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The physical meaning of orbital energies (Read 2295 times)
Gerrit-Jan Linker
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The physical meaning of orbital energies
18.09.08 at 08:20:14
 
The physical meaning of orbital energies
 
The result of a SCF calculation is a set or molecular orbitals with orbital energies.
 

  • Hartree Fock SCF orbital energies
    The HF–SCF equations FФi = εiФi imply that εi can be written as

     
    εi is the average value of the kinetic energy plus the Coulombic attraction to the nuclei for an electron in Фi plus the sum over all spin orbitals occupied in ψ of the Coulomb minus exchange interactions.  
     
    If Фi is an occupied spin orbital, the term [Ji,i – Ki,i] disappears. The latter sum represents the Coulomb minus exchange interaction of Фi with all of the N – 1 other occupied spin orbitals.  
     
    εi = hii + occj( <ij|ij> - <ij|ji> ). Usually the K integral is much smaller than the J integral except when i=j.  We approximate <ij|ji> ≈ δij<jj|jj>.
    εi = hii + occj( <ij|ij> - δij<jj|jj>)
    Фi is an occupied spin orbital, therefore j=i exists in the sum and εi = hii + occi (≠j)( <ij|ij>)
     
    When Фa is an occupied spin orbital, j=a does not exists in the sum and εa = haa + occj( <aj|aj>)
     
    Therefore for a virtual (unoccupied) spin orbital Фa, this cancellation does not occur, and one obtains the Coulomb minus exchange interaction of Фa with all N of the occupied spin orbitals. Thus, orbital energies of occupied orbitals pertain to interactions appropriate to a total of N electrons orbital energies of virtual orbitals pertain to a system with N electrons.  
     
    The unphysical situation for virtuals usually makes SCF virtual orbitals not very good for use in subsequent correlation calculations or for use in interpreting electronic excitation processes. To correlate a pair of electrons that occupy a valence orbital requires double excitations into a virtual orbital of similar size; the SCF virtual orbitals are too diffuse.  
     
    Note that in DFT this is different.  
     
    Source:
    Encyclopedia of Chemical Physics and Physical Chemistry. J.J. Moore.
     
  • DFT SCF orbital energies
    The lumo's are lower in energy than the HF SCF lumo energies leading to smaller homo-lumo gaps in DFT. The definition of a lumo is different in DFT than in HF.

 
HOMO: Highest Occupied Molecular Orbital
LUMO: Lowest Unoccupied Molecular Orbital
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« Last Edit: 02.11.13 at 09:37:28 by Gerrit-Jan Linker »  

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