TalentDFT

• Install Git, and try out some of the commands covered in Morten or Nicolas's lecture slides (computing.pdf talentdftguides.pdf) for your codes in the following problems.

• In your favorite programming language, make a program to construct a real symmetric $NxN$ matrix. Diagonalize it using the appropriate LAPACK or GSL routine, and write out some number of the lowest eigenvalues. (Suggestion: You might find it useful to use Mathematica (available on the ECT* computers) to diagonalize a small matrix that you can benchmark against.) This will help you test that you've linked to the GSL or LAPACK library.

• The code laguerre_general.f90.zip calculates the generalized Laguerre polynomials that appear in the definition of the HO wf's, see here. Use this subroutine to create a function or subroutine that calculates the HO wf's $R_{nl}(r)$.

• Check numerically that the constructed HO wf's are orthonormal. I.e., evaluate $\int r^2dr R_{nl}(r)R_{n'l'}(r)$. To do this, you will want to use Gaussian quadrature to discretize the integrals. Gaussian quadrature is discussed a bit in . (Gauss-Laguerre quadrature is ideal for this problem, but plain old Gauss-Legendre quadrature, which is widely available in canned routines you can easily find via Google, is sufficient.)

• Construct the matrix elements of the Coulomb potential given by

\begin{equation} \langle nl|V|n'l\rangle = \int_0^{\infty}r^2dr R_{nl}(r}\frac{1}{r}R_{n'l}(r) \end{equation}

• Construct the hamilton matrix $\langle nl|H|n'l\rangle$ keeping all HO basis states $n,n'<N_{max}$. Diagonalize the matrix for increasing $N_{max}$ values for different values of the oscillator length parameter $b$ (“oscl” in the code.) $b$ and $\hbar\omega$ are related by $b = \sqrt{(\hbar/(m\omega)}$. Plot your lowest eigenvalue for each $N_{max}$ as a function of $b$ (or $\hbar\omega$).

• Construct a subroutine that returns relative HO matrix elements of the Minnesota NN potential. The definition of the Minnesota potential and a sketch of how to proceed will be given on the black board.

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