Chem 301 Math 251

1
Chem 301 Math 251

• Chapter 2 Coordinate systems for Molecules

2
Cartesian Coordinate System

• Right handed system
• Written as ordered triple (x,y,z)

3

• Molecule with N atoms
• Each will have 3 coordinates (xi,yi,zi)
• 3N in total
• 3N degrees of freedom
• N3 matrix
• 3N1 vector
• Must label atoms
• 1st step draw molecule, label atoms and
  coordinate system

4
Water

• Can order in different ways
• Can orientate in different ways
• Cartesians lose Chemistry

5
Water

• Consider possible motions
• Each atom moves a distance of q in x direction
• Moved whole molecule
• Translation in x
• Can move the molecule in 3D
• 3 dof are Translations

6
Water

• Bring both Hs forward
• Rotated molecule about y axis
• Can do this about each axis
• 3 dof are Rotations

7
Carbon dioxide

• Rotate about x
• Rotate about y
• Rotate about z
• Only 2 Rotations
• Nonlinear molecules 3 Rotations
• Linear Molecules 2 rotations

8
What are other motions?

• Linear
• 3N Total
• 3 Translations
• 2 Rotations
• 3N-5 remain

• Nonlinear
• 3N Total
• 3 Translations
• 3 Rotations
• 3N-6 remain

• Vibrations
• chemically interesting motions
• Want to separate from translations and rotations

9
Total Kinetic Energy

• 1 particle
• 1D

• 3D

• N particles
• 3D

10
Total Kinetic Energy

• Can be expressed in matrix form

where
3N3N
3N1
11
Centre of mass

• Mass weighted average of all atomic positions
• (xC,yC,zC)

12
Example - Diatomic
B
A
Homonuclear (AB)
13
Example - Water
14
Centre of Mass Coordinate System

• Cartesian coordinate system centred at the centre
  of mass
• Must move with atom
• Will eliminate translations from our equations of
  motion
• Subtract position of COM from coordinates of each
  atom
• qCM,iqi qC

15
Centre of Mass Coordinate System

• Vector form

16
Centre of Mass Coordinate System

• Diatomic

17
Centre of Mass Coordinate System

• Water

18
Translational Motion

• Single particle in 1D
• Linear momentum

• N particles in 3D

• Linear momentum parallel to velocity of centre of
  mass

• In COM system will be 0

19
Rotational Motion

• Single particle rotating _at_ constant r

w angular velocity
I moment of inertia rotational equivalent of
mass

• Angular momentum

20
Rotational Motion

• N particles in 3D
• L and w are vectors
• In general will not be parallel

21
Inertia Tensor

• Diatomic

Reduced Mass
22
Inertia Tensor

• Diatomic

23
Inertia Tensor

• Diatomic

24
Inertia Tensor

• Diatomic

Only 2 rotational degrees of freedom Rotation
about z axis is undefined
25
Inertia Tensor

• Water

26
Inertia Tensor

• Water

27
Inertia Tensor

• Water

28
Inertia Tensor

• Water

29
Inertia Tensor

• Water

30
Inertia Tensor

• In both examples, Inertia Tensor was diagonal
• Not always the case
• There exists a coordinate system centred on the
  COM such that Inertia Tensor is diagonal
• Called principal axes
• In both examples, the COM system is the principal
  axes
• What if the Inertia Tensor is not diagonal?
• Need to diagonalize it
• How?

31
Inertia Tensor

• For a symmetric matrix, A, there exists an
  orthogonal transformation, P, such that

is diagonal
Need to find P Eigenvalue problem
32
Water - revisited
z
x
y
33
Water - revisited
z
x
y
34
Water - revisited
z
x
y
35
Water - revisited
z
x
y
36
Water - revisited
z
x
y
37
Water - revisited
z
x
y
38
Water - revisited
z
x
y
z is a principal axis x,y are not principal
axes A combination of x,y will make inertia
tensor diagonal
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inertia tensor diagonal Principal axes
40
SF4
Must use mass of isotope not natural abundance
average
41
SF4
42
SF4
43
Vibrational Motion

• Coordinate system
• Origin at the COM
• Translates with molecule
• Rotates with molecule
• Ignore translations and rotations (usually)
• Isolated molecules
• Only restoring force if molecule is distorted
  from eqm geometry

44
Vibrational Motion

• Taylor expansion

• Multiple variables

• Potential Energy

0
0
45
Vibrational Motion

• Potential Energy

FX - Cartesian force constant matrix - symmetric
46
Vibrational Motion

• Equations of motion

47
Vibrational Motion

• Equations of motion

(ijk)
(i?j, jk)
(i?j, ik)
48
Vibrational Motion

• Equations of motion

• Second derivative of each variable is linear
  combination of all variables

49
Vibrational Motion

• Equations of motion

Eigenvalue problem
50
Vibrational Motion

• Water

51
Vibrational Motion

• Water

52
Vibrational Motion

• Water

53
Vibrational Motion

• Water

54
Vibrational Motion

• Water

Find roots of 9th order polynomial !!!! (six
will be zero)
55
Vibrational Motion

• Cartesian coordinates
• Lose chemistry
• Give large polynomials to find roots
• Convert to internal symmetry coordinates

56
Vibrational Motion

• Internal coordinates
• Bond lengths
• Bond angles
• Dihedral Angles
• Angle between two planes
• Convert equations of motion to changes in these
  coordinates
• Coordinates are not orthogonal
• May have redundant coordinates

57
Internal Coordinates

• Bond Length

• Dihedral Angle

R1
R2

• Bond Angle

58
Internal Coordinates
R1 to R4 CH bond lengths R5 ?213 R6 ?214 R7
?215 R8 ?314 R9 ?315 R10 ?415 One extra
coordinate Need all of them can not just ignore
one
2
1
5
3
4
5 atoms 15 degrees of freedom 3 translation 3
rotations 9 vibrations
59
Internal Coordinates
R20 ?H1CCH4 R21 ?H1CCH5 R22 ?H1CCH6 R23
?H2CCH4 R24 ?H2CCH5 R25 ?H2CCH6 R26
?H3CCH4 R27 ?H3CCH5 R28 ?H3CCH6 Ten redundant
coordinates
8 atoms 24 degrees of freedom 3 translation 3
rotations 18 vibrations
R1 to R6 CH bond lengths R7 CC bond length R8
to R13 ?HCH R14 to R19 ?HCC
60
Z-matrix coordinates
Used in computational chemistry programs such as
Gaussian and Q-Chem 1st atom defines origin 2nd
atom defined as a distance from 1st atom 3rd atom
defined as a distance from one of previous atoms
and an angle with both 4th (and subsequent) atoms
defined as a distance from one of the previous
atoms, an angle with 2 and a dihedral angle with
3 Atoms do not have to be bonded Not unique!
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Z-matrix coordinates
O H1 O ROH H2 O ROH H1 HOH Variables ROH 0.96 HOH
104.0
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Z-matrix coordinates
C H1 C RCH H2 C RCH H1 HCH H3 C RCH H1 HCH H2
Ang H4 C RCH H1 HCH H3 Ang Variables RCH 1.0 HCH
109.4712 Ang 120.
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Z-matrix coordinates
Can use Dummy atoms to refer to a point where
there is no atom XC1 X RCCC2 X RCC C1 60.0 C3
X RCC C2 60.0 C1 180.C4 X RCC C3 60.0 C2 180.
C5 X RCC C4 60.0 C3 180.C6 X RCC C5 60.0 C4
180.H1 C1 RCH C6 120.0 C5 180.H2 C2 RCH C1
120.0 H1 0.H3 C3 RCH C2 120.0 H2 0.H4 C4 RCH C3
120.0 H3 0.H5 C5 RCH C4 120.0 H4 0.H6 C6 RCH C5
120.0 H5 0.VariablesRCC 1.3RCH 1.0
64
Back to Equations of Motion

• Assume we have dealt with redundant coordinates
• There exists a linear transformation from
  Cartesian coordinates X to Internal coordinates R

• R contains 3N-6 internal coordinates consisting
  of bond lengths, angles and dihedral angles (i.e.
  vibrational motion)
• And 6 external coordinates for translations and
  rotations

65
Back to Equations of Motion
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Back to Equations of Motion
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Back to Equations of Motion
Wilson GF method
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B matrix elements
3N-6 internal coordinates (3N-5 if linear)
To get elements of B find derivatives of each Rk
with respect to each qj
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B matrix elements
3 coordinates for translation of centre of mass
Rows 3N-5 to 3N-3 of B matrix
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B matrix elements
3 coordinates for rotation about of centre of
mass (2 if linear) (at eqm)
Rows 3N-2 to 3N of B matrix
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Diatomic
72
Diatomic
73
Diatomic
74
Diatomic
75
Diatomic
76
Diatomic
77
Diatomic
78
Diatomic
Potential energy Restoring force ? (distortion of
bond length)2 Hookes Law
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Diatomic
80
Diatomic
l 0 (5 times) and
In General Only need to solve the 3N-6 3N-6 GF
Eigenvalue problemLast 6 rows give zero root.
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Symmetry Coordinates

• Special linear combinations of internal
  coordinates
• Choose so that the coordinates behave according
  to the symmetry of the molecule
• Groups coordinates according to symmetry
• How to determine the combinations is discussed in
  Chemistry 305

82
Equations of Motion
S symmetry coordinates U Orthogonal
transformation matrix
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Equations of Motion
84
Symmetry Coordinates

• GS and FS are blocked
• Water
• Internal coordinates 33
• Symmetry coordinates 22 11
• Benzene
• Internal coordinates 3030
• Symmetry Coordinates 4(11), 6(22), 2(33) 2
  (44)

85
Water
R1
R2
kr - force constant for bond length kq - force
constant for bond angle krr - interaction
constant between 2 bonds krq - interaction
constant between bond and angle kqq - interaction
constant between 2 angles
86
Water
R1
R2
Formulae for the G element have been worked
out Do not need to do
87
Water
R1
R2
GF - Il 0 solve cubic get l 8.658695,
9.141301, 1.592337 mdyne Å-1 amu-1
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Water
R1
R2
89
Water
R1
R2
Block 1
90
Water
R1
R2
Block 2
Called wavenumber Has unit of 1/length (i.e. cm-1)
91
Water
R1
R2
What are the symmetry coordinates?
Symmetric stretch
Bend
Asymmetric stretch
S1 and S2 can mix S3 is a pure vibration In
general actual vibrations are combinations of
symmetry coordinates
92
Water
R1
R2
93
Experimentally

• Measure frequencies
• Calculate G based on eqm geometry and masses
• Geometry may have to be determined experimentally
  or computationally
• Use iterative procedure to determine F
• Usually more force constants than vibrations
• Use isotopomers