Many-Electron Atoms

1. Many-Electron Atoms Escuela Superior de Física y Matemáticas Instituto Politécnico Nacional Mexico City Juan Ignacio RodríguezHernández Part of the Course: Molecular Modelling September 2010

2. Many Electron Atoms For gold N=79, so we have 3*79=237 independent variables !!!

3. e= Atomic Units (a.u.)

4. Energy: Lenth: Mass: Atomic Units (a.u.) Charge:

5. Many electron atom Hamiltonianin a.u.

6. Spin angular momentum

7. Spin angular momentum

8. A postulate: Spin is an intrinsic variable

9. Orthonormality of spin funcitons

10. The Hamiltonian is not dependent of the spin operator (to first approximation): Spin orbitals

11. If φi form an orthonormalized set then so do the spin orbitals ψi ‘s Orthomalized Spin orbitals

12. Given a system whose Hamiltonian operator is time independent and whose lowest-energy eigenvalues is E0, if Ф is any normalized-well behaved function of the coordinates of the system’s particles that satisfies the boundary condiition of the problem, then Variational Theorem

13. The ground state function is the eigenfunction with the lowest eigenvalue (ground state energy) is the function that minimaze the (energy) functional: Variational Theorem Minimization Constrain:

14. Instead of solving: One minimizes: Hartree-Fock Approximationfor the many-electron atom

15. Spin Orbital concept: HF wave function: Slater Determinant

16. N is even and there is always a α and β spin orbitals for each spacial function : Closed shell restricted Hartree-Fock Approximation Function Vector Space HF (Slater determinant) space

17. Instead of solving: One minimizes: HF Aproximation

18. HF Approximation

19. HF Approximation

20. A necessary condition for the φi ‘s that minimize F[φi] is Minimizing the HF functional Variational derivatives

21. Fock operator Restricted-close shell HF Equations Hartree-Fock potenatial Orbital HF energies

22. We have separated the n-body problem!!! ● ONE-ELECTRON equations!!!! ● Non linear integral-differential equations ● Coupled equations Restricted-close shell HF Equations

23. Coulomb potential due to the other electrons !!!! Restricted-close shell HF Equations Does not have classical analogous

24. Coulomb operator Restricted-close shell HF Equations Exchange operator

25. Guess the φi ‘s • ConstructJ and Koperators • Solvethe HF equations • If the new set of orbitals thus obtained are the same than the previous ones under certain criterion, the process is said to converge. If not: • the HF equations are solved again using the new orbitals to calculate J and K and repeating the process until convergence. Solving the HF equation:Self-Consistent Field (SCF) method

26. Expanding the space orbitals in a basis set: K > N The Hartree-Fock-Roothaan equations Basis set of known and well-behaved functions

27. HF: A set of DIFFERENTIAL non-linear equations: HF-Roothann: A set of ALGEBRAIC non-linear equations: The HF-Roothaan equations

28. Fock operator matrix representation in the basis set: Overlap matrix: The HF-Roothaan equations

29. Coefficients matrix: Orbital energy matrix: The “unknown” matrices

30. The HF-Roothaan equations • It represents a system of nonlinear algebraic equations • In general, the basis functions are not orthogonal. So S is not always the identity matrix. • represents a generalized eigenvalue problem. The matrices C and represent the eigenvectors and eigenvalues, respectively • The better the quality of the basis set, the better the solution of HFR equaitons

31. Solving HF-Roothaan equations Step 1: Find matrix D so that Step 2: Define matrices C’ and F’HF Step 2: