PCh10 Lec 3

(4/29/20, bes)

Well...we are making great progress in the material. Great job turing in your lab reports!

Let's take a step back and look at the big picture...we have come a long way...

The Big Picture

As stated on the Wikipedia site, the time-independent Schrodinger equation looks like the following:

YES...sorry this is considered the easier, time-independent Schrodinger equation...the time-dependent Schrodinger equation is used in spectroscopy since there is a time-dependence of when you excite the system...it apply a pulse of RF in FT-NMR...or any light (UV, vis, IR, etc). Time-dependent quantum or "Quantum: Spectroscopy" is a graduate level class that most pchem PhD students will take.

A quote by Paul Dirac...

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

Parts of Schrodinger Equation

As we transition from the analytical solutions for the model QM systems and H-atom (Ch < 10) to the numerical solutions for the real-world chemical systems (Ch > 10), we can separate our discussions based on the 3 (interconnected) aspects of a the Schrodinger equation. I will outline these discussion below and then go into more details later.

Hamiltonian

Sec 15.3: Hartree-Fock Molecular Orbital Theory: A Direct Descendant of the Schrodinger Equation.

As we have previously done, the Hamiltonian for a multi-electron system is relatively easy to write out, but it is not possible to solve. So what part of the Hamiltonian is the tricky part...the electron-electron repulsion term. So let's just cancel this term from the Hamiltonian...and now call this Schrodinger equation the Hartree-Fock equation.

There are other ways to approximate the Hamiltonian and we might discuss this more later...done.

Wavefunctions

Sec 10.3: Wavefunctions Must Reflect Indistinguishability of Electrons.

One aspect of an electron wavefunction, that is to be used in a multi-electron system is that we cannot tell the electrons apart...ie, we cannot label electrons with subscripts 1, 2, etc. Another way with saying this is that electrons are indistinguishable. This leads to Postulate 6 in QM that requires all electron wavefunctions are antisymmetric as opposed to symmetric. If a 2-electron wavefunction is written as ψ(1,2) to indicate electron 1 and electron 2, then the wavefunction ψ(2,1) must also be an equivalent since the physical meaning of the wavefunction is ψ². If the wavefunction is antisymmetric then ψ(1,2)= - ψ(2,1), but ψ² are the same.

There is a method to make sure the wavefunction is antisymmetric that involves the use of "Slater Determinant." I am not going to go into details here of how to generate antisymmetric wavefunction, but I would like for you to be able to solve a 2x2 and 3x3 determinant...this is a basic math operation. Visit the determinant wikipedia page for basics and work Example problem 10.1 in your notes.

In the discussion below we discuss how to solve for the energy. In this conversation we discuss ways of varying the mathematical form of the wavefunction. This topic will be discussed in much more detail, but for now let me remind you of a wavefunction from last lecture related to the use of an effective nuclear charge (ζ) as opposed to the true nuclear charge (Z):

Q: What is the best value for ζ ?

A: (see below in Energy discussion) - the value that gives us the lowest energy...

Calculation of Energy

Sec 10.4: Using the Variational method to solve the Schrodinger equation.

Sec 10.5: The Hartree-Fock Self-Consistent Field Method.

As the name states the "variational method" means that you "vary" some parameter in the Schrodinger equation (mainly the wavefunction) and then calculate the energy. This method introduces some vocabulary...to start this method we select a trial wavefunction. Notice that we use the term wavefunction and not eigenfunction; the eigenfunction is the analytical solution and the wavefunction is the numerical solution...our hope of finding the true eigenfunction is long gone!

Variational method steps...

1) using the trial wavefunction (wave_1) in the Schrodinger equation we calculate an "energy_01"

2) vary some aspect of the trial wavefunction (wave_2) and calculate energy_02.

3) compare energy_01 with energy_02

if energy_01 > energy_02, then we accept the variation we made to the trial wavefunction (wave_2 is a better wavefunction than wave 1),

if energy_01 < energy_02, then we decline the variation we made to the trial wavefunction (wave_1 is a better wavefunction than wave 2),

the variational theorem says the best wavefunction is the one that generates to lowest energy.

4) vary the "best" trial wavefunction and calculate the energy_03

5) compare energies (energy_02/01 with energy_03), accept/decline variation,

6) continue steps 4 and 5 until any changes in the trial wavefunction results in the ~same energy, energy_n ~ energy_n+1.

at this point your energies have become "consistent"...or "consistent with itself" or self-consistent...

The above method is referred to as self-consistent field method using the variational method, and if you used a Hamiltonian approximation of Hartree-Fock, then it is called the H-F self consistent field method using the variational method.

Note: you can use the "self-consistent" criteria with other "methods" of changing/varying the trial wavefunction...and come up with all sorts of different "names" for calculation techniques. By the way, one reason i like drones and PID controllers is because they are applying a "self-consistent" strategy to maintain or change the position of the drone...i will control myself and not go off on a tangent.

END of Lecture