1. 4. One – and many electronic wave functions (multplicty) 5. The Hartree-Fock method and its limitations (Correlation Energy) 6. Post Hartree-Fock methods

2. The wave functions of a given state for an N-particle. If N=1 ParticlesSpinDescription ExamplesTypeEigenvalueStatisticsWave function , h BosonsnħBose-EinsteinSymmetric e, p, nFermions(n/2) ħFermi-DiracAnti- symmetric Spin symmetry restrictions For electrons (e), protons (p) and neutrons (n) the one paticle wave function u include both a space function  and spin function (  or  ).

3. Many-electron wave functions may be constructed as products of one-electron functions or spin-orbitals which equivalent to the formula for a determinant.

4. Multiplicity of the electronic system is related to the total spin: |S| of the electronic system. Number of odd electron s |S|mNameOccupancy Number of possible wave functions 001Singlet1 11/22Doublet2 213Triplet3 31 1/24Quadruplet4 Relationship between spin, multiplicity and quantum description

5. Variation of energy levels with increasing magnetic field strength (B) for system of different multiplicities

6. and Example: The singlet and triplet wave functions may be illustrated for a two electron system such as He or H 2. Single substitution involves the replacement of  1 in one of the 2 columns by  2

7. The linear combinations, having negative and positive signs, yield wave functions of singlet and triplet multiplicity respectively.

8. The doublet multiplicity demostrated on 3 electron system (Li).

9. 4. One – and many electronic wave functions (multplicty) 5. The Hartree-Fock method and its limitations (Correlation Energy) 6. Post Hartree-Fock methods

10. Variational Theorem (using a Slater determinant w.f) Hartree-Fock equation in integrated form And in the form of a matrix equation

11. Where the i th,j th elements of defined as. For an orthonormal set of functions{  i } =1 (i.e the unit matrix). Consequently = will be a diagonal matrix. Transformation of AO(  ) to MO(  ) for a system with 2M electrons and N different AO we obtain: in vector notation Hartree-Fock equation assumes the following form

12. From the coefficients of the M doubly occupied MO we may generate the density matrix(r) The elements of the Fock matrix F ij  are evaluated (h ij  ), (J ij  ) and (K ij  ) integrals, The total electronic energy (E)

13. A schematic illustration for the sequence transformation of AO to MO. Transformation from the 00 basis set (  ) to the MO basis set (  )

14. In matrix notation

15. A vector model of AO and MO orbitals assoiciated with H2 Convergence to the Hartree- fock limit (HFL) with increasing basis set size(N).

16. The concept of correlation energy and other components.

17. Make Decision Start Evaluate all molecular integrals Construct V-matrix from overlap Matrix S to orthogonalise {  }to{  } Set E > 0 From Density Matrix From the three matrices Calculate Energy: E n Initial coefficient matrix The simplest guess is Calculate the difference  E n =E n-1 -E n  =  E n -  E desired Form F-matrix Transform F   to F  Diagonalize F  Form coefficient matrix Print out  P and C ij End of SCF Print out E n “direct SCF” Flowchart for the iterative traditional Self-Consistent-Field (SCF) method.

18. Make Decision Start Evaluate all molecular integrals From Density Matrix From the three matrices Calculate Energy: E n Initial coefficient matrix The simplest guess is Calculate the difference  E n =E n-1 -E n  =  E n -  E desired Form F-matrix Transform F   to F  Diagonalize F  Form coefficient matrix Print out  P and C ij End of SCF Print out E n Set E > 0 Construct V-matrix from overlap Matrix S to orthogonalise {  }to{  } Flowchart for the iterative Direct Self- Consistent-Field (SCF) method.

19. 4. One – and many electronic wave functions (multplicty) 5. The Hartree-Fock method and its limitations (Correlation Energy) 6. Post Hartree-Fock methods

20. A breakdown of total energy for episulfide (C 2 H 4 S) to experimentally observable and quantum chemically calculable fraction. The convergence of ESCF to EHF with increasing basis set size for the ground state of LiH. J. Chem. Phys. 44, 1849 (1966)