Quantum dot

A common approximation for a ``quantum dot,'' a relatively small number of electrons trapped in a potential, is to model the trapping potential as a harmonic oscillator (which actually is the potential V(r) which you have currently in ``dft.m'') and to treat the electron-electron interactions within density-functional theory. To perform such a calculation, replace the code block in ``dft.m'',

%# Converge using steepest descents
W=lm(W,20); %# 20 iterations of simple lm() to get close to the minimum
[Wlm, Elm]=lm(W,50); %# 50 iterations of lm() from W, while recording results
               .
               .
               .
semilogy(its,Elm-18,its,Epclm-18,its,Ecg1-18,its,Ecg2-18,its,Ecg3-18);
%# Set W to best result for plotting
W=Wcg3;
pause;

which compares convergence of various minimization methods (now inappropriately assuming that the converged result is 18) with

%# Be sure to show lots of digits!
format long

%# Converge a little using steepest descents
W=sd(W,10);

%# Converge more with preconditioned line minimization
W=pclm(W,20);

%# Finish convergence with full preconditioned conjugate gradients
W=pccg(W,50,3);

Be sure to remove any ``pause;'' statements so that the code will also run through the part which gets the final wave functions and plots them.

Run your code. You should find a converged value of the total energy for this quantum dot of 8 electrons (4 states, with two electrons each) near Etot=43.337. You should also find states with similar s and p characters as to what you had before, but now with eigenvalues of approximately 5.509, 6.949, 6.949, 6.949.