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Unformatted text preview: XIII. The Hydrogen molecule We are now in a position to discuss the electronic structure of the simplest molecule: H 2 . For the low-lying electronic states of H 2 , the BO approximation is completely satisfactory, and so we will be interested in the electronic Hamiltonian 12 2 2 1 1 2 2 2 1 2 1 2 1 1 1 1 1 1 1 r R H B A B A AB el +-------- + - - = R r R r R r R r where 1 and 2 label the two electrons and A and B label the two nuclei. a. Minimal Atomic Orbital Basis It is not possible to solve this problem analytically, and so we want to follow our standard prescription for solving this problem: we define a basis set and then crank through the linear algebra to solve the problem in that basis. Ideally, we would like a very compact basis that does not depend on the configuration of the molecule; that is, we want basis functions that do not depend on the distance between the two nuclei, AB R . This will simplify the work of doing calculations for different bond lengths. The most natural basis functions are the atomic orbitals of the individual Hydrogen atoms. If the bond length is very large, the system will approach the limit of two non-interacting Hydrogen atoms, in which case the electronic wavefunction can be well approximated by a product of an orbital on atom A and an orbital on atom B and these orbitals will be exactly the atomic orbitals (AOs) of the two atoms. Hence, the smallest basis that will give us a realistic picture of the ground state of this molecule must contain two functions: A s 1 and B s 1 . These two orbitals make up the minimal AO basis for H 2 . For finite bond lengths, it is advisable to allow the AOs to polarize and deform in response to the presence of the other electron (and the other nucleus). However, the functions we are denoting A s 1 and B s 1 need not exactly be the Hydrogenic eigenfunctions; they should look similar to the 1s orbitals, but any atom-centered functions would serve the same purpose. Since the actual form of the orbitals will vary, in what follows, we will give all the expressions in abstract matrix form, leaving the messy integration to be done once the form of the orbitals is specified. b. Molecular Orbital Picture We are now in a position to discuss the basic principles of the molecular orbital (MO) method, which is the foundation of the electronic structure theory of real molecules. The first step in any MO approach requires one to define an effective one electron Hamiltonian, eff h . To this end, it is useful to split the Hamiltonian into pieces for electrons 1 and 2 separately and the interaction: ( ) ( ) B A B A h h R r R r R r R r---- - ---- - 2 2 2 2 2 1 1 1 2 1 2 1 1 1 2 1 1 1 12 12 1 r V The full Hamiltonian is then ( ) ( ) AB el R V h h H 1 2 1 12 + + + = where it should be remembered that within the BO approximation, AB R is just a number. For H 2 in a minimal basis, the simplest choice...
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