r2 r1 r Motion of Two Bodies w k Rc Each type of motion is best represented in its own coordinate system best suited to solving the equations involved Rotational Motion Motion of the C.M. Center of Mass Cartesian r r2 Translational Motion k Internal motion (w.r.t CM) Vibrational Motion Rc Internal coordinates r1 Origin 18_12afig_PChem.jpg

Motion of Two Bodies Centre of Mass Internal Coordinates: Weighted average of all positions Internal Coordinates: In C.M. Coordinates:

Kinetic Energy Terms Tanslational Motion: In C.M. Coordinates: ? ? ? Rotation and Vibration: Internal Coordinates: ? ? ?

Centre of Mass Coordinates

Centre of Mass Coordinates

Centre of Mass Coordinates Similarly

Centre of Mass Coordinates

Centre of Mass Coordinates Reduced mass

Hamiltonian Separable! C.M. Motion 3-D P.I.B Internal Motion Rotation Vibration

Rotational Motion and Angular Momentum rotational motion requires internal coordinates Linear momentum of a rotating Body p(t1) p(t2) Ds f Angular Velocity Parallel to moving body Always perpendicular to r Always changing direction with time???

Angular Momentum p v w r L f m Perpendicular to R and p Orientation remains constant with time

Rotational Motion and Angular Momentum Center of mass R As p is always perpendicular to r Moment of inertia Proxy for mass in rotational motion

Moment of Inertia and Internal Coordinates Center of mass R

Angular Momentum and Kinetic Energy Classical Kinetic Energy r Center of mass R

Rotational Motion and Angular Momentum Center of mass R Since r and p are perpendicular

Momentum Summary Classical QM Linear Momentum Energy Rotational (Angular) Momentum Energy

Angular Momentum

Angular Momentum

Angular Momentum in QM

Angular Momentum

Angular Momentum

Two-Dimensional Rotational Motion Polar Coordinates y r f How to we get: x

Two-Dimensional Rotational Motion Consider product rule product rule

Two-Dimensional Rotational Motion Consider product rule product rule

Two-Dimensional Rotational Motion

Two-Dimensional Rotational Motion

Two-Dimensional Rigid Rotor Assume r is rigid, ie. it is constant As the system is rotating about the z-axis

Two-Dimensional Rigid Rotor 18_05fig_PChem.jpg

Two-Dimensional Rigid Rotor 18_05fig_PChem.jpg

Two-Dimensional Rigid Rotor Periodic - Like a particle in a circular box m = quantum number 18_05fig_PChem.jpg

Two-Dimensional Rigid Rotor 18_05fig_PChem.jpg

Two-Dimensional Rigid Rotor 18.0 12.5 E 8.0 4.5 2.0 0.5 Only 1 quantum number is require to determine the state of the system.

Normalization

Normalization

Orthogonality For m = m’ For m ≠ m’ 18_06fig_PChem.jpg

Spherical Polar Coordinates ? 14_01fig_PChem.jpg

Spherical Polar Coordinates 14_01fig_PChem.jpg

The Gradient in Spherical Polar Coordinates Gradient in Spherical Polar coordinates expressed in Cartesian Coordinates 14_01fig_PChem.jpg

The Gradient in Spherical Polar Coordinates Gradient in Cartesian coordinates expressed in Spherical Polar Coordinates 14_01fig_PChem.jpg

The Gradient in Spherical Polar Coordinates 14_01fig_PChem.jpg

The Gradient in Spherical Polar Coordinates 14_01fig_PChem.jpg

The Laplacian in Spherical Polar Coordinates Radial Term Angular Terms OR OR 14_01fig_PChem.jpg

Three-Dimensional Rigid Rotor Assume r is rigid, ie. it is constant. Then all energy is from rotational motion only.

Three-Dimensional Rigid Rotor Separable? 18_05fig_PChem.jpg

Three-Dimensional Rigid Rotor k2= Separation Constant Two separate independent equations

Three-Dimensional Rigid Rotor Recall 2D Rigid Rotor 18_05fig_PChem.jpg

Three-Dimensional Rigid Rotor This equation can be solving using a series expansion, using a Fourier Series: Legendre polynomials Where 18_05fig_PChem.jpg

Three-Dimensional Rigid Rotor Spherical Harmonics

The Spherical Harmonics For l=0, m=0

The Spherical Harmonics For l = 0, m = 0 Everywhere on the surface of the sphere has value what is ro ? r = (ro, q, f)

The Spherical Harmonics Normalization: In Spherical Polar Coordinates Z r = (1, q, f) Y X The wavefunction is an angular function which has a constant value over the entire unit circle.

The Spherical Harmonics For l =1, m = 0 Along z-axis Z r = (1, q, f) Y The spherical Harmonics are often plotted as a vector starting from the origin with orientation q and f and its length is Y(q,f) X The wavefunction is an angular function which has a value varying as on the entire unit circle.

The Spherical Harmonics For l=1, m =±1 Complex Valued?? Along x-axis Along y-axis 18_05fig_PChem.jpg

The Spherical Harmonics XZ YZ 18_05fig_PChem.jpg

The Spherical Harmonics Are Orthonormal Example ODD

Yl,m are Eigenfuncions of H, L2, Lz

Dirac Notation Continuous Functions is complete Vectors Dirac Bra Ket

Dirac Notation

Dirac Notation Example Degenerate

Dirac Notation Example

Dirac Notation Example

3-D Rotational motion & The Angular Momentum Vector Rotational motion is quantized not continuous. Only certain states of motion are allowed that are determined by quantum numbers l and m. l determines the length of the angular momentum vector m indicates the orientation of the angular momentum with respect to z-axis 18_16fig_PChem.jpg

Three-Dimensional Rigid Rotor States 3 2 1 6.0 -1 -2 -3 E 2 1 3.0 -1 -2 1 1.0 -1 0.5 Only 2 quantum numbers are require to determine the state of the system.

Rotational Spectroscopy 19_01tbl_PChem.jpg

Rotational Spectroscopy J : Rotational quantum number 19_13fig_PChem.jpg

Rotational Spectroscopy Wavenumber (cm-1) Rotational Constant Line spacing v Dv Frequency (v)

Rotational Spectroscopy Predict the line spacing for the 16O1H radical. r = 0.97 A = 9.7 x 10-11 m mO = 15.994 amu = 2.656 x 10-26 kg mH = 1.008 amu = 1.673 x 10-26 kg 1 amu = 1 g/mol = (0.001 kg/mol)/6.022 x 10-23 mol-1 = 1.661 x 10-23 kg

Rotational Spectroscopy The line spacing for 1H35Cl is 21.19 cm-1, determine its bond length . mCl = 34.698 amu = 5.807 x 10-26 kg mH = 1.008 amu = 1.673 x 10-26 kg

? ? The Transverse Components of Angular Momentum Ylm are eigenfunctions of L2 and Lz but not of Lx and Ly Therefore Lx and Ly do not commute with either L2 or Lz!!!

Commutation of Angular Momentum Components FOIL product rule

Commutation of Angular Momentum Components FOIL product rule

Commutation of Angular Momentum Components

Commutation of Angular Momentum Components

Cyclic Commutation of Angular Momentum

Commutation with Total Angular Momentum

Commutation with Total Angular Momentum

Commutation with Total Angular Momentum Therefore they have simultaneous eigen functions, Yl,m Also note that: Therefore the transverse components do not share the same eigen function as L2 and Lz. This means that only any one component of angular momentum can be determined at one time.

Ladder Operators Consider: Note: Super operator Like an eigen equation but for an operator! Super operator

? ? Ladder Operators What do these ladder operators actually do??? Recall That: Raising Operator Similarly Lowering Operator

Ladder Operators Note: Similarly: Consider: Therefore is an Eigenfunction of with eigen values l and m+1 Which implies that

? Ladder Operators These are not an eigen relationships!!!! is not an normalization constant!!! These relationships indicate a change in state, by Dm=+/-1, is caused by L+ and L- Can these operators be applied indefinitely?? Not allowed Recall: There is a max & min value for m, as it represents a component of L, and therefore must be smaller than l. ie. ? Why is

More Useful Properties of Ladder Operators Recall This is an eigen equation of a physical observable that is always greater than zero, as it represents the difference between the magnitude of L and the square of its smaller z-component, which are both positive. This means that m is constrained by l, and since m can be changed by ±1

More Useful Properties of Ladder Operators Knowing that: Lets show that mmin & mmax are l & -l. Consider have to be determined in terms of

More Useful Properties of Ladder Operators Also note that: Similarly

Ladder Operators Recall

Ladder Operators Recall Since the minimum value cannot be larger than the maximum value, therefore .

Spin Angular Momentum Intrinsic Angular Momentum is a fundamental property like mass,and charge.

Coupling of Spin Angular Momentum

Spin and Magnetic Fields Paramagnetism ESR (EPR), NMR (NPMR) NQR Mossbauer Precession Zeeman Splitting

Nuclear Magnetic Resonance

Fig. 1. 19F-NMR spectrum (56.4 MHz, 26°C) of the XeF5 XeF5+SbF6- Fig. 1. 19F-NMR spectrum (56.4 MHz, 26°C) of the XeF5 cation (4.87 M XeF5 SbF6 in HF solution): (A) axial fluorine and (a) 129Xe satellites; (X) equatorial fluorines and (x) 129Xe satellites [30].