4. One – and many electronic wave functions (multplicty)

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 Spin symmetry restrictions For electrons (e), protons (p) and neutrons (n) the one paticle wave function u include both a space function f and spin function (a or b).

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. Relationship between spin, multiplicity and quantum description

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

6. Example:The singlet and triplet wave functions may be illustrated for a two electron system such as He or H2. Single substitution involves the replacement of f1 in one of the 2 columns by f2 and

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 ith,jth elements of defined as <fi|fj>. For an orthonormal set of functions{fi} =1 (i.e the unit matrix). Consequently = will be a diagonal matrix. Transformation of AO(h) to MO(f) 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 Fijh are evaluated (hijh), (Jijh) and (Kijh) integrals, The total electronic energy (E)

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

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. Start Evaluate all molecular integrals Construct V-matrix from overlap Matrix S to orthogonalise {h}to{c} Initial coefficient matrixThe simplest guess is Set E > 0 From Density Matrix From the three matrices Calculate Energy: En Print out En Calculate the difference DEn=En-1-En D=DEn-DEdesired Print out eP and Cij Make Decision Form F-matrix End of SCF Transform Fhto Fc Diagonalize Fc Form coefficient matrix “direct SCF” Flowchart for the iterative traditional Self-Consistent-Field (SCF) method.

18. Start Initial coefficient matrixThe simplest guess is Set E > 0 From Density Matrix Evaluate all molecular integrals Construct V-matrix from overlap Matrix S to orthogonalise {h}to{c} From the three matrices Calculate Energy: En Print out En Calculate the difference DEn=En-1-En D=DEn-DEdesired Print out eP and Cij Make Decision Form F-matrix End of SCF Transform Fhto Fc Diagonalize Fc Form coefficient matrix 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. The convergence of ESCF to EHF with increasing basis set size for the ground state of LiH. A breakdown of total energy for episulfide (C2H4S) to experimentally observable and quantum chemically calculable fraction. J. Chem. Phys. 44, 1849 (1966)