PY3004 Lecture 13: Molecular structure Lecture 13: Molecular structure o Hydrogen molecule ion (H 2 + ) o Overlap and exchange integrals o Bonding/Anti-bonding orbitals o Molecular orbitals PY3004 Schrödinger equation for hydrogen molecule ion Schrödinger equation for hydrogen molecule ion o Simplest example of a chemical bond is the hydrogen molecule ion (H 2 + ) . o Consists of two protons and a single electron. o If nuclei are far from each another, electron is localised on one nucleus. Wavefunctions are then those of atomic hydrogen. r ab r b r a + + - a b o ! a is the hydrogen atom wavefunction of electron belonging to nucleus a. Must therefore satisfy and correspondingly for other wavefunction, ! b . The energies are therefore PY3004 Schrödinger equation for hydrogen molecule ion Schrödinger equation for hydrogen molecule ion o If atoms are brought into close proximity, electron localised on b will now experience and attractive Coulomb force of nucleus a. o Must therefore modify Schrödinger equation to include Coulomb potentials of both nuclei: where o To find the coefficients c a and c b , substitute into Eqn. 2: (2) PY3004 Solving the Schrödinger equation Solving the Schrödinger equation o Can simplify last equation using Eqn. 1 and the corresponding equation for H a and H b . o By writing E a 0 ! a in place of H a ! a gives o Rearranging, o As E a 0 = E b 0 by symmetry, we can set E a 0 - E = E b 0 - E = -!E, (3)
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Schrödinger equation for hydrogen molecule ion · Lecture 13: Molecular structure ... Schrödinger equation for hydrogen molecule ion ... oSee Chapter 24 of Haken & Wolf for further
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