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Transcript
2.05•10-6
X”
C
H
C
Y”
3.05•10-6
χC-C (σ)
Set of 6
Centers
Z”
3.05•10-6
3.41•10-6
Y”
χC-H (σ)
H
Z”
X”
4.21•10-6
C
4.21•10-6
Set of 6
Centers
10.8•10-6
Y” H
”
Z
-6.5•10-6
C
X”
Set of 6
Centers
7.9•10-6
0.0•10-6
Y”
χC(delocalized π contribution)
At ring center
Z
X”
”
One set only
0.0•10-6
-33.0•10-6
-9.35•10-6
C
-9.35•10-6
χ⊥C-C 3.05x10-6
-9.35•10-6
Set of 6
Centers
Y
χ||C-C 2.05x10-6
0.6062
X
X
1.2124
χ⊥C-C, ⊥mol_plane
3.05x10-6
1.05
Z
1
H
XY
2 H
X
3H
X
H6
1
2
6
3
3
5
3
Z
4
3
C-H bond distance=1.087 A⁰
C-C bond length= 1.4A⁰
Angle C-C-C =120⁰
Angle C-C-H= 120⁰
X3
X
H
X
5
3
H
4X
3
The (locally) diagonal Tensors (in their respective X”,Y”,Z” frames) of the various parts of Benzene are all to be
transformed to a common Molecular axis system X,Y,Z.
The transformation matrices are obtained with the corresponding direction cosines.
Consider the C-C (σ) contribution of C-C (1-2); all the green dots represent the various local principal axis system of
the 6 different C-C bond susceptibilities. The C-C axis is the X” axis. Thus rotating the X” of the C-C (1-2) in the
molecular plane, by 30⁰ in the negative sense, by -30⁰, the X” becomes collinear with and along the X axis of the
molecular system. Since all the axes system are Right handed, rotating the C-C (1-2) PAS system about the
corresponding Z” axis, the C-C (1-2) PAS would coincide with Molecular Axis system.
Consider the coordinates of the origin of the X”,Y”,Z” axis system for the C-C (2-1) bond. The origin is at C atom No.2.
For that X”,Y”,Z” frame of reference at C2, the coordinates are (0,0,0). The same C2 atom coordinates in the
Molecular Axes System X,Y,Z are (-1.2124, 0.7,0).
Thus the transforming the coordinate (of the location of dipole placed at the center of C2-C1) bond from X”, Y”,Z”
reference system (0.7,0,0) to X,Y,Z system requires first to subject the coordinate set 0.7,0,0 to a rotation
corresponding to -30⁰ of the axis system about Z”, and then shift the coordinates corresponding to the shifted
origin.
Conventionally in matrix notation the coordinates of a vector are written as column vector, thus the radius vector
0.7,0,0 can be written as
Since this is column vector according to Matrix multiplication rule, this being the operand (the vector to be
transformed) will be on the RHS of the operator symbol, , chosen here for the matrix multiplication, the
elementary steps cumulatively called transformation.
Thus the first step is to rotate the coordinate system by -30⁰ and find the newly assigned values to this
column vector.
This transforming matrix consists of the direction cosines of the X”,Y”,Z” axes with X,Y,Z.
Angle X”, X is -30⁰ and so is angle Y”, Y. since it is rotation about Z”, perpendicular to molecular plane the situation
that both Z”, and Z are perpendicular to the same molecular plane makes the angle Z”,Z equal to 0.
The Direction Cosines, DCs, of X”, with X, Y, and Z would be as follows: DC of X” with X is Cos -30⁰; of X” with Y is Cos
60⁰, of X” with Z is Cos -90⁰. Similarly for Y”, the DCs with X, Y, Z would be Cos -120⁰, Cos -30⁰, Cos 90⁰. For Z” with X,
Y, Z would be Cos -90⁰, Cos 90⁰, Cos 0⁰.
=
To this values obtained after rotation the shift in the origin should be affected.
The origin in X”, Y”, Z” system has values 0, 0, 0.
This point has in the X, Y, Z system has values (-1.2124, 0.7, 0). Since a point 0,0,0 becomes (-1.2124, 0.7,0), all the
points in the X”, Y”, Z”after appropriate rotation must be added to the set (-1.2124, 0.7,0) to account for shift of origin.
+
=
Transforming a TENSOR of 3x3 matrix, unlike a column vector would require left multiplication as above followed
by a right multiplication with the transpose of the above matrix of DCs.
=
Consider the C-C (σ) contribution of C-C (2-3);
=
Coordinates of the origin of X”, Y”, Z” system at C3 expressed in X, Y, Z system are (-1.2124, -0.7, 0)
Coordinates of the midpoint of C2-C3 bond in X”, Y”, Z” are (0.7,0, 0) in that local X”, Y”, Z” system.
First is the Coordinate transformation for rotation:
=
=
Transforming Susceptibility tensors
=
=
Consider the C-C (σ) contribution of C-C (3-4);
=
=
Consider the C-C (σ) contribution of C-C (4-5);
=
=
Consider the C-C (σ) contribution of C-C (5-6);
=
=
Consider the C-C (σ) contribution of C-C (6-1);
=
=
C-C (1-2);
C-C (2-3);
C-C (3-4);
C-C (4-5);
C-C (5-6);
C-C (6-1)
Consider the C-C (σ) contribution
C-C (2-3);
C-C (1-2);
2.2999 -0.4330 0.0000
-0.4330 2.7999 0.0000
0.0000 0.0000 3.0500
C-C (3-4);
2.2999
0.4330
0.0000
0.0000
2.0500
0.0000
0.0000
0.0000
3.0500
C-C (4-5);
0.4330
2.7999
0.0000
0.0000
0.0000
3.0500
2.2999 -0.4330 0.0000
-0.4330 2.7999 0.0000
0.0000 0.0000 3.0500
C-C (6-1);
C-C (5-6);
3.0500
0.0000
0.0000
3.0500
0.0000
0.0000
0.0000
2.0500
0.0000
0.0000
0.0000
3.0500
2.2999
0.4330
0.0000
0.4330
2.7999
0.0000
0.0000
0.0000
3.0500
Consider the C-H (σ) contribution
1
H
XY
2 H
X
3H
X
H6
1
2
6
3
3
5
3
Z
4
3
C-H bond distance=1.087 A⁰
C-C bond length= 1.4A⁰
Angle C-C-C =120⁰
Angle C-C-H= 120⁰
X3
X
H
X
5
3
H
4X
3
3.41•10-6
Y”
χC-H (σ)
H
Z”
4.21•10
C1-H1
=
X”
C
-6
Set of 6
Centers
4.21•10-6
C2-H2
=
=
C3-H3
=
C4-H4
=
C5-H5
=
C6-H6
=
1
H
XY
2 H
X
H6
1
2
6
X3
X
C-H bond distance=1.087 A⁰
C-C bond length= 1.4A⁰
Angle C-C-C =120⁰
Angle C-C-H= 120⁰
χ⊥C-C 3.05x10-6
Y
χ||C-C 2.05x10-6
0.6062
X
X
1.2124
1.05
χ⊥C-C, ⊥mol_plane
3.05x10-6
Z
Coordinates of C atoms
Coordinates of C atoms
C1 =90⁰
C2 =150⁰
C3 =210⁰
C4 =270⁰
C5 =330⁰
C6 =390⁰≡30⁰
1.4A⁰
1.4A⁰
1.4A⁰
1.4A⁰
1.4A⁰
1.4A⁰
0.0000 A⁰
-1.2124 A⁰
-1.2124 A⁰
0.0000A⁰
1.2124A⁰
1.2124A⁰
1.4000A⁰
0.7000A⁰
-0.7000A⁰
-1.4000A⁰
-0.7000A⁰
0.7000A⁰
0.0000 A⁰
0.0000 A⁰
0.0000 A⁰
0.0000 A⁰
0.0000 A⁰
0.0000 A⁰
Midpoints of Cn+1-Cn
dm origin
-0.6062, 1.0500, 0.0000 [C2-C1]
-1.2124, 0.0000, 0.0000 [C3-C2]
-0.6062,-1.0500, 0.0000[C4-C3]
0.6062,-1.0500, 0.0000[C5-C4]
1.2124, 0.0000, 0.0000[C6-C5]
0.6062, 1.0500, 0.0000[C1-C6]
Midpoint of C-H, location of Dipole, DM origin
1.4+0.5(1.087)=1.9435
Midpoint of C-H, location of Dipole,
DM origin
0.0000 A⁰
1.9435 A⁰
0.0000 A⁰
-1.6831 A⁰
0.9718 A⁰
0.0000 A⁰
-1.6831 A⁰
-0.9718A⁰
0.0000 A⁰
0.0000A⁰
-1.9435A⁰
0.0000 A⁰
1.6831A⁰
-0.9718A⁰
0.0000 A⁰
1.6831A⁰
0.9718A⁰
0.0000 A⁰
1.9435 A⁰
C1-H1 =90⁰
1.9435 A⁰
C2 =150⁰
1.9435 A⁰
C3 =210⁰
1.9435 A⁰
C4 =270⁰
1.9435 A⁰
C5 =330⁰
1.9435 A⁰
C6 =390⁰≡30⁰
Proton Coordinates
1.4+1.087=2.4870
Proton Coordinates
0.0000 A⁰
2.4870A⁰
0.0000 A⁰
-2.1538 A⁰
1.2435A⁰
0.0000 A⁰
-2.1538 A⁰
-1.2435A⁰
0.0000 A⁰
0.0000A⁰
-2.4870A⁰ 0.0000 A⁰
2.1538 A⁰
-1.2435A⁰
0.0000 A⁰
2.1538 A⁰
1.2435A⁰
0.0000 A⁰
2.4870A⁰
2.4870A⁰
2.4870A⁰
2.4870A⁰
2.4870A⁰
2.4870A⁰
C1-H1 =90⁰
C2 =150⁰
C3 =210⁰
C4 =270⁰
C5 =330⁰
C6 =390⁰≡30⁰
1
H
XY
2 H
X
3H
X
H6
1
2
6
3
3
5
3
Z
4
3
H
4X
3
X3
X
H
X
5
3
C-H bond distance=1.087 A⁰
C-C bond length= 1.4A⁰
Angle C-C-C =120⁰
Angle C-C-H= 120⁰