(A1) and F(E) :
F(A,): F(E):
(K + 3F) K + BiF’ + 3iF ( - 2/3/4r(F’
+ F)
-2/3/4r(F’ rZH -
!ir’F’
+ F)
(5)
+ >ir’F )-
The normal frequencies are calculated by the secular equation 1GF - XI 1 = 0, where G and F are expressed by (2) and (5)) respectively, the observed value o in cm-r by the relation x = 4~r%~/N.
(6) and X is related with (7)
UREY-BRADLEY From (2)) expressed by to determine An additional showed that,
FORCE CONSTANTS
IN MOLECULES
95
(5)) and (6) it is apparent that the three normal frequencies are the four force constants K, H, F, and F’. Therefore it is impossible the four force constants uniquely from the three observed values. relation between t’he force constant’s is required. Shimanouchi F’ can be approximated by t#he relation: F
,l,ioF.
=
(8)
Using t#his, the three force constants can be readily calculated. The final expression for the normal ferquencies t.hus is XI = ~ly(K + 3F), x3 + x4 = (K + 3iF’ + (H x3x4 = ((K
+ :SF’
(3/d
+ .3iF)(%x ?;F’
+ PY) -
+ ?P~(%clx
+ 3,iF)(H
-
%(F
+ F)PX
+ 3pcly),
YiF’ + f,iF)
(9) -
9fs(F’
+ F)‘}
+ %xPv),
where F’ = -f,foF. From t,he observed values w1 , w3 , and wq , K, H, and F have been calculated with an IBM 650 Electronic Computor. The values used for c (velocity of light) and N (Avogadro’s number) were 2.997902 X 10” cm set? and 6.02544 X 10z3 Since this value of N is based on the physical scale of g mole-‘, respectively. atomic weights, the following values for the atomic weights were used; B1l, 11.012811; F1’, 19.004444; Cl’, 12.003844; S3’, 31.982274; N14, 14.007550; C135, of t’he spect#roscopic 34.98006; 016, 16.000000; and Br”, 78.94366. A summary data used is given in Table I. Although PO!- and A1C13 also are listed as planar XY, type molecules in Landolt-Bbrnstein Tables (9), these two systems have not been included since their existence as planar XY, species still remains doubtful. 3. RESULTS The values of force constants and repulsion constants thus calculated are summarized in Table II, where K, H, and F are the st,retching and bending force constant, and repulsion constant, respectively. The corresponding values obtained by Heat,h and Linnett (9) are listed in the table for comparison. The relationship between the t)wo is discussed lat’er. 4. DISCUSSION Some comparison of the present results with the calculations of Venkat,eswarlu and Sunderam (4) and Lindeman and Wilson (5) is possible. Although the force fields used were different, t,he G-matrix should have t#he same expression in each
JANZ
96
AND
RIIK4WA
TABLE
I
NORMAL FREQUENCIESOF PLANAR slv,
(CL%) B”Fa B”Cl3 B”Br3 SOS NO,co;BO;-
888 471 279 1068 1050 1060 910
~IOLE~ULES AND IONR
w:,
WP
(cm-‘)
icm-‘)
Ref.
480.4 243.0 151 560 720 680 700
1453.5 954.2 819.2 1332 1390 1415 1445
6 6 5 8 9 9 9
TABLE
II
FORCECONSTANTSAND REPULSIONCONSTANTSOF PLANARXI’,
TYPE MOLECULESAND IONS
Cmdyne/Al Stretching force constant k? K B”F, B”CI, B”Bra SOS X03co:BO;-
6.04 3.01 2.39 9.08 5.62 5.43 4.64
Bending force constant H k/3/3 @)
5.98 3.10 2.39 9.2 5.4 5.4 -
0.212 0.058 0.060 0.521 0.541 0.341 0.744
0.215 0.102 0.063 0.489 0.370 0.352 -
Repulsion constant F 2A@) 0.927 0.519 0.408 0.555 1.59 1.72 1.05
0.95 0.430 0.426 0.50 1.85 1.75 -
case. Inspectjion of the results shows that these expressions differ in each case. The discrepancy with t)he work of Lindeman and Wilson (5) may be att’ributed to misprints in the publication since t,heir values, by recalculation, are those gained from the use of t,he G-matrix in its correct form. The errata in this publicat,ion are listed be1ow.l The disagreement with t,he expression of Venkat’eswarlu and Sunderam (4) could not be resolved similarly owing to insufficient’ detail in t,he published work. In general t,he values for force const’ants depend on the force field assumed for the vibrational analysis and comparison of t’he values is not meaningful unless the same treatment has been used for the systems. Inspection of t,he Urey-Bradley potent,ial and the orbital valence force field shows that for t,he in-plane vibrations of the XI’, system, the force const’ants K, H, F, and F’ in the former are equiva1Errata in Ref. 5: (a) p. 246, (b) p. 246, (c) p. 246, (d) p. 244,
RS = (l/fi)(Aa
-
Aw)
Gss = W_X/~~)+ (MY/W + [(2/~2) + (Wd)
+ (~/~@)IPB
633 = GSS = (1/@)[3~x + (9/q/2)1 Table II: All of Blo and B” in the first line should be exchanged.
UREY-BRADLEY
FORCE
CONSTANTS
IN MOLECULES
97
lent, t’o k, , 443, 2A, and -B/R, , respectively, in the lat’ter. The values for the force constants gained in the present work and those of Heath and Linnett (2) accordingly are summarized in Table II for comparison. The agreement leaves lit’tle t’o be desired. The small difference may possibly be attribut,ed to the fact that F’ was estimated as equal to -jisF in t’he notation of the present, calculations, rat,her than -$foF as assumed in (8). The former values follow from the LennardJones equation :
Jr+”
(101
a”
”
for the repulsion energy, whereas t,he lat’ter has been used wit#hgood results by various investigators for a variety of molecular systems (10). The close agreement of the results (Table II) indicates that this difference is of a minor mat#ter in vibrational analyses. The force constant)s for a variety of polyatomic systems have been calculated by t,he Urey-Bradley pot’ential field (10-17). Since t’his force field is basically simpler and less laborious than the orbital valence force field, it seemed of interest to investigate the results for a possible correlation between the repulsion force TABLE THE Cl-Cl
III
REPULSION CONSTANT AND THE DISTANCE OF THE Two Molecule
Distance
CHLORINE ATOMS
Repulsion Constant
Ref.
CCl,-ccl, ccl, ccI~-ccl~ CHClz-CHCl, CHClt-COCl BCl,
2.898 2.87 2.87 2.87 2.87 3.00
0.60 0.65 0.65 0.64 0.65 0.52
11 1 1 is 12 This work
BClz-BC12 SiCl, SiCl:,-Sic13
3.00 3.30 3.30
0.34 0.30 0.30
1Q 1 15
TABLE THE O-O
IV
REPULSION CONSTANT AND THE DISTANCE OF THE Two
Molecule
Distance
SOI
2.48A
H,N+-SOaBO:NOaco;HCO?-
2.43 2.36 2.11 2.27 2.24
OXYGEN ATOMS
Repulsion Constant
Ref.
0.555 0.76 1.05 1.59 1.72 3.00
This woik 16 This work This work This work 17
JANZ
98
AND MKAWA
constants and the separat)ion of nonbonded atoms much in the manner of Linnett, and Heath (3) based on the orbit#al valency force field results. The collected results for the Cl.. .Cl and 0.. .O repulsion force const,ants and separation dist,ances are listed in Tables III and IV, respectively, and are illustrated graphically in Fig. 2. Inspection of the graph shows that. wit,h the exception of B&I, and NOa-, the repulsion force constant between either the chlorine at,oms or the oxygen at,oms can be correlated directly with the separation distance of t,he nonbonded atoms. Relative to the nitrate ion, it is known t,hat the vibrational frequencies vary with the stat’e of aggregation, and some experimental error in the N-O bond distance is conceivable. However, even with consideration of this uncertainty, the position of NOa- relat#ive to t’he oxygen correlat,ion in Fig. 2 remains excepGonal. The reason for t’his is not obvious, but it may be noted that a distinctive feature of the nitrate ion is the existence in close proximity of positive and negative centers of charge. The possible canonical st’ructures of the molecules and ions are shown in Fig. 3. The lowest energy configuration, indicated by stars (*) in Fig. 3, has no formal positive charge on the central atom in each case except for the nitrate ion. For the latter, the configuration of lowest energy has a positive charge in close proximit,y of the negative charge in t,he ion. Thus the most
0' 2.0
1
I
2.5
3.0
Distance FIG. 2. Empirical nonbonded atoms.
correlation
of repulsion
force
(A) constant
and separation
distance
of
UREY-BRADLEY
FORCE CONSTANTS
IN MOLECULES
99
FIG. 3. Canonical structures of SOa , SOa-, BO:-, NOs-9 CO’,-, and HCOz-
stable state can be reasonably accounted for in terms of resonance between three equivalent structures of type (a) with the small additional contribution of type 0 ?-
&+ -0
d
ko-
-0
(4
(b) . It follows tive charge in ment,‘* 1.39D, of the positive tures
zo_ @)
that there is a pronounced localization of the positive and the negathe nitrate ion. Support for this is seen in the marked dipole moof NzOs . The latter is understood by the pronounced localization and negative charge densities in the resonance hydrid of the st,rucO\+/O\+/O)J
0 a
\j/o\#o 1 o-
1 o-
much as in the nitrat’e ion. The great,ly shortened length of the N-O bond, and t’he deviation of the nitrate ion from the correlation of the repulsion force constant-nonbonded atom separation, illustrated in Fig. 2, may possibly be attributed in large part to this st’ructural feature which distinguishes the nitrate ion from the other species examined. No explanation of the low value for B&14 in t’he chlorine atom correlation is apparent at present.
JAN% ANI)
100
MIKAWA
This work was made possible in large part by grttn-ill-aid support from the National Science Foundation, Witshington, I1.C. C;rctteful acknowledgment is made also to Rensselaer Polytechnic Instit,ute, Computer Laboratory, for use of the IBM 650 Computer in these calculations. RECEIVEI)
: January
28,
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and the Chemical
Bond.”