Chapter 2:         Theory

 

 

 

2.1                       Born-Oppenheimer Approximation

 

The Schroedinger equation is the basic equation governing the interactions between atoms and molecules, and yet it can only be solved analytically for the hydrogen atom.  For other elements, systems of atoms, or molecules, we must use approximations to the Schroedinger equation.  The three important approximations made in the study of molecular systems are the Born-Oppenheimer approximation, Hartree-Fock (HF) approximation, and the assumption that the total wavefunction can be expressed as a Linear Combination of Atomic Orbitals (LCAO).   The Born-Oppenheimer approximation comes about from the large difference that exists between the mass of an electron and the mass of the nucleus.  The nucleus can be considered as stationary in space while the electrons moves around the nuclei.  In the Hartree-Fock approximation the motion of electrons are influenced by the average field that is generated by contributions from all the other electrons in the atomic or molecular system.  Any correlation between the motions of electrons is neglected in this approximation.  The third approximation relates to expressing the total wavefunction as a LCAO.  The true representation of the molecular orbital is approximated by the LCAO, which consists of an expansion in a set of basis functions.  The type of basis function used and the finite size of the basis set (section 2.3) used in the LCAO, are limiting factors in the approximation to the true representation of the molecular orbital.

The first approximation used in the study of atomic and molecular system is to simplify the Schroedinger equation by treating the motion of the electrons and nuclei

separately.  For a system of many atoms and electrons one can solve for the electronic

part and the nuclei part separately.  The total wavefunction can be written as the product of the electronic and nuclear parts of the wavefunction, with the electronic portion depending parametically on the position of the nuclei.  This is the essence of the Born-Oppenheimer approximation.  The general expression for the wavefunction can be written as

 

                                             (2.1)

                                                           

Where ‘r’ is the position vector of the electrons, ‘R’ is the position vector of the nuclei, and the subscripts ‘e’ and ‘n’ refers to the electrons and the nuclei, respectively.  The electronic wavefunction is a function of the position of the electrons and depends parametrically on the nuclei positions while the nuclei wavefunction is a function of only the position of the nuclei. 

In the study of atomic and molecular systems the distribution and behavior of the electrons will define the structure and properties of matter.   If we desire the total energies, we can simply add to the Hamiltonian (H) the energy of the nucleus to the energy of the electrons.  The Schroedinger equation can then be written in two parts, the electronic and the nuclear part.

 

                                                (2.2)

                                               

The electronic part of the Hamiltonian is a function of the position vector of both the electrons and nuclei, while the nuclear part of the Hamiltonian is dependant only on the nuclear position vector.

The electronic Hamiltonian, which does not contain the kinetic energy of the nuclei, can be written as a summation of the electronic kinetic energy, electron-nuclei coulomb interaction, electron-electron coulomb interactions, and nucleus-nucleus coulomb interactions. 

 

                                  (2.3)

                       

Where ‘Ne’ is the number of electrons, ‘Nn’ is the number of nuclei, ‘Z’ is the ionic charge, ‘e’ is the charge of an electron, me is the mass of an electron ‘T’ is the kinetic energy, and ‘V’ is the potential energy.  The second, third and forth term of the above equation are the total potential energies arising from coulomb interaction between electron-electron, electron-nuclei, and nuclei-nuclei pairs

The Hamiltonian for the nuclei is the sum of the kinetic energy of the individual nuclei and the electronic energy, which was found from the solution of the Schroedinger of the electronic Hamiltonian and can be written as

 

                                       (2.4)

 

2.2           Hartree-Fock Approximation

 

The total energy of the atomic or molecular system contains the electronic kinetic energy and the potential energies.  The total energy for a closed electronic configuration (even number of electrons) can be written as