Hartree-Fock Theory
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+.
Particles are electrons Born-Oppenheimer approximation 1927
Born-Oppenheimer approximation Born-Oppenheimer approximation
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 Assuming 12 This may be written Jij = = < 12I12> = (11I22)
Jij, Coulombic integral for 2 e Jij = < 12I12> = (11I22) 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 √I12I Kij is a direct consequence of the Pauli principle Kij = = < 12I21A> = (12I12)
Kij, Exchange integral
Interactions of 2 electrons
Jj and Kj operators
Slater rules
Slater rules
Electronic energy i is the index for an orbital: EE = εi εi = <iIhIi> + (2Jij-Kij)
Electronic energy – total energy The sum of the energies of all the electrons contains the ij repulsion twice: EE = εi εi = <iIhIi> + (2Jij-Kij)
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
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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