# 6: Electronic Structure

- Page ID
- 11556

Learning OBjectives

The subjects you should now be familiar with include

- The Hatree and Hartree-Fock models,
- Koopmans’theorem
- Atomic basis functions- Slater and Gaussian- and the notations used to describe them.
- Static and dynamic electron correlation.
- The CI, MPPT, CC, and DFT methods for treating correlation, as well as EOM or Greens function methods.
- The Slater-Condon rules.
- QM-MM methods.
- Experimental tools to probe electronic structures including methods for metastable states.
- Various contributions to spectroscopic line shapes and line broadening.

Electrons are the “glue” that holds the nuclei together in the chemical bonds of molecules and ions. Of course, it is the nuclei’s positive charges that bind the electrons to the nuclei. The competitions among Coulomb repulsions and attractions as well as the existence of non-zero electronic and nuclear kinetic energies make the treatment of the full electronic-nuclear Schrödinger equation an extremely difficult problem. Electronic structure theory deals with the quantum states of the electrons, usually within the Born-Oppenheimer approximation (i.e., with the nuclei held fixed). It also addresses the forces that the electrons’ presence creates on the nuclei; it is these forces that determine the geometries and energies of various stable structures of the molecule as well as transition states connecting these stable structures. Because there are ground and excited electronic states, each of which has different electronic properties, there are different stable-structure and transition-state geometries for each such electronic state. Electronic structure theory deals with all of these states, their nuclear structures, and the spectroscopies (e.g., electronic, vibrational, rotational) connecting them. In this Chapter, you were introduced to many of the main topics of electronic structure theory.

- 6.1: Theoretical Treatment of Electronic Structure
- The Born-Oppenheimer electronic energy, (E(r), as a function of the 3N coordinates of the N atoms in the molecule plays a central role. It is on this landscape that one searches for stable isomers and transition states, and it is the second derivative (Hessian) matrix of this function that provides the harmonic vibrational frequencies of such isomers. This chapter will introduce to the tools used to solve the electronic Schrödinger equation to generate E(R) and the electronic wave function.

- 6.3: The Hartree-Fock Approximation
- The Hartree approximation ignores an important property of electronic wave functions- their permutational antisymmetry.

- 6.6: The Slater-Condon Rules
- To form Hamiltonian matrix elements between any pair of Slater determinants constructed from spin-orbitals that are orthonormal, one uses the so-called Slater-Condon rules. These rules express all non-vanishing matrix elements involving either one- or two- electron operators.

- 6.8: High-End Methods for Treating Electron Correlation
- Although their detailed treatment is beyond the scope of this text, it is important to appreciate that new approaches are always under development in all areas of theoretical chemistry. In this Section, I want to introduce you to two tools that are proving to offer high precision in the treatment of electron correlation energies. These are the so-called quantum Quantum Monte-Carlo and r1,2- approaches to this problem.

- 6.9: Experimental Probes of Electronic Structure
- Visible and ultraviolet spectroscopies used study transitions between states of molecules/ions - these are called electronic transitions. When such transitions occur, the initial and final states generally differ in their electronic, vibrational, and rotational energies because any change to the electrons' orbital occupancy will induce changes in the Born-Oppenheimer energy surface which governs the vibrational and rotational character.

- 6.10: Molecular Orbitals
- Before moving on to discuss methods that go beyond the HF model, it is appropriate to examine some of the computational effort that goes into carrying out a HF SCF calculation on a molecule.

## Contributors and Attributions

Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry

Integrated by Tomoyuki Hayashi (UC Davis)