Unfortunately, we can't just discard electron to electron interactions. The magnitude of and are equivalent. So, what do we do? Solve the problem as is? Well, after you solve , you could quite possibly earn a Nobel Prize and/or a Fields Medal which makes this a wholly impractical method for this course. Instead we will use approximate methods for quantum mechanical problems more complicated than our earlier Hydrogen solution, and we currently have two primary methods for doing this.
The Thomas-Fermi Model is a semi-classical statistical method where we replace the exact potential and with an effective potential, , the screened coulomb potential.
Consider an electron gas with so many electrons that you don't have to count them one at a time, but instead just consider a continuous charge distribution. If we put a positive charge in the electron cloud then the positive charge would attract the electrons so the electron distribution would no longer be uniform. That said, the negative charges nearest the positive charge "screen" the electrons further out such that the fringe electrons don't feel as strong of an attraction.
Working out the details of this method requires a discussion of electron gasses, provided later in this text. Meanwhile, you should know that this hypothetical situation turns into a spherically distributed electron gas where the charge density, , is slowly varying. Although does change with , at each , we can treat as a uniform gas. Additionally, a point charge , placed in a uniform electron gas has a screened potential of:
This is sometimes called a Yukawa Potential. Also, is the Thomas-Fermi screening length and is related to the density of quantum states of the highest filled state, called the density of states at the Fermi Level.
Finally, for charge neutral atoms, where the number of electrons equals the atomic number, with a slowly varying electron gas in a spherically symetric central potential:
<FIGURE> "Electron Gas Potential" (Description)
The truth of the matter is that the Thomas-Fermi Model isn't a great solution for the multi-electron atom. Back in the 1920's it was quite successful and is still useful for simple approximations and the initial input to more advanced methods.
A better method is the iterative Hartree-Fock Method, or the Self-Consistent Field Method. For this method, we start by assuming that we can write:
This assumption further implies that the can say that . So, what is a reasonable ?
Now, this gives us:
is all the potential terms not involving electron-electron interaction. For the single atom, , but this method is generally applicable to more complex systems such as molecules. Hartree-Fock is the intellectual forbearer of several modern quantum chemical techniques.
<Source> "Primer on Calculus of Variation and Lagrange Multipliers"
The functional we are interested in is the energy, where . We want to find the extrema subject to the constraints for all . We will do this by using Lagrangian multipliers such that the integral we're interested in is:
We want to use calculus of variation to find the functions that make this function stationary (an extrema). Consider this function:
We want to vary the function to look for the stability condition. In other words, we want to change at an arbitrary constant .
Here is a small constant (we take the limit of as it goes to zero), and is an arbitrary function that deviates (this must be continuous over the limits of integration. This gives us:
The Maclaurin expansion in powers of is:
in the limit as goes to zero.
Let's return to our problem:
Writing this function out in detail gives us:
Note that you could alternatively carry through in this step, but in the end you don't need them. I chose to utilize instead of for purely aesthetic reasons. This gives us:
Looking at the term in the summation, , we can expand to:
For the and terms, there exists two conditions:
When , the terms inside the summation become , but integrating this will go to zero because and are orthogonal.
The only non-zero terms come about for
Thus, dividing through by gives us . Here, the term is again zero for , because the orthogonal functions come out from . The resulting term is:
Because depends on both and , we retain the wavefunctions inside the summation.
The functional variation now looks like:
Which can be solved as equations.
Where is the Hartree Term which is:
This looks like a Schrodinger equation, but is it? Yes, sort of... are Hartree wave functions. Remember that these are single electron since we started with . are Lagrangian multipliers for enforce normalization. However, we can use these and to study the system.
The Hartree equation has a physically intuitive form (if anything in quantum mechanics is physically intuitive). THe first term is kinetic energy, the second term is the electron-ion or any external potential energy (such as E or B), and the third term is the electron-electron potential energy defined by the sum:
Here is the charge density of the electron. So this sum can be thought of as an electron at interacting with some charge at .
What's wrong with this picture? Well, excluding our single particle assumption, we're putting several electrons into this system, which are fermions, but our wavefunction is not totally anti-symmetric. What we need to do is write using a Slater determinant.
Put this back into our original energy function and apply calculus of variation methods to the whole expression to find our answer. This was done by Fock (and Slater) in 1930. The resulting set of expressions is exactly the same except for one additional term.
The evaluation of is quite tricky. The expression for the expectation value is:
In many cases it is acceptable to approximate using the so-called "free electron exchange"
These equations are not trivial to solve. Notice that the operator depends on , but to get you need which requires solving the set of equations. This is where we solve by iterative, self-consistent methods. This technique is the basis for all of modern quantum mechanical methods.
Write , where is all the terms, not . It will depend on or or both.
Guess . You can use known solutions, random guesses, the Thomas-Fermi, or any other method of guessing you wish.
Solve the system of equations to find .
Calculate from .
If end. Otherwise return to step 2 and keep looping until to within a given tolerance.
This is called a self-consistent field loop (S.C.F.).
These type of SCF methods can be applied to all sorts of systems, but since this chapter is about many-electron problems, we will look at the Herman-Skillman atomic data, calculated from the Hartree-Fock method in 1963.