Question

Write down a generic molecular Hamiltonian (or, if you want a concrete example, the one for H2). Then using words and equations, show precisely what steps and approximations are made to ultimately come up with the harmonic oscillator Hamiltonian for vibration and the rigid rotator Hamiltonian for rotation of the molecule.

Answer 1

The system can be simplified using the concept of reduced mass which allows it to be treated as one rotating body. I he system can be entirely described by the fixed distance between the two masses instead of their individual radii of rotation. Relationships between the radii of rotation and bond length are derived from the COM given by: where l is the sum of the two radii of rotation: Ihrough simple algebra both radil can be tound in terms of their masses and bond length: Mi M2 and M1 M2 The kinetic energy of the system, T', is sum of the kinetic energy for each mass:

where and Using the angular velocity, the kinetic energy can now be written as: With the moment of inertia, (10) the kinetic energy can be further simplified: T=-. The moment of inertia can be rewritten by plugging in for R1 and R2:

Mi M2 Mi M2 (12) where MiM2 M1 M2 (13) is the reduced mass, μ. I he moment of Inertia and the system are now solely defined by a single mass, t, and a single length,l: (14) Angular Momentum Another important concept when dealing with rotating systems is the the angular momentum defined by: L Iw Looking back at the kinetic energy: (15) 2I 2I The angular momentum can now be described in terms of the moment of inertia and kinetic energy: Setting up the Schrödinger Equation

The wave functions for the rigid rotor model are found from solving the time-independent Schrödinger Equation: (16) Where the Hamiltonian Operator is: (17) where V2 is the Laplacian Operator and can be expressed in either Cartesian coordinates: (18) or in spherical coordinates: At this point it is important to incorporate two assumptions: » The distance between the two masses is fixed. This causes the terms in the Laplacian containing - to

or be zero . The orientation of the masses is completely described by θ and φ and in the absence of electric or magnetic fields the energy is independent of orientation. This causes the potential energy portion of the Hamiltonian to be zero The wave functions ψ(8,9) are customarily represented by Y(θ, d) and are called spherical harmonics. The Hamiltonian Operator can now be written: 21112 L sin θ with the Angular Momentum Operator being defned: (21) L-21T 80)sin The Schrodinger Equation now expressed: Solving the Schrodinger Equation

Solving the Schrödinger Equation The Schrodinger Equation can be solved separation of variables. Step 1: Let Y (8, d) Θ (θ) Φ (φ), and substitute: β Set the Schrodınger Equation equal to zero: using 2IE d2 φ +B sin Step 2: Because the terms containing Θ (9) are equal to the terms containing φ(d) they must equal the same constant in order to be defined for all values: sin θ d (26) -m Step 3: Solving for Φ is fairly simple and yields: (27) where m 0, +1 2, Solving for 6 is considerably more complicated but gives the quantized result:

1 imó (27) where m 0, +1, +2,.. Solving for 0 is considerably more complicated but gives the quantized result: (28) where J is the rotational level with J - 0,1,2,.. Step 4: The energy is quantized by expressing in terms of 3: (29) 2I the Step 5: Using the rotational constant, B - 2I BJ(J +1) energy is further simplified: E