# 9: The Electronic States of the Multielectron Atoms

Multi-electron systems, including both atoms and molecules, are central to the study of chemistry. While we can write the Schrödinger equations for a two-electron atom and for many-electron atoms, the Schrödinger equations for atoms (and molecules too) with more than one electron cannot be solved because of electron-electron Coulomb repulsion terms in the Hamiltonian. These terms make it impossible to separate the variables and solve the Schrödinger equation. Fortunately, reasonably good approximate solutions can be found, and an active area of research for physical chemists involves finding methods to make them even better.

In this chapter you will learn several key techniques for approximating wavefunctions and energies, and you will apply these techniques to multi-electron atoms such as helium. You also will learn how to use the theoretical treatment of the electronic states of matter to account for experimental observations about multi-electron systems. For example, the periodic trends in ionization potential and atomic size that are presented in introductory chemistry texts and reproduced here in Figure 9.1 arise directly from the nature of the electronic states of the atoms in the periodic table.

• 9.1: The Schrödinger Equation For Multi-Electron Atoms
As with the hydrogen atom, the nuclei for multi-electron atoms are so much heavier than an electron that the nucleus is assumed to be the center of mass. Fixing the origin of the coordinate system at the nucleus allows us to exclude translational motion of the center of mass from our quantum mechanical treatment.
• 9.2: Solution of the Schrödinger Equation for Atoms: The Independent Electron Approximation
In this section we will see a useful method for approaching a problem that cannot be solved analytically and in the process we will learn why a product wavefunction is a logical choice for approximating a multi-electron wavefunction.
• 9.3: Perturbation Theory
Perturbation theory is a method for continuously improving a previously obtained approximate solution to a problem, and it is an important and general method for finding approximate solutions to the Schrödinger equation.
• 9.4: The Variational Method
In this section we introduce the powerful and versatile variational method and use it to improve the approximate solutions we found for the helium atom using the independent electron approximation. One way to take electron-electron repulsion into account is to modify the form of the wavefunction. A logical modification is to change the nuclear charge, $$Z$$, in the wavefunctions to an effective nuclear charge $$Z_{eff}$$.
• 9.5: Single-electron Wavefunctions and Basis Functions
Finding the most useful single-electron wavefunctions to serve as building blocks for a multi-electron wavefunction is one of the main challenges in finding approximate solutions to the multi-electron Schrödinger Equation. The functions must be different for different atoms because the nuclear charge and number of electrons are different. The attraction of an electron for the nucleus depends on the nuclear charge, and the electron-electron interaction depends upon the number of electrons.
• 9.6: Electron Configurations, The Pauli Exclusion Principle, The Aufbau Principle, and Slater Determinants
The assignment of electrons to orbitals is called the electron configuration of the atom. We extend that idea to constructing multi-electron wavefunctions that obeys the Pauli Exclusion Principle, which requires that each electron in an atom or molecule must be described by a different spin-orbital. The mathematical analog of this process is the construction of the approximate multi-electron wavefunction as a product of the single-electron atomic orbitals.
• 9.7: The Self-Consistent Field Approximation (Hartree-Fock Method)
In this section we consider a method for finding the best possible one-electron wavefunctions that was published by Hartree in 1948 and improved two years later by Fock.
• 9.8: Configuration Interaction
The best energies obtained at the Hartree-Fock level are still not accurate, because they use an average potential for the electron-electron interactions. Configuration interaction (CI) methods help to overcome this limitation. The exact wavefunction must depend upon the coordinates of both electrons simultaneously. This independent-electron approximation can take to account such electron correlation effects. The method for taking correlation into account is called Configuration Interaction.
• 9.9: Chemical Applications of Atomic Structure Theory
In this section we examine how the results of the various approximation methods considered in this chapter can be used to understand and predict the physical properties of multi-electron atoms. Our results include total electronic energies, orbital energies and single-electron wavefunctions that describe the spatial distribution of electron density.
• 9.E: The Electronic States of the Multielectron Atoms (Exercises)
Exercises for the "Quantum States of Atoms and Molecules" TextMap by Zielinksi et al.
• 9.S: The Electronic States of the Multielectron Atoms (Summary)