Question

Please help me understand this?!?

2. For this problem, consider the Lition a. Draw a sketch to show the protons and electrons involved in this system b. Using atomic units, write the Born-Oppenheimer approximation for the Hamiltonian c. Using perturbation theory, calculate E© for the ground-state of Li in units of Hartree. d. Using perturbation theory, set up (but do not solve) the calculation for E of the ground- state of Li+

Im not sure how to approach this problem

Answer 1

Li⁺ has atomic number 3 with one electron removed from the outermost shell. Now it is a two electron system with 3 orotons clubbed inside nucleus.

a) Since this is a quantum chemistry problem, a sketch of protons amd electrons implies the interactions between the entities. Since the kinetic energy of protons is much less than the electrons, it is ignored and thus the interaction between two protons is also constant.

Now to proceed further to solve for this two electron system, the following picture should be considered

no nucleus (Harree protons) Ri Rz e, a elepon / el eepona e = e, M2 ez R2 - R = interaction between e, and n (distanu interaction between e anda interaction between e, and ez. (c) har

b) Born Oppenheimer approximation says that the nucleus being heavier than electrons moves relatively slow such that their kinetic energy can be ignored as compared to that of electrons.

So the hamiltonian for this system i.e.

H= 1/2m_e1. + 1/2m_e2 + 1/2m_n +1/r_{​​​​​​12} + 1/R1 + 1/R2

In this equation 1st three terms are the kinetic energies of electron 1, 2 and nucleus respectively and last three terms are the potential interaction terms as per the sketch above

By born Oppenheimer approximation, the kinetic energy term of nucleus is ignored.

So H= 1/2me1 + 1/2me2+ 1/r12+1/R1 + 1/ R2.