diff --git a/doc/Exercises/2024/FirstMidterm2024.pdf b/doc/Exercises/2024/FirstMidterm2024.pdf index dd0dc853..ec95b121 100644 Binary files a/doc/Exercises/2024/FirstMidterm2024.pdf and b/doc/Exercises/2024/FirstMidterm2024.pdf differ diff --git a/doc/Exercises/2024/FirstMidterm2024.tex b/doc/Exercises/2024/FirstMidterm2024.tex index d86ec10f..53c7aebf 100644 --- a/doc/Exercises/2024/FirstMidterm2024.tex +++ b/doc/Exercises/2024/FirstMidterm2024.tex @@ -183,7 +183,7 @@ \section*{Introduction} from the ground state, the spins of the various single-particle states should always sum up to zero. -\paragraph{Part a), setting up the basis.} +\paragraph{Part a), setting up the basis (10 pts)} We start with the helium atom and define our single-particle Hilbert space to consist of the single-particle orbits $1s$, $2s$ and $3s$, with their corresponding spin degeneracies. @@ -200,7 +200,7 @@ \section*{Introduction} states $|\Phi_{ij}^{ab}\rangle$ in a second quantization representation. -\paragraph{Part b) Second quantized Hamiltonian.} +\paragraph{Part b) Second quantized Hamiltonian (10 pts)} Define the Hamiltonian in a second-quantized form and use this to compute the expectation value of the ground state (defining the so-called reference energy and later our Hartree-Fock functional) of @@ -216,7 +216,7 @@ \section*{Introduction} matrix elements listed at the end of the midterm to find the value of $E$ as function of $Z$ and compute $E$ as function of $Z$. -\paragraph{Part c) Limiting ourselves to one-particle-one excitations.} +\paragraph{Part c) Limiting ourselves to one-particle-one excitations (10 pts)} Hereafter we will limit ourselves to a system which now contains only one-particle-one-hole excitations beyond the chosen state $|c\rangle$. Using the possible Slater determinants from exercise a) for the helium @@ -244,7 +244,7 @@ \section*{Introduction} The exact energy with our Hamiltonian is $-2.9037$ atomic units for helium. This value is also close to the experimental energy. -\paragraph{Part d) Moving to the Beryllium atom.} +\paragraph{Part d) Moving to the Beryllium atom (10 pts)} We repeat parts b) and c) but now for the beryllium atom. Define the ansatz for $|c\rangle$ and limit yourself again to @@ -287,7 +287,7 @@ \section*{Introduction} \end{equation*} brings us into the new basis $\psi$. The new basis is orthonormal and $C$ is a unitary matrix. -\paragraph{Part e) Hartree-Fock.} +\paragraph{Part e) Hartree-Fock (10 pts)} Minimizing with respect to $C^*_{p\alpha}$, remembering that $C^*_{p\alpha}$ and $C_{p\alpha}$ (and that the indices contain all single-particle quantum numbers including spin) are independent and @@ -310,7 +310,7 @@ \section*{Introduction} in the original basis (in our case the hydrogen-like wave functions) while roman letters refer to the new basis. -\paragraph{Part f) The Hartree-Fock matrices.} +\paragraph{Part f) The Hartree-Fock matrices (20 pts)} The Hartree-Fock equations with a variation of the coefficients $C_{p\alpha}$ lead to an eigenvalue problem whose eigenvectors are the coefficients $C_{p\alpha}$ and eigenvalues are the new single-particle @@ -324,7 +324,7 @@ \section*{Introduction} coefficients $C_{p\beta}$ etc. is $C_{p\beta}=1$ for $p=\beta$ and zero else. -\paragraph{Part g) Writing a Hartree-Fock code.} +\paragraph{Part g) Writing a Hartree-Fock code (30 pts)} The final stage is to set up an iterative scheme where you use the new wave functions determined via the coefficients $C_{p\alpha}$ to solve iteratively the Hartree-Fock equations till a given self-consistency