Hartree-Fock Theory The orbitalar approach

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

• The spinorbitals should be solution of SZ and S2. This is not the case for UHF.

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