Solving the Schrödinger equation for a molecule first requires specifying the Hamiltonian and then finding the wavefunctions that satisfy the equation. Since the wavefunctions involve the coordinates of all the nuclei and electrons that comprise the molecule, the complete molecular Hamiltonian consists of several terms. The nuclear and electronic kinetic energy operators account for the motion of all of the nuclei and electrons. The Coulomb potential energy terms account for the interactions between the nuclei, the electrons, and the nuclei and electrons. 10.1: The Born-Oppenheimer Approximation The Born-Oppenheimer approximation is one of the basic concepts underlying the description of the quantum states of molecules. This approximation makes it possible to separate the motion of the nuclei and the motion of the electrons. 10.2: The Orbital Approximation and Orbital Configurations To describe the electronic states of molecules, we construct wavefunctions for the electronic states by using molecular orbitals. These wavefunctions are approximate solutions to the Schrödinger equation with each electron described by a product of a spin-orbitals Since electrons are fermions, the electronic wavefunction must be antisymmetric with respect to the permutation of any two electrons. A Slater determinant containing the molecular spin orbitals produces the antisymmetric wavefunction. 10.3: Basis Functions The molecular spin-orbitals that are used in the Slater determinant usually are expressed as a linear combination of some chosen functions, which are called basis functions. This set of functions is called the basis set. The fact that one function can be represented by a linear combination of other functions is a general property. All that is necessary is that the basis functions span-the-space, which means that the functions must form a complete set and must be describing the same thing. 10.4: The Case of H₂⁺ One can develop an intuitive sense of molecular orbitals and what a chemical bond is by considering the simplest molecule, H₂⁺, which consists of two protons held together by the electrostatic force of a single electron. Clearly the two protons, two positive charges, repeal each other. 10.5: Homonuclear Diatomic Molecules The LCAO-MO method that we used for H₂⁺ can be applied qualitatively to homonuclear diatomic molecules to provide additional insight into chemical bonding. A more quantitative approach also is helpful, especially for more complicated situations, like heteronuclear diatomic molecules and polyatomic molecules. Quantitative theories are described in subsequent sections. 10.6: Semi-Empirical Methods- Extended Hückel Hückel Molecular Orbital Theory is one of the first semi-empirical methods to be developed to describe molecules containing conjugated double bonds. This theory considered only electrons in pi orbitals and ignored all other electrons in a molecule and was successful because it could address a number of issues associated with a large group of molecules at a time when calculations were done on mechanical calculators. 10.7: Mulliken Populations Mulliken populations can be used to characterize the electronic charge distribution in a molecule and the bonding, antibonding, or nonbonding nature of the molecular orbitals for particular pairs of atoms. To develop the idea of these populations, consider a real, normalized molecular orbital composed from two normalized atomic orbitals. 10.8: The Self-Consistent Field and the Hartree-Fock Limit In a modern ab initio electronic structure calculation on a closed shell molecule, the electronic Hamiltonian is used with a single determinant wavefunction. This wavefunction, Ψ , is constructed from molecular orbitals, ψ that are written as linear combinations of contracted Gaussian basis functions. 10.9: Correlation Energy and Configuration Interaction The Hartree-Fock energy is not as low as the exact energy. The difference is due to electron correlation effects and is called the correlation energy. The Hartree-Fock wavefunction does not include these correlation effects because it describes the electrons as moving in the average potential field of all the other electrons. The instantaneous influence of electrons that come close together at some point is not taken into account. 10.E: Theories of Electronic Molecular Structure (Exercises) Exercises for the "Quantum States of Atoms and Molecules" TextMap by Zielinksi et al. 10.S: Theories of Electronic Molecular Structure (Summary) In general, electronic wavefunctions for molecules are constructed from approximate one-electron wavefunctions. These one-electron functions are called molecular orbitals. The expectation value expression for the energy is used to optimize these functions, i.e. make them as good as possible. The criterion for quality is the energy of the ground state. According to the Variational Principle, an approximate ground state energy always is higher than the exact energy. 10.10: Electronic States The electronic configuration of an atom or molecule is a concept imposed by the orbital approximation. While a single determinant wavefunction generally is adequate for closed-shell systems (i.e. all electrons are paired in spatial orbitals), the best descriptions of the electronic states, especially for excited states and free radicals that have unpaired electrons, involve configuration interaction using multiple determinants.