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The calculations of orbital valency force field constants of methane, silane and germane
&#UrM~ of Molecular Structure, 37 (1977) 139-143 Q&Pier Scientific Publishing Company, Amsterdam -
Printed in The Netherlands
TIIE C~C~A~ONS OF ORBITAL VALENCY FORCE FIELD CONSTANTS OF ME~NE, SILANE AND GERM~E
HYUNYONG Department
KIM of Chemistry.
University of Missouri. Colum bta. Missouri 65201
(II S.A )
(Received 4 May 1976)
ABSTRACT The Orbital Valency Force Field treatment of Tyson, Claasen and Kim has been applied to the XY,, XY,Z, and XY,Z, isotopes of methane, silane, and germane. Four harmonic force constants of XH, species predict the harmonic frequencies of isotopic species within the range of their uncertainties. Four constants determined from the fundamentals of XH, species and small anh~monieity corrections in isotopic masses were used to c&&ate fundamentis of various isotopic species, and they are compared with the observed values. Good agreement obtamed between the calculated and observed values, and the facility which their treatment offers for calculations of isotopically substituted species, suggest that their method can be used to predict the as yet unknown frequencies of isotopic species, thereby assisting with spectral assignments. XNTRODUCTION
‘#Q+sonet al. [l] have recently expressed the Orbital Valency Force Field (OVFF) of XY,, type molecules in Cartesian coordinates and pointed out that the use of Cartesian coordinates makes the calculation of isotopically substituted species extremely simple. This paper applies their method to the cases of methane, silane, and germane in order to test its usefulness as a means of predicting the as yet unknown frequencies of isotopic species, thereby-assisting with spectral assignments. A set of four OVFF valence force constants, determined from the observed frequencies of XHs (X = C, Si or Ge) by the weighted least-squares procedure, were used to calculate the frequencies of various debterium and tritium substituted species. The OVFF model was first proposed by Heath and Linnett [2] _ This valence force lhodel differs from the Urey-Bradley model in the treatment of the potential energy associated with bond bending. Instead of expressing tbe‘angulair~_$isplatiements ti_ terms of the interbond angle Q, they are expressed in tirms of an angle between the actual positions of the ligand atoms and thb‘prefered orbital position when the orbital overlap would be ma&nui&The p&ential energy expression for an XY4 type molecule is
140
This is a familiar Urey-Bradley expression except for the third term. The angle /T? is the angle between the line from the centraI atom-to the ith Iigand and the axis of the ;ith.orbital. In the OVFF’ mode& it is assumed that a set of rigid orbit& forming the-T, symmetry”about the central atom follows the motion of the ligand atoms such that E# is minimum. The angle 6, as they are defined has no redundancy relation. This is not the case with the interbond angle Q. Therefore, the value of the constant D is physically more significant within the framework of the valence force field models. For XY,, type molecules, the OVFF model has been shown to be superior to other models using the interbond angle in terms of fitting the observed fundamentais [2,31CALCULATIONS
The kinetic and potential energies are expressed in terms of mass-weighted Ctisian coordinates, s, = mf”xi m=s’.S 28 = S’ M-‘/2 AM-“2s where the matrix A is the force constant matrix evatuated in Cartesian coordinates. The elements of A are carefully evaluated in ref. 1 for the common symmetry types of XY, molecules. The matrix A is symmetric and the matrix M-‘12 is diagonal, having the dimension of 3N X 3N. The product matrix M-l” AM-II2 is diagonalized by similarity transformation, and the eigenvalue Ai corresponds to 4n2C2vf. Eigenvalues corresponding to either rotational or translational modes are zero. The matrix A is a function of the geometry and force constants of the molecule and is independent of mass. Therefore, the same matrix is used for isotopic species regardless of the molecular symmetry. For each isotopic species, only the diagonal matrix M- ‘j2 needs to be changed. This makes it extremely simple to calculate the vibrational frequencies of isotopically substituted species when the vibrational analysis of the parent species is done. In order to properly test a valence force model, harmonic f&quencies must he usd. However, the molecules for whiqh reliable harmonic frequencies can be obtained are limited to XH.,, X&I,, and_.CT+ The hqrmonic frequencies of Ci-&, CD,, and CT, used in this work are thoseevaluated by Jones-and McDowell f43 from the observed combination and overtone frequencies. Values of wj and w4 for Sii and Sil& evahzated_by B&l and h?[cKean f5], and for Gels and C&L), eva+ated by Chahn~rs &d McIceaq [6j:were e. Values of w E and w2 for SiH+ SiD,, Gef-Lp,an@ GeD4-wee evaluated.fqllowing Dennison’s method 171, in which_*e hqrmonic_frequency is express@ as (Yk j, W~el%? Yk is the observed fundamental,-and the al’IharmoQ= wk = vk (I+ icity constant ek is evaluated using the7productrule. A set of OVFF con&a&, E L), F, and P! are c&ul&ed from the
141
harmonic frequencies of each XH, species by the weighted least-squares procedure. Using these constants, harmonic frequencies of XD, and CTJ species are predicted. Table 1 lists the harmonic frequencies evaluated from the observed values and the calculated frequencies. The corresponding valence force constants are listed in Table 2. Agreement is quite satisfactory, considering that only 4 valence constants are used. Most of the calculated values agree within 5 cm-* which is in the range of uncertainty involved in evaluating the harmonic frequencies. Jones and McDowell [4] calculated harmonic frequencies of CY3Z and CYzZz isotopes of methane (Y and Z are H, D, and T) using 5 CY, force constants. We have cakulated harmonic frequencies* of these mixed isotopes with the same 4 valence constants of CH+ Out of 63 normal frequencies, the largest deviation is 16 cm-’ (less than 1%) and the average deviation is 4 cm-*. There is a practical advantage in using the observed fundamentals as it is much more convenient to compare the predicted and observed values when TABLE
1
Obsexved and calculated harmonic frequencies (in cm-‘) QJ,@,)
w :(@I
WJfa)
w,V:)
CH*
Oh% Cal.
3143 3143
1573 1573
3154 3154
1357 1357
CD,
Obs. Cal.
2224 2223 (-1)
1113 1112 k-1 1
2333 2324 (-9)
1027 1029 (+2)
DW.
1817 1815 P-2)
910 908 P-2)
1990 1975 F-15)
880 884 (+4)
SIH,
ObS. Cd.
2244 2244
1028 1028
2312 2312
940 940
SiD,
ObS. Cal.
1587 1587 (0)
727 727 (0)
1673 1671 (-2)
689 690 (*II
Cd.
1296
594
1393
581
ObS. Cd.
2179 2179
964 964
2224 2224
836 836
Oh
Cal. DJPV.
1541 1641 (0)
682 682 (0)
1581 1586 (+5)
605 601 (-4 1
Cd.
1258
557
1306
499
DeV.
CT,
ObS. Cd.
IkV.
-T.
*Tables of obsexved and cahht.ed frequencies of CY ,Z and CY $2, isotopic methane,and XY, isotopic species and XY,Z ~otopic sibme and gexmane (where X = C, Si, Cie; Y - 2, = H, D, T) axe available as SUP. 26047 (8 pages) ftom E.L.L.D.
142 TABLE
2
Least-squares harmonic force constants (in mdyn A-‘)
CH* SiH, GH,
K
D
F
F’
4.7419 29143 2.7695
1.3778 0.6696 0.6034
0.2689 0.0128 0.0067
0.1896 0.0601 0.0628
they are used for spectral assignment. The least-squaresforce constants, K, D, I?, and F’ are calculated using the observed fund~en~s, and the fundamentals of the isotopically substituted species are then predicted. The observed values used are those compiled by Shimanouchi 183. We have found large deviations in the stretching fundamentals, while agreement in the bending fundamentals is quite satisfactory. We then calculated the leastsquares force constants of XI& species and compared them with those of the XI& species. It was found that the ratio of K&u was nearly constant; 1.024 for methane, 1.020 for siiane, and 1.020 for germane. The least squaresforce constants of XY, fundamentals are listed in Table 3. In our method, it is much more convenient to make a small anharmonicity correction in the masses,rather than in the force constants. The massesof deuterium and tritium were scaled by 1.020 and 1.040, respectively, and the calculations repeated. The calculated fundamentals of silane and germane are remarkable in their agreement with the observed values. The largest deviation is 7 cm-’ and the average deviation is less than 3 cm- ‘. The anharmonicity corrections on mass, correct respectively for the stretching frequencies of v , and u3of Xl&, uz and v4 of XYJZ, and ZJ~ and us of XY& isotopes. For methane, the same corrections on mass are not sufficient to account for the large anharmonicity found in the symmetrical stretching fundamentals. The calculated values of ~1 in CY4 and v2’s in mixed isotopes deviate from the observed values by almost 2% Otherwise, the agreement in methane is good, the largest deviation being 16 cm-’ and average deviation less than 5 cm”‘. It should TABLE
3
Least-squares force constants (in mdyn /1-’ )
CH, CR CT. SiH, SiD,
@Ha err,
K
D
F
F’
4.421 4.529 4.599
1.441 1.434 1.443 0.537 0.559 0.525 0.536
0.148 0.178 0.184 0.060 0.067 0.035 0.035
0.209 0.207 .0.209 0.037
2.576
2.629 2474 2.524
0.039 ao5v 0.050
143
also be noted that some of the observed frequencies are not accurate due to experimental difficulties or the accidental shifts of Fermi resonance [9]. CONCLUSION
The four valence force constants of the OVFF model calculated from the harmonic frequencies of XI& species predict the harmonic frequencies of XY+ XY3Z, and XYz& isotopes within the range of uncertainties in their evaluations. Similar constants which fit the fundamentals of Xi% molecules and small anharmonicity corrections in isotopic masses predict the observed fundamentals of ~otopic~ly substituted species remarkably well. Particularly, the near perfect agreement obtained for the bond-bending modes are gratifying because the OVFF model is designed to describe better the bond-bending motions. No valence force field model with 4 constants matches the agreement obtained in this work between the observed and calculated values C9,lOl. This suggests that the OVFF model deserves more attention than it has received, at least for the XY, type molecules. The OVFF model has not been used very widely, mainly due to its algebraic complexity. Now, the method presented in ref. 1 makes the OVFF applications very simple. Furthermore, the use of Cartesian coordinates makes the calculations of the isotopically substituted species very convenient. The good agreement shown between the observed and calculated ~nd~en~s and the facility in calculating the isotopic species as demonstrated in this work would make the above method quite useful in predicting the as yet unknown frequencies of isotopic species. The physical meaning of the constant anharmonicity correction factors used for deuterium and tritium is not yet clear. Shimanouchi et al. [9] used a similar constant factor in their force constant calcuiations. it needs to be further tested with molecules having many different central atoms. REFERENCES 1 J. Tyson, H. H. Ciaassen and H. Kim, J. Chem. Phys., 54 (1971) 3142. 2 D. C. Heath and J. W. Linnett, ‘Dana Faraday Sot., 44 (1948) 873,878,884; Trans. Faraday Sot., 45 (1949) 264. 3 H. Kim, P. S. Souder and H. H. CIeassen. J. Mol. Spectrosc., 26 (1968) 46. 4 L. H. Jones and R. S. McDowell. J. Mol. Spectrosc., 3 (1959) 632. 5 D. F. Rail and D. C. McKean, Spectrochrm. Acta, 18 (1962) 1019. 6 A_ A. Chalmers and D. C. McKean. Spectrochim Acta, 21(1965) 1941. 7 I). M. Dennison, Rev. Mod. Phys.. 12 (1940) 175. 8 T. Sh~~noucbi, Tabtea of Molecuiar Vibrational Frequencies, NSRDS-NBS 6 and NSRDS-NBS II, National Bureau of Standards, 1967. 9 T. Shimanouchi, I. Nakagawa, J. Hiraishi and M. Iahii, J. Mol. Spectrosc., 19 (1966) 78 10 J. L. Duncan and I. M. Mills, Spectrochim. Acta, 29 (1964) 523.
Printed in The Netherlands
TIIE C~C~A~ONS OF ORBITAL VALENCY FORCE FIELD CONSTANTS OF ME~NE, SILANE AND GERM~E
HYUNYONG Department
KIM of Chemistry.
University of Missouri. Colum bta. Missouri 65201
(II S.A )
(Received 4 May 1976)
ABSTRACT The Orbital Valency Force Field treatment of Tyson, Claasen and Kim has been applied to the XY,, XY,Z, and XY,Z, isotopes of methane, silane, and germane. Four harmonic force constants of XH, species predict the harmonic frequencies of isotopic species within the range of their uncertainties. Four constants determined from the fundamentals of XH, species and small anh~monieity corrections in isotopic masses were used to c&&ate fundamentis of various isotopic species, and they are compared with the observed values. Good agreement obtamed between the calculated and observed values, and the facility which their treatment offers for calculations of isotopically substituted species, suggest that their method can be used to predict the as yet unknown frequencies of isotopic species, thereby assisting with spectral assignments. XNTRODUCTION
‘#Q+sonet al. [l] have recently expressed the Orbital Valency Force Field (OVFF) of XY,, type molecules in Cartesian coordinates and pointed out that the use of Cartesian coordinates makes the calculation of isotopically substituted species extremely simple. This paper applies their method to the cases of methane, silane, and germane in order to test its usefulness as a means of predicting the as yet unknown frequencies of isotopic species, thereby-assisting with spectral assignments. A set of four OVFF valence force constants, determined from the observed frequencies of XHs (X = C, Si or Ge) by the weighted least-squares procedure, were used to calculate the frequencies of various debterium and tritium substituted species. The OVFF model was first proposed by Heath and Linnett [2] _ This valence force lhodel differs from the Urey-Bradley model in the treatment of the potential energy associated with bond bending. Instead of expressing tbe‘angulair~_$isplatiements ti_ terms of the interbond angle Q, they are expressed in tirms of an angle between the actual positions of the ligand atoms and thb‘prefered orbital position when the orbital overlap would be ma&nui&The p&ential energy expression for an XY4 type molecule is
140
This is a familiar Urey-Bradley expression except for the third term. The angle /T? is the angle between the line from the centraI atom-to the ith Iigand and the axis of the ;ith.orbital. In the OVFF’ mode& it is assumed that a set of rigid orbit& forming the-T, symmetry”about the central atom follows the motion of the ligand atoms such that E# is minimum. The angle 6, as they are defined has no redundancy relation. This is not the case with the interbond angle Q. Therefore, the value of the constant D is physically more significant within the framework of the valence force field models. For XY,, type molecules, the OVFF model has been shown to be superior to other models using the interbond angle in terms of fitting the observed fundamentais [2,31CALCULATIONS
The kinetic and potential energies are expressed in terms of mass-weighted Ctisian coordinates, s, = mf”xi m=s’.S 28 = S’ M-‘/2 AM-“2s where the matrix A is the force constant matrix evatuated in Cartesian coordinates. The elements of A are carefully evaluated in ref. 1 for the common symmetry types of XY, molecules. The matrix A is symmetric and the matrix M-‘12 is diagonal, having the dimension of 3N X 3N. The product matrix M-l” AM-II2 is diagonalized by similarity transformation, and the eigenvalue Ai corresponds to 4n2C2vf. Eigenvalues corresponding to either rotational or translational modes are zero. The matrix A is a function of the geometry and force constants of the molecule and is independent of mass. Therefore, the same matrix is used for isotopic species regardless of the molecular symmetry. For each isotopic species, only the diagonal matrix M- ‘j2 needs to be changed. This makes it extremely simple to calculate the vibrational frequencies of isotopically substituted species when the vibrational analysis of the parent species is done. In order to properly test a valence force model, harmonic f&quencies must he usd. However, the molecules for whiqh reliable harmonic frequencies can be obtained are limited to XH.,, X&I,, and_.CT+ The hqrmonic frequencies of Ci-&, CD,, and CT, used in this work are thoseevaluated by Jones-and McDowell f43 from the observed combination and overtone frequencies. Values of wj and w4 for Sii and Sil& evahzated_by B&l and h?[cKean f5], and for Gels and C&L), eva+ated by Chahn~rs &d McIceaq [6j:were e. Values of w E and w2 for SiH+ SiD,, Gef-Lp,an@ GeD4-wee evaluated.fqllowing Dennison’s method 171, in which_*e hqrmonic_frequency is express@ as (Yk j, W~el%? Yk is the observed fundamental,-and the al’IharmoQ= wk = vk (I+ icity constant ek is evaluated using the7productrule. A set of OVFF con&a&, E L), F, and P! are c&ul&ed from the
141
harmonic frequencies of each XH, species by the weighted least-squares procedure. Using these constants, harmonic frequencies of XD, and CTJ species are predicted. Table 1 lists the harmonic frequencies evaluated from the observed values and the calculated frequencies. The corresponding valence force constants are listed in Table 2. Agreement is quite satisfactory, considering that only 4 valence constants are used. Most of the calculated values agree within 5 cm-* which is in the range of uncertainty involved in evaluating the harmonic frequencies. Jones and McDowell [4] calculated harmonic frequencies of CY3Z and CYzZz isotopes of methane (Y and Z are H, D, and T) using 5 CY, force constants. We have cakulated harmonic frequencies* of these mixed isotopes with the same 4 valence constants of CH+ Out of 63 normal frequencies, the largest deviation is 16 cm-’ (less than 1%) and the average deviation is 4 cm-*. There is a practical advantage in using the observed fundamentals as it is much more convenient to compare the predicted and observed values when TABLE
1
Obsexved and calculated harmonic frequencies (in cm-‘) QJ,@,)
w :(@I
WJfa)
w,V:)
CH*
Oh% Cal.
3143 3143
1573 1573
3154 3154
1357 1357
CD,
Obs. Cal.
2224 2223 (-1)
1113 1112 k-1 1
2333 2324 (-9)
1027 1029 (+2)
DW.
1817 1815 P-2)
910 908 P-2)
1990 1975 F-15)
880 884 (+4)
SIH,
ObS. Cd.
2244 2244
1028 1028
2312 2312
940 940
SiD,
ObS. Cal.
1587 1587 (0)
727 727 (0)
1673 1671 (-2)
689 690 (*II
Cd.
1296
594
1393
581
ObS. Cd.
2179 2179
964 964
2224 2224
836 836
Oh
Cal. DJPV.
1541 1641 (0)
682 682 (0)
1581 1586 (+5)
605 601 (-4 1
Cd.
1258
557
1306
499
DeV.
CT,
ObS. Cd.
IkV.
-T.
*Tables of obsexved and cahht.ed frequencies of CY ,Z and CY $2, isotopic methane,and XY, isotopic species and XY,Z ~otopic sibme and gexmane (where X = C, Si, Cie; Y - 2, = H, D, T) axe available as SUP. 26047 (8 pages) ftom E.L.L.D.
142 TABLE
2
Least-squares harmonic force constants (in mdyn A-‘)
CH* SiH, GH,
K
D
F
F’
4.7419 29143 2.7695
1.3778 0.6696 0.6034
0.2689 0.0128 0.0067
0.1896 0.0601 0.0628
they are used for spectral assignment. The least-squaresforce constants, K, D, I?, and F’ are calculated using the observed fund~en~s, and the fundamentals of the isotopically substituted species are then predicted. The observed values used are those compiled by Shimanouchi 183. We have found large deviations in the stretching fundamentals, while agreement in the bending fundamentals is quite satisfactory. We then calculated the leastsquares force constants of XI& species and compared them with those of the XI& species. It was found that the ratio of K&u was nearly constant; 1.024 for methane, 1.020 for siiane, and 1.020 for germane. The least squaresforce constants of XY, fundamentals are listed in Table 3. In our method, it is much more convenient to make a small anharmonicity correction in the masses,rather than in the force constants. The massesof deuterium and tritium were scaled by 1.020 and 1.040, respectively, and the calculations repeated. The calculated fundamentals of silane and germane are remarkable in their agreement with the observed values. The largest deviation is 7 cm-’ and the average deviation is less than 3 cm- ‘. The anharmonicity corrections on mass, correct respectively for the stretching frequencies of v , and u3of Xl&, uz and v4 of XYJZ, and ZJ~ and us of XY& isotopes. For methane, the same corrections on mass are not sufficient to account for the large anharmonicity found in the symmetrical stretching fundamentals. The calculated values of ~1 in CY4 and v2’s in mixed isotopes deviate from the observed values by almost 2% Otherwise, the agreement in methane is good, the largest deviation being 16 cm-’ and average deviation less than 5 cm”‘. It should TABLE
3
Least-squares force constants (in mdyn /1-’ )
CH, CR CT. SiH, SiD,
@Ha err,
K
D
F
F’
4.421 4.529 4.599
1.441 1.434 1.443 0.537 0.559 0.525 0.536
0.148 0.178 0.184 0.060 0.067 0.035 0.035
0.209 0.207 .0.209 0.037
2.576
2.629 2474 2.524
0.039 ao5v 0.050
143
also be noted that some of the observed frequencies are not accurate due to experimental difficulties or the accidental shifts of Fermi resonance [9]. CONCLUSION
The four valence force constants of the OVFF model calculated from the harmonic frequencies of XI& species predict the harmonic frequencies of XY+ XY3Z, and XYz& isotopes within the range of uncertainties in their evaluations. Similar constants which fit the fundamentals of Xi% molecules and small anharmonicity corrections in isotopic masses predict the observed fundamentals of ~otopic~ly substituted species remarkably well. Particularly, the near perfect agreement obtained for the bond-bending modes are gratifying because the OVFF model is designed to describe better the bond-bending motions. No valence force field model with 4 constants matches the agreement obtained in this work between the observed and calculated values C9,lOl. This suggests that the OVFF model deserves more attention than it has received, at least for the XY, type molecules. The OVFF model has not been used very widely, mainly due to its algebraic complexity. Now, the method presented in ref. 1 makes the OVFF applications very simple. Furthermore, the use of Cartesian coordinates makes the calculations of the isotopically substituted species very convenient. The good agreement shown between the observed and calculated ~nd~en~s and the facility in calculating the isotopic species as demonstrated in this work would make the above method quite useful in predicting the as yet unknown frequencies of isotopic species. The physical meaning of the constant anharmonicity correction factors used for deuterium and tritium is not yet clear. Shimanouchi et al. [9] used a similar constant factor in their force constant calcuiations. it needs to be further tested with molecules having many different central atoms. REFERENCES 1 J. Tyson, H. H. Ciaassen and H. Kim, J. Chem. Phys., 54 (1971) 3142. 2 D. C. Heath and J. W. Linnett, ‘Dana Faraday Sot., 44 (1948) 873,878,884; Trans. Faraday Sot., 45 (1949) 264. 3 H. Kim, P. S. Souder and H. H. CIeassen. J. Mol. Spectrosc., 26 (1968) 46. 4 L. H. Jones and R. S. McDowell. J. Mol. Spectrosc., 3 (1959) 632. 5 D. F. Rail and D. C. McKean, Spectrochrm. Acta, 18 (1962) 1019. 6 A_ A. Chalmers and D. C. McKean. Spectrochim Acta, 21(1965) 1941. 7 I). M. Dennison, Rev. Mod. Phys.. 12 (1940) 175. 8 T. Sh~~noucbi, Tabtea of Molecuiar Vibrational Frequencies, NSRDS-NBS 6 and NSRDS-NBS II, National Bureau of Standards, 1967. 9 T. Shimanouchi, I. Nakagawa, J. Hiraishi and M. Iahii, J. Mol. Spectrosc., 19 (1966) 78 10 J. L. Duncan and I. M. Mills, Spectrochim. Acta, 29 (1964) 523.