**Pre-requisites:** Physical Chemistry

Physicists at the end of the nineteenth century believed that most of the fundamental physical laws had been worked out. They expected only minor refinements to get “an extra decimal place” of accuracy. As it turns out, the field of physics was transformed profoundly in the early twentieth century by Einstein's discovery of relativity and by the development of quantum mechanics. While relativity has had fairly little impact on chemistry, all of theoretical chemistry is founded upon quantum mechanics. The development of quantum mechanics was initially motivated by several observations, which demonstrated the inadequacy of classical physics (among these the ``ultraviolet catastrophe'' and the photoelectric effect).

This course begins with the describing the failure of classical mechanics in giving correct explanation to atomic and molecular phenomena such as atomic spectra, black-body radiation and photoelectric effect, and the events and contributions that led to the development of quantum mechanics, starting from Planck’s energy density equation through Einstein’s explanation of the photoelectric effect and de Broglie’s thesis on wave-particle duality, up to Schrodinger’s development of the electronic wave equation and the practical approximations to solve it.

This leads to the field of quantum mechanics, as a new subject with its own set of theoretically and experimentally justifiable postulates. This course introduces the problems in classical mechanics and dives into the methods developed to apply the quantum mechanical set of equations for solving fundamental and applied problems. It will involve learning and using computational quantum mechanics software packages to abstract theory in context.

The 16 week course will introduce the basic concepts behind the development of quantum mechanics theory required to understand how the electronic/nuclear structure and composition drives material properties and behavior, present the computational models and methods from first-principles quantum mechanics (QM) and its approximations to characterize material systems, and demonstrate their application for studying and solving fundamental problems in chemistry and physics.

The course is meant as a foundational introduction to the field of computational quantum mechanics that exposes students to the tools of modern computational physics, chemistry and computational materials science relevant to atomistic descriptions of the structures and properties of chemical, biological, and materials systems. It is aimed at experimentalists and theorists in chemistry, materials science, chemical engineering, applied physics, biochemistry, physics, geophysics, and from other engineering programs with an interest in characterizing, optimizing or designing materials and understanding material properties and phenomena.

After successful course completion, the attendant will:

- Achieve a conceptual and theoretical understanding of electronic structure methods from quantum mechanics
- Acquire the practical knowledge and skills about first-principles quantum chemistry methods, specifically on ab initio methods and higher-level approximations such as density functional theory (DFT).
- Learn how to use quantum chemistry codes to characterize material systems.

**Prerequisites**

- Basic knowledge of C/C++, Python and Unix is required. Will also be used during the course lectures as well as in class assignments. Mathematica or Matlab are a plus.
- Background required in differential equations, statistical and classical mechanics, and linear algebra.
- The students must have successfully completed a course on Physical Chemistry.

This course deals with ab initio (first-principles) computations that are derived directly from theoretical principles, with no inclusion of experimental data. For the most part, this refers to an approximate quantum mechanical calculation. We will discuss these approximations, which are mostly mathematical approximations, such as using a simpler functional form for a function or getting an approximate solution to a differential equation.

1. History and Formulations of Quantum Mechanics (2 sessions)

- History of Quantum Mechanics
- The wave equation
- The Fundamental Postulates of Quantum Mechanics

We start with the Hartree Fock calculation (abbreviated HF), in which the primary approximation is called the central field approximation that does not explicitly take into account the Coulombic electron-electron repulsion. However, it's net effect is included in the calculation. This is a variational calculation, meaning that the approximate energies calculated are all equal to or greater than the exact energy. Because of the central field approximation, the energies from HF calculations are always greater than the exact energy and tend to a limiting value called the Hartree Fock limit.

2. Methods for Approximating the Solution of the Schrödinger Wave Equation (2 sessions)

- Introduction
- The Variation Method
- Perturbation Theory
- Time-independent Rayleigh-Schroedinger Perturbation Theory
- Perturbation Treatment for Nondegenerate Energy Levels
- Perturbation Treatment of a Degenerate Energy Level

- Time-Dependent Perturbation Theory
- Interaction Between Radiation Electric Field and an Atomic or Molecular System

The second approximation in HF calculations is that some functional form, which is only known exactly for a few one-electron systems, must describe the wave function. The functions used most often are linear combinations of Slater type orbitals exp(-ax) or Gaussian type orbitals exp(-ax2), abbreviated STO and GTO. The wave function is formed from linear combinations of atomic orbitals or more often from linear combinations of basis functions. The exact set of basis functions used is often specified by an abbreviation, such as STO-3G or 6-311++g.

3. The Hartree-Fock Self-Consistent Field Method (3 sessions)

- The Hartree-Fock Self-Consistent Field Method for Atoms.
- Spin-Orbit Coupling
- The Born-Oppenheimer Approximation
- The Linear Combination of Atomic Orbitals Method
- The Hartree-Fock-SCF Method For Molecules
- LCAO-MO-SCF Method
- The Electron Correlation Energy
- Configuration Interaction

A number of types of calculations begin with a HF calculation then correct for the explicit electron-electron repulsion, referred to as correlation. Some of these methods are Mohler-Plesset perturbation theory (MPn, where n is the order of correction), the Generalized Valence Bond (GVB) method, Multi-Configurations Self Consistent Field (MCSCF), Configuration Interaction (CI) and Coupled Cluster theory (CC). As a group, these methods are referred to as correlated calculations.

4. Computational Methods in Quantum Chemistry (3 session)

- The Zero Differential Overlap Approximation
- Hueckel MO Theory of Conjugated Molecules
- The Fock Matrix Elements in the ZDO Approximation
- The Pariser-Parr-Pople-SCFrc-MOMethod
- The Extended Hueckel Theory
- All Valence Electrons Semiempirical SCF MO Methods
- Mohler-Plesset (MP) Perturbation Treatment of Many-Electron Systems
- The Coupled Perturbed Hartree-Fock (CPHF) Theory
- The Coupled Cluster (CC) Theory

A method that avoids making the HF mistakes in the first place is called Quantum Monte Carlo (QMC). There are several flavors of QMC, including variational, diffusion and Green's functions. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo integration. These calculations can be very time consuming, but they are probably the most accurate methods known today.

5. Quantum Monte Carlo (2 sessions)

- Quantum manybody problem
- Metropolis
- Statistical foundations of Monte Carlo methods
- Variational Quantum Monte Carlo
- Diffusion Quantum Monte Carlo
- Path Integral Monte Carlo
- Typical QMC calculation

An alternative ab initio approximate method is Density Functional Theory (DFT), in which the total energy is expressed in terms of the total electron density, rather than the wavefunction. In this type of calculation, there is an approximate Hamiltonian and an approximate expression for the total electron density.

6. Density Functional Theory (2 sessions)

- Theoretical motivation
- Foundations
- Kohn-Sham self-consistent field methodology
- Exchange-correlation functionals
- DFT vs MO theory
- Performance

7. Quantum Mechanical Studies of Hydrogen Bonding (1 sessions)

- Electrostatic Theory of the H-bond
- Potential Function Treatments of the H-bond
- Studies on H-bonding Using Variational Methods
- Perturbation Theory and the H-bond

8. Application to other molecular, surface and bulk systems (1 sessions)

**Simulation packages and tools**

We will be using different modeling and simulation tools to demonstrate and apply the theory and methods described during the lectures, among these:

- SeQuest general purpose electronic structure code for periodic systems, slabs or finite molecules (Sandia National Laboratory, SNL, http://dft.sandia.gov/Quest/)
- Quantum Espresso general purpose electronic structure code for periodic systems, slabs or finite molecules (http://www.quantum-espresso.org)
- Octopus time-dependent DFT code (http://www.tddft.org/programs/octopus/wiki/index.php/Main_Page)
- GAMESS general ab initio quantum chemistry package (http://www.msg.chem.iastate.edu/gamess/)
- Jaguar (commercial package http://www.schrodinger.com/Jaguar/)

A brief tutorial on the use of a QM package will be offered. The student will be responsible for learning other tools that he/she chooses to use for the same level of calculations as those listed here.

**Other useful tools:**

- VMD visual molecular dynamics (http://www.ks.uiuc.edu/Research/vmd/)
- Ovito molecular visualizer (http://www.ovito.org)
- Avogadro molecule builder (http://avogadro.cc/wiki/Main_Page)

Attend lectures, read the class notes ahead of time (if applicable), read the suggested scientific literature recommended during class sessions, practice simulation runs on local cluster, and solve all the assignments, including the class project individually (unless explicitly stated otherwise). Teamwork is encouraged, but you must hand in your own results for all assignments.

30% | Homework assignments |

20% | Midterm |

20% | Final |

20% | Project |

Homework will be handed out each week during the first 8 weeks, and it will use generally available computer software implementing the basic methods on applications aimed at exposing the students to understanding how to use atomistic simulations to solve problems. Each calculation requires making decisions on the specific approaches and parameters relevant and how to analyze the results. The lowest homework score will be dropped from the final grade.

No late assignments will be accepted.

The midterm will require each student to submit a proposal for a project using the methods of this course to solve a research problem that can be completed in the final 8 weeks. The homework for the last 8 weeks is to turn in a one-page report on progress with the project.

The final is a research report describing the calculations and conclusions.

**References**

- Essentials of Computational Chemistry: Theories and Models, Christopher J. Cramer, Wiley
- Computational Methods in Quantum Chemistry, Ahmed A. Hasanein, Myron Wyn Evan