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An approximate method for the calculation of mean amplitudes of vibration in complex molecules
252
Journal of Molecular Structure
Elsevier Publishing Company, Amsterdam. Printed in the Netherlands
An approximate method for the calculation of mean amplitudes of vibration in complex molecules
Calculations of mean amplitudes of vibration from spectroscopic data have become very common recently but, especially for complicated molecules, the calculations are not easy and mistakes are often made. For all cases with two or more frequencies in a symmetry class, either data additional to the vibrational frequencies or an approximation method must be used. For the case of the two frequency problem one such method has been developed1*2 by which elements of the C matrix (matrix of the mean-square amplitudes of the symmetry coordinates) can be immediately calculated. A similar approximation has recently been made allowing solution of many 3 x 3 problems 3. However even when the z matrix is known the mean amplitudes, especially between nonbonded atoms, are not easily obtained (even for such a simple molecule as ZXY, (C,,) the correct equation has never been published). It is possible to circumvent this by a general computer program but we know of only one such in existence4. For molecules where the approximations mentioned above are valid, it is possible to assign the vibrational frequencies. If this is so then the molecule may be split up into groups of 3 atoms, YXZ, where Y and Z may or may not be alike, and the vibrational frequencies of these “pseudo-molecules” estimated. The mean amplitudes of these systems may readily be obtained from their E elements, the equations being relatively simple. Formation of ZXY ‘6pseudo-molecules”
For a molecule in which the frequency assignment can be at least partially made one knows the values of, for instance, the XZ and XY stretching frequencies, the ZXY deformation mode etc. These frequencies are then taken over for the ZXY pseudomolecule. In the case of several frequencies being ascibed to the same vibration (sym, asym, degenerate etc.) they are averaged arithmetically, after appropriately weighting degeneracies. For the case where Z = Y both a sym- and asym-v(XY) are needed and are appropriately chosen. For instance: FC103
A, v(FC1) 715 E v(Cl0,)
v(Cl0,)
1061
1315 6(C103) 591
6(ClOs) 550 6(FClO) 405
gives two problems
J. Mol.
FClO
v(FC1) 715
OCIO
v, 1061
v(CI0)
v,, 1315
Structure, 3 (1969) 252-255
3(1061-l-2-1315) = 1230 6(ClO,) 3(550+2-591)
6(FClO) 405
= 577
SHORT
253
COMMUNICATIONS
G-matrices
For most molecules the atoms in the pseudo-molecules may be taken as in the original: however, if there is a hydrogen atom present it vibrates so fast it must be considered part of the atom it is joined to; e.g. HSiCI, , gives one HSiCl problem and one (HSi)CI, problem. Using this the G-matrices may be easily set up -from the following equations: XY,-fragments:
Sym: G, I = 2px cos2 A+py G 12
= 4~, sin2 A+2p, = YXY angle)
-,/?pxsin2A
=
Asym: Gs3 = 9G22 ZXY fragments. Here an estimate of the bond lengths as well as the bond angle is needed: G,, = /L, cos B
61
=Px+PY
G 13
=
-p,
sin B/r(XY)
G 23
=
-p,
sin B/r&Z)
G33
=
--
PY
r(XY)2
(B = ZXY
G 22
=Px+Pz
1
cc2
+
+ r(XZ)2
angle: r(XY),
( r(XY)2
r(XZ)
1
2cos B
+r(XZ)2
-r(XY)r(XZ)
“*
= XY, XZ bond lengths)
Mean amplitudes
For the XY, pseudo-molecule there is one 2 x 2 problem (symmetric) and a single asymmetric frequency. For the former the approximation that the-element L,, in the eigenvector matrix ((G - F)L = LA) is zerol*’ allows the Z matrix elements to be written as:
-&I = %A
x 12
=
Z22
Gl2d1
=
(A2
det G+d,G1z2)/G1t
where 16.8575 coth
AI
‘i)
vi
(vi in cm- ‘, Tin OK) The asym. Z element is exactly given as A,G,, mean amplitudes are then given by: u(XY)2 = t(-L
(o*71Fg
(v, = v,; v, = 6,; ~3 = vas). The
+E33)
u(Y * - Y)” = 2Z,, sin’ A +Js
z12 sin 2A +I&, cos’ A. J. Mol. Struciure, 3 (1969) 252-255
254
SHORT
COMMUNICATIONS
For the ZXY fragments all three frequencies are in the same symmetry class, so tbe approximation LIZ = Li 3 = Lz3 = O3 is chosen. This is valid if the lowest frequency is the deformation. The .Z elements are then: =11 =
AlGll
I:22 =
@W12~f~2A)lG~i
x33
=
z 12 =
AlG132/Gx1
dlGl2
Xl3 z23
2
=
@IG,~GI~+A~@/GH
+A2B2/G, ,A+A,det
(Ai defined as above; v1 = v(XY);
diGl3
G/A
vp = v(X2);
v3
=
G(XYZ))
where A = (GIIGZ -G122); B = (G11G23-G12G13); (GIZGZS- G,,G,,) then det G = AG33-BG23-kCG,3. TABLE MEAN
and if C =
1
AMPLITUDES
OF
VIBRATlON
AT
25”
FOR
A
REPRESENTATXVE
SELECTION
OF ZxY,
AND
my,
MOLECULL?s (& a,
“Best values”
from ref. 5; b, from XY,
fragments;
OCHl
0.0805 0.0804 0.0804
OCFz
0.0436 0.0454 0.0457
0.0362
SCF,
0.0457 0.0486 0.0486
0.0385
HONO%
CJNO:!
FCIO,
0.0390 0.0385 0.0390 0.0388 0.0384 0.0390 0.0356 0.0360 0.0353
0.0377
o.OG9
0.050 0.052
0.056 0.058
0.053 0.059
0.049
0.054
0.050 0.051
0.0485 0.0434
0.056 0.062
0.0452 0.0446
0.0516
0.0476 0.0502 0.0489
0.0880
0.0415
0.056 0.055 0.156 0.155
0.125 0.131
0.152 0.151
0.0536
0.139 0.146
0.094 0.092
0.0887
0.0476
0.065 0.066
0.0418
0.0480
0.063 0.062
0.0449
0.0887 0.0888 0.0887
J. Mol. Strucrure, 3 (1969) 252-255
0.096
0.094 0.053 0.054
6 0381
iSiH,
0.0481 0.0496 0.0483
0.121 0.120
0.0364
0.0893 0.0893 0.0893
CISnBr,
fragments.
0.0374
FGeH,
HSiBrs
c, from ZXY
0.134 0.124
0.132 0.125
0.104 0.106
SHORT
255
COMMUNICATIONS
The mean amplitudes are then given by:
u(XY)”
= z:,,
u(xz)”
= z22
u(Y - - 2)” = (D2&l+E2&,+F2C,,+2DEC,2+2DFa!&+2EF~,,)/r2(Y~~ where D = r(XY)
- r(XZ)
cos B; E= r(XZ)
- r(XY)
Z)
cos B; F= r(XY)r(XZ)
sin B.
Examples In Table 1 are shown a selection of examples of ZXY,(C,) and ZXYJ(C,,) molecules calculated as above and compared with the best available values taken from ref. 5. The agreement is seen to be very good especially when the central atom is heavier than the outer. For the set the average deviation on bonded mean amplitudes is 2* “/,; non-bonded 5%. It should be noted that in normal electron diffraction studies where the mean amplitudes are used as additional parameters in the least-squares refinement of the data they are generally measured with an error of 10 ok or so. The use of values even from this approximation could hence help the evaluation of electron diffraction data considerably.
Conclusion Using the method outlined above it is now possible to calculate mean amplitudes of vibration with little trouble for any molecule or molecule part, for which a vibrational frequency assignment can be given, being especially accurate for vibrations of lighter atoms around a heavier one. We thank Professor Dr. 0. Glemser for his encouragement in our work, and the Fonds der Chemischen Industrie for financial support.
Anorganisch-chemisches der Universitiit, Giittingen (Germany)
Institut
A. MILLER C. J. PEACOCK H. SCHULZE U.
1 2 3 4 5
HEIDBORN
C. J. PEACOCK AND A. MULLER, J. Mol. Spectry., 14 (1968) 393. A. MUELLER,2. Phys. Chem. (Leipzig), 238 (1968) 116. C.J. PEACOCK AND A. MOLLER,J. Mol. Spectry., submitted. S. J. CYVIN, private communication. A. MULLER, B. KREBS, A. FALNNI, 0. GLEMSER, S. J. CYVIN, J. BRUNVOLL, B. N. CYVIN, I. ELWBREDD,G.HAGEN AND B.VIZI, Z.Nururfirrch., 23a (1968)1656.
Received October 2nd, 1968 J. Mol. Structure, 3 (1969) 252-255
Journal of Molecular Structure
Elsevier Publishing Company, Amsterdam. Printed in the Netherlands
An approximate method for the calculation of mean amplitudes of vibration in complex molecules
Calculations of mean amplitudes of vibration from spectroscopic data have become very common recently but, especially for complicated molecules, the calculations are not easy and mistakes are often made. For all cases with two or more frequencies in a symmetry class, either data additional to the vibrational frequencies or an approximation method must be used. For the case of the two frequency problem one such method has been developed1*2 by which elements of the C matrix (matrix of the mean-square amplitudes of the symmetry coordinates) can be immediately calculated. A similar approximation has recently been made allowing solution of many 3 x 3 problems 3. However even when the z matrix is known the mean amplitudes, especially between nonbonded atoms, are not easily obtained (even for such a simple molecule as ZXY, (C,,) the correct equation has never been published). It is possible to circumvent this by a general computer program but we know of only one such in existence4. For molecules where the approximations mentioned above are valid, it is possible to assign the vibrational frequencies. If this is so then the molecule may be split up into groups of 3 atoms, YXZ, where Y and Z may or may not be alike, and the vibrational frequencies of these “pseudo-molecules” estimated. The mean amplitudes of these systems may readily be obtained from their E elements, the equations being relatively simple. Formation of ZXY ‘6pseudo-molecules”
For a molecule in which the frequency assignment can be at least partially made one knows the values of, for instance, the XZ and XY stretching frequencies, the ZXY deformation mode etc. These frequencies are then taken over for the ZXY pseudomolecule. In the case of several frequencies being ascibed to the same vibration (sym, asym, degenerate etc.) they are averaged arithmetically, after appropriately weighting degeneracies. For the case where Z = Y both a sym- and asym-v(XY) are needed and are appropriately chosen. For instance: FC103
A, v(FC1) 715 E v(Cl0,)
v(Cl0,)
1061
1315 6(C103) 591
6(ClOs) 550 6(FClO) 405
gives two problems
J. Mol.
FClO
v(FC1) 715
OCIO
v, 1061
v(CI0)
v,, 1315
Structure, 3 (1969) 252-255
3(1061-l-2-1315) = 1230 6(ClO,) 3(550+2-591)
6(FClO) 405
= 577
SHORT
253
COMMUNICATIONS
G-matrices
For most molecules the atoms in the pseudo-molecules may be taken as in the original: however, if there is a hydrogen atom present it vibrates so fast it must be considered part of the atom it is joined to; e.g. HSiCI, , gives one HSiCl problem and one (HSi)CI, problem. Using this the G-matrices may be easily set up -from the following equations: XY,-fragments:
Sym: G, I = 2px cos2 A+py G 12
= 4~, sin2 A+2p, = YXY angle)
-,/?pxsin2A
=
Asym: Gs3 = 9G22 ZXY fragments. Here an estimate of the bond lengths as well as the bond angle is needed: G,, = /L, cos B
61
=Px+PY
G 13
=
-p,
sin B/r(XY)
G 23
=
-p,
sin B/r&Z)
G33
=
--
PY
r(XY)2
(B = ZXY
G 22
=Px+Pz
1
cc2
+
+ r(XZ)2
angle: r(XY),
( r(XY)2
r(XZ)
1
2cos B
+r(XZ)2
-r(XY)r(XZ)
“*
= XY, XZ bond lengths)
Mean amplitudes
For the XY, pseudo-molecule there is one 2 x 2 problem (symmetric) and a single asymmetric frequency. For the former the approximation that the-element L,, in the eigenvector matrix ((G - F)L = LA) is zerol*’ allows the Z matrix elements to be written as:
-&I = %A
x 12
=
Z22
Gl2d1
=
(A2
det G+d,G1z2)/G1t
where 16.8575 coth
AI
‘i)
vi
(vi in cm- ‘, Tin OK) The asym. Z element is exactly given as A,G,, mean amplitudes are then given by: u(XY)2 = t(-L
(o*71Fg
(v, = v,; v, = 6,; ~3 = vas). The
+E33)
u(Y * - Y)” = 2Z,, sin’ A +Js
z12 sin 2A +I&, cos’ A. J. Mol. Struciure, 3 (1969) 252-255
254
SHORT
COMMUNICATIONS
For the ZXY fragments all three frequencies are in the same symmetry class, so tbe approximation LIZ = Li 3 = Lz3 = O3 is chosen. This is valid if the lowest frequency is the deformation. The .Z elements are then: =11 =
AlGll
I:22 =
@W12~f~2A)lG~i
x33
=
z 12 =
AlG132/Gx1
dlGl2
Xl3 z23
2
=
@IG,~GI~+A~@/GH
+A2B2/G, ,A+A,det
(Ai defined as above; v1 = v(XY);
diGl3
G/A
vp = v(X2);
v3
=
G(XYZ))
where A = (GIIGZ -G122); B = (G11G23-G12G13); (GIZGZS- G,,G,,) then det G = AG33-BG23-kCG,3. TABLE MEAN
and if C =
1
AMPLITUDES
OF
VIBRATlON
AT
25”
FOR
A
REPRESENTATXVE
SELECTION
OF ZxY,
AND
my,
MOLECULL?s (& a,
“Best values”
from ref. 5; b, from XY,
fragments;
OCHl
0.0805 0.0804 0.0804
OCFz
0.0436 0.0454 0.0457
0.0362
SCF,
0.0457 0.0486 0.0486
0.0385
HONO%
CJNO:!
FCIO,
0.0390 0.0385 0.0390 0.0388 0.0384 0.0390 0.0356 0.0360 0.0353
0.0377
o.OG9
0.050 0.052
0.056 0.058
0.053 0.059
0.049
0.054
0.050 0.051
0.0485 0.0434
0.056 0.062
0.0452 0.0446
0.0516
0.0476 0.0502 0.0489
0.0880
0.0415
0.056 0.055 0.156 0.155
0.125 0.131
0.152 0.151
0.0536
0.139 0.146
0.094 0.092
0.0887
0.0476
0.065 0.066
0.0418
0.0480
0.063 0.062
0.0449
0.0887 0.0888 0.0887
J. Mol. Strucrure, 3 (1969) 252-255
0.096
0.094 0.053 0.054
6 0381
iSiH,
0.0481 0.0496 0.0483
0.121 0.120
0.0364
0.0893 0.0893 0.0893
CISnBr,
fragments.
0.0374
FGeH,
HSiBrs
c, from ZXY
0.134 0.124
0.132 0.125
0.104 0.106
SHORT
255
COMMUNICATIONS
The mean amplitudes are then given by:
u(XY)”
= z:,,
u(xz)”
= z22
u(Y - - 2)” = (D2&l+E2&,+F2C,,+2DEC,2+2DFa!&+2EF~,,)/r2(Y~~ where D = r(XY)
- r(XZ)
cos B; E= r(XZ)
- r(XY)
Z)
cos B; F= r(XY)r(XZ)
sin B.
Examples In Table 1 are shown a selection of examples of ZXY,(C,) and ZXYJ(C,,) molecules calculated as above and compared with the best available values taken from ref. 5. The agreement is seen to be very good especially when the central atom is heavier than the outer. For the set the average deviation on bonded mean amplitudes is 2* “/,; non-bonded 5%. It should be noted that in normal electron diffraction studies where the mean amplitudes are used as additional parameters in the least-squares refinement of the data they are generally measured with an error of 10 ok or so. The use of values even from this approximation could hence help the evaluation of electron diffraction data considerably.
Conclusion Using the method outlined above it is now possible to calculate mean amplitudes of vibration with little trouble for any molecule or molecule part, for which a vibrational frequency assignment can be given, being especially accurate for vibrations of lighter atoms around a heavier one. We thank Professor Dr. 0. Glemser for his encouragement in our work, and the Fonds der Chemischen Industrie for financial support.
Anorganisch-chemisches der Universitiit, Giittingen (Germany)
Institut
A. MILLER C. J. PEACOCK H. SCHULZE U.
1 2 3 4 5
HEIDBORN
C. J. PEACOCK AND A. MULLER, J. Mol. Spectry., 14 (1968) 393. A. MUELLER,2. Phys. Chem. (Leipzig), 238 (1968) 116. C.J. PEACOCK AND A. MOLLER,J. Mol. Spectry., submitted. S. J. CYVIN, private communication. A. MULLER, B. KREBS, A. FALNNI, 0. GLEMSER, S. J. CYVIN, J. BRUNVOLL, B. N. CYVIN, I. ELWBREDD,G.HAGEN AND B.VIZI, Z.Nururfirrch., 23a (1968)1656.
Received October 2nd, 1968 J. Mol. Structure, 3 (1969) 252-255