Schrodinger equation hydrogen molecule, Hamiltonian

So, let s get a bit more chemical and imagine the formation of an H2 molecule from two separated hydrogen atoms, Ha and Hb, initially an infinite distance apart. Electron 1 is associated with nucleus A, electron 2 with nucleus B, and the terms in the electronic Hamiltonian / ab, ba2 and are all negligible when the nuclei are at infinite separation. Thus the electronic Schrodinger equation becomes... 

In principle, it should be possible to obtain the electronic energy levels of the molecules as a solution of the Schrodinger equation, if inter-electronic and internuclear cross-coulombic terms are included in the potential energy for the Hamiltonian. But the equation can be solved only if it can be broken up into equations which are functions of one variable at a time. A simplifying feature is that because of the much larger mass of the nucleus the motion of the electrons can be treated as independent of that of the nucleus. This is known as the Bom-Oppen-heimer approximation. Even with this simplification, the exact solution has been possible for the simplest of molecules, that is, the hydrogen molecule ion, H + only, and with some approximations for the H2 molecule. 

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form ... 

Then it turned out that a promising approach (the test was for the hydrogen molecule) is to start with an accurate solution to the Schrodinger equation and go directly toward the expectation value of the Breit-Pauli Hamiltonian with this wave function (i.e., to abandon the Dirac equation), and then to the QED corrections. This Breit-Pauli... 

In the hydrogen atom, the electron is assumed to move in a circle around the nucleus, attracted by the Coulomb electrostatic force. The eigenstate of the electronic motion is therefore determined by the Schrodinger equation for the rotational motion. The Hamiltonian operator is in the same form as that for the rotational motions of molecules mentioned in the last section. Following Eq. (1.57), the Hamiltonian operator is represented as... 

The hydrogen molecule, like the He atom, poses a real problem the Hamiltonian operator contains a term representing the repulsion between the electrons, and the presence of this term makes an exact solution of the Schrodinger equation impossible. In the case of the helium atom we turned to the hydrogen atom for guidance in the choice of approximate wavefunctions. In the case of the hydrogen molecule we turn to the hJ ion and assume that the wavefunction may be approximated by the product of two molecular orbitals... 

The Schrodinger equation has not been solved exactly for electrons in molecules larger than the H2 ion the interactions of multiple electrons become too complex to handle. However, the eigenfunctions of the Hamiltonian operator provide a complete set of functions, and as mentioned in Sect. 2.2.1, a linear combination of such functions can be used to construct any well-behaved function of the same coordinates. This suggests the possibility of representing a molecular electronic wavefunction by a linear combination of hydrogen atomic orbitals centered at the nuclear positions. In principle, we should include the entire set of atomic orbitals... 

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