The Born Oppenheimer - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides, Slides of Chemistry

Download The Born Oppenheimer - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! ∑ ∑∑∑∑∑∑∑ − = += − = +== === ++−∇−∇−= 1 1 1 21 1 1 2 1 1 2 1 2 e 2 2 1 I 2 )( 2m )( 2M Ĥ N NN N I N IJ IJ JI en i en ij ij N I en i iI I en i NN I R eZZ r e r eZiI hh KE of nuclei Unlike the atomic systems, we now have the kinetic energy of the nuclei. We can no longer place ‘the’ nucleus at the origin and consider all coordinates relative coordinates. Thus, the total wave function is a function of both nuclei and electrons: ),...,,,...,( 2121 NNne QQQqqqΨ Again the lower case coordinates, q, refer to the electrons and the upper case coordinates, Q, refer to the nuclei. docsity.com The Born-Oppenheimer Approximation In the BO approximation, we assume the nuclei are FIXED as the electrons carry out their motion. ∑ ∑∑∑∑∑∑∑ − = += − = +== === ++−∇−∇−= 1 1 1 21 1 1 2 1 1 2 1 2 e 2 2 1 I 2 )( 2m )( 2M Ĥ N NN N I N IJ IJ JI en i en ij ij N I en i iI I en i NN I R eZZ r e r eZiI hh 0T̂N = This allows us to define an Electronic Hamiltonian, which does not contain the nuclear kinetic energy operator. ∑ ∑∑∑∑∑∑ − = += − = +== == ++−∇−= 1 1 1 21 1 1 2 1 1 2 1 2 e 2 el )( 2m Ĥ N NN N I N IJ IJ JI en i en ij ij N I en i iI I en i R eZZ r e r eZih We are making the approximation that there is no coupling between their motions. docsity.com NUCel total NUCel N NUCel el ETH ΨΨ=ΨΨ+ΨΨ ˆˆ ΨNUC does not depend on coordinates of electrons Here we assume no coupling between the electronic state and the nuclear motion (again the BO-approximation) NUCel total NUC N elel el NUC ETH ΨΨ=ΨΨ+ΨΨ ˆˆ kinetic energy of nuclei assumed to be zero in this approximation NUCelelel el NUC EH ΨΨ=ΨΨ ˆ elelel el EH Ψ=Ψˆ Divide by ΨNUC Electronic Schrodinger equation docsity.com Thus, if the nuclear kinetic energy is zero, then the Schrodinger equation for the molecule becomes: ):()():(ˆ Ii el I el Ii elel QqQEQqH nnn Ψ=Ψ Molecular Electronic Hamiltonian Electronic wave function for the nth state Electronic energy for the nth state n refers to the quantum state of the molecule Hel, Ψel and Eel all depend on the nuclear coordinates, Q, as parameters. They are parametrically dependent upon what frozen nuclear framework we have chosen. Thus, we solve for the electronic wave function, with the nuclei fixed. docsity.com Consider a diatomic. Eel Bond distance R1 elelelel nnn REH Ψ=Ψ )(ˆ 1 A B docsity.com Consider a diatomic. elelelel nnn REH Ψ=Ψ )(ˆ 4 R1 R2 R3 R4 A B Eel Bond distance docsity.com This leads to pictures involving more than one electronic state of the sort: Potential energy curve for each state is different docsity.com NOTE: ∑ ∑∑∑∑∑∑ − = += − = +== == ++−∇−= 1 1 1 21 1 1 2 1 1 2 1 2 e 2 el )( 2m Ĥ N NN N I N IJ IJ JI en i en ij ij N I en i iI I en i R eZZ r e r eZih Sometimes the last term is not included in the electronic Hamiltonian. This term corresponds to the nuclear-nuclear repulsion energy and is an ‘electronic’ energy term. Thus it is often dropped, such that: ∑∑∑∑∑ − = +== == +−∇−= 1 1 1 2 1 1 2 1 2 e 2 el )( 2m Ĥ en i en ij ij N I en i iI I en i r e r eZi Nh docsity.com Consider a diatomic where we calculate the electronic energy for various bond distances. Eel Bond distance This procedure, gives us what is called the potential energy surface. of a molecule which relates the energy of a molecular system to its nuclear geometry. The potential energy surface is a fundamental concept. Many important properties of molecular systems can be derived if one knows the potential energy surface of the system. docsity.com Bond distance Eel The Potential Energy Surface. •Relates the potential energy of a molecular system to its nuclear geometry. •It does not include the kinetic energy of the nuclei •It is often given the symbol, U = U(QI) or V = V(QI) •The potential energy surface is a fundamental concept. Many important properties of molecular systems can be derived if one knows the potential energy surface of the system. docsity.com What is the Physical Reasoning Behind the BO approximation? The B-O approximation takes advantage of the fact that the mass of the electron is much smaller than the mass of the nuclei. me : Mp+ = 1:1280electron to proton mass ratio (all other nuclei will have even larger ratios) Thus, the nuclei will move much, much slower than the electrons. Why? Thus, 22 2 1 2 1 NNee VMVm = Consider the classical picture, where a well equilibrated system suggests that the electrons and nuclei have the same kinetic energy: 2 2 e N N e V V M m = Another way of thinking of it is, that as the nuclei move (vibrate), we assume the electronic system can instantaneously adjust. docsity.com