[{"categories":[{"LinkTitle":"Blogs","RelPermalink":"/categories/blogs/"}],"content":" Bottom-up synthesis of Density Functional Theory (DFT) From the Born-Oppenheimer approximation to the Kohn \u0026amp; Sham equation. Disclaimer The content of this article is based on various internet resources1. Furthermore, I have attempted to provide a simplified approach to the re-derivation of the equations. Therefore, there may be potential hiccups and mathematicals glichs in some simplifications. Here, I apologize in advance for any mistakes. However, I would welcome any corrections from you.\n1: There was a Turkish website from Istanbul University with in-depth derivations, if I remember correctly. Unfortunately, I stupidly forgot to keep track of the URL. Thousand apologies for that 😅.\n💪 Motivation\nGerbrand Ceder Pr @ University of California, Berkeley \u0026 Samsung Distinguished Chair in Nanoscience and Nanotechnology Research “Today we know less than 1% of known compounds in the world; less than 1% of the materials properties.” Gerbrand Ceder @ The Material Desing Revolution the World Economic Forum (2014, Davoz). 1. Introduction Go to TOC\nDefinition:\nA mathematical model about a sufficiently large set of natural phenomena is called an “ab initio model ” or a “first principles model” if the starting point of the model can not be further reduced to more fundamental concepts (at least not within the language of the considered natural phenomena).\nBy Pr. Stefaan Cottenier @ Ghent University. Furhtermore, the main idea is to tame the following beast:\n\\[ \\begin{equation} \\hat{\\mathcal{H}}\\Psi = E\\Psi \\end{equation} \\]Where the Hamiltonian is given in atomic unit for \\(N_e\\) electrons and \\(N_n\\) nucleus as following:\n\\[ \\begin{equation} \\mathcal{H} = -\\sum_i^{N_e} \\frac{\\nabla^2_{\\overrightarrow{r_i}}}{2m_i}-\\sum_i^{N_n} \\frac{\\nabla^2{\\overrightarrow{R_i}}}{2M_i} + \\frac{1}{2}\\sum_{i =1}^{N_e}\\sum_{i\\neq j}^{N_e} \\frac{1}{|\\overrightarrow{r_i}-\\overrightarrow{r_j}|}+ \\frac{1}{2}\\sum_{i =1}^{N_n}\\sum_{i\\neq j}^{N_n} \\frac{Z_iZ_j}{|\\overrightarrow{R_i}-\\overrightarrow{R_j}|} - \\frac{1}{2}\\sum_{i =1}^{N_e}\\sum_{j = 1}^{N_n} \\frac{Z_j}{|\\overrightarrow{r_i}-\\overrightarrow{R_j}|} \\end{equation} \\]The right-hand side of \\((2)\\) are respectively: electron and nucleus kinetic energy, electron-electron and nuclei-nuclie Coulomb interaction and the electron-nuclie Coulomb interaction.\n2. How to tame Schrödinger equation ? Go to TOC\n2.1 Born-Oppenheimer approximation The basic idea is the following: ”We consider the nuclei as stationary objects, so we freeze the nuclear positions.”. We say that this approximation is adiabatic (We move from \\(4(Ne + Nn) \\longrightarrow 4Ne\\)).\n\\[ \\begin{equation}\\Psi(\\overrightarrow{r_i},\\overrightarrow{\\sigma_i},\\overrightarrow{R_i},\\overrightarrow{\\sigma_i^N}) \\longrightarrow \\Psi(\\overrightarrow{r_1},\\overrightarrow{\\sigma_1},\\dots , \\overrightarrow{r_{N_e}},\\overrightarrow{\\sigma_{N_e}}; \\underbrace{\\left[\\overrightarrow{R_1}, \\dots , \\overrightarrow{R_{N}}\\right]}_{Paramaters})\\end{equation} \\]The previous wavefunction should be anti-symmetric. The problem is reduced into solving the Schrödinger equation for the following Hamiltonian:\n\\[ \\begin{equation}\\mathcal{H} = -\\sum_i^{N_e}\\frac{\\nabla_{r_i}^2}{m_i} - \\sum_{\\{i,j\\}\\longrightarrow \\{N_e, N_n\\}}\\frac{Z_j}{|\\overrightarrow{R_j}-\\overrightarrow{r_i}|} + \\frac{1}{2}\\sum_{i}^{N_e}\\sum_{i\\neq j}^{N_e}\\frac{1}{|\\overrightarrow{r_j}-\\overrightarrow{r_i}|} + \\underbrace{\\frac{1}{2}\\sum_{i}^{N_n}\\sum_{i\\neq j}^{N_n}\\frac{Z_jZ_i}{|\\overrightarrow{R_j}-\\overrightarrow{R_i}|}}_{Constant}\\end{equation} \\] ℹ️ Note: that \\(\\textbf{BO}\\)-approximation is a major simplification, but it is way too hard to solve! more work is needed.\n2.2 Hartree-Fock method (post-HF method) Go to TOC\n💡 The main idea: For all the possible sets of the wavefunctions that can describe our system, we search for the one associated to the ground state. With the following restrictions:\nJust the anti-symmetric wavefunctions;\nUse Salter determinant;\nThe energy minimization principal.\n⚠️ At the end this is an approximation (work well for chemistry topics, not for solids).\nThe first step is visualizing the terms in the reduced Schrödinger equation after \\(\\textbf{BO}\\) approximation. After all, we end up with:\n\\[ \\begin{equation} \\mathcal{H} = \\sum_i^{N_e}\\underbrace{\\left[-\\frac{\\nabla_{r_i}^2}{m_i} - \\sum_j^{N_n}\\frac{Z_j}{|\\overrightarrow{R_j}-\\overrightarrow{r_i}|} \\right]}_{\\hat{h}_1(\\overrightarrow{r_i})\\equiv\\ single-particle } + \\frac{1}{2}\\sum_{i}^{N_e}\\underbrace{\\sum_{i\\neq j}^{N_e}\\frac{1}{|\\overrightarrow{r_j}-\\overrightarrow{r_i}|} }_{\\hat{h}_2(r_i,r_j)\\equiv\\ effective\\ potential} \\end{equation} \\]Notice that we omitted the constant term, the reason will be more obvious in the following derivations. The Hartree-Fock method is a variational, wavefunction-based approach. Although it is a many-body technique, the approach followed is that of a single-particle picture, i.e. the electrons are considered as occupying single-particle orbitals making up the wavefunction. Each electron feels the presence of the other electrons indirectly through an effective potential. Each orbital, thus, is affected by the presence of electrons in other orbitals. Now to build our wavefunction, instinctively we write our function as the product of the orbitals occupied by the electrons, and for simplicity we drop the spin degrees of freedom:\n\\[ \\begin{equation} \\Psi(r_1,\\dots,r_n) = \\varphi_1(r_1)\\times \\dots\\times \\varphi_N(r_N) \\end{equation} \\]❗ But wait, this expression violate the first restriction mentioned above ! We need another tools that respect the anti-symmetric feature of the wavefunction.\n2.2.1 Slater expression Go to TOC\nOne way to overcome the raised problem is the so called \\(Slater\\) determinant:\n\\[ \\begin{equation} \\Psi(r_1,\\dots,r_n) = \\frac{1}{\\sqrt{N_e!}}\\begin{pmatrix} \\varphi_1(r_1) \u0026 \\dots \u0026 \\varphi_N(r_1)\\\\ \\vdots \u0026 \\vdots \u0026 \\vdots \\\\ \\varphi_1(r_N) \u0026 \\dots \u0026 \\varphi_N(r_N) \\end{pmatrix} \\end{equation} \\]Now, we will try to write more compact formula for our wavefuction:\n\\[ \\begin{equation*} \\Psi(\\{\\overrightarrow{r}\\}) = \\frac{1}{\\sqrt{N_e!}} \\sum_i^{N_e} (-1)^{P(i)}\\varphi_{i_1}(r_1)\\dots \\varphi_{i_N}(r_N) \\end{equation*} \\] 🗣 The notation here is ambiguous, and not obvious! By far, the \\((-1)^{P(i)}\\) factor result from the expansion of the determinant. For which it is +1 if we have a term with an even permutation (e.g: \\(3\\longrightarrow 1 \\longrightarrow 2\\)) or -1 for an odd permutation (e.g: \\(3\\longrightarrow 2 \\longrightarrow 1\\)). 2.2.2 Energy minimization Go to TOC\nAs the \\(\\textbf{HF}\\) method relies on a variational principal: \\[ \\begin{equation} \\bra{\\Psi}\\mathcal{H}_{HF}\\ket{\\Psi} = \\bra{\\Psi}\\sum_i\\hat{h}_1(r_i)\\ket{\\Psi} +\\frac{1}{2}\\bra{\\Psi}\\sum_{i\\neq j}\\hat{h}_{2}(r_i,r_j)\\ket{\\Psi} \\end{equation} \\]We start by evaluating the expected value of the single particle (first term on the right side ):\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_n\\hat{h}_1(r_n)\\ket{\\Psi} =\u0026 \\frac{1}{N!}\\sum_n^{N}\\sum_j^{N!}\\sum_i^{N!} (-1)^{P(j)}(-1)^{P(i)}\\times\\\\ \u0026\\langle\\varphi_{j_1}(r_1)\\dots \\varphi_{j_N}(r_N)|\\hat{h}_1(r_n)|\\varphi_{i_1}(r_1)\\dots \\varphi_{i_N}(r_N)\\rangle \\end{align*} \\]The trick that works with me is to write down \\(\\textcolor{green}{first}\\) the orbitals that the Hamiltonian is mainly! applied on. Then we write the remaining terms as single \\(bra-ket\\) products :\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_n\\hat{h}_1(r_n)\\ket{\\Psi} =\u0026 \\frac{1}{N!}\\sum_n^{N}\\sum_j^{N!}\\sum_i^{N!} (-1)^{P(j)}(-1)^{P(i)}\\times\\\\ \u0026 \\langle\\varphi_{1_j}(r_j)|\\varphi_{1_i}(r_i)\\rangle\\times \\dots\\\\ \u0026 \\times \\textcolor{green}{\\langle\\varphi_{j_n}(r_n)|\\hat{h}_1(r_n)|\\varphi_{i_n}(r_n)\\rangle}\\times \\dots \\\\ \u0026 \\times \\langle\\varphi_{N_j}(r_N)|\\varphi_{i_N}(r_N)\\rangle \\end{align*} \\]With the \\(anzats\\): \\(\\langle\\varphi_j|\\varphi_i\\rangle=\\delta_{ij}\\)\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_n\\hat{h}_1(r_n)\\ket{\\Psi} =\u0026 \\frac{1}{N!}\\sum_n^{N}\\sum_j^{N!}\\sum_i^{N!} (-1)^{P(j)}(-1)^{P(i)}\\times\\\\ \u0026 \\delta_{1_j1_i}\\times \\dots\\times\\langle\\varphi_{j_n}(r_n)|\\hat{h}_1(r_n)|\\varphi_{i_n}(r_n)\\rangle\\times \\dots \\times \\delta_{N_jN_i} \\end{align*} \\]Since we end up with one sum(\\(i=j\\)) \\(N!\\) times over all the orbitals, we have always the same \\(+\\) sign in front of the product:\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_n\\hat{h}_1(r_n)\\ket{\\Psi} =\\frac{1}{N!}\\sum_n^{N} \\sum_i^{N!}\\langle\\varphi_{i_n}(r_i)|\\hat{h}_1(r_n)|\\varphi_{i_n}(r_i)\\rangle \\end{align*} \\]In conclusion the first term in \\(\\textbf{HF}\\) energy is the expected total energy of all the single particles treated individually.\n\\[ \\begin{equation*} \\bra{\\Psi}\\sum_i\\hat{h}_1(r_i)\\ket{\\Psi} =\\sum_i^{N} \\langle\\varphi_{i}(r_i)|\\hat{h}_1(r_i)|\\varphi_{i}(r_i)\\rangle \\end{equation*} \\]Let\u0026rsquo;s tackle the second terms:\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_{n\\neq m}\\hat{h}_{2}(r_n,r_m)\\ket{\\Psi} =\u0026 \\frac{1}{N!}\\sum_{n\\neq m}\\sum_j^{N!}\\sum_i^{N!} (-1)^{P(j)}(-1)^{P(i)}\\times\\\\ \u0026\\langle\\varphi_{j_1}(r_1)\\dots \\varphi_{j_N}(r_N)|\\hat{h}_2(r_n, r_m)|\\varphi_{i_1}(r_1)\\dots \\varphi_{i_N}(r_N)\\rangle \\end{align*} \\]We use the same tricks as before:\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_{n\\neq m}\\hat{h}_{2}(r_n,r_m)\\ket{\\Psi} =\u0026 \\frac{1}{N!}\\sum_{n\\neq m}\\sum_j^{N!}\\sum_i^{N!} (-1)^{P(j)}(-1)^{P(i)}\\times\\\\ \u0026 \\delta_{1_j1_i}\\times \\dots \\times \\textcolor{green}{\\langle\\varphi_{j_n}(r_n)\\varphi_{j_m}(r_m)|\\hat{h}_2(r_n, r_m)|\\varphi_{i_n}(r_n)\\varphi_{i_m}(r_m)\\rangle}\\\\ \u0026\\times \\dots \\times \\delta_{N_jN_i} \\end{align*} \\]Which is reduced into ( Notice the order \\(n \u003c m\\).):\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_{n\u003c m}\\hat{h}_{2}(r_n,r_m)\\ket{\\Psi} =\u0026 \\frac{1}{N!}\\sum_{n\u003c m}\\sum_i^{N!}(-1)^{P(i)}\\langle\\varphi_{i_n}(r_n)\\varphi_{i_m}(r_m)|\\hat{h}_2(r_n, r_m)|\\varphi_{i_n}(r_n)\\varphi_{i_m}(r_m)\\rangle \\end{align*} \\]Here we have one permutation operators \\(P\\) acting on the order of the set of orbitals \\(\\{1, 2, \\dots, n, \\dots, m, \\dots, N\\}\\). It is not obvious, but clearly we are expecting something to pop-up for sure! And the idea is to separate all the \\(even\\) and the \\(odd\\) permutations in the previous set:\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_{n\u003c m}\\hat{h}_{2}(r_n,r_m)\\ket{\\Psi} =\u0026 \\frac{1}{N!}\\sum_{n\u003c m}\\sum_i^{N!}\\\\ \u0026\\langle\\varphi_{i_n}(r_n)\\varphi_{i_m}(r_m)|\\hat{h}_2(r_n, r_m)|\\varphi_{i_n}(r_n)\\varphi_{i_m}(r_m)\\rangle \\\\ \u0026- \\langle\\varphi_{i_m}(r_n)\\varphi_{i_n}(r_m)|\\hat{h}_2(r_n, r_m)|\\varphi_{i_m}(r_n)\\varphi_{i_n}(r_m)\\rangle \\end{align*} \\]For both \\(even\\) or \\(odd\\) terms we have \\(N!\\) times repeated quantities. Hence we find:\n\\[ \\begin{align*} \\bra{\\Psi}\\sum_{n\u003c m}\\hat{h}_{2}(r_n,r_m)\\ket{\\Psi} =\u0026 \\sum_{n\u003c m}^N\\langle\\varphi_{n}(r_n)\\varphi_{m}(r_m)|\\hat{h}_2(r_n, r_m)|\\varphi_{n}(r_n)\\varphi_{m}(r_m)\\rangle \\\\ \u0026- \\langle\\varphi_{m}(r_n)\\varphi_{n}(r_m)|\\hat{h}_2(r_n, r_m)|\\varphi_{m}(r_n)\\varphi_{n}(r_m)\\rangle \\end{align*} \\]At the end we get the following expression for our many-body problem:\n\\[ \\begin{equation} E_{HF} = \\sum_{i}^{N}\\bra{\\varphi_i}\\hat{h}_1\\ket{\\varphi_i} + \\frac{1}{2}\\sum_{i}^{N}\\sum_{j}^{N}\\left[\\bra{\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_i\\varphi_j}-\\bra{\\varphi_j\\varphi_i}\\hat{h}_2\\ket{\\varphi_i\\varphi_j}\\right] \\end{equation} \\]2.2.3 Hartree-Fock equation Go to TOC\nIn this section we will apply the variational principale on the last equation and the aim is to minimize the total energy:\n\\[ \\begin{equation} \\delta E_{HF} = 0 \\end{equation} \\]We use the Lagrange multipliers and the orthogonality of the set of our wvefunctions basis as following:\n\\[ \\begin{gather} \\delta \\left[ E_{HF} +\\sum_{i,j}\\left(-\\lambda_{i,j} +\\lambda_{i,j}\\right) \\right] = 0 \\\\ \\delta \\left[ E_{HF} -\\sum_{i,j}\\lambda_{i,j} \\left(\\langle\\varphi_i|\\varphi_j\\rangle+\\delta_{i,j}\\right) \\right] = 0\\\\ \\delta F = 0 \\end{gather} \\]The \\(F\\) stand for \\(Fock\\) operator in the last equation. Now we will consider small change in our wavefunction basis (\\(\\varphi \\longrightarrow \\varphi + \\delta \\varphi\\)). We first treat the the term with the \\(Langrange\\) multipliers:\n\\[ \\begin{gather*} \\delta \\left[\\sum_{i,j}\\lambda_{i,j} \\left(\\langle\\varphi_i|\\varphi_j\\rangle+\\delta_{i,j}\\right) \\right] = \\sum_{i,j}\\lambda_{i,j} \\left[\\langle\\delta\\varphi_i|\\varphi_j\\rangle+\\langle\\varphi_i|\\delta\\varphi_j\\rangle \\right] \\end{gather*} \\]We can write the second term as:\n\\[ \\begin{gather*} \\langle\\varphi_i|\\delta\\varphi_j\\rangle =\\underset{\\mathbb{R}^2}{\\int} \\varphi_i^*(r_i)\\delta\\varphi_j(r_j) dr^2 = \\left[\\underset{\\mathbb{R}^2}{\\int} \\varphi_i(r_i)\\delta\\varphi_j^*(r_j) dr^2\\right]^*=\\langle\\delta\\varphi_i|\\varphi_j\\rangle^* \\end{gather*} \\]We move forward to differentiate the single electrons expectation value:\n\\[ \\begin{gather*} \\delta\\left[\\sum_i^N\\bra{\\varphi}\\hat{h}_1\\ket{\\varphi_i}\\right] = \\sum_i^N\\left[\\bra{\\delta\\varphi}\\hat{h}_1\\ket{\\varphi_i} + \\bra{\\varphi}\\hat{h}_1\\ket{\\delta\\varphi_i}\\right] \\end{gather*} \\]Since \\(\\hat{h}_1\\) is hermitian operator i.e \\(\\left(\\hat{h}_1\\right)^{\\dagger}=\\hat{h}_1\\):\n\\[ \\begin{equation*} \\delta\\left[\\sum_i^N\\bra{\\varphi_i}\\hat{h}_1\\ket{\\varphi_i}\\right] = \\sum_i^N\\left[\\bra{\\delta\\varphi}\\hat{h}_1\\ket{\\varphi_i} + C.C\\right] \\end{equation*} \\]Where \\(C.C\\) stand for \\(Complex\\ Conjugate\\). Then we work with second term of \\(Hartree-Fock\\) energy (\\(E_{HF}\\)), step by step we will start by the electrons coulomb interaction:\n\\[ \\begin{gather*} \\delta\\left[ \\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_i\\varphi_j}\\right] = \\delta\\left[\\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_{i}(r_i)\\varphi_{j}(r_j)}\\hat{h}_2\\ket{\\varphi_{i}(r_i)\\varphi_{j}(r_j)}\\right] \\\\ = \\textcolor{green}{\\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\delta\\varphi_{i}(r_i)\\varphi_{j}(r_j)}\\hat{h}_2\\ket{\\varphi_{i}(r_i)\\varphi_{j}(r_j)}} \\\\ + \\textcolor{orange}{\\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_{i}(r_i)\\delta\\varphi_{j}(r_j)}\\hat{h}_2\\ket{\\varphi_{i}(r_i)\\varphi_{j}(r_j)}} \\\\ + \\textcolor{green}{\\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_{i}(r_i)\\varphi_{j}(r_j)}\\hat{h}_2\\ket{\\delta\\varphi_{i}(r_i)\\varphi_{j}(r_j)}} \\\\ + \\textcolor{orange}{\\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_{i}(r_i)\\varphi_{j}(r_j)}\\hat{h}_2\\ket{\\varphi_{i}(r_i)\\delta\\varphi_{j}(r_j)}} \\end{gather*} \\]We notice that we have \\(C.C\\) that pop-up automatically as in the previous treatment:\n\\[ \\begin{equation*} \\delta\\left[ \\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_i\\varphi_j}\\right] = \\sum_{i}^{N}\\sum_{j}^{N}\\left[\\bra{\\delta\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_i\\varphi_j} + \\bra{\\varphi_i\\delta\\varphi_j}\\hat{h}_2\\ket{\\varphi_i\\varphi_j} + C.C \\right] \\end{equation*} \\]After this we are arrive to the exchange term, and with the same treatment:\n\\[ \\begin{align*} \\delta\\left[ \\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_j\\varphi_i}\\hat{h}_2\\ket{\\varphi_i\\varphi_j}\\right] \u0026= \\delta\\left[\\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_{i}(r_j)\\varphi_{j}(r_i)}\\hat{h}_2(r_i, r_j)\\ket{\\varphi_{j}(r_i)\\varphi_{i}(r_j)}\\right] \\\\ \u0026= \\sum_{i}^{N}\\sum_{j}^{N} \\textcolor{green}{\\bra{\\delta\\varphi_{i}(r_j)\\varphi_{j}(r_i)}\\hat{h}_2(r_i, r_j)\\ket{\\varphi_{j}(r_i)\\varphi_{i}(r_j)} }\\\\ \u0026 + \\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_{i}(r_j)\\delta\\varphi_{j}(r_i)}\\hat{h}_2(r_i, r_j)\\ket{\\varphi_{j}(r_i)\\varphi_{i}(r_j)} \\\\ \u0026 + \\sum_{i}^{N}\\sum_{j}^{N}\\textcolor{green}{\\bra{\\varphi_{i}(r_j)\\varphi_{j}(r_i)}\\hat{h}_2(r_i, r_j)\\ket{\\delta\\varphi_{j}(r_i)\\varphi_{i}(r_j)} }\\\\ \u0026 + \\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_{i}(r_j)\\varphi_{j}(r_i)}\\hat{h}_2(r_i, r_j)\\ket{\\varphi_{j}(r_i)\\delta\\varphi_{i}(r_j)} \\end{align*} \\]I highlighted the corresponding lines to show the trick!\n\\[ \\begin{align*} \\bra{\\delta\\varphi_{i}(r_j)\\varphi_{j}(r_i)}\\hat{h}_2(r_i, r_j)\\ket{\\varphi_{j}(r_i)\\varphi_{i}(r_j)} \u0026=\\underset{\\mathbb{R}^{6N}}{\\int} \\delta\\varphi_{i}^*(r_j)\\varphi_{j}^*(r_i)\\hat{h}_2(r_i, r_j)\\varphi_{j}(r_i)\\varphi_{i}(r_j) dr_idr_j \\\\ \u0026= \\left[\\underset{\\mathbb{R}^{6N}}{\\int} \\delta\\varphi_{i}(r_j)\\varphi_{j}(r_i)\\hat{h}_2(r_i, r_j)\\varphi_{j}^*(r_i)\\varphi_{i}^*(r_j) dr_idr_j \\right]^* \\\\ \u0026= \\left[\\underset{\\mathbb{R}^{6N}}{\\int} \\varphi_{j}^*(r_i)\\varphi_{i}^*(r_j)\\hat{h}_2(r_i, r_j) \\delta\\varphi_{i}(r_j)\\varphi_{j}(r_i) dr_idr_j \\right]^* \\end{align*} \\]Since \\(\\{r_i;r_j\\}\\) are \\(dummy\\) variables, the integral value doesn\u0026rsquo;t change if we swap the indices. Also, the operator \\(\\hat{h}_2\\) is invariant by changing the order (\\(\\hat{h}_2\\) represent a coulomb interaction ):\n\\[ \\begin{align*} \\underset{\\mathbb{R}^{6N}}{\\int} \\varphi_{j}^*(r_i)\\varphi_{i}^*(r_j)\\hat{h}_2(r_i, r_j) \\delta\\varphi_{i}(r_j)\\varphi_{j}(r_i) dr_idr_j \u0026= \\underset{\\mathbb{R}^{6N}}{\\int} \\varphi_{i}^*(r_j)\\varphi_{j}^*(r_i)\\underbrace{\\hat{h}_2(r_j, r_i)}_{\\equiv\\ \\hat{h}_2(r_i,r_j)} \\delta\\varphi_{j}(r_i)\\varphi_{i}(r_j) dr_idr_j\\\\ \u0026= \\bra{\\varphi_{i}(r_j)\\varphi_{j}(r_i)}\\hat{h}_2(r_i,r_j)\\ket{\\delta\\varphi_{j}(r_i)\\varphi_{i}(r_j)} \\end{align*} \\]Hence, we can write in more condensed way as before:\n\\[ \\begin{align*} \\delta\\left[ \\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_i}\\right] \u0026= \\sum_{i}^{N}\\sum_{j}^{N} \\bra{\\delta\\varphi_{i}(r_j)\\varphi_{j}(r_)}\\hat{h}_2(r_i, r_j)\\ket{\\varphi_{j}(r_i)\\varphi_{i}(r_j)}\\\\ \u0026+ \\sum_{i}^{N}\\sum_{j}^{N} \\bra{\\varphi_{i}(r_j)\\delta\\varphi_{j}(r_i)}\\hat{h}_2(r_i, r_j)\\ket{\\varphi_{j}(r_i)\\varphi_{i}(r_j)} \\\\ \u0026+ C.C \\end{align*} \\]Now we will omit the notation \\(\\varphi_i(r_j)\\) :\n\\[ \\begin{equation*} \\delta\\left[ \\sum_{i}^{N}\\sum_{j}^{N}\\bra{\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_i}\\right] =\\sum_{i}^{N}\\sum_{j}^{N} \\left[ \\bra{\\delta\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_i} + \\bra{\\varphi_i\\delta\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_i} + C.C \\right] \\end{equation*} \\]We are almost there! Lets write all the terms:\n\\[ \\begin{align*} \\delta F \u0026= \\sum_i^N\\Big[ \\bra{\\delta\\varphi_i}\\hat{h}_1\\ket{\\varphi_i} \\\\ \u0026+ \\textcolor{green}{\\frac{1}{2}} \\sum_{j}^{N}\\big[ \\bra{\\delta\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_i\\varphi_j} + \\bra{\\varphi_i\\delta\\varphi_j}\\hat{h}_2\\ket{\\varphi_i\\varphi_j}\\\\ \u0026- \\bra{\\delta\\varphi_i\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_i} - \\bra{\\varphi_i\\delta\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_i} \\big]\\\\ \u0026- \\sum_j^N \\lambda_{i,j}\\langle{\\delta\\varphi_i}|\\varphi_j\\rangle \\Big]\\\\ \u0026+ C.C \\end{align*} \\]It is not obvious, and the following argument that I rise here may be naive! but the factor of half disappear. Why? We can justify it by the interchanging the indices \\(i\\ \\\u0026\\ j\\) order, or we say at some point of the summation; the states are reached twice! At this point, it is more proper and urgent! to write the last equation in its integral form:\n\\[ \\begin{align*} \\delta F \u0026= \\sum_i^N \\Big\\{ \\underset{\\mathbb{R}^{6N}}{\\int} \\delta \\varphi^*_i(r_i)\\hat{h}_1(r_i)\\varphi_i(r_i)dr_i^2\\\\ \u0026+ \\sum_j^N \\Big[\\underset{\\mathbb{R}^{6N}}{\\int} \\delta \\varphi^*_i(r_i)\\varphi^*_j(r_j)\\hat{h}_2(r_i,r_j)\\varphi_i(r_i)\\varphi_j(r_j)dr_idr_j \\\\ \u0026- \\underset{\\mathbb{R}^{6N}}{\\int} \\delta \\varphi^*_i(r_j)\\varphi^*_j(r_i)\\hat{h}_2(r_i,r_j)\\varphi_j(r_i)\\varphi_i(r_j)dr_idr_j \\Big] \\\\ \u0026- \\sum_j^N \\lambda_{i,j}\\underset{\\mathbb{R}^{6N}}{\\int}\\delta \\varphi^*_i(r_i)\\varphi_j(r_j)dr_idr_j \\Big\\}\\\\ \u0026 + C.C \\end{align*} \\]Now we have our final \\(action\\) expression (\\(F\\)) of our system, we take the derivative of \\(F\\) with respect to some \\(k^{th}\\) bra (\\(\\bra{\\delta\\varphi_k} \\equiv \\varphi^*_k\\)). Notice that \\(\\frac{\\delta C.C}{\\delta\\varphi} = 0\\) (the reason for choosing the \\(bra\\) over the \\(ket\\) will be obvious at the end!).\n\\[ \\begin{align*} \\frac{\\delta F}{\\delta \\varphi_k^*} \u0026= \\underset{\\mathbb{R}^{3N}}{\\int} \\hat{h}_1(r_k)\\varphi_k(r_k)dr_k\\\\ \u0026+ \\sum_j^N \\Big[\\underset{\\mathbb{R}^{6N}}{\\int} \\varphi^*_j(r_j)\\hat{h}_2(r_k,r_j)\\varphi_k(r_k)\\varphi_j(r_j)dr_kdr_j \\\\ \u0026- \\underset{\\mathbb{R}^{6N}}{\\int} \\varphi^*_j(r_k)\\hat{h}_2(r_k,r_j)\\varphi_j(r_k)\\varphi_k(r_j)dr_kdr_j \\Big] \\\\ \u0026- \\sum_j^N \\lambda_{j}\\underset{\\mathbb{R}^{3N}}{\\int}\\varphi_j(r_j)dr_j \\end{align*} \\]The idea of factoring the \\(ket\\) is just to re-write our expression in Schrodinger-equation like form, and it\u0026rsquo;s not rigorous from the mathematical point of view! Hence, Factoring out the \\(\\ket{\\varphi_k}\\) gives:\n\\[ \\begin{gather*} \\underset{\\mathcal{R}^{3N}}{\\int} \\Big\\{ \\hat{h}_1(r_k) + \\sum_j^N \\Big[\\underset{\\mathcal{R}^{3N}}{\\int} \\varphi^*_j(r_j)\\hat{h}_2(r_i,r_j)\\varphi_j(r_j)dr_j - \\underset{\\mathbb{R}^{3N}}{\\int} \\varphi^*_j(r_i)\\hat{h}_2(r_i,r_j)\\varphi_k(r_j)dr_j \\Big] \\Big\\}\\varphi_k(r_k)dr_k= \\\\ = \\underset{\\mathcal{R}^{3N}}{\\int}\\sum_j^N \\lambda_{j}\\varphi_j(r_j)dr_j = \\underset{\\mathcal{R}^{3N}}{\\int}\\sum_j^N \\lambda_{k}\\varphi_k(r_k)dr_k \\end{gather*} \\]Now we can go back to \\(bra-ket\\) notation without confusion:\n\\[ \\begin{equation*} \\Big[\\hat{h}_1 + \\sum_{j}^{N}( \\hat{J}_j - \\hat{K}_j )\\Big]\\ket{\\varphi_k}= \\sum_k^N \\lambda_{k}\\ket{\\varphi_k} \\end{equation*} \\]This is the why we choose \\(bra\\) over \\(ket\\), to end-up with system of Schrodinger-like equations! for which \\(\\hat{J}_i\\) is an integral operator stand for the classical interaction of an electron distributions given by \\(|\\varphi_j|^2\\) and is called the \\(\\textbf{direct\\ term}\\). While \\(\\hat{K}_j\\) called the \\(\\textbf{exchange operator}\\), has no classical analogue and it is a direct result of the antisymmetry property of the wavefunction. The Fock operator is given by:\n\\[ \\begin{equation} \\hat{F} = \\hat{h}_1 + \\sum_{j}^{N}( \\hat{J}_j - \\hat{K}_j ) \\end{equation} \\]The problem is reduced into finding the eigenvalues of the following set of equations:\n\\[ \\begin{equation} \\hat{F}\\ket{\\varphi_k}=\\varepsilon_k\\ket{\\varphi_k} \\end{equation} \\]For which the solutions \\(\\varepsilon_k\\) hold the new identity of the Lagrange multipliers.\n🗣 Although it’s tempting to interpret the eigenvalues as the energy levels of an \\(\\textbf{interacting system}\\), this is in fact \\(\\underline{not\\ justified}\\) because the single-electron picture is \\(\\textbf{not correct}\\). However, if interpreted correctly the \\(\\textbf{Hartree-Fock}\\) eigenvalues do correspond to certain physical entities. For instance, there exist \\(\\textbf{Koopman’s theorem}\\), stating that each eigenvalue of the \\(\\textbf{Fock}\\) operator gives the energy required to remove an electron from the corresponding single-electron state. And, similarly the energy required to add an electron to orbital \"\\(n\\)\" can be proven to be given by \\(+\\ \\varepsilon_n\\). 2.2.4 KOOPMAN\u0026rsquo;S THEOREM \u0026amp; The Ionization energy Go to TOC\nIn this section we use \\(Koopman's\\ theorem\\) to give an insight interpretation to the eigenenergies \\(\\varepsilon_k$\\) we find in the previous equation. We will consider two system with (N-1) and N electrons respectively. For which the system of \\(N-1\\) electron is missing one electron on some orbital \u0026ldquo;n\u0026rdquo;. Also, beside omitting the spin orbitals degrees of freedom, we postulate that the orbitals are unchanged during this process. We start by writing the HF energy of both system:\n\\[ \\begin{gather*} E_{HF}^{N-1} = \\sum_{k\\neq n}^N \\bra{\\varphi_k}\\hat{h}_1\\ket{\\varphi_k} + \\frac{1}{2} \\sum_{k \\neq n}^{N}\\sum_{j \\neq n}^{N} \\Big[\\bra{\\varphi_k\\varphi_j}\\hat{h}_2\\ket{\\varphi_k\\varphi_j} - \\bra{\\varphi_k\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_k}\\Big] \\\\ E_{HF}^{N} = \\sum_{k}^N \\bra{\\varphi_k}\\hat{h}_1\\ket{\\varphi_k} + \\frac{1}{2} \\sum_{k }^{N}\\sum_{j}^{N} \\Big[\\bra{\\varphi_k\\varphi_j}\\hat{h}_2\\ket{\\varphi_k\\varphi_j} - \\bra{\\varphi_k\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_k}\\Big] \\end{gather*} \\]Notice that for first line we do not have the \u0026ldquo;n\u0026rdquo; orbital in our sum. Now, we evaluate the difference \\(E_{HF}^{N-1}-E_{HF}^{N}\\) term by term through the full expansion of both summations:\n\\[ \\begin{align*} \\Delta E_{HF} \u0026= -\\bra{\\varphi_n}\\hat{h}_1\\ket{\\varphi_n} \\\\ \u0026-\\frac{1}{2}\\sum_{j}^{N}\\Big[ \\bra{\\varphi_n\\varphi_j}\\hat{h}_2\\ket{\\varphi_n\\varphi_j} - \\bra{\\varphi_n\\varphi_j}\\hat{h}_2\\ket{\\varphi_j\\varphi_n} \\Big]\\\\ \u0026- \\frac{1}{2}\\sum_{k}^{N}\\Big[ \\bra{\\varphi_k\\varphi_n}\\hat{h}_2\\ket{\\varphi_k\\varphi_n} - \\bra{\\varphi_k\\varphi_n}\\hat{h}_2\\ket{\\varphi_n\\varphi_k} \\Big] \\end{align*} \\]Since the indices \\(k\\) and \\(j\\) are irrelevant (\\(k\\equiv j\\)), and by make use of the identity \\(\\bra{f_1f_2}\\hat{A}\\ket{f_3f_4} = \\bra{f_2f_1}\\hat{A}\\ket{f_4f_3}\\), we get:\n\\[ \\begin{gather*} \\Delta E_{HF} = - \\bra{\\varphi_n}\\hat{h}_1\\ket{\\varphi_n} - \\sum_{j}^{N} \\Big[ \\bra{\\varphi_j\\varphi_n}\\hat{h}_2\\ket{\\varphi_j\\varphi_n} - \\bra{\\varphi_j\\varphi_n}\\hat{h}_2\\ket{\\varphi_n\\varphi_j}\\Big] \\end{gather*} \\]The last equation is nothing less that the negative eigenenergy of the \\(\\textbf{Fock}\\) operator on the orbital \u0026ldquo;n\u0026rdquo;:\n\\[ \\begin{gather*} - \\bra{\\varphi_n}\\hat{F}\\ket{\\varphi_n} = - \\varepsilon_n \\end{gather*} \\] 🗣 \\(\\textbf{Koopmans’ theorem}\\), states that each eigenvalue of the \\(\\textbf{Fock}\\) operator gives the energy required to remove an electron from the corresponding single-electron state. And, similarly the energy required to add an electron to orbital \"n\" can be proven to be given by \\(+\\ \\varepsilon_n\\). 2.2.5 Hartree-Fock-Roothaan Go to TOC\nOne of the attempt to solve the \\(\\textbf{HF}\\) equation was done in 1951 by \\(\\textbf{Roothaan}\\). Clemens C. J. Roothaan was a Dutch physicist and chemist known for his development of the self-consistent field theory of molecular structure2. For a quick read brows the Roothaan equation .\n2: Wikipedia-Bibliography 3 The many-body Problem as electron density dependence Go to TOC\nIn this section we will rewrite the Hamiltonian as function of the electron density. Again we will treat the Hamiltonian as a spin-less problem. This is the core idea behind density dunctional theory where the function we are interested in take another function and return an output a.k.a number (\\(f \\longrightarrow I[f]\\)).\nBut first we need to write the later entity \\(n(r)\\). So to start we define \\(n(r_1)\\) the electron density of finding an electron at position \\(r_1\\) while the rest of electrons set are at positions \\(\\{r_2,r_3, \\dots, r_N\\}\\) as following:\n\\[ \\begin{gather} \\hat{n}(r_1) = \\delta (r-r_1) \\\\ n(r_1) = \\bra{\\Psi(r_1, r_2, \\dots, r_N)}\\hat{n}(r_1)\\ket{\\Psi(r_1, r_2, \\dots, r_N)} \\\\ n(r_1) = \\underset{\\mathbb{R}^3}{\\int} \\dots \\underset{\\mathbb{R}_N^3}{\\int} \\Psi^*(r_1, r_2, \\dots, r_N) \\delta (r-r_1) \\Psi(r_1, r_2, \\dots, r_N) dr_1dr_2\\dots dr_N \\end{gather} \\]First thing to notice is that we can write the product of the \\(bra-ket\\) function as the norm squared \\(|\\Psi(r_1, r_2, \\dots, r_N)|^2\\) (the operator \\(\\hat{n}(r_1)\\) does not acts on the wavefunction) which yields:\n\\[ \\begin{gather*} n(r_1) = \\underset{\\mathbb{R}_N^3}{\\int} \\dots \\underbrace{\\Big(\\underset{\\mathbb{R}^3}{\\int} \\delta (r-r_1) |\\Psi(r_1, r_2, \\dots, r_N)|^2 dr_1\\Big)}_{}dr_2\\dots dr_N\\\\ n(r_1) = \\underset{\\mathbb{R}_2^3}{\\int}\\dots\\underset{\\mathbb{R}_N^3}{\\int} |\\Psi(r, r_2, \\dots, r_N)|^2 dr_2\\dots dr_N \\end{gather*} \\]By following the same reasoning the electron density of many-body system \\(n(r)\\) is given by the sum of all individual electrons densities(the definition \\(n(r)= \\sum_i^N n(r_i)\\), is justified by seeing the operator of \\(\\hat{n}(r)\\) as a linear one):\n\\[ \\begin{gather*} n(r)= \\underset{\\mathbb{R}^{3(N-1)}}{\\int} |\\Psi(r, r_2, \\dots, r_N)|^2 dr_2\\dots dr_N + \\underset{\\mathbb{R}^{3(N-1)}}{\\int} |\\Psi(r_1, r, \\dots, r_N)|^2 dr_1dr_3\\dots dr_N + \\dots \\end{gather*} \\]Since the propriety of the Norm squared of the wavefunction does not change with its arguments shuffled(\\(|\\Psi(r_1, \\dots r_i,r_j,\\dots, r_N)|^2 = |\\Psi(r_1, \\dots r_j,r_i,\\dots, r_N)|^2\\)). It is obvious that we end-up having \\(N\\) equal terms, because of the indistinguishable nature of electrons:\n\\[ \\begin{gather} n(r)= N \\underset{\\mathbb{R}^{3(N-1)}}{\\int} |\\Psi(r, r_2, \\dots, r_N)|^2 dr_2\\dots dr_N \\end{gather} \\]If we integrate the last equation over the remaining spaces we get the electrons number(here we consider a normalized wavefunction):\n\\[ \\begin{equation} N = \\int drn(r) \\end{equation} \\]3.1 The electrons-ions interaction Go to TOC\nPreviously we had established the electrons-ions interaction, now we will introduce the electrons density \\(n(r)\\). We start as always by taking the expectation values of the later coulomb interaction:\n\\[ \\begin{gather*} \\bra{\\Psi(r_1, \\dots, r_N)}\\hat{V}_{e-n}\\ket{\\Psi(r_1, \\dots, r_N)}\\\\ =\\sum_i^{N} \\underset{\\mathbb{R}^{3N}}{\\int} \\Psi^*(r_1, \\dots, r_N)v_{e-n}(r_i)\\Psi(r_1, \\dots, r_N)dr_1\\dots dr_N\\\\ = -\\sum_i^{N}\\sum_j^{N_n}\\underset{\\mathbb{R}^{3N}}{\\int} \\Psi^*(r_1, \\dots, r_N)\\frac{Z_j}{|r_i-R_j|}\\Psi(r_1, \\dots, r_N)dr_1\\dots dr_N \\end{gather*} \\]The operator \\(\\hat{V}_{e-n}\\) allows for the appearance of the Norm squared of the wavefunction. Then we expand the summation over the electrons:\n\\[ \\begin{align*} \\langle\\hat{V}_{e-n}\\rangle_{\\Psi} \u0026= -\\sum_j^{N_n} \\Big[\\underset{\\mathbb{R}^{3}}{\\int} \\frac{Z_j}{|r_1-R_j|}|\\Psi(r_1, \\dots, r_N)|^2dr_1\\dots dr_N+ \\dots\\\\ \u0026 + \\underset{\\mathbb{R}^{3}}{\\int} \\frac{Z_j}{|r_N-R_j|}|\\Psi(r_1, \\dots, r_N)|^2dr_1\\dots dr_N\\big]\\\\ \u0026=\\sum_j^{N_n} \\Big[\\underset{\\mathbb{R}^{3}}{\\int} \\frac{Z_j}{|r_1-R_j|}dr_1\\underset{\\mathbb{R}^{3(N-1)}}{\\int} |\\Psi(r, \\dots, r_N)|^2dr_2\\dots dr_N+ \\dots\\\\ \u0026+\\underset{\\mathbb{R}^{3}}{\\int} \\frac{Z_jdr_1}{|r_N-R_j|} dr_N\\underset{\\mathbb{R}^{3(N-1)}}{\\int} |\\Psi(r_1, \\dots, r_{N-1}, r)|^2dr_1\\dots dr_{N-1}\\Big] \\end{align*} \\]The factor of each individual coulomb interaction integral is the same! Beside the same trick of shuffling the arguments and changing re-arranging the indices, we use the dummy variables \\(r_i \\longrightarrow r\\) we rewrite:\n\\[ \\begin{gather*} \\langle\\hat{V}_{e-n}\\rangle_{\\Psi} = -N\\sum_j^{N_n}\\underset{\\mathbb{R}^{3}}{\\int} \\frac{Z_j}{|r-R_j|}dr\\underbrace{\\underset{\\mathbb{R}^{3(N-1)}}{\\int} \\Psi(r, \\dots, r_{N})dr_2\\dots dr_{N}}\\_{n(r)/N} \\end{gather*} \\]We notice the appearance of the electron density that we define it previously, hence:\n\\[ \\begin{equation} \\langle\\hat{V}_{e-n}\\rangle_{\\Psi} = \\int V_{e-n}(r)n(r)dr \\end{equation} \\]3.2 The electrons-electrons interaction Go to TOC\nUsing the same manner that we did in the last section we will introduce the electron density. Note that it is not quite the same quantity \\(n(r)\\)! Within the electrons-electrons interaction. And the starting point is always the avearage value:\n\\[ \\begin{gather*} \\bra{\\Psi(r_1, \\dots, r_N)}\\hat{V}_{e-e}\\ket{\\Psi(r_1, \\dots, r_N)}\\\\ = \\frac{1}{2}\\sum_i^{N}\\sum_{j\\neq i}^{N}\\underset{\\mathbb{R}^{3N}}{\\int} \\Psi^*(r_1, \\dots, r_N)v_{e-e}(r_i,r_j)\\Psi(r_1, \\dots, r_N)dr_1\\dots dr_N\\\\ = \\frac{1}{2} \\sum_i^{N} \\sum_{j\\neq i}^{N} \\underset{\\mathbb{R}^{3N}}{\\int} \\frac{|\\Psi^*(r_1, \\dots, r_N)|^2}{|r_i-r_j|}dr_1\\dots dr_N \\end{gather*} \\]We will expend over the \\(i\\) summation:\n\\[ \\begin{gather*} 2\\langle\\hat{V}_{e-e}\\rangle_{\\Psi}=\\sum_{j\\neq i}^{N_n}\\Big[\\underset{\\mathbb{R}^{3}}{\\int}dr_1 \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^*(r_1, \\dots, r_N)|^2}{|r_1-r_j|}dr_2\\dots dr_N \\\\ + \\underset{\\mathbb{R}^{3}}{\\int}dr_2 \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^{*}(r_1, \\dots, r_N)|^2}{|r_2-r_j|}dr_1\\dots dr_N + \\dots \\\\ + \\underset{\\mathbb{R}^{3}}{\\int}dr_N \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^{*}(r_1, \\dots, r_N)|^2}{|r_N-r_j|}dr_1\\dots dr_{N-1}\\Big] \\end{gather*} \\]Lets expand the first term over the summation \\(j\\neq i\\): \\[ \\begin{gather*} \\sum_{j\\neq i}^{N_n}\\underset{\\mathbb{R}^{3}}{\\int}dr_1 \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^*(r_1, \\dots, r_N)|^2}{|r_1-r_j|}dr_2\\dots dr_N=\\underset{\\mathbb{R}^{3}}{\\int}dr_1 \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^*(r_1, \\dots, r_N)|^2}{|r_1-r_2|}dr_2\\dots dr_N \\\\ + \\underset{\\mathbb{R}^{3}}{\\int}dr_1 \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^*(r_1, \\dots, r_N)|^2}{|r_1-r_3|}dr_2\\dots dr_N + \\dots \\\\ + \\underset{\\mathbb{R}^{3}}{\\int}dr_1 \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^*(r_1, \\dots, r_N)|^2}{|r_1-r_N|}dr_2\\dots dr_N\\\\ = \\underset{\\mathbb{R}^{6}}{\\int}\\frac{dr_1dr_2}{|r_1-r_2|} \\underset{\\mathbb{R}^{3(N-2)}}{\\int}|\\Psi^*(r_1, \\dots, r_N)|^2dr_3\\dots dr_N \\\\ + \\underset{\\mathbb{R}^{6}}{\\int}\\frac{dr_1dr_3}{|r_1-r_3|} \\underset{\\mathbb{R}^{3(N-2)}}{\\int}|\\Psi^*(r_1, \\dots, r_N)|^2dr_2\\dots dr_N + \\dots\\\\ + \\underset{\\mathbb{R}^{6}}{\\int}\\frac{dr_1dr_{N}}{|r_1-r_N|} \\underset{\\mathbb{R}^{3(N-2)}}{\\int}|\\Psi^*(r_1, \\dots, r_N)|^2dr_2\\dots dr_{N-1} \\end{gather*} \\]We apply a dummy variable changing \\(r_1\\longrightarrow r\\) and \\(r_j\\longrightarrow r'\\):\n\\[ \\begin{gather*} \\sum_{j\\neq i}^{N_n}\\underset{\\mathbb{R}^{3}}{\\int}dr_1 \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^*(r_1, \\dots, r_N)|^2}{|r_1-r_j|}dr_2\\dots dr_N = \\underset{\\mathbb{R}^{6}}{\\int}\\frac{drdr'}{|r-r'|} \\underset{\\mathbb{R}^{3(N-2)}}{\\int}|\\Psi^*(r, r', \\dots, r_N)|^2dr_3\\dots dr_N + \\dots\\\\ + \\underset{\\mathbb{R}^{6}}{\\int}\\frac{drdr'}{|r-r'|} \\underset{\\mathbb{R}^{3(N-2)}}{\\int}|\\Psi^*(r, \\dots, r')|^2dr_3\\dots dr_N \\end{gather*} \\]Basically the integral over the \\(\\mathbb{R}^{3(N-2)}\\) space is the same, and we have \\((N-1)\\) term. And again the shuffling propriety of the wave function:\n\\[ \\begin{gather*} \\sum_{j\\neq i}^{N_n}\\underset{\\mathbb{R}^{3}}{\\int}dr_1 \\underset{\\mathbb{R}^{3(N-1)}}{\\int}\\frac{|\\Psi^*(r_1, \\dots, r_N)|^2}{|r_1-r_j|}dr_2\\dots dr_N = (N-1)\\underset{\\mathbb{R}^{6}}{\\int}\\frac{drdr'}{|r-r'|} \\underset{\\mathbb{R}^{3(N-2)}}{\\int}|\\Psi^*(r, r', \\dots, r_N)|^2dr_3\\dots dr_N \\end{gather*} \\]We treat each \\(N\\) remaining term the same, and we get:\n\\[ \\begin{gather*} \\langle\\hat{V}_{e-e}\\rangle_{\\Psi} = \\frac{N(N-1)}{2}\\underset{\\mathbb{R}^{6}}{\\iint}\\frac{drdr'}{|r-r'|} \\underset{\\mathbb{R}^{3(N-2)}}{\\int}|\\Psi^*(r, r', \\dots, r_N)|^2dr_3\\dots dr_N \\end{gather*} \\]Now we define the new pair electron density as following:\n\\[ \\begin{equation} n^{(2)}(r,r') = N(N-1)\\underset{\\mathbb{R}^{3(N-2)}}{\\int}|\\Psi^*(r, r', \\dots, r_N)|^2dr_3\\dots dr_N \\end{equation} \\] ℹ️ Which describe the density of two pair of electrons, one of them is seen as the center of the mass of the left \\(N-1\\) electrons system. Finally the electrons-electrons interaction is reduced into:\n\\[ \\begin{equation*} \\langle\\hat{V}_{e-e}\\rangle_{\\Psi} =\\iint drdr' v_{e-e}(r,r')n^{(2)}(r, r') \\end{equation*} \\]Here we will re-write the new raised quantity as an expansion of the usual density:\n\\[ \\begin{equation*} n^{(2)}(r,r') = n(r)\\times n(r') + n^{(2)}_{corr}(r,r') \\end{equation*} \\]Which lead to:\n\\[ \\begin{equation} \\langle\\hat{V}_{e-e}\\rangle_{\\Psi} =\\iint drdr' v_{e-e}(r,r')n(r)n(r') + E_c \\end{equation} \\]The \\(E_c\\) is a correction term.\n3.3 The Kinetic Energy Go to TOC\nYet, we did not write our Energy as a functional of the density \\(n(r)\\). We still have the Kinetic term which is hard to deal with since we have the operator \\(\\nabla^2\\) that act on the wavefunction (\\(\\Psi\\)). In order to tackle the kinetic energy, we make one of the key assumptions of density functional theory. We assume that the density can be written as the sum norm squares of a collection of single-particle orbitals:\n\\[ \\begin{equation*} n(r) = \\sum_i^N |\\varphi_i(r)|^2 \\end{equation*} \\]These orbitals are called \\(\\textbf{Kohn-Sham orbitals}\\) and they are initially completely unspecified in much the same way as in the orbitals in the Slater determinant in the \\(\\textbf{Hartree-Fock}\\) formalism. The above form cannot really be considered an approximation. It simply says that instead of the full many-particle system we consider an auxiliary system of single-particle orbitals that have the same ground state density as the real system. Hence the Kinetic energy can be written in term to the KS orbitals to a correction term; as following:\n\\[ \\begin{equation} T = -\\frac{1}{2}\\sum_i^N\\int \\varphi_i(r)\\nabla^2_r \\varphi_i(r)dr + E_x \\end{equation} \\] 💡 The idea behind the \\textbf{KS-orbitals} is to let introduce the density indirectly especially when it comes to the variation principal. The global idea is that the Kinetic energy had been linked to the density through \\(\\textbf{KS-orbitals}\\). 3.4 Introducing the Exchange Correlation term Go to TOC\nIn the following we will refer to the paper of \\(\\textsc{Kohn and Sham}\\)3. As it was reported in the later paper, the ground state energy has the following form:\n\\[ \\begin{equation} E[n(r)]= -\\frac{1}{2}\\sum_i^N\\int \\varphi_i(r)\\nabla^2_r \\varphi_i(r)dr + \\iint \\frac{drdr'}{|r-r'|}n(r)n(r') + \\int V_{e-n}(r)n(r)dr + E_{xc}[n(r)] \\end{equation} \\]Where:\n\\[ \\begin{equation} E_{xc}[n(r)] = \\int dr\\varepsilon_{xc}[n(r)]n(r) \\end{equation} \\]Where \\(\\varepsilon_{xc}\\) is the exchange and correrrelation energy rep electron of a uniform electron gas of density \\(n(r)\\)3.\nThe origin of this term is the difference between a system of N interacting and noninteracting particles. More specifically:\n\\(\\textbf{Exchange}\\) energy is the Pauli repulsion, omitted in the Hartree term; \\(\\textbf{Correlation}\\) energy is the repulsion between electrons. 3 Walter Kohn and Lu Jeu Sham. Self-consistent equations including exchange and correlation effects. Physical review, 140(4A):A1133, 1965 3.5 THE HOHENBERG-KOHN THEOREM Go to TOC\nTheorem I:\n📜 The external potential \\(V_{ext}(r)\\) is a unique functional of the density \\(n(r)\\). Proof\nLet assume that exist two different external potential \\(V_{ext}^{(1)}(r)\\) and \\(V_{ext}^{(2)}(r)\\) that gives rise to the same electrons density for non-degenerate ground state \\(n(r)\\). Hence, we have two wavefunction for two Hamiltonian:\n\\[ \\begin{gather*} \\langle \\hat{H}^{(1)}\\rangle_{\\psi^{(1)}} = \\langle \\hat{T}\\rangle_{\\psi^{(1)}} + \\langle \\hat{V}_{e-e}\\rangle_{\\psi^{(1)}} + \\int n(r)V_{ext}(r)dr \\\\ \\langle \\hat{H}^{(1)}\\rangle_{\\psi^{(1)}} \u003c \\langle \\hat{T}\\rangle_{\\psi^{(2)}} + \\langle \\hat{V}_{e-e}\\rangle_{\\psi^{(2)}} + \\int n(r)V_{ext}(r)dr\\\\ \\langle \\hat{T}^{(1)}\\rangle_{\\psi^{(1)}} + \\langle \\hat{V}_{e-e}\\rangle_{\\psi^{(1)}} \u003c \\langle \\hat{T}^{(2)}\\rangle_{\\psi^{(2)}} + \\langle \\hat{V}_{e-e}\\rangle\\_{\\psi^{(2)}} \\end{gather*} \\]In the other hand:\n\\[ \\begin{gather*} \\langle \\hat{H}^{(2)}\\rangle_{\\psi^{(2)}} = \\langle \\hat{T}\\rangle_{\\psi^{(2)}} + \\langle \\hat{V}_{e-e}\\rangle_{\\psi^{(2)}} + \\int n(r)V_{ext}(r)dr \\\\ \\langle \\hat{H}^{(2)}\\rangle_{\\psi^{(2)}} \u003c \\langle \\hat{T}\\rangle*{\\psi^{(1)}} + \\langle \\hat{V}_{e-e}\\rangle_{\\psi^{(1)}} + \\int n(r)V_{ext}(r)dr\\\\ \\langle \\hat{T}^{(2)}\\rangle_{\\psi^{(2)}} + \\langle \\hat{V}_{e-e}\\rangle_{\\psi^{(2)}} \u003c \\langle \\hat{T}^{(1)}\\rangle_{\\psi^{(1)}} + \\langle \\hat{V}_{e-e}\\rangle_{\\psi^{(1)}} \\end{gather*} \\]Which is a \\(\\textbf{Contradiction}!!!\\)\nTheorem II:\n📜 A universal functional for the energy $E[n(r)]$ can be defined in terms of the density. The exact ground state is the global minimum value of this functional. Proof:\nLet \\(E\\) be the exact energy of the ground state of our system of \\(N\\) interacting electron (non-degenerate state), then:\n\\[ \\begin{gather*} E= \\underset{\\psi}{\\text{min}} \\langle \\hat{T} + \\hat{V}_{e-e} + \\hat{V}_{ext}\\rangle_{\\psi}\\\\ E= \\underset{\\psi}{\\text{min}} \\Big[ \\langle \\hat{T} + \\hat{V}_{e-e}\\rangle_{\\psi}+ \\langle\\hat{V}_{ext}\\rangle_{\\psi}\\Big]\\\\ E= \\underset{n(r)}{\\text{min}} \\Big[ \\iint drdr' v_{e-e}(r,r')n(r)n(r') + \\int n(r)V\\_{ext}(r)dr+ G[n(r)]\\Big] \\end{gather*} \\]Where \\(G[n(r)]\\) is a universal function of the density:\n\\[ \\begin{equation} G[n(r)] = \\underset{n(r)}{\\text{min}} \\Big[ T[n(r)] + E_{xc}[n(r)] \\Big] \\end{equation} \\]4 Kohn and Sham Equation: Derivation attempt! Go to TOC\nWe will try to re-derive the \\(\\textbf{KS}\\) equation by following the same procedures used in \\(\\textbf{HF}\\) approach. The idea is as simple as the minimization over the wavefunction! But an approximation is needed. As we discussed previously we already write the Energy as functional of the density (i.e: \\(E[n(r)]\\)); but it was shown that the kinetic energy raise a problem, that it can not be written in terms of the density \\(n(r)\\), so we consider the \\(\\textbf{KS}\\) orbitals that builds the density from ground i.e: \\(n(r)=\\sum_i^N |\\varphi(r)|^2\\), hence the kinetic term can be minimized through the set of these orbitals! that we postulate theirs orthogonality constraints:\n\\[ \\begin{gather*} \\frac{\\delta E[n]}{\\delta \\varphi^*_k(r)} =\\frac{\\delta}{\\delta \\varphi^*_k(r)} \\Big[ \\int n(r)v_{e-N}(r)dr + \\iint n(r)v_{e-e}(r,r')n(r')drdr' + G[n] \\Big] = 0\\\\ \\frac{\\delta}{\\delta \\varphi^*_k(r)} \\Big[ \\int n(r)v_{e-N}(r)dr + \\iint n(r)v_{e-e}(r,r')n(r')drdr' + G[n]\\\\ -\\sum_i^N\\varepsilon_i\\Big( \\int\\varphi^*_i(r)\\varphi_i(r)dr-1\\Big)\\Big] = 0 \\end{gather*} \\]Where \\(\\varepsilon\\) stand for the \\(\\textbf{Lagrange multiplier}\\). The two first terms on the left hand side are easy to manipulate!\n\\[ \\begin{gather*} \\underbrace{\\int v_{e-N}(r)\\varphi_k(r)dr + \\iint v_{e-e}(r,r')n(r')\\varphi_k(r)drdr'}*{\\int\\Theta(r)\\varphi_k(r)dr} + \\frac{\\delta G[n]}{\\delta \\varphi^*_k(r)} =\\\\ \\frac{\\delta }{\\delta \\varphi^*_k(r)} \\sum_i^N\\varepsilon_i\\Big( \\int\\varphi^*_i(r)\\varphi_i(r)dr-1\\Big) \\end{gather*} \\]We will try to tackle the universal function \\(G[n]\\):\n\\[ \\begin{gather*} \\frac{\\delta}{\\delta \\varphi^*_k(r)} G[n] = \\frac{\\delta}{\\delta \\varphi^*_k(r)} \\Big[T[n(r)] + E_{xc}[n(r)]\\Big] = \\frac{\\delta}{\\delta \\varphi^*_k(r)}\\Big[-\\frac{1}{2}\\sum_i^N\\int\\varphi^*_i(r)\\nabla^2_{r_i}\\varphi_i(r)dr\\Big] + \\frac{\\delta E_{xc}[n(r)]}{\\delta \\varphi^*_k(r)}\\\\ -\\frac{1}{2}\\int \\nabla^2_{r_k}\\varphi_k(r)dr + \\int \\frac{\\delta E_{xc}[n(r)]}{\\delta n(r')} \\frac{\\delta n(r')}{\\delta \\varphi^*_k(r)}\\delta(r-r')dr'=-\\frac{1}{2}\\int \\nabla^2_r\\varphi_k(r)dr + \\int \\frac{\\delta E_{xc}[n(r)]}{\\delta n(r)} \\varphi_k(r)dr = \\varepsilon_k\\int \\varphi_k(r)dr \\end{gather*} \\]For the \\(\\textbf{Lagrange multiplier}\\) we get:\n\\[ \\begin{gather*} \\frac{\\delta }{\\delta \\varphi^*_k(r)} \\sum_i^N\\varepsilon_i\\Big( \\int\\varphi^*_i(r)\\varphi_i(r)dr-1\\Big) = \\varepsilon_k\\int \\varphi_k(r)dr \\end{gather*} \\]The final expression called \\(\\textbf{KS-equation}\\) has the foollowing form:\n\\[ \\begin{gather} \\int \\Theta(r)\\varphi_k(r)dr-\\frac{1}{2}\\int\\nabla^2_{r_k}\\varphi(r)dr + \\frac{\\delta E_{xc}[n(r)]}{\\delta n(r)} \\varphi_k(r)dr \\end{gather} \\]Which is reduced into:\n\\[ \\begin{gather} \\Big[-\\frac{1}{2}\\nabla^2_{r_k} + \\Theta(r) + V_{xc}(r)\\Big] \\varphi_k(r) = \\varepsilon_k\\varphi_k(r) \\end{gather} \\]The problem is now can be solved exactly if we know the function \\(V_{xc}=\\frac{\\delta E_{xc}[n(r)]}{\\delta n(r)}\\)!\n✍️ The key for deriving the \\textbf{KS-equation} is knowing the following : \\[ \\begin{gather*} \\frac{\\delta h[f(x)]}{\\delta f(x)} = \\int \\frac{h[f(x)]}{g(x')}\\frac{g(x')}{f(x)} dx \\end{gather*} \\]Also, the ratio \\(\\frac{g(x')}{f(x)}\\) in our case is evaluated to be \\(f(x)\\delta(x-x')\\), where \\(f(x)\\) is the \\textbf{KS-wavefunction} \u0026ldquo;\\(\\varphi(r)\\)\u0026rdquo;.\n5 Solving KS equation Go to TOC\nIn this section we will see how we can solve \\(\\textbf{KS-equation}\\), but first we will write down the matrix formula equivalent to the Schrodinger-like equation; using the Planewave expansion. Without further ado, we start by defining our wavefunction (i.e: the \\(\\textbf{KS-orbitals}\\)) as following:\n\\[ \\begin{gather*} \\varphi_i(\\overrightarrow{r})= \\frac{1}{\\sqrt{\\Omega} }\\sum_{\\overrightarrow{q}} c_{i,q} e^{j\\overrightarrow{q}\\cdot\\overrightarrow{r}} \\end{gather*} \\]Where, \\(\\Omega\\) is the volume of the real space where we are working in, and \\(c_{i,q}\\) are the Fourier coefficients of the wavefunction. Note, that we are moving to the reciprocal space \\({\\overrightarrow{q}}\\) instead of the real one, which will make \\(\\textbf{Fourier transform}\\) the pillars of this work! For our Hamiltonian that is described by \\(\\textbf{KS}\\) equation will be rewritten as follwing:\n\\[ \\begin{gather*} \\hat{H}_{KS} = \\hat{T}_{s} + \\hat{V}_{eff} \\end{gather*} \\]Where, \\(\\hat{T}_{s}\\) is the single-particle kinetic energy and \\(\\hat{V}_{eff}\\). Where \\(\\hat{V}_{eff} = \\hat{\\Theta} + \\hat{V}_{xc}\\). Is the effective potential which contains the external, \\(\\textbf{Hartree}\\) (i.e: \\(\\hat{\\Theta}\\)) and \\(\\textbf{exchange-correlation}\\) (i.e: \\(\\hat{V}_{xc}\\)) parts. So far we can write the wavefunction in the basis set of \\(\\{\\overrightarrow{q}\\}\\) as following:\n\\[ \\begin{gather*} \\varphi_i(\\overrightarrow{r})= \\frac{1}{\\sqrt{\\Omega} }\\sum_{\\overrightarrow{q}} c_{i,q} \\ket{\\overrightarrow{q}} \\end{gather*} \\]Now to get the eigenvalues \\(\\varepsilon_i\\) we evaluate the following product:\n\\[ \\begin{gather*} \\bra{\\overrightarrow{q'}}\\hat{H}_{KS}\\varphi_i(r) = \\bra{\\overrightarrow{q'}}\\Big[\\hat{T}_{S} + \\hat{V}_{eff}\\Big]\\varphi_i(r)\\\\ \\frac{1}{\\sqrt{\\Omega} }\\sum_{\\overrightarrow{q}} c_{i,q}\\bra{\\overrightarrow{q'}}\\hat{H}_{KS} \\ket{\\overrightarrow{q}} = \\frac{1}{\\sqrt{\\Omega} }\\sum_{\\overrightarrow{q}} c_{i,q}\\varepsilon_i\\underbrace{\\langle\\overrightarrow{q'}|\\overrightarrow{q}\\rangle}_{\\delta_{q';q}} \\end{gather*} \\]Which yield to :\n\\[ \\begin{gather*} \\underline{\\underline{H}}\\cdot \\underline{\\underline{C}}=\\varepsilon_i\\cdot\\underline{\\underline{C}} \\end{gather*} \\]where \\(\\underline{\\underline{H}}\\) is the Hamiltonian in matrix representation and \\(\\underline{\\underline{C}}\\) is a vector of coefficients. Now, we will work out the kinetic energy term:\n\\[ \\begin{gather*} \\bra{\\overrightarrow{q'}}\\hat{T}_{s}\\ket{\\overrightarrow{q}} = -\\frac{1}{2}\\bra{\\overrightarrow{q'}}\\Big(\\frac{d^2}{dr^2}\\underbrace{\\ket{\\overrightarrow{q}}}_{\\equiv e^{j\\overrightarrow{q}\\cdot\\overrightarrow{r}}}\\Big) =\\frac{1}{2}||\\overrightarrow{q}||^2\\delta_{q';q} \\end{gather*} \\]In the planewave representation, the kinetic energy term assumes an extremely simple, diagonal form. Now we proceed with effective potential, since \\(V_{eff}\\) has the periodicity of the lattice and therefore the only allowed Fourier components are those with the wavevectors in the reciprocal space of the crystal:\n\\[ \\begin{gather*} V_{eff}(\\overrightarrow{r})= \\sum_n \\bar{V}_{eff}(\\overrightarrow{G}_n)e^{j\\overrightarrow{G}_n\\cdot\\overrightarrow{r}}\\\\ \\text{Where:\\ } \\bar{V}_{eff}(\\overrightarrow{G}_n) = \\frac{1}{\\Omega_{cell}}\\int_{\\Omega_{cell}}V_{eff}(\\overrightarrow{r})e^{-j\\overrightarrow{G}_n \\cdot\\overrightarrow{r}}d\\overrightarrow{r} \\end{gather*} \\]Performing the same product as previously gives:\n\\[ \\begin{gather*} \\bra{\\overrightarrow{q'}}\\hat{V}_{eff}\\ket{\\overrightarrow{q}}=\\sum_n \\bar{V}_{eff}(\\overrightarrow{G}_n) \\langle\\overrightarrow{q'}|\\overrightarrow{G}_n+\\overrightarrow{q}\\rangle= \\sum_n \\bar{V}_{eff}(\\overrightarrow{G}_n) \\delta_{||\\overrightarrow{q'}-\\overrightarrow{q}||;\\overrightarrow{G}_n } \\end{gather*} \\]By make use of the change in variable, by make \\(\\overrightarrow{q} = \\overrightarrow{k}+\\overrightarrow{G}_m\\) and \\(\\overrightarrow{q'} = \\overrightarrow{k}+\\overrightarrow{G}_m'\\), so the wave vectors (\\(q\\ \\\u0026\\ q'\\)) differ by a reciprocal lattice vector. which brings the Schrodinger-equation like to the \\(k\\) space:\n\\[ \\begin{gather*} \\sum_m H_{m',m}(\\overrightarrow{k})c_{i,m}(\\overrightarrow{k})=\\varepsilon_ic_{i,m'}(\\overrightarrow{k}) \\end{gather*} \\]In summary, we reduced the problem into the First \\(\\textbf{Brillouin zone}\\) (The primitive cell in the reciprocal space), thanks to \\(\\textbf{Bloch's theorem}\\). Now it is important to see how the \\(V_{eff}\\) can be written in the \\textbf{Fourier Space}. For the sake of notation we denote \\(n(\\overrightarrow{r})\\) the real electron density and \\(\\rho(\\overrightarrow{G})\\) it conjugate in the reciprocal space.\n5.1 The electron-electron interaction (Hartee Energy term) Go to TOC\nWe use the same definition of the planewave expension for the electron density:\n\\[ \\begin{align*} \\iint v_{e-e}(\\overrightarrow{r'}, \\overrightarrow{r})n(\\overrightarrow{r})n(\\overrightarrow{r'})d\\overrightarrow{r}d\\overrightarrow{r'} = \\frac{1}{2}\\sum_{\\overrightarrow{G'}, \\overrightarrow{G}}\\iint d\\overrightarrow{r}d\\overrightarrow{r'} \\frac{\\rho(\\overrightarrow{G'})\\rho(\\overrightarrow{G})}{||\\overrightarrow{r'}-\\overrightarrow{r}||}e^{j\\overrightarrow{G}\\cdot\\overrightarrow{r}}e^{j\\overrightarrow{G'}\\cdot\\overrightarrow{r'}} \\end{align*} \\]We write : \\(\\overrightarrow{u} = \\overrightarrow{r}-\\overrightarrow{r'} \\longleftrightarrow d\\overrightarrow{u} = \\overrightarrow{r}\\), and in the first exponential we add and subtract \\(\\overrightarrow{r'}\\):\n\\[ \\begin{align*} \\frac{1}{2}\\sum_{\\overrightarrow{G'}, \\overrightarrow{G}}\\iint d\\overrightarrow{u}d\\overrightarrow{r'} \\frac{\\rho(\\overrightarrow{G'})\\rho(\\overrightarrow{G})}{||\\overrightarrow{u}||}e^{j\\overrightarrow{G}\\cdot\\overrightarrow{u}}e^{j\\overrightarrow{r'}\\cdot\\Big(\\overrightarrow{G'}+\\overrightarrow{G}\\Big)} \\end{align*} \\]We separate the independent integral and write \\(\\underset{\\mathbb{R}^3}{\\int d\\overrightarrow{u}} =2\\pi\\int_{0}^{\\infty}\\int_{0}^{\\pi} u^2 du sin(\\theta) d\\theta\\):\n\\[ \\begin{align*} \\pi\\sum_{\\overrightarrow{G'}, \\overrightarrow{G}} \\rho(\\overrightarrow{G'})\\rho(\\overrightarrow{G}) \\underbrace{\\int e^{j\\overrightarrow{r'}\\cdot\\Big(\\overrightarrow{G'}+\\overrightarrow{G}\\Big)} d\\overrightarrow{r'} }_{\\Omega_{cell}\\times \\delta(\\overrightarrow{G'}-\\overrightarrow{G}) }\\int_{0}^{\\infty} \\Big[\\int_{0}^{\\pi} \\underbrace{e^{j G\\cdot u cos(\\theta)} u sin(\\theta) d\\theta}_{\\equiv - \\frac{e^{jx}}{G}dx \\ |\\ x = G\\cdot u cos(\\theta)\\ |\\ 1 \\langle x \\langle -1}\\Big]d u\\\\ \\iint v_{e-e}(\\overrightarrow{r'}, \\overrightarrow{r})n(\\overrightarrow{r})n(\\overrightarrow{r'})d\\overrightarrow{r}d\\overrightarrow{r'} = \\pi\\Omega_{cell}\\sum_{\\overrightarrow{G'}, \\overrightarrow{G}} \\rho(\\overrightarrow{G'})\\rho(\\overrightarrow{G})\\times \\delta(\\overrightarrow{G'}-\\overrightarrow{G}) \\int_{0}^{\\infty} \\underbrace{\\frac{e^{jG\\cdot u}}{jG}|_{-1}^{1}}_{2\\times \\frac{sin(Gu)}{G}} du \\end{align*} \\]The last integral factor it is not a proper integral!\n\\[ \\begin{equation*} \\int_0^\\infty \\frac{sin(x)}{G^2}dx = \\underset{a\\longrightarrow \\infty}{\\lim} \\frac{cos(X)}{G^2}\\Big|_{a}^0 =\\left\\{\\begin{array}{c} 0 \\\\ 2/G^2 \\\\ \\end{array}\\right. \\end{equation*} \\]Rigorously the limit does not exist (not unique a.k.a \\(\\lim_{x\\longrightarrow a} f(x) \\neq f(a) \\))!\n\\[ \\begin{align*} \\iint v_{e-e}(\\overrightarrow{r'}, \\overrightarrow{r})n(\\overrightarrow{r})n(\\overrightarrow{r'})d\\overrightarrow{r}d\\overrightarrow{r'} =4\\pi\\Omega_{cell}\\sum_{\\overrightarrow{G'}, \\overrightarrow{G}} \\frac{\\rho(\\overrightarrow{G'})\\rho(\\overrightarrow{G})}{||\\overrightarrow{G}||^2}\\times \\delta(\\overrightarrow{G'}-\\overrightarrow{G}) \\end{align*} \\]A further simplification: \\(G = -\\nu G'\\), because the density is a physical measurable quantity, \\(\\mathcal{F}\\{\\Phi\\}(\\omega)=\\mathcal{F}\\{\\Phi\\}(-\\omega)\\), where \\(\\mathcal{F}\\) is Fourier transform and \\(\\Phi\\) physical entity i.e the negative and the positive frequency components are equal [see chapter 8 of Aschroft and Mermin’s \u0026ldquo;Solid State Physics\u0026rdquo;].\nAnd by considering the propriety : \\(\\rho(\\overrightarrow{G}) = \\rho(\\overrightarrow{G'})\\)\n\\[ \\begin{gather} \\iint v_{e-e}(\\overrightarrow{r'}, \\overrightarrow{r})n(\\overrightarrow{r})n(\\overrightarrow{r'})d\\overrightarrow{r}d\\overrightarrow{r'} =4\\pi\\Omega_{cell} \\sum_{\\overrightarrow{G} }\\frac{\\rho^2(\\overrightarrow{G})}{||\\overrightarrow{G}||^2} \\end{gather} \\]5.2 The electron-electron interaction (Hartee Potential term) Go to TOC With the same analogy:\n\\[ \\begin{align*} \\int v_{e-n}(\\overrightarrow{r})n(\\overrightarrow{r})d\\overrightarrow{r} \u0026= -\\frac{1}{2}\\sum_{\\overrightarrow{G}} \\int d\\overrightarrow{r} \\frac{\\rho(\\overrightarrow{G})}{||\\overrightarrow{r}-\\overrightarrow{R}||} e^{j\\overrightarrow{G}\\cdot(\\overrightarrow{r}-\\overrightarrow{R})}e^{j\\overrightarrow{G}\\cdot\\overrightarrow{R}}\\\\ \u0026 = -\\frac{1}{2}\\sum_{\\overrightarrow{G}} \\underset{\\mathbb{R}^3}{\\int} du e^{j\\overrightarrow{G}\\cdot\\overrightarrow{R}} \\frac{\\rho(\\overrightarrow{G})}{G||\\overrightarrow{u}||}e^{j G\\cdot u cos(\\theta)}d(-G u cos(\\theta))\\\\ \u0026= -\\pi\\sum_{\\overrightarrow{G}}e^{j\\overrightarrow{G}\\cdot\\overrightarrow{R}} \\rho(\\overrightarrow{G}) \\underbrace{\\int_0^\\infty \\Big[\\frac{e^{j G\\cdot u cos(\\theta)}}{jG}\\Big|_\\pi^0\\Big]du}_{\\text{We accept blindly \"4/$G^2$\" as a result!}} \\end{align*} \\]And we ends up by having:\n\\[ \\begin{gather*} \\int v_{e-n}(\\overrightarrow{r})n(\\overrightarrow{r})d\\overrightarrow{r} = - 4\\pi\\sum_{\\overrightarrow{G}}\\frac{\\rho(\\overrightarrow{G})}{||\\overrightarrow{G}||^2} e^{j\\overrightarrow{G}\\cdot\\overrightarrow{R}} \\end{gather*} \\]5.3 The exchange-correlation term Go to TOC\nAs we have seen before that the last contribution of the effective potential in \\(\\textbf{KS}\\) Hamiltonian is due to the change of exchange-correlation energy function over a small variation in the electron density (i.e: \\(V_{xc}(\\overrightarrow{r}) = \\frac{\\delta E[n(\\overrightarrow{r})]}{\\delta n(\\overrightarrow{r})}\\)). We will use the expression given by Walter Kohn et al. 3 for the later term, and since we can exchange the derivation of a function over the integral like the case with regular functions, note that we could derive first and perform the Fourier decomposition, and we ends up by having two terms:\n\\[ \\begin{align*} V_{xc}(\\overrightarrow{r}) \u0026= \\frac{\\delta }{\\delta n(\\overrightarrow{r})} \\Big[ \\int d\\overrightarrow{r} \\rho(\\overrightarrow{r})\\bar{\\varepsilon}_{xc}[n(\\overrightarrow{r})] \\Big]\\\\ \u0026= \\frac{\\delta }{\\delta n(\\overrightarrow{r})} \\Big[\\sum_{\\overrightarrow{G},\\overrightarrow{G'}} \\int d\\overrightarrow{r} \\rho(\\overrightarrow{G})e^{j\\overrightarrow{G}\\cdot\\overrightarrow{r}}\\bar{\\varepsilon}_{xc}(\\overrightarrow{G'})e^{j\\overrightarrow{G'}\\cdot\\overrightarrow{r}} \\Big]\\\\ \u0026= \\frac{\\delta }{\\delta n(\\overrightarrow{r})} \\Big[\\sum_{\\overrightarrow{G},\\overrightarrow{G'}} \\rho(\\overrightarrow{G})\\bar{\\varepsilon}_{xc}(\\overrightarrow{G'}) \\int e^{j(\\overrightarrow{G}+\\overrightarrow{G'})\\cdot\\overrightarrow{r}}d\\overrightarrow{r}\\Big]\\\\ \u0026= \\Omega_{cell}\\frac{\\delta }{\\delta n(\\overrightarrow{r})} \\Big[\\sum_{\\overrightarrow{G},\\overrightarrow{G'}} \\rho(\\overrightarrow{G})\\bar{\\varepsilon}_{xc}(\\overrightarrow{G'}) \\times \\delta(\\overrightarrow{G}-\\overrightarrow{G'})\\Big] = \\Omega_{cell}\\frac{\\delta }{\\delta n(\\overrightarrow{r})} \\Big[\\sum_{\\overrightarrow{G}} \\rho(\\overrightarrow{G})\\bar{\\varepsilon}_{xc}(\\overrightarrow{G}) \\Big] \\end{align*} \\]3 Walter Kohn and Lu Jeu Sham. Self-consistent equations including exchange and correlation effects. Physical review, 140(4A):A1133, 1965 At this stage we shall say that we manage to perform Bottom-up synthesis of Density Functional Theory (DFT).\nThank you for reaching this points. ","permalink":"https://anouardelenda.github.io/blogs/dft/","tags":[{"LinkTitle":"Density Functional Theory","RelPermalink":"/tags/density-functional-theory/"},{"LinkTitle":"Kohn and Sham","RelPermalink":"/tags/kohn-and-sham/"},{"LinkTitle":"Schrödinger Equation","RelPermalink":"/tags/schr%C3%B6dinger-equation/"},{"LinkTitle":"Born-Oppenheimer Approximation","RelPermalink":"/tags/born-oppenheimer-approximation/"},{"LinkTitle":"Hartree-Fock Equation","RelPermalink":"/tags/hartree-fock-equation/"},{"LinkTitle":"Electronic Density","RelPermalink":"/tags/electronic-density/"},{"LinkTitle":"Wavefunction","RelPermalink":"/tags/wavefunction/"}],"title":"Bottom-up synthesis of Density Functional Theory (DFT)"},{"categories":[{"LinkTitle":"Publications","RelPermalink":"/categories/publications/"}],"content":" Abstract Compared to bulk solids, defects in low-dimensional materials and, specifically, 2D systems are expected to have a stronger effect, detrimental or beneficial, on their properties. Owing to their geometry, defects in 2D materials can easily be formed due to the interaction with the environment or under impacts of energetic particles, such as ions and electrons. At the same time, many concepts of defect production under irradiation in bulk systems are not applicable for 2D materials or require substantial modifications. Various aspects of the physics and chemistry of defects in 2D materials have been addressed, and the results of these investigations are presented in hundreds of research papers and review articles. However, the challenges and open questions that still remain in the field have received relatively little attention. These topics were recently addressed at the symposium “Defect-mediated engineering of nanomaterials for energy and quantum applications” organized by the Beilstein-Institut. Following the discussions at the symposium, here, we present the challenges and open questions in our understanding of the behavior of defective 2D materials, interaction of energetic particles with low-dimensional targets, and defect-mediated engineering of the properties of 2D systems. We further discuss possible solutions to these problems or suggest “work-arounds”, which should accelerate the progress in the field.\n","permalink":"https://anouardelenda.github.io/publications/beilstein/","tags":[{"LinkTitle":"2D Materials","RelPermalink":"/tags/2d-materials/"},{"LinkTitle":"Defects","RelPermalink":"/tags/defects/"},{"LinkTitle":"Electron Irradiation","RelPermalink":"/tags/electron-irradiation/"},{"LinkTitle":"Ion Bombardment","RelPermalink":"/tags/ion-bombardment/"}],"title":"Beilstein: Defects and defect-mediated engineering of two-dimensional materials: challenges and open questions"},{"categories":[],"content":"Dislocations in layered carbon nanomaterials Affiliation:🏛️Institut des Matériaux de Nantes Jean Rouxe, Nantes, France 🗺️ .\nReserach team 👥 Materials physics and nanostructures (PMN ) team\nSupervisors: Dr. Christopher Ewels , RD CNRS \u0026amp; Dr. Colin Bousige , RD CNRS.\nStarting date: 12 Nov 2024\nDuration: (03) Three years\nStatus: 🟩🟩🟩|🟩⬜⬜|⬜⬜⬜\nIn materials engineering, defects such as dislocations (e.g. screws and edges) are of fundamental importance. The control of these defects has the potential to drive significant advancements in the field of materials science. However, these defects are not yet fully understood, especially in the case of anisotropic materials. My research focuses on understanding the behaviour of these defects and their impact on layered materials such as graphite. To this end:\nI am focusing on dislocations defects (initially screws) and their impact on layered materials (primarily graphite). I use both Gaussian (AIMPRO ) and Plan-wave (VASP ) DFT-based code I perform large scale molecular dynamics simulations (LAMMPS ) using machine learning interatomic potential. ","permalink":"https://anouardelenda.github.io/portfolio/phd/","tags":[{"LinkTitle":"PhD","RelPermalink":"/tags/phd/"}],"title":"My Phd's context"},{"categories":[{"LinkTitle":"Publications","RelPermalink":"/categories/publications/"}],"content":"Influence of screw dislocations on stacking order in graphite Gabriel R. Francas,1,Anouar-Akacha Delenda2, Jacob W. Martin1, Colin Bousige3, Irene Suarez-Martinez1, Nigel A. Marks1, Chris Ewels2\n1 Dept. of Physics and Astronomy, Curtin University, Perth, Australia.\n2 Nantes Université, CNRS, Institut des Matériaux de Nantes Jean Rouxel, IMN, Nantes, F-44000, France.\n3 Universite Claude Bernard Lyon 1, CNRS, LMI UMR 5615, Villeurbanne F-69100, France.\nCarbon Journal Volume 247, February 2026, 120995\nAbstract This work examines how screw dislocations disrupt ideal stacking in graphenic materials, using a machine learning interatomic potential to model screw dislocation dipoles within different periodic cells. A novel tool is developed to assess local interlayer registration to quantify and visualise regions exhibiting stacking order. Using molecular dynamics simulations, we demonstrate that single screws exhibit greater stability in rhombohedral stacking compared to AA stacking, whereas double screws exhibit greater stability in Bernal AB stacking. The investigation reveals several mechanisms through which the ideal stacking configuration is achieved despite the presence of screw dislocations, including shearing, bond length distortion, and buckling. Furthermore, an upper threshold for the density of screw dislocations is calculated, beyond which ideal stacking cannot be realised, potentially offering an explanation for certain forms of turbostratic carbon. The findings also indicate that single screw dislocations hinder Bernal AB stacking in graphite, whereas double screw dislocations support Bernal stacking, which is significant for understanding the formation processes of graphite.\n","permalink":"https://anouardelenda.github.io/publications/screwstacking/","tags":[{"LinkTitle":"Screw Dislocation","RelPermalink":"/tags/screw-dislocation/"},{"LinkTitle":"Graphite","RelPermalink":"/tags/graphite/"},{"LinkTitle":"AB Graphite","RelPermalink":"/tags/ab-graphite/"},{"LinkTitle":"ABC Graphite","RelPermalink":"/tags/abc-graphite/"},{"LinkTitle":"Stacking","RelPermalink":"/tags/stacking/"},{"LinkTitle":"Bernal","RelPermalink":"/tags/bernal/"},{"LinkTitle":"Rhombohedral Graphite","RelPermalink":"/tags/rhombohedral-graphite/"},{"LinkTitle":"Graphitisation","RelPermalink":"/tags/graphitisation/"}],"title":"Influence of screw dislocations on stacking order in graphite"},{"categories":[],"content":"Table of contents I. Conferences I.1 The Carbon conference 2025, Saint-Malo, France (June 29th → July 4th 2025) I.2 The NanoteC24 conference, Nantes, France (27th → 30th August 2024) I.3 THE FIRST INTERNATIONAL CONFERENCE ON RENEWABLE ENERGY ADVANCED TECHNOLOGIES AND APPLICATIONS (ICREATA'21), Adrar, Algeria 25th → 27th October 2021 II. Workshops II.1 The Pyroman workshop (PNRB), Rennes, France (28th → 29th June 2025) II.2 Exploring chemical reactions in VASP (Online 🌐) (06th → 08th November 2024) N.B:\nWebsites for conferences and workshops are generally not maintained for more than one year. So, if some links does not work, it is probably because the platform is no longer maintained.\nI. Conferences I.1 The Carbon conference 2025, Saint-Malo, France (🗺️ ) The Carbon Conference is an international event organised every year in a different location around the world, including Europe, Asia and the United States. It gather scientists specialising in carbon materials from academia and industry gather to exchange views on all aspects of this fascinating element. Topics covered include the various synthesis processes, applications and characterisation of carbon materials, and an understanding of their properties. During this conference I participated with a poster titled: Modelling screw dislocations in graphite.\nPoster Participation cetificate I.2 The NanoteC24 conference, Nantes, France (🗺️ ) The NanoteC is a student friendly international meeting focusing on nanocarbon materials: their production, treatment, properties, in all their monocolour (black) glory! This three days meeting has been running annually,almost continuously since 1998, and in 2024 it took place in Nantes at the Institute of Materials, one of its regular haunts here many previous memorable NanoteCs have taken place. During this conference I participated with a poster titled: Development of a reactive Neural Network Potential for borophene on silver and gold. A fruitful work of the internship conducted at Institut de Lumière Matère (ILM), Theoretical Physical Chemistry team and Laboratory des Matériaux et Interfaces (LMI), Matériaux à Basse Dimensionnalité (MBD) team, during my last master in the University Claude Bernard Lyon 1, France. Villeurbane, France.\nPoster Participation cetificate Best poster award I.3 THE FIRST INTERNATIONAL CONFERENCE ON RENEWABLE ENERGY ADVANCED TECHNOLOGIES AND APPLICATIONS (ICREATA'21), Adrar, Algeria (🗺️ ) The video-conference aim to promote an exchange of recent advances and developments among scientists, scholars, engineers and in the various areas of renewable energy technologies include solar photovoltaic and thermal systems, wind farm, hydroelectricity power, biomass, biofuels, geothermal systems, Hybrid energy, Energy Storage, Hydrogen and Fuel Cells, etc.\nParticipation cetificate II. Workshops II.1 The Pyroman workshop (PNRB), Rennes, France (🗺️ ) Bulk Carbon Materials: processing conditions, structure \u0026 properties While a large body of contemporary research focuses on crystalline/ordered nanoforms of carbon (graphene, nanotubes, etc.), most current applications – as structural materials in aeronautics/space or nuclear applications, as electrodes for energy storage or catalysis, as adsorbents for gas/pollutants separation and storage, etc. – still rely on bulk and disordered solids. Describing accurately and unambiguously the inner “structure” - with this word taken in all the possible meanings and length scales - of bulk carbon materials, and relating the very details of this structure to the materials’ processing conditions and properties remains extremely challenging. Nowadays, the ever growing capacity of structural determination and imaging tools, down to atomic scale, as well as the advent of machine learning, brings a new momentum to this topic. More accurate description and modeling of carbon materials structure and properties are at hand. Meanwhile, the processing methods have undergone significant advances, in terms of diversity and of process control efficiency, thanks in particular to in-situ diagnostics.\nII.2 Exploring chemical reactions in VASP (Online 🌐) This workshop is provide by VASP every year. For more information and similar events check the News .\nParticipation cetificate ","permalink":"https://anouardelenda.github.io/portfolio/confsworkshops/","tags":[{"LinkTitle":"Carbon Conference","RelPermalink":"/tags/carbon-conference/"},{"LinkTitle":"Pyroman","RelPermalink":"/tags/pyroman/"},{"LinkTitle":"Nanotec","RelPermalink":"/tags/nanotec/"},{"LinkTitle":"Conferences","RelPermalink":"/tags/conferences/"},{"LinkTitle":"Workshops","RelPermalink":"/tags/workshops/"}],"title":"Conferences \u0026 Workshops"},{"categories":[{"LinkTitle":"Publications","RelPermalink":"/categories/publications/"}],"content":"A Portable Data Set for Borophene Growth Modeling with Reactive Neural Network Potentials Colin Bousige1, Anouar-Akacha Delenda2, Abdul-Rahman Allouche2, and Pierre Mignon2\n1 Universite Claude Bernard Lyon 1, CNRS, LMI UMR 5615, Villeurbanne F-69100, France.\n2 Universite Claude Bernard Lyon 1, CNRS, iLM UMR 5306, Villeurbanne F-69100, France.\nThe Journal of Physical Chemistry C 2025 129 (41), 18760-18771\nAbstract In this study, we develop and validate machine learned interaction potentials (MLIPs) for reactive simulation of borophene on metal substrates. A versatile training data set is constructed to accurately represent both extended and reactive borophene structures. It should be portable to train any MLIP architecture. Indeed, three generations of MLIPs, namely n2p2, DeePMD and NNMP, are trained and validated against density functional theory (DFT) calculations. Our results demonstrate the capability of the MLIPs to accurately reproduce DFT-calculated structures, energies, and forces. We finally show that it is possible to use these MLIPs to simulate the growth of borophene on a silver substrate.\n","permalink":"https://anouardelenda.github.io/publications/portabledsborml/","tags":[{"LinkTitle":"Boron","RelPermalink":"/tags/boron/"},{"LinkTitle":"Chemical Structure","RelPermalink":"/tags/chemical-structure/"},{"LinkTitle":"Neural Networks","RelPermalink":"/tags/neural-networks/"},{"LinkTitle":"Potential Energy","RelPermalink":"/tags/potential-energy/"},{"LinkTitle":"Silver","RelPermalink":"/tags/silver/"}],"title":"A Portable Data Set for Borophene Growth Modeling with Reactive Neural Network Potentials"},{"categories":[],"content":" M.Sc : Physics of Optics and Photonics (Lyon, France) 🏛️ Université de Claude Bernard Lyon 1. Villeurbane, France (🗺️ )\nFrom September 2023 to June 2024\nMore details The Master at the university of Claude Bernard Lyon 1, is a new track launched recently (~2022). The program is mostly dedicated to Optics and Photonics (OPHO). The program offers various courses mainly: Quantum and non-linear optics: Quantification of the Electromagnetic fields, Light nature and Photons statistics, Non-linear Optics process (SFG, SHG, THG ...) etc. Ultrafast optics: Ultrafast Dynamic, Autocorrelation; Frequency Resolved Optical Gating (FROG) and probe-pump measurements, Femtosecond Chirped Pulse Amplification (CPA) etc. Artificial intelligence: Supervised Machine Learning (Regression \u0026 Classification), Neural Network and Convolutional Neural Network ... Alongside this, other courses are proposed that I have enrolled for, such as Optics for Nano Structures Solids and Imaging Live cells ... M.Sc : Condensed Matter and Nanophysics (Strasbourg, France) 🏛️ Université de Strasbourg, Institut Physique et Chimiques Des Materiaux De Strasbourg (IPCM) (🗺️ )\nFrom September 2022 to June 2023\nMore details The Master has been renamed as \"Physics of quantum and soft condensed matter (PhyQS)\". The program is highly diverse and unique. The lectures cover materials' electronic structure, optical, magnetic properties, and their applications on the nanometer scale, while tracking recent advancements in the fields of nanophysics. The program has three main pillars hovering around solid-state physics: Advanced Quantum Mechanics: Quantum theory of diffusion; Landauer-Büttiker approach for the conductance of coherent systems; Kubo formalism and linear response theory for quantum electronic transport; graphene... Advanced Statistical Physics: Non-equilibrium statistical physics and transport phenomena: Brownian motion (concept of “coarse-graining”); Markov processes (Fokker-Planck and Langevin equations...); linear response theory (fluctuation-dissipation theorem \u0026 Kramers-Kronig relation). Light and Matter Interaction: Linear response of a material medium to an electromagnetic excitation; semi-classical approximations and optically induced transitions between electronic states of matter (Franck–Condon) ... In addition, I enrolled for additional lectures within the attended program: Theory and Computational Modeling of the Electronic Structure of Materials; Low Dimensional Nanostructures and Biophysics ... B.Sc and M.Sc : Materials Physics (Algiers, Algeria) 🏛️ University of Science and Technology - Houari Boumediene (USTHB), Algiers, Algeria (🗺️ )\nFrom 2019 to 2021 (Bachelors + Graduate/Master)\nMore details Here marks the beginning of my journey. After three years of studying fundamental physics, I decided to pursue a master's degree in Materials Physics. During this phase, I acquired a robust foundation in solid-state physics. Furthermore, the track I selected was focused on Semiconductors and Dielectric Materials. The program aimed to prepare future materials science researchers, encompassing both experimental and theoretical aspects. The master's program offered a range of lectures, and I will highlight a few of them: Physics of Condensed Matters Elastic Waves in Fluids and Solids Dielectric and Magnetic Properties of Materials Advanced Optics and Laser Physics Transport Phenomena in Semiconductors Materials Characterization ","permalink":"https://anouardelenda.github.io/portfolio/academic_background/","tags":[],"title":"Academic background"},{"categories":[],"content":"Identification of borophene allotropes from STM images by Machine Learning: from the development of a neural network interatomic potential to building the image classification tool. February 2024 → July 2024 (Master2)\n🧪🔬🖥️ Institut de Lumière Matère (ILM) , team: Theoretical Physical Chemistry 🧪🔬🖥️ Laboratory des Matériaux et Interfaces (LMI) , team: Matériaux à Basse Dimensionnalité (MBD) 🗺️ VILLEURBANNE, FRANCE. 📑 Thesis More details Borophene is a 2D material that has honeycomb structures similar to graphene. Borophene exists in various structures called allotropes. Furthermore, to characterize such materials, a Scanning Tunneling Microscope (STM) is generally used. However, the process is not straightforward and involves comparing experimental data to theoretically simulated images generated from Density Functional Theory (DFT). Although DFT is accurate, it is time-consuming. The intended purpose is to utilize an existing Neural Network Potential (NNP) to create a database of structures and simulated STM images for training and testing a classification neural network. Borophene Allotropes\nHigh dimensional neural network architecture\nEnergies \u0026 Forces-normes validations\nCharacterization Of Ultrashort Laser Pulse Centred at 800 nm February 2023 → June 2023 (Master2)\n🧪🔬🖥️ Institut de physique et chimie des Matériaux de Strasbourg (IPCMS) , department: Ultrafast Optics and Nanophotonics (DON) 🗺️ Strasbourg, FRANCE. 📑 Thesis More details In order to study the kinetics of molecules and probe their optical properties, we employ spectroscopy, which enables us to perform transient absorption and observe the signatures of rapid photo-reactions (e.g., carbon-carbon double bond isomerization) occurring on picosecond or sub-picosecond timescales. However, achieving this requires a well-characterized laser pulse, which means full access to temporal and spectral profiles. Unfortunately, electronic measurement does not accommodate the rapid oscillation of the electric field (few cycles). Here, we call for indirect optical characterization, such as using Frequency Resolved Optical Gating (FROG). For this work, we deployed Time Domain Ptychography techniques to solve a reconstruction problem, in particular, spectral phase retrieval. I implemented the Time Domain Ptychography algorithm to characterize the ultrashort pulses. Frequency Resolved Optical Gating (FROG)\nRetrievement of the pulse profile/phase.\nThe retrieved 2D FROG map difference between measured \u0026 reconstructed spectra.\nPower Loss Analysis On Based Silicon Solar cells March 2021 → June 2021 (Master2)\n🧪🔬🖥️ Research Center for Semiconductor Technology and Energetics (CRTSE) Division: Semiconductor Conversion Device Development 🗺️ Algiers, Algeria. 📑 Absract More details In order to study the kinetics of molecules and probe their optical properties, we employ spectroscopy, which enables us to perform transient absorption and observe the signatures of rapid photo-reactions (e.g., carbon-carbon double bond isomerization) occurring on picosecond or sub-picosecond timescales. However, achieving this requires a well-characterized laser pulse, which means full access to temporal and spectral profiles. Unfortunately, electronic measurement does not accommodate the rapid oscillation of the electric field (few cycles). Here, we call for indirect optical characterization, such as using Frequency Resolved Optical Gating (FROG). For this work, we deployed Time Domain Ptychography techniques to solve a reconstruction problem, in particular, spectral phase retrieval. I implemented the Time Domain Ptychography algorithm to characterize the ultrashort pulses. Manufactured m-Si solar cell (before screen printing) Manufactured m-Si solar cell (after screen printing) with Al leads (contacts) Dissipated Power in the shunt, the series resistance and in the forward bias diode. Current Analysis of losses due to reflection and parasitic absorption. ","permalink":"https://anouardelenda.github.io/portfolio/internships/","tags":[],"title":"Internships"},{"categories":[{"LinkTitle":"Projects","RelPermalink":"/categories/projects/"}],"content":"Model Selection for Atomization Energy Prediction Ultrafast Ptychography Linear tetrahedron method for electronic density calculations LearnPlatform COVID-19 Impact on Digital Learning ","permalink":"https://anouardelenda.github.io/portfolio/projects/","tags":[{"LinkTitle":"Projects","RelPermalink":"/tags/projects/"}],"title":"Projects"},{"categories":[{"LinkTitle":"Projects","RelPermalink":"/categories/projects/"}],"content":"Overview The Ptychography is a microscopy tool to investigate the crystal structure, this idea started in the lately sixties (1969) by Walter Hoppe. And by the growth of the computational power it start to gain more attention in 1982. The main concept addressed by this method is to reconstruct a crystal structure. Emerging the Ptychography into spectroscopy can be done by changing the problem from spatial-reciprocal space representation to time-frequency description.\nRead more: Here I built a python package to deal with daily retrieved spectroscopy data, a 2D map of retrieved intensity w.r.t a given wavelength range. The aim is to characterize the ultrafast lazer pulse by access to the intensity profile and the phase of the pulse.\n","permalink":"https://anouardelenda.github.io/projects/fastptyco/","tags":[{"LinkTitle":"Spectroscopy","RelPermalink":"/tags/spectroscopy/"},{"LinkTitle":"Ptychography","RelPermalink":"/tags/ptychography/"},{"LinkTitle":"Ultrafast","RelPermalink":"/tags/ultrafast/"},{"LinkTitle":"Optical Pulse","RelPermalink":"/tags/optical-pulse/"},{"LinkTitle":"Characterization","RelPermalink":"/tags/characterization/"},{"LinkTitle":"Spectral Phase","RelPermalink":"/tags/spectral-phase/"}],"title":"FASTPtyco"},{"categories":[{"LinkTitle":"Projects","RelPermalink":"/categories/projects/"}],"content":"LearnPlatform COVID-19 Impact on Digital Learning This was a Data Analysis challenge in Kaggle platform, during the the COVID pendamic. Where the challenge was to explore (1) the state of digital learning in 2020 and (2) how the engagement of digital learning relates to factors such as district demographics, broadband access, and state/national level policies and events.\nProblem Statement: The COVID-19 Pandemic has disrupted learning for more than 56 million students in the United States. In the Spring of 2020, most states and local governments across the U.S. closed educational institutions to stop the spread of the virus. In response, schools and teachers have attempted to reach students remotely through distance learning tools and digital platforms. Until today, concerns of the exacaberting digital divide and long-term learning loss among America’s most vulnerable learners continue to grow.\nRead more ","permalink":"https://anouardelenda.github.io/projects/dacovid19/","tags":[{"LinkTitle":"Data Analysis","RelPermalink":"/tags/data-analysis/"},{"LinkTitle":"COVID19","RelPermalink":"/tags/covid19/"},{"LinkTitle":"Education","RelPermalink":"/tags/education/"},{"LinkTitle":"Digital Learning","RelPermalink":"/tags/digital-learning/"},{"LinkTitle":"Engagement","RelPermalink":"/tags/engagement/"},{"LinkTitle":"Clickbait","RelPermalink":"/tags/clickbait/"},{"LinkTitle":"Kaggle","RelPermalink":"/tags/kaggle/"}],"title":"LearnPlatform COVID-19 Impact on Digital Learning"},{"categories":[{"LinkTitle":"Projects","RelPermalink":"/categories/projects/"}],"content":"linearLinear tetrahedron method for electronic density calculations This Projects aim to reproduce the results of Chadi and Cohen about the density of state (DOS). For this The Linear Tetrahedron Method were used. Here we just showed up the code, the provided data (Mesh grid of reciprocal space) are not shown. For this we took some basic example of the Germanium with the same fitted parameters from the previous papers for the Tight Binding approach.\nRead more : You can find more information and detailed description of the LTM with python code to solve taigh binfing model for various systems (Si, C etc).\n","permalink":"https://anouardelenda.github.io/projects/ltm/","tags":[{"LinkTitle":"Tight Banding","RelPermalink":"/tags/tight-banding/"},{"LinkTitle":"Linear Tetrahedron Method","RelPermalink":"/tags/linear-tetrahedron-method/"}],"title":"LTM"},{"categories":[{"LinkTitle":"Projects","RelPermalink":"/categories/projects/"}],"content":"Overview In this work, I aim to span a various Machine Learning models; especially the regression ones. And the goal is to select the most promissing model that can predict in a smart fashion the Atomization energy of a given molecular structure. The Data can be found in the DataBase called GDB−13, also it is available under the name roboBohr as csv file format in kaggle\u0026rsquo;s Dataset energy-molecule. The idea was a pivot of discussion in several works [M. Rupp et al , Lorenz C. Blum and J-L Reymond and lately Burak Himmetoglu ]. Moreover, the present notebook consist of learning about the suitable models that can be deployed; and to investigate how far I can achieve in term of performance. I will start as usual, by pre-processing the data (cleaning, droping NAN values \u0026hellip; etc), then I tackle very interesting section in data transformation fields mainly the so called Singular Value Decomposition (SVD). After that, I begin testing the models and norrowing the bands of these smart objects. The chosen models will undergo a testing performance according to various scores metrics. An adequate way for biasing the flow work is cited. A comparison between some strategies will be highlighted. After that, I try to tune the Hyperparamaters, and disscusse a features engineering attempt.\nRead more: for further detailed read a well public explained notebook is available on Kaggle platform.\n","permalink":"https://anouardelenda.github.io/projects/mlatomseng/","tags":[{"LinkTitle":"Machine Learning","RelPermalink":"/tags/machine-learning/"},{"LinkTitle":"Atomization Energy","RelPermalink":"/tags/atomization-energy/"},{"LinkTitle":"Energy-Molecule","RelPermalink":"/tags/energy-molecule/"},{"LinkTitle":"Singular Value Decomposition","RelPermalink":"/tags/singular-value-decomposition/"},{"LinkTitle":"Features Engineering","RelPermalink":"/tags/features-engineering/"},{"LinkTitle":"Hyperparamaters","RelPermalink":"/tags/hyperparamaters/"}],"title":"Model Selection for Atomization Energy Prediction"},{"categories":[],"content":"📝 Latest Blogs 19-04-2026 :\nNew blog post on Density Functional Theory published 📄 Latest Publications Published on 31 Mar 2026\nDefects and defect-mediated engineering of two-dimensional materials: challenges and open questions - Beilstein J. Nanotechnol. 2026 📬 Contact Feel free to reach out at :\nanouar-akacha.delenda@cnrs-imn.fr anouar-akacha.delenda@etu.univ-nantes.fr delenda.akacha@gmail.com ","permalink":"https://anouardelenda.github.io/next/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/manifest.json","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.de/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.es/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.fr/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.hi/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.jp/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.nl/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.pl/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.ru/","tags":[],"title":""},{"categories":[],"content":"","permalink":"https://anouardelenda.github.io/search/_index.zh-cn/","tags":[],"title":""}]