New developments in Molecular Orbital Theory – C.C.J. Roothaan

1. New developments in Molecular Orbital Theory – C.C.J. Roothaan Applied Quantum Chemistry 20131028 HochanJeong

2. Introduction the molecular wave function is constructed from the wave functions of the individual atoms. • Each electron is assigned to a one-electron wave function or molecular orbital It is the purpose of this paper to build a rigorous mathematical framework for the MO method.

3. Introduction • Assumptions • We shall be concerned only with the electronic part of the molecular wave functions. ; the nuclei are considered to be kept in fixed positions; • The magnetic effects due to the spins and the orbital motions of the electrons will be neglected throughout this paper.

4. General considerations An electron -> one wave function -> extends over the whole molecule give each electron a wave function depending on the space coordinates of that electron only, called a molecular orbital(MO) uthelectron the subscript I labels the different MO's x, y, z : space coordinates.

5. General considerations Molecular Spin Orbital ( MSO ) the subscripts k and i label the different MSO's general spin functions

6. General considerations Antisymmetrized product of MSO’s (AP) The total N-electron wave function is now built up as AP

7. General considerations Antisymmetrized product of MSO’s (AP) • when BR is any operator which acts symmetrically on the superscripts of an AP (that is, which acts symmetrically on all the N electrons), then

8. General considerations Antisymmetrized product of MSO’s (AP) • A wave function of the type (6) has several interesting properties. • 1. all the MSO's must be linearly independent -> otherwise determinant = 0 2. only the two MO’s can be the same( opposite spins ) -> pauli principle

9. General considerations Antisymmetrized product of MSO’s (AP)

10. General considerations Antisymmetrized product of MSO’s (AP)

11. General considerations Antisymmetrized product of MSO’s (AP)

12. General considerations Antisymmetrized product of MSO’s (AP)

13. General considerations Antisymmetrized product of MSO’s (AP)

14. General considerations Antisymmetrized product of MSO’s (AP) the energy of a closed-shell AP

15. General considerations Antisymmetrized product of MSO’s (AP) the energy of a closed-shell AP nuclear field orbital energies Hi the coulomb imtegralsJij The exchaegeietegralsKij

16. General considerations Antisymmetrized product of MSO’s (AP) These operators are linear and hermitian.

17. General considerations Antisymmetrized product of MSO’s (AP)

18. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE Best AP : the AP for which the energy reaches its absolute minimum. - Minimize E varying the MO’s

19. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE

20. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE H, J, K -> Hermitian Operator Same results for 2 brakets

21. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE To solve eq. 27 - > lagrangian multipliers Resulting restrictions on the variations

22. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE

23. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE

24. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE Taking the complex conjugate of the second one of Eqs. (31), and subtracting it from the first one, we obtain Conclusionly, 2 equations for eq.31 are complex conjugate

25. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE total electron irlteractiorl, operator G ; HartreeFock ham-iltonian operator F

26. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE our set of "best" MO's satisfies the simpler equations Fock’s equations they state that the MO's which give the best AP are all eigenfunctions of the same hermitianoperator F, which in turn is defined in terms of these MO's.

27. THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE The general procedure for solving Fock'sequations is one of trial and error. - assume a set of functions - calculate G & F - solve eq. (44) for the n lowest eigenvalues - compare the resulting functions with the assumed function. - a new set of function is chosen and procedure is repeated - calculation ends when the assumed one agrees with resulting one Hartree-Fock self consistent field (SCF) method.

28. THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE For atoms, the problem of solving Fock's equations is greatly simplifedby the central symmetry. For molecules, because of the absence of central symmetry, the situation is less fortunate We therefore have to use approximations to the best MO's. by representing all the electrons of the molecule by LCAO MO's, as given by X„'s are normalized AO's,

29. THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE

30. THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE it is useful to define for every one-electron operator M the corresponding matrix elements M„, evaluated with the set of AO's,

31. THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE

32. THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE

33. THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE We vary the vectors c, by infinitesimal amounts dci, and find for the variation of the energy Similar to that of the previous section

34. THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE

35. THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE LCAO self cortsistent field method. - to solve eq 59, assume C -> get F -> eigenvalue -> compare resulting C repeated