**Computational Physics II FYS4411/FYS9411**, Department of Physics, University of Oslo, Norway

Date: **Jan 11, 2019**

Copyright 1999-2019, Computational Physics II FYS4411/FYS9411. Released under CC Attribution-NonCommercial 4.0 license

The aim of this project is to use the coupled cluster method method to evaluate the ground state energy of quantum dots with $N=2$, $N=6$, $N=12$ and $N=20$ electrons. These are so-called closed shell systems.

We consider a system of electrons confined in a pure two-dimensional isotropic harmonic oscillator potential, with an idealized total Hamiltonian given by

$$
\begin{equation}
\label{eq:finalH} \tag{1}
\hat{H}=\sum_{i=1}^{N} \left( -\frac{1}{2} \nabla_i^2 + \frac{1}{2} \omega^2r_i^2 \right)+\sum_{i<j}\frac{1}{r_{ij}},
\end{equation}
$$

$$
\hat{H}_0=\sum_{i=1}^{N} \left( -\frac{1}{2} \nabla_i^2 + \frac{1}{2} \omega^2r_i^2 \right),
$$

and the repulsive interaction between two electrons given by

$$
\hat{H}_1=\sum_{i<j}\frac{1}{r_{ij}},
$$

with the distance between electrons given by $r_{ij}=\vert \boldsymbol{r}_1-\boldsymbol{r}_2\vert$. We define the modulus of the positions of the electrons (for a given electron $i$) as $r_i = \sqrt{r_{i_x}^2+r_{i_y}^2}$.

The aim of this project is to develop a coupled cluster doubles (CCD) code, where $2p-2h$ excitations are included only.

We will start with a two-electron problem and compare our results to those of Taut, see reference [1] below.

The ansatz for the ground state is given by

$$
\vert \Psi_0\rangle = \vert \Psi_{CC}\rangle = e^{\hat{T}} \vert
\Phi_0\rangle = \left( \sum_{n=1}^{N} \frac{1}{n!} \hat{T}^n
\right) \vert \Phi_0\rangle,
$$

$$
\begin{align*}
\hat{T} &= \hat{T}_1 + \hat{T}_2 + \ldots + \hat{T}_N
\\ \hat{T}_n &= \left(\frac{1}{n!}\right)^2
\sum_{\substack{ i_1,i_2,\ldots i_n \\ a_1,a_2,\ldots
a_n}} t_{i_1i_2\ldots i_n}^{a_1a_2\ldots a_n}
a_{a_1}^\dagger a_{a_2}^\dagger \ldots a_{a_n}^\dagger
a_{i_n} \ldots a_{i_2} a_{i_1}.
\end{align*}
$$

The energy is given by

$$
E_{\mathrm{CC}} = \langle\Phi_0\vert \overline{H}\vert
\Phi_0\rangle,
$$

where $\overline{H}$ is a similarity transformed Hamiltonian

$$
\begin{align*}
\overline{H}&= e^{-\hat{T}} \hat{H}_N e^{\hat{T}}
\\ \hat{H}_N &= \hat{H} - \langle\Phi_0\vert \hat{H} \vert
\Phi_0\rangle.
\end{align*}
$$

$$
\begin{equation}\label{eq:amplitudeeq} \tag{2}
0 = \langle\Phi_{i_1 \ldots i_n}^{a_1 \ldots a_n}\vert
\overline{H}\vert \Phi_0\rangle.
\end{equation}
$$

$$
\begin{equation}\label{eq:bch} \tag{3}
\overline{H}= \hat{H}_N + \left[ \hat{H}_N, \hat{T} \right]
+ \frac{1}{2} \left[\left[ \hat{H}_N, \hat{T} \right],
\hat{T}\right] + \ldots + \frac{1}{n!} \left[
\ldots \left[ \hat{H}_N, \hat{T} \right], \ldots \hat{T}
\right] +\dots
\end{equation}
$$

and simplified using the connected cluster theorem

$$
\overline{H}= \hat{H}_N + \left( \hat{H}_N \hat{T}\right)_c
+ \frac{1}{2} \left( \hat{H}_N \hat{T}^2\right)_c + \dots +
\frac{1}{n!} \left( \hat{H}_N \hat{T}^n\right)_c +\dots
$$

We will discuss parts of the the derivation below.

We will now approximate the cluster operator $\hat{T}$ to include only $2p-2h$ correlations. This leads to the so-called CCD approximation, that is

$$
\hat{T}\approx
\hat{T}_2=\frac{1}{4}\sum_{abij}t_{ij}^{ab}a^{\dagger}_aa^{\dagger}_ba_ja_i,
$$

meaning that we have

$$
\vert \Psi_0 \rangle \approx \vert \Psi_{CCD} \rangle =
\exp{\left(\hat{T}_2\right)}\vert \Phi_0\rangle.
$$

$$
\hat{H}=\hat{H}_N+E_{\mathrm{ref}},
$$

with

$$
\hat{H}_N=\sum_{pq}\langle p \vert \hat{f} \vert q \rangle
a^{\dagger}_pa_q + \frac{1}{4}\sum_{pqrs}\langle pq \vert \hat{v}
\vert rs \rangle a^{\dagger}_pa^{\dagger}_qa_sa_r,
$$

we obtain that the energy can be written as

$$
\langle \Phi_0 \vert
\exp{\left(-\hat{T}_2\right)}\hat{H}_N\exp{\left(\hat{T}_2\right)}\vert
\Phi_0\rangle = \langle \Phi_0 \vert \hat{H}_N(1+\hat{T}_2)\vert
\Phi_0\rangle = E_{CCD}.
$$

This quantity becomes

$$
E_{CCD}=E_{\mathrm{ref}}+\frac{1}{4}\sum_{abij}\langle ij \vert
\hat{v} \vert ab \rangle t_{ij}^{ab},
$$

$$
\langle \Phi_{ij}^{ab} \vert
\exp{\left(-\hat{T}_2\right)}\hat{H}_N\exp{\left(\hat{T}_2\right)}\vert
\Phi_0\rangle = 0.
$$

$$
0 = \langle ab \vert \hat{v} \vert ij \rangle +
\left(\epsilon_a+\epsilon_b-\epsilon_i-\epsilon_j\right)t_{ij}^{ab}+\frac{1}{2}\sum_{cd} \langle ab \vert \hat{v} \vert
cd \rangle t_{ij}^{cd}+\frac{1}{2}\sum_{kl} \langle kl \vert \hat{v}
\vert ij \rangle t_{kl}^{ab}+\hat{P}(ij\vert ab)\sum_{kc} \langle kb
\vert \hat{v} \vert cj \rangle t_{ik}^{ac} \nonumber
$$

$$
\begin{equation} +\frac{1}{4}\sum_{klcd} \langle kl \vert \hat{v} \vert cd \rangle
t_{ij}^{cd}t_{kl}^{ab}+\hat{P}(ij)\sum_{klcd} \langle kl \vert
\hat{v} \vert cd \rangle t_{ik}^{ac}t_{jl}^{bd}-\frac{1}{2}\hat{P}(ij)\sum_{klcd} \langle kl \vert \hat{v} \vert
cd \rangle t_{ik}^{dc}t_{lj}^{ab}-\frac{1}{2}\hat{P}(ab)\sum_{klcd}
\langle kl \vert \hat{v} \vert cd \rangle t_{lk}^{ac}t_{ij}^{db},
\label{eq:ccd} \tag{4}
\end{equation}
$$

where we have defined

$$
\hat{P}\left(ab\right)= 1-\hat{P}_{ab},
$$

$$
\hat{P}(ij\vert ab) = (1-\hat{P}_{ij})(1-\hat{P}_{ab}).
$$

$$
t_{ij}^{ab}=-t_{ji}^{ab}=-t_{ij}^{ba}=t_{ji}^{ba}.
$$

The two-body matrix elements are also anti-symmetrized, meaning that

$$
\langle ab \vert \hat{v} \vert ij \rangle = -\langle ab \vert
\hat{v} \vert ji \rangle= -\langle ba \vert \hat{v} \vert ij
\rangle=\langle ba \vert \hat{v} \vert ji \rangle.
$$

The non-linear equations for the unknown amplitudes $t_{ij}^{ab}$ are solved iteratively.

In order to develop a program, chapter 8 of the recent Lecture Notes in Physics (volume 936) is highly recommended as literature. All material is available from the source site. Example of CCD codes are available from the program site. These can be used to benchmark your own program.

We will use our Hartree-Fock basis from project 1 to define matrix elements and the single-particle energies to be used in the CCD equations. The Hartree-Fock basis defines the so-called reference energy

$$
\begin{equation}
E_{\mathrm{ref}} = \sum_{i\le F} \sum_{\alpha\beta}
C^*_{i\alpha}C_{i\beta}\langle \alpha | h | \beta \rangle +
\frac{1}{2}\sum_{ij\le F}\sum_{{\alpha\beta\gamma\delta}}
C^*_{i\alpha}C^*_{j\beta}C_{i\gamma}C_{j\delta}\langle
\alpha\beta|\hat{v}|\gamma\delta\rangle.
\label{_auto1} \tag{5}
\end{equation}
$$

$$
\begin{equation}
\langle pq \vert \hat{v} \vert rs\rangle_{AS}=
\sum_{{\alpha\beta\gamma\delta}}
C^*_{p\alpha}C^*_{q\beta}C_{r\gamma}C_{s\delta}\langle
\alpha\beta|\hat{v}|\gamma\delta\rangle_{AS},
\label{_auto2} \tag{6}
\end{equation}
$$

where the coefficients are those from the last Hartree-Fock iteration and the matrix elements are all anti-symmetrized. You can extend your Hartree-Fock program to write out these matrix elements after the last Hartree-Fock iteration. Make sure that your matrix elements are structured according to conserved quantum numbers, avoiding thereby the write out of many zeros.

To test that your matrix elements are set up correctly, when you read in these matrix elements in the CCD code, make sure that the reference energy from your Hartree-Fock calculations are reproduced.

Set up a code which solves the CCD equation by encoding the equations as they stand, that is follow the mathematical expressions and perform the sums over all single-particle states. Compute the energy of the two-electron systems using
all single-particle states that were needed in order to obtain the Hartree-Fock limit. Compare these with Taut's results for $\omega=1$ a.u. Since you do not include singles you will not get the exact result. If you wish to include singles, you will able to obtain the exact results in a basis with at least ten major oscillator shells.

Perform also calculations with $N=6$, $N=12$ and $N=20$ electrons and compare with reference [2] of Pedersen et al below.

The next step consists in rewriting the equations in terms of matrix-matrix multiplications and subdividing the matrix elements and operations in terms of two-particle configuration that conserve total spin projection and projection of the orbital momentum. Rewrite also the equations in terms of so-called intermediates, as detailed in section 8.7 of Lietz et al. This section gives a detailed description on how to build a coupled cluster code and is highly recommended.

Rerun your calculations for $=2$, $N=6$, $N=12$ and $N=20$ electrons using your optimal Hartree-Fock basis. Make sure your results from 2b) stay the same.

Calculate as well ground state energies for $\omega=0.5$ and $\omega=0.1$. Try to compare with eventual variational Monte Carlo results from other students, if possible.

The final step is to parallelize your CCD code using either OpenMP or MPI and do a performance analysis. Use the $N=6$ case. Make a performance analysis by timing your serial code with and without vectorization. Perform several runs and compute an average timing analysis with and without vectorization. Comment your results.

Compare thereafter your serial code(s) with the speedup you get by parallelizing your code, running either OpenMP or MPI or both. Do you get a near $100\%$ speedup with the parallel version? Comment again your results and perform timing benchmarks several times in order to extract an average performance time.

M. Taut, Phys. Rev. A

**48**, 3561 - 3566 (1993).M. L. Pedersen, G. Hagen, M. Hjorth-Jensen, S. Kvaal, and F. Pederiva, Phys. Rev. B

**84**, 115302 (2011)

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Give a short description of the nature of the problem and the eventual numerical methods you have used.

Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.

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