Visualizing Electron Density Of Multi-Electron Atoms and Molecules
As we all learned in Quantum
Physics it is fairly simple to reach an analytical solution for
Schrodinger's equation in the hydrogen atom. The result can be beautifully
seen in a project done by Joey Fox in the Computational Physics group.
However, when it comes to more complicated atoms that contain more than one electron or molecules that are composed from more than one atom it is impossible to devise an analytical solution.
The simple example that demonstrates this is the helium atom. The helium atom consists of two protons and two electrons. By marking the electrons as 1,2 we get the following Hamiltonian :
Without the last term of the Hamiltonian, we would have been able to separate it according to the different coordinates and solve it. The last term which represents the electron repulsion causes the problem to be unsolvable.
For a molecule with N electrons and M atoms we get the following Hamiltonian:
Obviously it is also analytically unsolvable. You may notice that some terms are missing in the Hamiltonian. The kinetic energy of the atoms and their electric repulsion potential are assumed to be constant, therefore these terms were removed.
In order to find a numerical solution for the problem we look at the Hatree-Fock approximation. The Hatree-Fock approximation is based on Pauli's Exclusion Principle, which requires identical electrons to be in an asymmetrical state. According to the approximation, we can represent the wave-function of the electrons in the molecule as a Slater determinant composed of single electron functions:
If we assume that each single electron function is known up to a certain number variables we can use the variation principle to reach a solution. According to the variation principle we change the variables of the single electron functions until we reach a minimal energy.
Another part of the Hartree-Fock method is representing the Hamiltonian so it becomes calculable. The last term of the molecular Hamiltonian is the problematic one and can be represented as a term called Hartee-Fock potential (VHF) :
Overtime, since Hartree-Fock
approximation was proposed in 1935, there have been many methods of dealing with
the VHF. The goal was to create an effective potential which is
calculable and as accurate as possible. This site does not contain a description
about these methods, as it is a very long one.
However, the important thing is, that Hartree-Fock approximation enabled scientists to calculate reasonable solutions for the electron wave-function in complicated atoms and molecules.
A. Honnenberg-Kohn Theorems :
In many cases the Hartree-Fock method fails, from unreasonable differences between calculated energy to the one measured up to physically impossible solutions. There is another method, which deals with same problem but from a different approach, called Density Function Theory (DFT). Instead of solving the problem using single electron wave-function, this method uses one function which represents the entire electron density of the molecule represented as .
According to a proof given by P. Honnenberg and W. Kohn there will always exist such a function describing the electrons' density in the molecule. Moreover they showed that the Hamiltonian is a unique functional of the density of their ground state. This allows us to perform iterative calculation of only 3 degrees of freedom of the density instead of 3N degrees in the Hartree-Fock method.
In addition to this proof they divided the calculation of the energy to two parts - one universal and the other specific for a certain molecule.
Eee - electronic interaction,
easily calculable when the density is given.
T - non-interacting kinetic energy.
VNe - a potential that which is due to the electron-nuclei interaction. This is the only part which is different from one molecule to another.
In their second proof Honnenberg and Kohn show that the ground state of the system can be derived using the variation method, in the same manner as done in Hartree-Fock methods.
B. Constrained Search :
Much later articles showed that the search for the ground state can constrained. The Levy constrained-search (1979) formulation shows that the ground state density can be done in two separate levels. First, going through all the possible densities. Second, For each electron density the method seeks the electronic wave functions that give a minimal energy under the constraint of that their total electron density match the density which is currently under investigation. The ground state density will be the one that gives minimal energy
Adding to this formulation to fact that we use a universal term, we get :
Khon-Sham Theorems :
Even though it seems that the calculation of the density functional is now quite doable, it turns out that there are many problems unknown, and many attempts to find analytical expressions for them failed. In their articles, Kohn and Sham approached the problem by taking out the terms that are calculable and putting all the incalculable term into one effective potential. This is done especially by separating the problematic kinetic energy term T to Ts which is the non-interacting kinetic energy and Tc which is the residual kinetic energy which completes it to the true one T=Ts+Tc. Eventually we get :
It is easy to see that if we try to use the variation method here we get a self-consistent method. By changing the wave-functions which compose the electron-density, we change the Hamiltonian by which the energy is calculated. Therefore the solution for the problem has to begin with a guess and continue in an iterative way until reaching convergence.
4. Basis Functions :
In the first part of the project (Lithium Atom, Hydrogen, Methane and Ethylene molecules) the calculations were done using a basis function called STO (Slater Type Orbital) which is given as:
N - normalization factor.
n - the energy level of the shell represented.
The most common method, which
is also used here, for using STO is STO-3G. In this method each shell (orbital
with different n,l,m) is represented by 3 STO functions. The STO function as
set as a basis determined empirically and while the linear combination of the different
shells composing the problem is set by using the variation method.
* Most of the description here is based on "A Chemist's Guide To Density Functional Theory" - Wolfram Koch, Max C. Holthausen.