theory can also be used to simulate the behavior of molecules In carrying out simulations, 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 In the present Chapter, I want to provide you with an introduction to the tools that we use to solve the electronic Schrodinger equation to generate e(r) and the electronic wave function Y(rR). In essence, this treatment will focus on orbitals of atoms and molecules and how we obtain and interpret them For an atom, one can approximate the orbitals by using the solutions of the hydrogenic Schrodinger equation discussed in the Background Material. Although such ns are not proper solutions to the actual N-electron Schrodinger equation(believe it or not, no one has ever solved exactly any such equation for n> 1)of any atom, they can be used as perturbation or variational starting-point approximations when one may be satisfied with qualitatively accurate answers. In particular, the solutions of this one- electron Hydrogenic problem form the qualitative basis for much of atomic and molecular orbital theory. As discussed in detail in the background Material, these orbitals are labeled by n, I, and m quantum numbers for the bound states and by I and m quantum numbers and the energy E for the continuum states Much as the particle-in-a-box orbitals are used to qualitatively describe T electrons in conjugated polyenes or electronic bands in solids, these so-called hydrogen like orbitals provide qualitative descriptions of orbitals of atoms with more than a single electron. By introducing the concept of screening as a way to represent the repulsive 22 theory can also be used to simulate the behavior of molecules. In carrying out simulations, 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. In the present Chapter, I want to provide you with an introduction to the tools that we use to solve the electronic Schrödinger equation to generate E(R) and the electronic wave function Y(r|R). In essence, this treatment will focus on orbitals of atoms and molecules and how we obtain and interpret them. For an atom, one can approximate the orbitals by using the solutions of the hydrogenic Schrödinger equation discussed in the Background Material. Although such functions are not proper solutions to the actual N-electron Schrödinger equation (believe it or not, no one has ever solved exactly any such equation for N > 1) of any atom, they can be used as perturbation or variational starting-point approximations when one may be satisfied with qualitatively accurate answers. In particular, the solutions of this one￾electron Hydrogenic problem form the qualitative basis for much of atomic and molecular orbital theory. As discussed in detail in the Background Material, these orbitals are labeled by n, l, and m quantum numbers for the bound states and by l and m quantum numbers and the energy E for the continuum states. Much as the particle-in-a-box orbitals are used to qualitatively describe p￾electrons in conjugated polyenes or electronic bands in solids, these so-called hydrogen￾like orbitals provide qualitative descriptions of orbitals of atoms with more than a single electron. By introducing the concept of screening as a way to represent the repulsive