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Minimizing parameters combing several functions is generally an improvement (at the most, it is useless).What is better than a function? Several ones.What is stronger than a turkishman? Turkishmen.We can use functions localized on bonds, on vacancies… We can use any function not necessarily localized on atoms: for solids, plane waves are useful. There is strictly no requirement to start by searching functions close to hydrogenoids.It is however natural to search for solutions resembling them. There is no general solution for the Schrödinger except for hydrogenoids.In a strict sense, semi-empical,not ab-initio since an approximate (fitted) term is introduced in the hamiltonian. One can not compare results using different functionals*.Disadvantages: less reliable than IC or VB.Advantages : much less expensive than IC or VB.Hybrid methods: B3-LYP (Becke, three-parameters, Lee-Yang-Parr) n(r) = 8/(3h3) pFermi3 and T(n)=c ∫ n(r)5/3 drįirst theorem on Existence : demonstrationĮxchange correlation functionals VXC.Equating #of electrons in coordinate space to that in phase space gives: We fill out a sphere of momentum space up to the Fermi value, 4/3  pFermi3. It is postulated that electrons are uniformely distributed in space.Thomas-Fermi model (1927): The kinetic energy for an electron gas may be represented as a functional of the density. Its use goes back to the calculus of variations where one searches for a function which minimizes a certain functional. In other words, it is a function that takes a vector as its argument or input and returns a scalar. In mathematics, a functional is traditionally a map from a vector space to the field underlying the vector space, which is usually the real numbers.What is a functional? A function of another function:.IC: increasing the space of configuration Diexcitation promotion of i and j to k and l.Interaction Configuration: mono, di, tri, tetra excitations… MCSCF: The OM are optimized simultaneously with the IC (each one adapted to the state).Usually they are those for the ground state.OM/IC: In general the OM are those calculated in an initial HF calculation.To satisfy the Pauli principle, functions are determinants or linear combinations of determinants build from spinorbitals. There behavior for dissociation is then correct. It consists in describing electronic states of a molecule from AOs by eigenfunctions of S2, Sz and symmetry operators.Importance of correlation effects on Energy UHF: Variational solutions are not eigenfunctions of S2 Unpaired electrons: gu Singlet and triplet states bielectronic termsĪ combination of Slater determinant then may be an eigenfunction of S2 Kij is a direct consequence of the Pauli principle.|(1,2.)|2 does not depend on the ordering of particles 1,2.:.

Jij, Coulombic integral involved in two electron pairs When 1 are developed on atomic orbitals 1, bilectronic integrals appear involving 4 AOs (pqIrs).This integral is positive (it is a repulsion).Jij = (11I22) means the product of two electronic density  Coulombic integral.Let consider 2 electrons, one in orbital 1, the other in orbital 2, and calculate the repulsion.Since we get closer to a real solution, the energy decreases.

The process is iterated up to convergence.

Each one electron operator is the sum of one electron terms + bielectronic repulsions.The use of the term "orbital" was first used by Mulliken in 1925. A MO is a wavefunction associated with a single electron.