Question

Physical chemistry question!!!! Please help!!!!!

Write out the complete wave functions for a 2p_0 orbital and a 2p_1 orbital. Show that the full wave function for a 2p_1 orbital is normalized. Find the most probable position and the average position of an electron in a 2p_0 orbital.

Answer 1

Firstly we should know the Hartee-Fock Method:

The Most accurate version of independent Particle Approx

the equation is $\hat{}$ $\hat{h}\left ( i \right )\phi \left ( i \right )=\epsilon \left ( i \right )\phi \left ( i \right )$ where

$\hat{h}\left ( i \right )=\hat{t}\left ( i \right )+v_{e-n}\left ( i \right )+v_{e-e}$

where $\hat{h}\left ( i \right )=\hat{t}\left ( i \right )+v_{e-n}\left ( i \right )+v_{e-e}$

$\hat{h}\left ( i \right )=\hat{t}\left ( i \right )+v_{e-n}\left ( i \right )+v_{e-e} \hat{t}\left ( i \right )$

here

$\hat{t}\left ( i \right )$ =Kinetic engery of nuclei

V_e-e=replusion of electron

V_e-n=attraction of electron and nuclei

as a solution we get

• $\phi \left ( i \right )$ Orbital

• $\epsilon \left ( i \right )$ orbital energies

since $\hat{h}$ is similar to Hamiltonian of hydrogen atom hence the solution also be similar

the angular part of wave function will be the same therefore we can classify orbital as 1s,2s,2p_0,2p_1,2p-1 etc

wave function created from the occupied orbital by a product or as a deteminant(Hatree-Fock); the occupied orbital are selected according to increasing value of the orbital energy( so Called Aufbau principle)

some important terms:

• Shell:collection of the orbital with the same n quantum number

• Subshell: collection of the orbital with common n and l quantum numbers which are degenerate according to the discussion above orbital 1s and 2s from subshellsalone, while 2p₀,2p_1.es 2p-1(or 2p_x,2p_y,2pz) orbitrals from subshell 2p subshell 3d has five components, 4f seven etc

• Configuration : defines the occupation of subshell

3. Find the most probable postion and averge postion

answer quantum mechanics tell u the probability for an electron is a certain orbital to be in certain position or momentum state inverting this to find the probability for the electron to be in certain orbital given its position momentum stat requires some priori idea of the orbital are the most probable this is stated succinctly by bayers theorem

$P(n,k,m\mid \vec{x})\alpha P(\vec{x}\mid n,k,m)P(n,k,m)$

The coefficient in this proportionality is fixed by requiring the norm of the probability to be unity. Without some good model for these a priori probabilities, the problem is un-answerable.

Luckily, a good model for these might be taken from thermodynamics. If the atom is in a bath of temperature TT, the a priori probability to be in the state ∣n,k,m〉∣n,k,m〉 is

P(n,k,m)∝exp(−E(n,k,m)kBT)