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Unformatted text preview: Physical Chemistry Course Number: C362 11 The Born-Oppenheimer approximation, the Many Electron Hamiltonian and the time-independent molecular Schr¨odinger Equation 1. We have now seen a few problems where we have solved the time-independent Schr¨odinger Equation. Lets now consider a general molecular system com- prising N nuclei and n electrons. 2. We may write down the Hamiltonian for a molecular system comprising N nuclei and n electrons. In general this Hamiltonian will contain the following terms (note we are using lower case for electrons upper case for nuclei): (a) The nuclear kinetic energy: − ¯ h 2 2 N summationdisplay I =1 1 M I ∇ 2 I (11.1) (b) The electronic kinetic energy term. − ¯ h 2 2 m n summationdisplay i =1 ∇ 2 i (11.2) (c) The electron-nuclear attraction. − N summationdisplay I =1 n summationdisplay i =1 Ze 2 r i,I (11.3) Here r i,I ≡ | r i − r I | , where r i and r I are both vectors. (d) The nuclear-nuclear repulsion. + N summationdisplay I =1 N summationdisplay J =1 ′ Z I Z J e 2 r I,J (11.4) The summation here is restricted so there is no double counting (that is you should not have the ( I, J ) and ( J, I ) terms) and that I negationslash = J . (e) The electron-electron repulsion. + n summationdisplay i =1 n summationdisplay j =1 ′ e 2 r i,j (11.5) Here again the summation is restricted so there no double counting (that is you should not have the ( i, j ) and ( j, i ) terms) and that...
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- Winter '11
- Quantum Chemistry, Ri, Schr¨ dinger