new developments in molecular orbital theory – c.c.j. roothaan
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New developments in Molecular Orbital Theory – C.C.J. Roothaan. Applied Quantum Chemistry 20131028 Hochan Jeong. Introduction. the molecular wave function is constructed from the wave functions of the individual atoms . - PowerPoint PPT PresentationTRANSCRIPT
New developments in Molecular Or-bital Theory – C.C.J. Roothaan
Applied Quantum Chemistry20131028 Hochan Jeong
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 rigor-ous mathematical framework for the MO method.
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 or-bital motions of the electrons will be neglected throughout this paper.
Introduction
An electron -> one wave function -> extends over the whole molecule
General considerations
give each electron a wave function depending on the space coordinatesof that electron only, called a molecular orbital(MO)
uth electron
the subscript I labels the different MO'sx, y, z : space coordinates.
Molecular Spin Orbital ( MSO )
General considerations
the subscripts k and i label the different MSO's
general spin functions
Antisymmetrized product of MSO’s (AP)
General considerations
The total N-electron wave function is now built up as AP
Antisymmetrized product of MSO’s (AP)
• when BR is any operator which acts symmetrically on the super-scripts of an AP (that is, which acts symmetrically on all the N elec-trons), then
General considerations
Antisymmetrized product of MSO’s (AP)
• A wave function of the type (6) has several interesting proper-ties.
• 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
General considerations
Antisymmetrized product of MSO’s (AP)
General considerations
Antisymmetrized product of MSO’s (AP)
General considerations
Antisymmetrized product of MSO’s (AP)
General considerations
Antisymmetrized product of MSO’s (AP)
General considerations
Antisymmetrized product of MSO’s (AP)
General considerations
Antisymmetrized product of MSO’s (AP)
the energy of a closed-shell AP
General considerations
Antisymmetrized product of MSO’s (AP)
the energy of a closed-shell AP
General considerations
nuclear field orbital energies Hi
the coulomb imtegrals JijThe exchaege ietegrals Kij
Antisymmetrized product of MSO’s (AP)
General considerations
These operators are linear and hermitian.
Antisymmetrized product of MSO’s (AP)
General considerations
Best AP : the AP for which the energy reaches its absolute mini-mum.
- Minimize E varying the MO’s
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
H, J, K -> Hermitian OperatorSame results for 2 brakets
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
To solve eq. 27 - > lagrangian multipliers
Resulting restrictions on the variations
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
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
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
total electron irlteractiorl, operator G ;
Hartree Fock ham-iltonian operator F
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
our set of "best" MO's satisfiesthe simpler equations
Fock’s equations
they state that the MO's which give the best AP are all eigenfunctions of the same hermitian operator F, which in turn is defined in terms of these MO's.
THE HARTREE-FOCK SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND-STATE
The general procedure for solving Fock's equations 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.
THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE
For atoms, the problem of solving Fock's equationsis greatly simplifed by 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,
THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE
THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE
it isuseful to define for every one-electron operator M the corresponding matrix elements M„, evaluated with the set of AO's,
THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE
THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE
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
THE LCAO SELF-CONSISTENT FIELD METHOD FOR A CLOSED-SHELL GROUND STATE
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
