3. Discussion of the Hartree-Fock Equations

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N N N1∑ε2 ∑∑ ( ). (3.9)E = − J −KHF i ij iji= 1 i= 1 j=1The reason is because there is a “double counting” of electron-electron repulsions in Eq.(3.8): the energy of an orbital is the kinetic energy plus electron-nuclear attraction energyplus the electron-electron repulsion (with exchange corrections) to all the other orbitals inthe system. However, in the total energy formula, we count the electron repulsion betweenorbitals i and j only once—not once for ψ ( )ir interacting with ψ ( )jr and a second time forψ r interacting with ψ ( r ), as in Eq. (3.8).( )jiBased on the preceding argument, and using the fact J ij> K ij, we see that if youassume the total energy is the sum of the orbital energies, you obtain an energy that is toobig. Indeed, the energy so computed is usually extremely poor.At first, the result in Eq. (3.9) may be counterintuitive: shouldn’t the electronicenergy of a molecule be the sum of the energies of the composing electrons? Equation (3.9)says this is not the case, but gives us little insight into why it is not the case. To make thisa bit more clear, suppose his kind students enroll your professor in the local chapter ofMisanthropes Anonymous.B. The energy it takes to excite an electron from an occupiedorbital, ψ ( )iz , to a unoccupied orbital, ψ ( )az , is not equal to thedifference in the orbital energies, ε a− ε i.We can approximate the Hartree-Fock wave function for an excited state of thesystem by replacing an occupied orbital with an unoccupied orbital in the Hartree-Fockwave function, obtaining the Slater determinantΦ ≡ ψ … ψ ψ ψ ψ … ψ . (3.10)ai 1 i− 1 a i+ 1 i+2 NNote that this Slater determinant is not stationary with respect to the energy: because ofthe self-consistency requirement, exciting an electron changes and Fock operator, whichcauses the orbitals to change. When we skip the self-consistent procedure that produces this“revision” in the orbitals due to the excitation, we say that have neglected orbitalrelaxation. Neglecting orbital relaxation, the excitation energy can be approximated withi a a ˆ aE → ≈ Φ H Φ − Φ HˆΦ (3.11)exc.i i HF HFand, using the formula for the energy of a Slater determinant, Eq. (3.7), we obtain4
E ≈ E −Ei→a aexc i HF⎛k≠i; l≠ i k= a, l≠i k≠ i;l=a⎞ N N N N N1 1 1≈ hk 2 ( Jkl Kkl ) ha 2 ( Jal Kal ) 2 ( Jka Kka)∑ + ∑∑ − + + ∑ − + ∑ − ⎜k= 1 k= 1 l= 1 l= 1 k=1⎜⎝k≠i k≠i l≠i l≠i k≠i⎠⎟⎛N N N⎞1− ⎜∑hk +2 ∑∑( Jkl −Kk)l⎜⎝k= 1 k= 1 l=1 ⎠⎟⎛ ⎛N⎞ ⎛N⎞⎞1 1h a 2 ( Jal Kal) ( Jai Kai) 2 ( Jka Kka) ( Jia Kia)+ − − − + − − −⎜∑⎟ ⎜∑⎝ l= 1 ⎠ ⎝k=1⎠⎟l= i k=i= NNk==l i1 1−hi −2∑( Jki −Kki) −2∑( Jil −Kil) −( Jii −Kii)k= 1 l=1⎝⎜k≠i l≠i⎟⎠⎛N⎞ ha + ( Jka −Kka) −( Jai −Kai)∑k=1= N⎜−hi −∑( Jki −Kki)−0⎜⎝ k=1⎠⎟= εa −εi −( Jai −K(3.12)ai)The excitation energy is less than the difference in orbital energies because excitingan electron from an occupied orbital leaves a “vacancy” in the electron distribution thesystem—a place where electrons are well stabilized (as evidenced by the occupied orbital inthat vicinity) but which is, in the excited state, vacant. The excited state orbital is“attracted” to this region, which accounts for the J ai− K aiterm:2( )* *ψi r ψa( r′ ) ψ ( ) ( )ar ψi r ψi ( r′ ) ψa( r′)−( Jia−Kia) ≡− ∫∫drdr′ + δσσd d ′.a i− ′ ∫∫r r (3.13)r r r −r′Note that because Kai> 0 unless the spins of the orbitals, ψ ( )az and ψ ( )iz , are different,we conclude that if the α -spin and β -spin orbitals are the same, as they are for a tripletstate, then flipping the spin of the electron (to form a triplet excited state) gives a smallerexcitation energy. This predicts that triplet excited states will be more stable than singletexcited states, which is consistent with Hund’s maximum multiplicity rule.C. The orbital energy of occupied orbitals are approximations tothe ionization potentials of the system.We can approximate the Hartree-Fock wave function for the stationary states of thecation by removing an occupied orbital from the Hartree-Fock wave function,Φ ≡ ψ … ψ ψ ψ … ψ(3.14)freei 1 i− 1 i+ 1 i+2 N5
Page 2 and 3: desired, the unoccupied) orbitals i
Page 6: where the notation indicates than a
Page 9 and 10: There aren’t many compounds that

3. Discussion of the Hartree-Fock Equations

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