CH-331 Agenda

Lecture 1: The Rise of Quantum Mechanics 

Classical mechanics is unable to explain certain phenomena observed in nature including the emission of blackbody radiators that is sensitive to the temperature of the radiator. This distribution follow's Planck distribution which can be used to derive two other experimental Laws (Wien's and Stefan-Boltzmann laws). A key finding is that the energy given off by a blackbody was not continuous, but given off at certain specific wavelengths, in regular increments. Photoelectric effect was introduced with an analogy to ionization energy given.

Lecture 2: The Rise of Quantum Mechanics

Classical mechanics is unable to explain certain phenomena observed in nature. The photoelectron effect has several experimental observations that break with classical predictions. Einstein proposed a solution that light is quantized given with each quantum of light is called a photon. And the energy is proportional to its frequency. This was an impressive argument in that it said light is not always a wave, but can be a particle. This duality also applies to matter. Hydrogen atom emission spectra consist of "lines" rather than a continuum expected of classical mechanics. These lines were separated into different classes.

Lecture 3: The Rise of Quantum Mechanics

Hydrogen atom emission spectra consist of "lines" rather than a continuum expected of classical mechanics. These lines were separated into different classes. Rydberg showed that a simple single equation can predict the energies of these transitions by introducing two integers of unknown origin. While the photoelectron effect demonstrated that light can be wave-like and particle-like (e.g., "photon"), de Broglie demonstrated that matter also exhibits wave-like and particle-like behavior. The Bohr atom was introduced as the first successful description of a quantum atom from basic principles (either as a particle or as a wave, both were discussed).

Lecture 4: Bohr atom and Heisenberg Uncertainty

The Bohr atom was the first successful description of a quantum atom from basic principles (either as a particle or as a wave, both were discussed). From a particle perspective, stable orbits are predicted from the result of opposing forces (Coloumb's force vs. centripetal force). From a wave perspective, stable "standing waves" are predicted. The Bohr atom predicts quantized energies. Heisenberg's Uncertainly principle argues that trajectories do not exist in quantum mechanics.

Lecture 5: Wave Equations

Schrödinger Equation is a wave equation that is used to describe quantum mechanical system and is akin to Newtonian mechanics in classical mechanics. The Schrödinger Equation is an eigenvalue/eigenvector problem. To use it we have to recognize that observables are associated with linear operators that "operate" on the wavefunction.

Lecture 6: Schrödinger Equation

The Schrödinger Equation has solutions called wavefunctions. The time-dependent Schrödinger Equation results in time-dependent wavefunctions with both spatial aspect and a temporal aspects. The time-independent Schrödinger Equation results in time-independent wavefunctions with only a spatial aspect. Which one we use dependents if their is an explicit time-dependence in the Hamiltonian. It is important to recognize that wavefunctions ALWAYS have a temporal part (we typically ignore though).

Lecture 7: Operators, Free Particles and the Quantum Superposition Principle

Wavefunctions have a probabilistic interpretation, more specifically, the wavefunction squared (or to be more exact, the Ψ∗Ψ is a probability density). To get a probability, we have to integrate \(Ψ^∗Ψ\) over an interval. The probabilistic interpretation means \(Ψ^∗Ψ\) must be finite, nonnegative and not infinite. and that the wavefunctions must be normalized. We the introduced the particle in the box, which is "easy" to solve the Schrödinger Equation to get oscillatory wavefunctions.

Lecture 8: Topical Overview of PIB and Postulates QM

This lecture focused on gaining an intuition of wavefunctions with an emphasis on the particle in the box. Specifically, we considered the four principal properties of continuous distributions and applied it to the particle in the box. We want to develop an intuition behind how the energy and wavefunctions change in PIB when mass is increased, when box length is increased and when quantum number n is increased. We ended the discussion discussing that eigenstates of an operator are orthogonal.

Lecture 9: PIB and orthogonality

Another overview of particle in a box. Identifying key intuition basics including how the energy of the particles change with length of box, mass of particle and quantum number. We want to develop an intuition behind how the energy and wavefunctions change in PIB when mass is increased, when box length is increased and when quantum number n is increased. We ended the discussion discussing that eigenstates of an operator are orthogonal. We discussed how integrating over an odd integrand will be zero (assuming correct limits of integration).

Lecture 10: Expectation values, 2D-PIB and Heisenberg Uncertainty Principle

We extend the 1D particle in a box to the 2-D and 3D cases. From this we identified a few interesting phenomena including multiple quantum numbers and degeneracy where multiple wavefunctions share the identical energy. We were able to provide a quantitative backing in using the Heisenberg Uncertainty principle from wavefuctions in terms of the standard deviations

Lecture 11: Vibrations

We will started the lecture on the five postulates of quantum mechanics. We first introduce bra-key notation as a means to simplify the manipulation of integrals. Three aspects were addressed: (1) Introduction of the commutator which is meant to evaluate is two operators commute. Not every pair of operators will commute meaning the order of operations matter. (2) Redefine the Heisenberg Uncertainty Principle now within the context of commutators to identify if any two quantum measurements can be simultaneously evaluated. (3) We introduction of vibrations, including the harmonic oscillator potential were qualitatively shown (via Java application). We review the classical picture of vibrations including the classical potential, bond length, and bond energy and introduced the quantum harmonic oscillator as an approximation of the true potential. Solving the resulting (time-independent) Schrödinger equation with a parabolic potential to obtain the eigeinstates, energies, and quantum numbers (v) results is beyond this course, so they are given. Key aspect of these solution are the fundamental frequency and zero-point energy.

Exam 1:

• Exam covers lectures 1 through 11; readings through (and not including 4.6) and content in all worksheets that do not address harmonic oscillator explicitly.

Lecture 12: Vibrational Spectroscopy of Diatomic Molecules

We review the classical picture of vibrations including the classical potential, bond length, and bond energy and introduced the quantum harmonic oscillator as an approximation of the true potential. Solving the resulting (time-independent) Schrödinger equation with a parabolic potential to obtain the eigeinstates, energies, and quantum numbers (v) results is beyond this course, so they are given. Key aspects of these solution are the fundamental frequency and zero-point energy.

Lecture 13: Harmonic Oscillators and Rotation of Diatomic Molecules

We introduced a qualitative discussion of IR spectroscopy and then focused on "selection rules" for what vibrations are "IR-active." The two criteria we got discussed were (1) the vibration requires a changing dipole moment and (2) that \(\Delta v = \pm 1\) required for the transition (within harmonic oscillators). These selection rules can be derived from the concept of a transition moment and symmetry.

Lecture 14: Harmonic Oscillators and IR Spectroscopy

Discussion involving quantum harmonic oscillators, harmonic oscillators eigenstates, anhamonicity, Morse potential.

Lecture 15: 3D Rotations and Microwave Spectroscopy

Symmetry (and direct product tables for odd/even functions) were discussed and showed Harmonic Oscillator wavefunctions alternated between even and odd due to Hermite polynomial component, which affects the transition moment integral so only transitions in the IR between adjacent wavefunctions will be allowed (i.e., no harmonics). This is an approximation and the Taylor expansion of an arbitrary potential shows that anharmonic terms must be used. We introduced the Morse oscillator & rotations. We continue our discussion of the solutions to the 3D rigid rotor: The wavefunctions (the spherical harmonics), the energies (and degeneracies) and the TWO quantum numbers (\(J\) and \(m_J\)) and their ranges. We discussed that the components of the angular momentum operator are subject to the Heisenberg uncertainty principle and cannot be know to infinite precision simultaneously, however the magnitude of angular momentum and any component can be. This results in the vectoral representation.

Lecture 16: 3D Rotations and Microwave Spectroscopy

The potential, Hamiltonian and Schrödinger equation for the Hydrogen atom is introduced. The solution of which involves radial and angular components. The latter is just the spherical harmonics derived from the rigid rotor systems. The radial component is a function of four terms: a normalization constant, associated Laguerre polynomial, a nodal function, and an exponential decay. We also discussed that the energy is a function of only one quantum number and that there is a degeneracy to address.

Lecture 17: Hydrogen-like Solutions

The potential, Hamiltonian and Schrödinger equation for the Hydrogen atom is introduced. The solution of which involves radial and angular components. The latter is just the spherical harmonics derived from the rigid rotor systems. The radial component is a function of four terms: a normalization constant, associated Laguerre polynomial, a nodal function, and an exponential decay. We also discussed that the energy is a function of only one quantum number and that there is a degeneracy to address. While there are three quantum numbers in the solutions to the corresponding Schrodinger equation, that the energy for a hydrogen atom is only a function of n. We continued our discussion of the radial component of the wavefunctions as a product of four terms that crudely results in an exponentially decaying amplitude as a function of distance from the nucleus scaled by a pair of polynomials. We discussed the volume and shell element in spherical space and introduce the radial distribution function.

Lecture 18: Orbital Angular Momentum, Spectroscopy and Multi-Electron Atoms

Angular moment of an electron is described by the \(l\) quantum number. The \(m_l\) quantum number designates the orientation of that angular moment wrt the z-axis. The degeneracy can be partial broken by an applied magnetic fields. There is not always do a one-to-one correspondence between quantum numbers and orbitals. Basic electronic spectroscopy was reviewed and specifically selection rules. The impossible to solve He system was discussed requiring approximations; a poor one was introduced.

Lecture 19: Variational Method, Effective Charge, and Matrix Representation

Three aspects were addressed: (1) We continued discussing the complications of electron-electron repulsions and showed ignoring it is really pretty poor. (2) We can qualitatively address them by introducing an effective charge within a shielding and penetration perspective. (3) We motivated variational method by arguing the energy of a trial wavefunction will be lowest when it most likely resembles the true wavefunction (the same for the corresponding energies).

No Lecture: (Veterans' Day)

• Worksheet 7A: The Probabilistic Interpretation of Atomic Orbitals

Exam 2: Model Systems & Atoms

• Exam covers lectures 12 through 21 (but the basics of the content on the first exam should be mastered). The exception is that Perturbation theory is NOT on the exam.
• Practice Exam Questions

Lecture 20: Linear Variational Theory and Perturbation Theory

Three aspects were addressed: (1) We continued discussing the complications of electron-electron repulsions and showed ignoring it is really pretty poor. (2) We can qualitatively address them by introducing an effective charge within a shielding and penetration perspective. (3) We motivated variational method by arguing the energy of a trial wavefunction will be lowest when it most likely resembles the true wavefunction (the same for the corresponding energies).

Lecture 21: Linear Variational Theory and Perturbation Theory

The variational method approach requires postulating a trial wavefunction and calculating the energy of that function as a function of the parameters of that trail wavefunction. Then we can minimize the energy as a function of these parameters and the closer the wavefunction "looks" like the true wavefunction, the closer the trail energy matches the true energy. Several example trial wavefunctions for the He atom are discussed. We introduce the matrix representation of Quantum mechanics.

Lecture 22: Perturbation Theory

This lecture reviews the basic steps in variational method, the linear variational method and the linear variation method with functions that have parameters that can float (e.g., a linear combination of Gaussians with variable widths in ab initio chemistry calculations). The latter two will be more applicable in the discussions of molecules using atomic orbitals as the basis set (th LCAO approximation). The final approximation, perturbation theory is introduced, but not used in an example.

Lecture 23: Electron Spin, Indistinguishability and Slater Determinants

The basic steps perturbation theory is discussed including its application to the energy and wavefunctions. A reminder of the orbital approximation was discussed (where an N-electron wavefunction can be described as N 1-electron orbitals that resemble the hydrogen atom wavefunctions). A consequence of the orbital approximation is the ability to construct electron configurations which are filled by the aufbau principle. However, the aufbau principle is only a guideline and not a hardfast rule.

Lecture 24: Coupling Angular Momenta and Atomic Term Symbols

This lecture address two unique aspects of electrons: spin and indistinguishability and how they couple into describing multi-electron wavefunctions. The spin results in an angular momentum that follows the same properties of orbital angular moment including commutators and uncertainty effect. The Slater determinant wavefunction is introduced as a way to consistently address both properties.

Lecture 25: Molecules and Molecular Orbital Theory

The consequence of indistinguishability in electronic structure calculations. The Hartree and Hartree-Fock (HF) calculations were introduced within the Self-Consistent-Field (SCF) approach (similar to numerical evaluation of minima). The Hartree method treats electrons via only as an average repulsion energy and the HF approach using Slater determinant wavefunctions introduces an exchange energy term. Ionization energy and electron affinities are discussed within the context of Koopman's theorem.

No lecture: Thanksgiving

Happy Turkey Day

No Lecture: Thanksgiving

Happy Turkey Day

Lecture 26: Populating Molecular Orbitals: σ and π Orbitals

Last lecture address how the different orbital angular momenta of multi-electron atoms couple to break degeneracies predicted from the "Ignorance is Bliss" approximation (i.e., the hydrogen atom). Total angular momenta are introduced along with multiplicity. Atomic term symbols are discussed along with all three of Hund's rules to identify the most stable combination of angular momenta for a specific electron configuration.

Lecture 27: Molecular Orbitals and Diatomics

The application of term symbols to describe atomic spectroscopy is demonstrated. The corresponding selection rules are discussed. The Born-Approximation is introduced to help solve the N-bodies Schrödinger equation of molecules. This introduces the concept of a potential energy curve (surface). The LCAO is introduced as a mechanism to solve for Molecular Orbitals (MOs).

Lecture 28: Molecular Orbitals with higher Energy Atomic Orbitals

The application of term symbols to describe atomic spectroscopy is demonstrated. The corresponding selection rules are discussed. The Born-Approximation is introduced to help solve the N-bodies Schrödinger equation of molecules. This introduces the concept of a potential energy curve (surface). The LCAO is introduced as a mechanism to solve for Molecular Orbitals (MOs).

Lecture 29: (Unit IV: Diatomic Molecules)

From this LCAO-MO approach arises the Coulomb, Exchange (similar to HF calculations of atoms), and Overlap integrals. The concept of bonding and anti-bonding orbitals results.The application of LCAO toward molecular orbitals is demonstrated including linear variational theory and secular equations. Bond order, bond length and bond energy is emphasized for H2 species. Simple MO theory does not predicted He dimers. The MOs of first row diatomics is discussed including both π and σ MOs. The MO diagram is presented. Bond order, bond length, and bond energies are emphasized. The flip over of pi/sigma MO is demonstrated and the paramagnetism of oxygen is a natural conclusion of MO theory.

Final Exam:

• Comprehensive Exam covers lectures 1 through 28