1. Computational Spectroscopy III. Spectroscopic Hamiltonians (e) Elementary operators for the harmonic oscillator (f) Elementary operators for the asymmetric rotor (g) Implementation of complex Hamiltonians (h) Matrix diagonalization methods (i) Symmetry and wavefunction diagnostics (j) Selection rules Updated: April 10, 2008

2. The harmonic oscillator problem The harmonic potential is Approximates a realistic potential for small amplitude vibrations. Is a starting point for solving the motion in a real vibrational potential. The Schrodinger equation is Energy levels are where the harmonic frequency is the reduced mass is and the quantum number is n = 0, 1, 2, 3, … Realistic Potential 4(x)4(x) 3(x)3(x) 2(x)2(x) 1(x)1(x) 0(x)0(x) (e) Elementary operators for the harmonic oscillator

3. Dimensionless coordinates x - “multiply by x” operator for x coodinate. Dimension of length = [L]. p - operator for the momentum in the x direction. Dimension of momentum = [M][L]/[T] H has dimensions of energy = [M][L] 2 /[T] 2 P, Q, and H are dimensionless. The Shroedinger equation becomes

4. Ladder operators Define the ladder operators a and a +. The ladder operators are NOT Hermitian. a is the lowering operator since it has the effect of transforming the harmonic oscillator wavefunction  n (x) into a multiple of the next lower function  n-1 (x). Likewise, a + is the raising operator. The x and p operators can be expressed in terms of the ladder operators a and a +. Therefore ANY Hamiltonian that can be writen in terms of the x and p operators can be expressed in terms of the ladder operators a and a +. In this context, a and a + will be our elementary operators. n(x)n(x)  n-1 (x)  n-2 (x)  n+1 (x)  n+2 (x) a+a+ a+a+ a+a+ a+a+ a a a a

5. Matrices for Ladder Operators To make the matrix for an operator, we first need to define our basis set. The eigenfunctions of a Hermitian operator are orthogonal, are (or can be) normalized, and form a complete set. Let’s use the harmonic oscillator wave functions as our basis: {  0,  1,  2,  3,  4, …} The matrix of a + is the set of elements This gives n=0 1 2 3 4 m=0 1 2 3 4 n=0 1 2 3 4 m=0 1 2 3 4

6. Programing a + and a function aa=aplusm(n) % APLUSM calculates the matrix of dimension n for the raisning operator a+. aa=zeros(n,n); % start with an nxn matrix filled with zeros. for k=1:n-1 aa(k+1,k)=sqrt(k); % Fill the elements along the diagonal above the principal diagonal. end

7. Matrices for x, p, x 2, p 2, and H Once the matices for a and a + have been calculated ( aplus, a ) and the constants, , k, and  are defined, just type in the algebraic formulae for the other operators! omega = sqrt(k/mu) x = sqrt(hbar/(2*mu*omega)*(aplus+a) p = i*sqrt(hbar*mu*omega/2)*(aplus-a) xsq = x*x psq = p*p H = (psq/(2*mu))+0.5*k*xsq For an anharmonic Hamiltonian, e.g., Hanh=H+cx 3 +dx 4 : Hanh = H + c*x^3 + d*x^4

8. Rigid Body Rotation: Diatomic Molecules Center of mass transformation (f) Elementary operators for the asymmetric rotor Dick Zare’s book, Angular Momentum - Spatial Aspects of Chemistry will be helpful for this part of the course.

9. Rigid Body Rotation: Diatomic Molecules Transformation to spherical polar coordinates The Schroedinger equation is (1) For the rotation of a diatomic molecule, we have in Cartesian coordinates Transform into spherical polar coordinates, we get for fixed r: Then solve eq (1) for (2) (3)

10. Angular Momentum The linear momentum of a mass m is given interms of the velocity: The angular momentum is given in terms of it position r = (x, y, z) and its linear momentum p = (p x, p y, p z ) as (4) The operators for the components of the momentum are Using eq (5) in eq (4), we get the angular momentum operators (5) (6)

11. Angular Momentum: Commutation Relationships The comutator of operators (matrices) A and B is defined We already know that a coordinate does not commute with its corresponding linear momentum: For the components of the angular momentum, we get These commutation relationships are considered to be so fundamental that we define a general angular momentum j as one whose components obey these rules: (7) (8)

12. Angular momentum eigenvectors Define We can show that j 2 commutes with its components A theorem in quantum mechanics says that we can find functions that are simultaneously eigenfunctions of two different Hermitian operators ONLY if those two operators commute. Let’s choose j 2 and j z to be our two commuting operators, and we will denote the eigenfunctions as. We can show that the eigenvalue relationships are where and (2j+1 values of m ) (9) (11) (10)

13. Ladder operators for angular momentum Define the ladder operators: The action of the ladder operators on the angular momentum eigenfunctions is We can rearrange eq (12) to get (12) m min = -j and m max = +j (13) (14)

14. Matrix elements of angular momentum operators If we remember that the angular momentum eigenfunctions are normailized and orthognonal, the above results may be summarized as follows: where we have used the “bra”-“ket” notation”: (14)

15. Angular momentum eigenfunctions: The spherical harmonics The eigenfunctions are called the spherical harmonics: These are familiar to us as the angular parts of the hydrogen atom orbitals, and they are also the rotational wavefunctions for diatomic and linear molecules. They are orthogonal and normalized.

16. Rotation of Diatomic & Linear Molecules The rotational Hamiltonian is This is easier to deal with if we set in the definitions of the angular momentum operators and include all of the units in a single rotational constant: The rotational eigenfunctions are ten the spherical harmonics: The rotational energy levels are the eigenvalues, Bj(j+1) where j=0, 1, 2, 3, …

17. The vector model of angular momentum Figure from p 13 of Dick Zare’s book. As is common in theoretical work, Dick has set The angular momentum vector has a fixed length and a fixed projection on the z-axis, which means that the angle  is fixed.

18. Symmetric Tops The 3 principal moments of inertia are given by where i labels the atoms in the molecule. When two of the 3 principal monents of inertia are equal, the molecule is called a symmetric top. Prolate Symmetric Top Oblate Symmetic Top (stick-like)(frisbee-like) (18)

19. Symmetric Top Rotational Energy Levels The rotational Hamiltonian for a rigid body is For example consider a prolate top where B=C. We also use the relation to rewrite the Hamiltonian Now we know that and commute so we can find eigenfunctions which are simultaneous eigenfunctions of both operators and that are therefore also eigenfunctions of H: (19) (20) (21) (22)

20. Symmetric Top Rotational Energy Levels It is convenient to set which means that we will include the units of in the rotational constants to give (for the prolate symmetric rotor): We have still used a little slight of hand here since J a is the projection of J on an axis (a = z) fixed to the molecule, rather than a space-fixed axis Z as we did in the derivation of the angular momentum operators. However, we still have to consider the space-fixed axes and a full derivation shows that the symetric rotor wavefunctions have 3 quantum numbers: J, K, and M. M is the quantum number for projection of the angular momentum on a space- fixed axis (Z) and K is the quantum number for the projection on a molecule-fixed axis (z). The rotational energy levels do not depend on M, so often we supress it. Dick Zare derives the explicit form of the symmetric rotor wavefunctions in terms of the three Euler angles, , , and . (23) (24) (25)

21. Symmetric Top Rotational Energy Levels Prolate top A > B = C Oblate top A = B > C G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold (1945), p. 25. Energy J=2

22. Asymmetric rotor molecules are between the prolate and oblate limits. The closeness of a given molecule to the prolate or oblate limts is measured by Ray’s asymmetry parameter  (kappa) defined as The prolate limit is  = -1 and in the oblate limit  = 1. K is no longer a good quantum number, but K prolate = K -1 and K oblate = K +1 can be defined from the correlation diagram at right. The asymmetric rotor levels are labeled as Most molecules are near the prolate limit, and so their spectra resemble prolate symmetric rotor spectra with additional “asymmetry” splittings. Asymmetric Rotor Rotational Energy Levels (Kappa) Zare, p 276

23. Symmetric Top Selection Rules G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold (1945), p. 417. Example: a prolate symmetric rotor

24. A Parallel Infrared Band of a Prolate Symmetric Rotor G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold (1945), p. 418.

25. A Perpendicular Infrared Band of a Prolate Symmetric Rotor G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold (1945), p. 425.