update

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