| Themes > Science > Physics > Molecular Physics > Molecular Hamiltonian & Born_Oppenheimer Approximation > Molecular H - How to solve via B.O. Approx and treating R parametrically |
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We desire to calculate energy levels (eignvalues) and wavefunctions or molecular orbitals (eigenfns) of molecules. Lets start with AB-type molecule where atoms are hydrogen like. The complete 10-term hamiltonian for the kinetic and pot'l energy of the 4 bodies (2 nuclei, 2 electrons): |
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Note that the terms from left to right are: K.E. of elec#1, K.E. of elec#2, K.E. of nucA, K.E. of nucB, Pot. E elec1/nucA, pot. E. elec1/nucB, pot. E elec2/nucA, pot. E elec2/nucB, pot. E. elec1/elec2 (repulsive), and pot. E. nucA/nucB (repulsive) Born Oppenheimer Approximation Due to relatively faster kinetic motion of electrons than nuclei, the nuclear K.E. and electron K.E. may be separated: ® y total = y elec y nuc. Etot = Eele + Enuc so we may obtain the electronic (plus nuclear repulsion) hamiltonian by dropping nuclear k.E. terms to obtain |
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The internuclear distance (and orientation of nuclei for polyatomics), i.e. bond distances and bond angles are fixed, and treated as parameters for the electronic hamiltonian. The above schrodinger is solved parametrically w.r.t. R, such that the true energy (binding energy) is Etot(R). This total energy (binding E) that takes into account electronic and nuclear motion is plotted as a function of nuclear positions and is called the Potential Energy Curve ( in 2-D it is a potential energy surface = PES). |
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