Hartree-Fock Theory The orbitalar approach


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Hartree-Fock Theory


The orbitalar approach



MO approach

  • A MO is a wavefunction associated with a single electron. The use of the term "orbital" was first used by Mulliken in 1925.



MO approach



Generalizing the LCAO approach: A linear combination of atomic orbitals or LCAO

  • It was introduced in 1929 by Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by Pauling for H2+.



Schrödinger equation for LCAO



Particles are electrons

  • Born-Oppenheimer approximation

  • 1927



Born-Oppenheimer approximation

  • Born-Oppenheimer approximation





Vibrations, nuclear motion











Approximation of independent particles

  • HHartree= i Fi

  • The hamiltonian, HHartree , is the sum of Fock operators only operating on a single electron i.

  • The wave function is the determinant of orbitals, (i) to satisfy the Pauli principle. Fi (i)=εi (i)



Approximation of independent particles

  • Fi = Ti + k Zik/dik+ j Rij

  • Each one electron operator is the sum of one electron terms + bielectronic repulsions



Self-consistency

  • Given a set of orbitals i, we calculate the electronic distribution of j and its repulsion with i.

  • This allows expressing

  • and solving the equation to find new i allowing to recalculate Rij. The process is iterated up to convergence. Since we get closer to a real solution, the energy decreases.



Jij, Coulombic integral for 2 e

  • Let consider 2 electrons, one in orbital 1, the other in orbital 2, and calculate the repulsion <1/r12>.

  • Assuming 12

  • This may be written

  • Jij = = < 12I12>

  • = (11I22)



Jij, Coulombic integral for 2 e

  • Jij = < 12I12> = (11I22) means the product of two electronic density  Coulombic integral.

  • This integral is positive (it is a repulsion). It is large when dij is small.

  • When 1 are developed on atomic orbitals 1, bilectronic integrals appear involving 4 AOs (pqIrs)



Jij, Coulombic integral involved in two electron pairs



Particles are electrons! Pauli Principle

  • electrons are indistinguishable:

  • |(1,2,...)|2 does not depend on the ordering of particles 1,2...:

  • |  (1,2,...)|2 = |  (2,1,...)|2



Particles are fermions! Pauli Principle







Kij, Exchange integral for 2 e

  • Let consider 2 electrons, one in orbital 1, the other in orbital 2, and calculate the repulsion <1/r12>.

  • Assuming √I12I

  • Kij is a direct consequence of the Pauli principle

  • Kij = = < 12I21A>

  • = (12I12)



Kij, Exchange integral



Interactions of 2 electrons



Unpaired electrons: gu Singlet and triplet states bielectronic terms



Jj and Kj operators



Slater rules



Slater rules



Electronic energy

  • i is the index for an orbital:

  • EE = εi εi = <iIhIi> + (2Jij-Kij)



Electronic energy – total energy

  • The sum of the energies of all the electrons contains the ij repulsion twice:

  • EE = εi εi = <iIhIi> + (2Jij-Kij)



Restricted Hartree-Fock

  • Closed-shell system: an MO is doubly occupied or vacant. The spatial functions are independent from the spin. Assuming k and l with spin , the expression of a Fock matrix-element, Fkl, is



Restricted Hartree-Fock



Koopmans theorem



Koopmans theorem



What is wrong with Koopmans theorem?





Open-Shell – spin contamination



ROHF

  • The same spatial function is taken whatever the spin is.

  • Robert K. Nesbet

  • This allows eigenfunctions of spin operators.

  • This allows separating spatial and spin functions.

  • It is not consistent with variational principle (that does better without this constraint).



Spin operator







Relations in Angular momentum

  • L2 is the norm and Lx, Ly, Lz are the projections.

  • [Jx, Jy]= i Jz [Jx, J2]= 0

  • [Jy, Jz]= i Jx eigenfunctions of J2 are j(j+1)

  • [Jz, Jx]= i Jy eigenfunctions of Jz are m

  • J2Ij,m> = j(j+1) Ij,m> and Jz Ij,m> = m Ij,m>



Introducing J+ = Jx+iJy and J- = Jx-iJy

  • J2 = Jx2 + Jy2 + Jz2 = Jz2 + ½ (J+J- + J-J+ )



spin

  • For an electron: s =1/2 → S2=s (s +1) =3/4 and ms=±1/2.

  • A spin vector is either  |s,ms> = |1/2,1/2> or : |s,ms> = |1/2,-1/2>



spin

  • For several electrons: the total spin is the vector sum of the individual spins.

  • For the projection Sz=Ms =s ms

  • For the norm S2 = S12+S22+2S1S2

  • S2=[S1z2+S1z+S1-S1+]+[S2z2+S2z+S2-S2+]+[(S1+S2- ++S1-S2+)+2S1zS2z]

  • For 2 electrons, there are 4 spin functions: and that are solutions of Sz.

  • Are they solution of S2?



 is solution of S2



 is not a solution of S2



± are solutions of S2



Use of indentation operators upon vector sums



S2 as an operator on determinant

  • S2(D)= P(D) + [(n-n)2 +2n +2n ] D

  • P is an operator exchanging the spins :



A single Slater determinant is not necessarily an eigenfunction of S2



A combination of Slater determinant then may be an eigenfunction of S2



UHF: Variational solutions are not eigenfunctions of S2



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