MS310 Quantum Physical Chemistry Ch 7. A Quantum Mechanical Model for the Vibration and Rotation of Molecules Schrödinger eq. for the Q.M. harmonic oscillator Described by energy spectrum and energy eigen- functions of molecules having translational, vibrational, and rotational degrees of freedom Schrödinger eq. for rotation in 2-D and 3-D Angular monentum to consider orbitals… MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 7.1 Solving the Schrödinger equation for the Q.M Harmonic Oscillator Example of vibration in Q.M : chemical bond Bonding electron in the simple potential, and equilibrium distance is determined by bond length. Real potential : anharmonic oscillator(not ideal) At 300K, 1 or 2 state of vibration occupied → can approximate V(x) as a harmonic oscillator MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Schrödinger equation is given by Find the form of solution E term can be ignored because of x2 >> E when y2 → ∞ Multiply the 2dψ/dx both side and use the product rule MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Text p.104 MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Assume right term is much smaller than left term Solution : ‘Gaussian’ form Therefore, we can assume the solution MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Schrödinger equation is rewritten by Hermite equation is already solved : hermite polynomials Solution is given by Even state(n:even) : ψ(-x)=ψ(x) : even function Odd state(n:odd) : ψ(-x)=-ψ(x) : odd function MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Eigenvalue is given by There are 2 different phenomenon to classical H.O 1) energy of ground state is not zero : ZPE 2) particle can be found in the classical forbidden region Probability in the interval ∆x : MS310 Quantum Physical Chemistry

Probability density of 12th state of H.O Time-dependent solution : standing wave Probability density of 12th state of H.O MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 7.2 Solving the Schrödinger equation for rotation in 2-dimensions Neglect the coupling, hamiltonian operator is sum of individual operators for the degrees of freedom for the molecule And, total energy also can divide to each energy Finally, total wavefunction is product of eigenfunctions of each operator MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Set V=0 : no vibration(make easier problem) Rotation : internal motion → motion of reduced mass Laplacian in 2-dimension Fixed r : radial term canceled Solution is given by angular term It means clockwise and counterclockwise rotation MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Ex) 7.4 Normalize the rotational wavefunctions in 2-dimension Sol) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Boundary condition : ‘quantization’ of angular momentum Angular momentum must be periodic function because of φ+2π= φ always satisfies. → indistinguishable values φ and φ+2nπ Use Euler’s relation condition of ml : 0, ±1, ±2, ±3, … : quantization of angular momentum MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Energy of the rotation State of +ml and –ml : same energy and orthogonal each other → 2-fold degenerate with ml ≠ 0 level l : angular momentum vector, : angular momentum operator MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Φ+(φ) , Φ-(φ) : eigenfunction of both of hamiltonian and momentum operator Eigenvalue of momentum operator : +ℏml and -ℏml Then, we can obtain the similar form as the C.M Probability of angular motion : same for all region MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 7.3 Solving the Schrödinger equation for rotation in 3-dimension 3-dimensional rigid rotor : similar than 2-dimensional problem Laplacian in spherical coordinate is given by Rigid rotor : ‘fixed r’ → r term canceled Like the 2-dimension problem, we can write the Schrödinger equation the MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Define the β=2μr02E/ℏ2 Equation is changed by Use the separation of variable : Y(θ,φ) = Θ(θ)Φ(φ) Solve it by the left part and right part is ‘constant’ MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Right part : similar to 2-dimensional problem : set c = ml2 Equation can be change to two ODEs. Second equation : same as the 2-dimensional problem φ part of Y(θ,φ) : depends on ml MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Solve the first equation : Legendre’s equation Set z = cos θ and use it, equation change to Case of ml = 0 : Legendre’s equation Use the power series, write the solution P(z) instead of Θ(z) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Recurrence relation is given by If β ≠ integer, this series will not terminate. However, it cannot be solution because it diverge at z=1. Why? By the ratio test, series diverges at z=1 and it cannot the solution of wavefunction! If β = integer, well-behaved wavefunction exists and eigenvalue of equation is given by β = l(l+1) (set n=l) P(z) is called the Legendre polynomials. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Case of ml ≠ 0 : Associated Legendre polynomial Write the solution as the Plml(z) set Plml(z) = (1 - z2)m/2F(z) and equation is given by MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Finally, equation is changed to Solution is given by ml must be | ml | ≤ l : if more than l times of differentiation, wavefunction becomes zero and it is not allowed state. Therefore, quantum number is given by β = l(l+1), l = 0,1,2,3… ml = -l, -(l-1), … , -1, 0, 1, …, l-1, l Wavefunction Ylml(θ,φ) is given by MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Energy of angular momentum And this notation satisfies, too. Case of 2-dimensional rotation : 2-fold degenerecy In this problem(3-dimentional rotation) : 2l+1 degenerecy There are 2l+1 ml values per one l value, and these states have same energy! MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 7.4 The quantization of angular momentum Energy of angular motion is given by Difference between |l2| and E : divide by 2I Therefore, hamiltonian and operator also satisfy same relationship. Total energy quantized → |l|2 quantized We can write for operator Therefore, value of |l| is given by commute hamiltonian, not Then, has 3 component : lx,ly,lz, obtained by the l = r x p MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Can calculate this formula Angular momentum operator in spherical coordinate is Commutator relation is given by lx, ly, lz are not commute. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry How can obtain the component of angular momentum? → see lz : simplest form(only depends on φ) Ylml(θ,φ) is eigenfunction of lz → Ylml(θ,φ) is eigenfunction of both and lz Therefore, we can choose and lz can solve the problem easily. Also, we can know the length of angular momentum l and value of z-component lz, but we cannot know the value of x and y component. Why z component is special? → no special! We can choose another direction and it also commute to . It means z component is simple only in the spherical coordinate and ‘only 1’ component of angular momentum is commute with . MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 7.5 The spherical harmonic functions We see the spherical harmonic functions MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Spherical harmonic functions : ‘complex’ Make the function ‘real’ by the linear combination Real wavefunctions(we called it ‘orbital’) are orthonormal, too. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Shape of p and d orbitals. We can see the each orbital is perpendicular. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Superposition of p and d orbital Magnitude of pz and py orbital MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 7.6 The classical harmonic oscillator Example of oscillator : two masses connect by spring Introduce the center-of-mass coordinate and relative position MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Differentiate by time, we can obtain the center-of-mass velocity and relative velocity Total energy of 2-mass system Change the total energy by the vCM and v. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry vCM : velocity of ‘whole system’ : independent to motion of internal system v : relative velocity : dependent to motion of internal system Restoring force act to ‘internal system’ → restoring force act to reduced mass MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Therefore, we can divide this motion by two motions. (whole motion : motion of center of mass + motion of reduced mass) Our focus is motion by restoring force. If oscillator is harmonic oscillator, force is given by F = -kx Solution is given by MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry If initial condition is x(0)=0,v(0)=v0 Potential & kinetic energy of harmonic oscillator Classical harmonic oscillator : ‘continuous energy spectrum’ MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 7.7 Angular motion and the classical rigid rotor Rotation of 2-particle : centered at center of mass Consider the constant r = r1+r2 Kinetic energy of system is given by MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Centripetal acceleration Angular velocity and angular acceleration is given by MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Direction of angular velocity and angular acceleration : right-hand rule Case of constant acceleration I = μr2 : moment of inertia MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Angular momentum l is defined by x : cross product Magnitude or Angular momentum is φ : angle between p and r Kinetic energy is given by Classical rigid rotor : continuous energy MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 7.8 Spatial quantization See the angular momentum. First, we see the semiclassical description angular momentum cannot lie on the z-axis. Why? | ml | ≤ l is condition of ml and magnitude of l is given by Therefore, if the case of ml = l (extreme case) → z-component cannot be same as the magnitude of angular momentum. Angular momentum lie on the z-axis : x, y component = 0 → know 3 component simultaneously But it cannot be possible because commutator is not zero! MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry If we know the total angular momentum and z-component, then we cannot know the x and y component and only we know the → cone has an open end Finally, we can see the l=2 case(d orbital, too), vector model of angular momentum Vector of angular momentum only have certain orientation in space. → spatial quantization c.f) classical case : possible l values make the surface of sphere, not a cone MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Summary Quantum mechanics is used to study the vibration and rotation of a diatomic molecules. Vibrational degree of freedom modeled by the harmonic oscillator was considered. The harmonic oscillator has a discrete energy spectrum like the particle in the box in Q.M The Q.M model for rotational motion providing a basis for understanding the orbital motion of electrons around the nucleus of an atom as well as the rotation of a molecule about its principal axes was formulated. MS310 Quantum Physical Chemistry