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lecture33_umn - Lecture 33 The Potential Energy Surface...

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Lecture 33 December 4, 2009 The Potential Energy Surface Revisited Transition-state Structures and Chemical Kinetics The One-dimensional Schrodinger Equation for Molecular Vibration
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The Born-Oppenheimer Approximation H in eq. 17-2 contains pairwise attraction and repulsion terms: no particle is moving independently of all of the others correlation ’: interdependency of motion In order to simplify the problem, we may invoke the so-called Born-Oppenheimer approximation. (17-2) H = " h 2 2 m e i # $ i 2 " h 2 2 m k k # $ k 2 " e 2 Z k 4 %& 0 r ik k # i # + e 2 4 0 r ij i < j # + e 2 Z k Z l 4 0 r kl k < l #
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The Born-Oppenheimer Approximation Nuclei of molecular systems are moving much, much more slowly than the electrons. Electronic ‘relaxation’ with respect to nuclear motion is instantaneous. Decouple the two motions, compute electronic energies for fixed nuclear positions. Nuclear kinetic energy term is taken to be independent of the electrons. Correlation in the attractive electron-nuclear potential energy term is eliminated. Repulsive nuclear-nuclear potential energy : constant for a given geometry.
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The Born-Oppenheimer Approximation The electronic Schrödinger equation (17-13) el ’ because of BO approximation H el includes only first, third, fourth terms eq. 17-2 V N is the nuclear-nuclear repulsion energy electronic coordinates q i are independent variables nuclear coordinates q k are parameters Eigenvalue of the electronic Schrödinger equation: ‘electronic energy’ H el + V N ( ) ! el q i ; q k ( ) = E el ! el q i ; q k ( )
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The Potential Energy Surface Without the Born-Oppenheimer approximation we would lack the concept of a potential energy surface : The PES is the surface defined by E el over all possible nuclear coordinates . We would lack the concepts of equilibrium and transition state geometries , since these are defined as critical points on the PES.
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Born-Oppenheimer approx. allows us to separate electronic and nuclear motion
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PES Revisited PES is a hypersurface defined by the potential energy of a collection of atoms over all possible atomic arrangements; 3 N – 6 internal coordinate dimensions, N is the number of atoms 3. Dimensionality from the three-dimensional Cartesian space.
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lecture33_umn - Lecture 33 The Potential Energy Surface...

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