Ψ ( r1 , r2 ,..., rn ) = ψ 1 (r1 )ψ 2 (r2 )... ψ n (rn )
From Quantum Mechanism to Computational Chemistry ‘Orbital’ Approximation (single-electronic Approximation): • The Pauli principle needs that the total electronic wave function must be antisymmetric. • The wave function in Slater determination is antisymmetric. ψ (r )ψ (r )...ψ (r )
Pµν =
OCC i =1
∑ 2Cµ Cν
i
i
S µν = < φµ | φν >
Ab Initio
Hartree-Fock-Rootheen Equation
ε = 2∑ H ii + ∑ ∑ (2 J ij − K ij )
i i j n n n
= 2∑ ε i − ∑ ∑ (2 J ij − K ij )
HeΨi = EeΨi
Zα Z β Zα 1 1 2 H e =− ∑ ∇i + ∑ ∑ + ∑∑ − ∑∑ 2 i α β >α Rαβ α i i> j r i r ij iα
From Quantum Mechanism to Computational Chemistry ‘Orbital’ Approximation (single-electronic Approximation): • Consider each electron to move in some sort of "average potential" which incorporates the interactions with all the nuclei and an "averaged interaction" with the other electrons. • The wavefunction is taken to be a product of one electron wavefunctions (Molecular Orbital, MO):
– The Møller-Plesset Perturbation Theory (MP) – Configuration Interaction (CI) – Coupled Claser (CC)
Take into consideration electron correlation
Ab Initio
µ
Ab Initio
Hartree-Fock-Rootheen Equation
FC = SCE
1 Fµν = H µν + ∑ Pλσ ( µν | λσ ) − ( µλ | νσ ) 2 λσ
ˆ |φ 〉 〈φµ | h H µν = ν ZC 1 2 = 〈φµ | − ∇ | φν 〉 + 〈φµ | −∑ | φν 〉 2 C rC
1 1 2 1 N 1
Ψ SD
1 ψ 1 (r2 )ψ 2 (r2 )...ψ N (r2 ) = N ! ................................ ψ 1 (rN )ψ 2 (rN )...ψ N (rN )
Ab Initio
Hartree-FockTheory Or self-consistent field method (SCF)
From Quantum Mechanism to Computational Chemistry Time-independent Schrödinger Equation HΨ = EΨ Relativistic Approximation
Born-Oppenheimer Approximation
Ab Initio
Møller-Plesset Perturbation Theory
• Adding electron correlation effects by means of Rayleigh– Schrödinger perturbation theory (RS-PT). • Usually to second (MP2), third (MP3) or fourth (MP4) order. • The unperturbed Hamilton operator is taken as a sum over Fock operators. Then
• Assumes that the exact, N-body wave function of the system can be approximated by a single Slater determinant. • MO’s were assumed to be a linear combination of a finite number of basis functions. (LCAO)
i i j
n
n
n
=
（ ε ∑ii+ H ii )
2.0∑ P µν ( H µν + Fµν )
µ ,ν
Solve HFR equation iteratively
Ab Initio
Closed Shell and Open Shell
• Closed
Shell: RHF • Open Shell: UHF, ROHF
• MP0 = E(MP0) =
∑ε
i =1
N
i
• MP1 = MP0 + E(MP1) = E(HF)
• E ( MP 2)
= ∑∑
i < j a <b
occ vir
[< φiφ j | φaφb > − < φiφ j | φbφa >]2
εi + ε j − εa − εb
Ab Initio
α β
RHF singlet
UHF doublet
Based on Hartree-Fock-Roothaan Equation
SE
Post SCF
FC=SCE
VB MM
DFT
Ab Initio
Ab Initio
• Ab initio translated from Latin means “from first principles.” This refers to the fact that no experimental data is used and computations are based on quantum mechanics. • Different Levels of Ab Initio Calculations
Electronic Correlation
• The interaction between electrons in the electronic structure of a quantum system. • Correlation Energy
Ab Initio
Electronic Correlation
Configuration Interaction
• Configuration, for example, (1s)2(2s)2(2p)1... • Generally, using the linear combination of Slater determinants descripts configuration wave function. (For the closed shell grounded state, one SD is enough.) • Interaction means the mixing (interaction) of different electronic configurations (states). • CI wave function:
Computational Chemistry and Software
Computational chemistry: A branch of chemistry that uses principles of computer science to assist in solving chemical problems. • Use the results of theoretical chemistry • Incorporate into efficient computer programs • Calculate the structures and properties of molecules and solids. --- Wikipedia （维基百科）
• Coulomb correlation: Because single determinant wavefunction • Fermi correlation: Preventing two parallel-spin electrons from being found at the same point in space • Dynamic correlation: is the correlation of the movement of electrons (CI) • Static correlation: is important for molecules where the ground state is well described only with more than one (nearly-)degenerate determinant (MCSCF)