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.
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 12This 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
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 √I12IKij is a direct consequence of the Pauli principle Kij = = < 12I21A> = (12I12)
Kij, Exchange integral
Interactions of 2 electrons
Unpaired electrons: gu Singlet and triplet states bielectronic terms
is solution of S2
is not a solution of S2
A combination of Slater determinant then may be an eigenfunction of S2
UHF: Variational solutions are not eigenfunctions of S2
Electronic correlation VB method and polyelectronic functionsICDFT
Electronic correlation
Importance of correlation effects on Energy
Valence Bond
Heitler-London 1927
The resonance, ionic functions
H2 dissociation
MO behavior of H2 dissociation
The Valence-Bond method It consists in describing electronic states of a molecule from AOs by eigenfunctions of S2, Sz and symmetry operators. There behavior for dissociation is then correct. These functions are polyelectronic. To satisfy the Pauli principle, functions are determinants or linear combinations of determinants build from spinorbitals.
Covalent function for electron pairs
Interaction Configuration OM/IC: In general the OM are those calculated in an initial HF calculation. Usually they are those for the ground state.MCSCF: The OM are optimized simultaneously with the IC (each one adapted to the state).
Interaction Configuration: mono, di, tri, tetra excitations… Monexcitation: promotion of i to kDiexcitation promotion of i and j to k and l
IC: increasing the space of configuration
What is a functional? A function of another function: 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. In other words, it is a function that takes a vector as its argument or input and returns a scalar. Its use goes back to the calculus of variations where one searches for a function which minimizes a certain functional. E = E[(r)] E() = T() + VN-e() + Ve-e()
Thomas-Fermi model (1927): The kinetic energy for an electron gas may be represented as a functional of the density. It is postulated that electrons are uniformely distributed in space. We fill out a sphere of momentum space up to the Fermi value, 4/3 pFermi3 . Equating #of electrons in coordinate space to that in phase space gives: n(r) = 8/(3h3) pFermi3 and T(n)=c ∫ n(r)5/3 dr T is a functional of n(r).
DFT Two Hohenberg and Kohn theorems :
First theorem: on existence
First theorem on Existence : demonstration
Second theorem: Variational principle
Kohn-Sham equations
Kohn-Sham equations
Kohn-Sham equations
Exchange correlation functionals VXC[(r)]
Exchange correlation functionals SCF-X
Exchange correlation functionals VXC[(r)]
Hybrid methods: B3-LYP (Becke, three-parameters, Lee-Yang-Parr)
DFT - Advantages : much less expensive than IC or VB.
- adapted to solides, metal-metal bonds.
- Disadvantages: less reliable than IC or VB.
- One can not compare results using different functionals*. In a strict sense, semi-empical,not ab-initio since an approximate (fitted) term is introduced in the hamiltonian.
DFT good for IPs
DFT good for Bond Energies
DFT good for distances
DFT polarisabilities (H2O)
DFT dipole moments (D)
Basis sets There is no general solution for the Schrödinger except for hydrogenoids. It is however natural to search for solutions resembling them.There is strictly no requirement to start by searching functions close to hydrogenoids. We can use any function not necessarily localized on atoms: for solids, plane waves are useful. We can use functions localized on bonds, on vacancies…
What is stronger than a turkishman? TurkishmenWhat is better than a function? Several ones.Minimizing parameters combing several functions is generally an improvement (at the most, it is useless).Basis set associated with hydrogenoids: minimum basis set. more: extended basis set.
SCF limit Increasing the number of (independent) functions leads to improve the energy (variational principle). This improvement saturates. The limit is called SCF limit. This limit can be estimated by interpolation.
SCF convergence for H+
Variation of the Slater exponent Hydrogen =1.24 or =1.30 smaller than =1.0Diffuse orbitals: small “soft”contracted orbitals: large “hard” =Z/na0 < r> = a0 n(n+1/2)/Z = (n+1/2)/Z
Correlation The SCF energy is always above the exact energy. The difference is called the correlation energy, a term coined by Löwdin.A certain amount of electron correlation (Fermi correlation) is already considered within the HF approximation, found in the electron exchange term describing the correlation between electrons with parallel spin. The charge or spin interaction between 2 electrons is sensitive to the real relative position of the electrons that is not described using an average distribution. A large part of the correlation is then not available at the HF level. One has to use polyelectronic functions (VB method) or post Hartree-Fock methods (CI).
Incomplete Basis sets Some functions may be redundant. Therefore, it is better to express the functions on a basis set of orthogonal and normalized functions.
Basis set superposition error, BSSE Since basis sets are incomplete, there is an error when calculating A + B → C.Indeed, it seems fair to use the same basis set for A, B and C. However in C, the orbitals of B contribute to stabilize A if it the basis set to describe A is incomplete. The same is true the other way round. Each monomer "borrows" functions from other nearby components, effectively increasing its basis set and improving the calculation of derived properties such as energy Thus A and B should be better described using the orbitals centered on the other fragment than alone. It follows that A+B is underestimated relative to C.
Basis set superposition error, BSSE (counterpoise method). The energy gain for the reaction is therefore overestimated.For a diatomic formation, the solution is to calculate A and B using the full basis set for A+B. (B being a dummy atom when A is calculated). Ghost orbitals are orbitals localized where there is no nucleus (no potential).For an interaction between 2 large fragments, there is a problem of choosing the geometry for A: that of lowest energy for A or that in the fragment A-B. The method is estimating the BSSE correction in the fragment of A-B and assuming that it is the correction for A.
Basis set superposition error, BSSE (Chemical Hamiltonian approach) The (CHA) replaces the conventional Hamiltonian with one designed to prevent basis set mixing a priori, by removing all the projector-containing terms which would allow basis set extension.Though conceptually different from the counterpoise method, it leads to similar results.
Basis set transformation, orthogonalization
Orthogonal basis sets Advantage: - Leads to easier calculations (no rectangle terms). - objectivity (canonic or Löwdin)Inconvenient: - Interpretation becomes difficult; it is not possible to talk of AO occupancy if they are delocalized.
hydrogenoids Solving Schrödinger equation for hydrogenoids leads to spatial functions:(r,)= Nn,l rl Pn,l(r) exp(-Zr/n) R() Nn,l is a normalization function (it contains the dimension a0-3/2) rl is a power of r and Pn,l(r) a polynom of degree n-l-1exp(-Zr/n) makes the summation in the universe finite. R() is an spherical harmonic function.
Slater-type orbitals (STOs ) 1930
Slater-type orbital They are solution for a spherical potential V’ different from V allowing the same eigenfunctions and the same eigenvalues than the Schrödinger equation for the atom.V' = -Z/r + n* (n*-1)/2r2or 2r2[E-V']-2r2[E-V’] = n*(n*-1) Let verify for 2s= N r e-Zr/2:
Slater-type orbital Why this simplification?In LCAO, we make combinations of AOs.It is therefore useless to start by imposing the polynoms.What changes?The hydrogenoids are orthogonal <1sI2s>=0. (eigenfunctions associated with different quantum numbers).The Slater orbitals are not orthogonal. 1s+2s resembles the hydrogenoid 1s (no node) and 2s-1s resembles the 2s orbital (the combination makes the nodal surface appear.
Double zeta Using several Slater functions allows representing better different oxidation states. When there is an electron transfer, an atom could be A+, A° or A-. For a metal, often several atomic configurations are close in energy: s2d8, s1d9 or d10. This correspond to different exponents for the s and d AOs.Double and triple zeta functions 2 or 3 AOs adapted to one oxidation state each and allowing variation in a linear combination and flexibility.As soon as the reference to oxidation states disappears (Gaussian contractions) the terminology becomes less justified and just qualify the number of independent functions.
Gaussian functions
Gaussian functions fitting Slater functions with =1
Fit of a Slater-type orbital by STO-NG
Fit of a Slater-type orbital by STO-NG
Correspondance for ≠1
Minimal basis set and split valence basis set. STO-3G has been a long time used; with improvement of computing facilities, this is not the case nowadays in spite of the simplicity of using minimal basis. One way to improve accuracy is taking more functions.Releasing all contractions (N functions instead of 1 linear combination) is expensive. We can split the Gaussian into two sets. The partition may make groups or isolate the outermost primitive. The first procedure perhaps involves larger energy contribution but the second one is more chemical. The flexibility in reaction is necessary for the electron participating to the transformation (chemical reaction). If only the outermost primitive is isolated, we have the split-valence basis set named N-X1G by Pople.
Split valence basis set; N-X1G Core orbitals are represented by a single orbital with N primitives.Valence orbitals are represented by 2 orbitals: one orbital with X primitives and one diffuse orbital.
Forget about Slater : Minimal basis set Huzinaga and Dunning
Alternative partitioning for extended basis sets; Huzinaga and Dunning
cc-pVDZ and others
Polarized Basis sets
Basis sets for anions, 6-31++G**
Rydberg functions These orbitals are much more diffuse than the others associated with the valence. They are associated with loosely tight electrons occupying atomic orbitals with a quantum number n+1
Pseudopotentials
Pseudopotentials
Step calculations To save calculation efforts, on can use different accuracy for optimization of geometry and calculation of properties on the optimized geometry.UHF/3-21G(d)means that the geometry was optimized using UHF/3-21G(d) and the final result was calculated using UB3LYP/6-31G(d)
Counterintuitive effect Hij-ESij is usually negative since Hij is larger than -ESij. For high S values and low lying orbitals this can be not true.Then since interacting terms change sign, the out-of-phase combination become lower in energy than the in-phase combination.The change of sign also imposes some negative population of atomic orbitals and some values exceeding 2. Mulliken population are less reliable when the basis set is extended, since there are large S values between two functions that have nearly the same localization.
How many AOs? How many occupied MOs? How many vacant MOs? For C2H4
How many AOs? How many occupied MOs? How many vacant MOs? For C2H4
Do'stlaringiz bilan baham: |
