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Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_esluglxmto/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_esluglxmto/Manifest.toml` Instantiating... === Precompiling... ===PlutoUI& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_esluglxmto/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_esluglxmto/Manifest.toml` Instantiating... === Precompiling... ===HypertextLiteral& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_esluglxmto/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_esluglxmto/Manifest.toml` Instantiating... === Precompiling... ===PlutoTeachingTools& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_esluglxmto/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_esluglxmto/Manifest.toml` Instantiating... === Precompiling... ===enabledìinstantiated÷restart_recommended_msgrestart_required_msginstall_time_ns*m busy_packagescell_inputs$7142e1bd-200f-4395-94bf-0f0b3bb7ecd0cell_id$7142e1bd-200f-4395-94bf-0f0b3bb7ecd0code md""" This motivates a separation of variables ansatz: ```math \Psi(x,t) = \sum_{n = 0}^{\infty} c_n(t) \varphi_n(x). ``` We will see later that this postulate is justified. To start, let's just consider one term of the sum, i.e. $\Psi(x,t) = c(t) \varphi(x)$. - From the TDSE ```math i c'(t) \varphi(x) - c(t) \mathcal{H} \varphi(x) = 0 ``` - Assuming that $\varphi(x)$ and $c(t)$ are non-zero, we can divide ```math \begin{align} \frac{i c'(t)}{c(t)} = \frac{\mathcal{H} \varphi(x)}{\varphi(x)} && ∀ \ t>0, x ∈ \mathbb{R}^3. \end{align} ``` - The left only depends on $t$, and the right only on $x$. As a result, we need ```math \frac{i c'(t)}{c(t)} = \frac{\mathcal{H} \varphi(x)}{\varphi(x)} = E \in \mathbb R ``` for some constant $E$. """metadatashow_logsèdisabled®skip_as_script«code_folded$4378e396-1e6a-414f-8a8c-5f9ee7d8cadecell_id$4378e396-1e6a-414f-8a8c-5f9ee7d8cadecodeOmd""" The dynamics of the system are now given by the time-dependent Schrödinger equation (TDSE) ```math i \frac{\partial \Psi}{\partial t} = \mathcal{H} \Psi \tag{TDSE} ``` In general, solving this equation is quite complicated. However, since it is linear, we know that superpositions of multiple solutions are also a solution. """metadatashow_logsèdisabled®skip_as_script«code_folded$b102f351-f5b1-47c0-8737-f813b5b745f6cell_id$b102f351-f5b1-47c0-8737-f813b5b745f6code1md""" ## The Schrödinger equation and quantum mechanics At the microscopic level, the physics and chemistry of materials is governed by the interaction of electrons and nuclei. At this scale, the regime of quantum mechanics applies. In quantum mechanics the state of a system is described by the complex-valued square-integrable wave function $\Psi : (x,t) ↦ \Psi(x,t) \in \mathbb C$, where $x$ corresponds to the degrees of freedom (e.g. position or spin) of the particles (e.g. electrons) in the system. Without going into too many details we want to briefly discuss the key ideas how quantum mechanics describes physical systems. For simplicity, take $x \in \mathbb R^3$, i.e. one particle in 3D. Part of the meaning of $\Psi$ appears through its modulus squared. Indeed, for all $t$, $|\Psi(x,t)|^2$ corresponds to the probability distribution of finding the particle at position $x$ at time $t$. Further information appears through its Fourier transform, as $|\widehat \Psi (p,t)|^2$ corresponds to the probability distribution of finding the particle at momentum $p$ at time $t$, where the Fourier transform is given by ```math \widehat \Psi (p,t) = \frac{1}{(2\pi)^{3/2}} \int_{\mathbb R^3} \Psi (x,t) e^{- i \ p \cdot x}dx. ``` For those already familiar with quantum mechanics, note that we set $\hbar = 1$. """metadatashow_logsèdisabled®skip_as_script«code_folded$e52eb9de-c5b3-424e-b2d5-e1b926f76fd4cell_id$e52eb9de-c5b3-424e-b2d5-e1b926f76fd4codemd""" To solve this problem, we perform separation of variables ```math u(x,t) = \sum_{n = 1}^{\infty} c_n(t) v_n(x). ``` We will see later that this postulate is justified. To start, let's just consider one term of the sum, i.e. $u(x,t) = c(t) v(x)$. - From the heat equation ```math c'(t) v(x) - c(t) \Delta v(x) = 0 ``` - Assuming that $v(x)$ and $c(t)$ are non-zero, we can divide ```math \begin{align} \frac{c'(t)}{c(t)} = \frac{\Delta v(x)}{v(x)} && ∀ \ t>0, x ∈ \Omega. \end{align} ``` - The left only depends on $t$, and the right only on $x$. As a result, we need ```math \frac{c'(t)}{c(t)} = \frac{\Delta v(x)}{v(x)} = μ \in \mathbb R ``` for some $μ$. """metadatashow_logsèdisabled®skip_as_script«code_folded$52b4d68c-d6f1-11ee-18b4-db386a5995adcell_id$52b4d68c-d6f1-11ee-18b4-db386a5995adcodeJbegin using HypertextLiteral using PlutoUI using PlutoTeachingTools endmetadatashow_logsèdisabled®skip_as_script«code_folded$6748d907-6820-42af-b94c-6f6b9ef23948cell_id$6748d907-6820-42af-b94c-6f6b9ef23948code=md"Show outline $(@bind show_outline CheckBox(default=true))"metadatashow_logsèdisabled®skip_as_script«code_folded$53e49738-8673-4ecd-9e86-b1431200b132cell_id$53e49738-8673-4ecd-9e86-b1431200b132code٬md""" ## Opening remarks **[Slides of the presentation on moodle](/)**: These summarise the following paragraphs and the content of the course in a short presentation. """metadatashow_logsèdisabled®skip_as_script«code_folded$4db1319a-0b7c-42d5-ba03-00f106542c80cell_id$4db1319a-0b7c-42d5-ba03-00f106542c80code|md""" - One can show that the operator $- \Delta$ with homogeneous Dirichlet boundary conditions has a sequence of eigenvalues $0 < λ_1 ≤ λ_2 ≤ … → ∞$ and orthonormal eigenfunctions $\{ \phi_n \}_{n=1}^∞$, i.e. ```math - \Delta \phi_n = λ_n \phi_n ``` with ```math \langle \phi_n, \phi_m \rangle = ∫_\Omega \phi_n(x) \phi_m(x) dx = δ_{mn}. ``` - These form a complete basis of $H^1_0 (\Omega)$, the relevant Hilbert space for this problem. This justifies the ansatz ```math u(x,t) = \sum_{n=1}^∞ c_n(t) \phi_n(x) ``` we made before, where $\phi_n$ are now eigenfunctions of the Laplace operator. """metadatashow_logsèdisabled®skip_as_script«code_folded$189b99a5-776f-4ad6-b05a-66b9801a97c4cell_id$189b99a5-776f-4ad6-b05a-66b9801a97c4codeBmd""" ## The connection between linear and eigenvalue problems: the heat equation Let $\Omega \subset \mathbb R^n$ with a Lipschitz boundary. We seek $u : \Omega \times \mathbb R^+ → \mathbb R$ such that ```math \begin{align} \frac{∂ u}{∂ t} - \Delta u &= 0 && \text{in } \Omega \times \mathbb R^+ \tag{Heat equation} \\ u(\cdot,0) &= u_0 && \text{in } \Omega \tag{Initial conditions} \\ u &= 0 && \text{on } ∂ \Omega \times \mathbb R^+ \tag{Dirichlet b.c.} \end{align} ``` Where the initial conditions $u_0 : \Omega → \mathbb R$ are of appropriate regularity. """metadatashow_logsèdisabled®skip_as_script«code_folded$6a2f33fb-1561-4469-a392-2dc71caf74b4cell_id$6a2f33fb-1561-4469-a392-2dc71caf74b4codemd""" Less importantly for this course the above argument also shows that eigenvalue problems are closely linked to the dynamics (or time propagation) of differential operators. Understanding eigenvalue problems and the spectral properties of operators therefore also gives us understanding about PDE problems in general. Another example that drives this message home in a clearer fashion is the classic heat equation, sketched below. """metadatashow_logsèdisabled®skip_as_script«code_folded$33c563af-3436-4275-becd-5333d480b4f4cell_id$33c563af-3436-4275-becd-5333d480b4f4codesmd""" - With this we obtain, ```math \begin{align} E(t) &= \frac1{2} \int_{\mathbb R^3} | \nabla \Psi(x,t) |^2 dx + \int_{\mathbb R^3} V(x) |\Psi(x,t)|^2 dx \\ & \stackrel{IBP}{=} - \frac1{2} \int_{\mathbb R^3} \overline{\Psi(x,t)} \Delta \Psi(x,t) dx + \int_{\mathbb R^3} \overline{\Psi(x,t)} V(x) \Psi(x,t) dx \\ &= \int_{\mathbb R^3} \overline{\Psi(x,t)} \ \mathcal{H} \ \Psi(x,t) dx \\ &= \langle \Psi, \mathcal{H} \Psi \rangle_{L^2(\mathbb R^3)} \end{align} ``` Where we used integration by parts to go from the first to the second line. - In this, the *Hamiltonian* ```math \mathcal{H} \coloneqq - \frac1{2} \Delta + V(x) ``` gives the total energy of the system when applied to the wavefunction $\Psi$, where - $- \Delta / 2$ is the kinetic energy operator - $V(x)$ is the potential energy, applied point-wise in real space. """metadatashow_logsèdisabled®skip_as_script«code_folded$9cb722ec-a7b2-42ef-8f3a-ebeef6bf9746cell_id$9cb722ec-a7b2-42ef-8f3a-ebeef6bf9746codemd""" For many technologically relevant applications, the available materials are what keeps engineered solutions from becoming faster, more efficient or less costly. In particular questions related to the energetic transition (better solar cells or battery devices), communication (data transmission and storage) or computing (quantum materials, energy efficiency or power of CPUs) are examples that come to mind. Nowadays the discovery of new materials with better-suited physical or chemical properties features simulations at center stage to predict properties of fictitious materials without the need to make them in the lab --- which would be more costly and time-intensive. Given the scale at which such materials simulations are performed (thousands to millions) and the need to employ complex quantum-mechanical models to properly describe important physical effects, obtaining a balance between physical soundness, computational cost and predictive power needs to be found. In practice this means that for simulating "real" materials of practical interest the full problem of quantum mechanics cannot be treated, as we will see. Instead one needs to rely on **more approximate**, thus simpler, **physical models**, but also employing **numerical** instead of analytical techniques for solving the underlying equations. In this course we will focus primarily on an extremely popular model of the field, namely **density-functional theory**. An approximate (mean-field) version of the full Schrödinger equation of quantum mechanics. """metadatashow_logsèdisabled®skip_as_script«code_folded$6f347a85-610d-4ea4-98ef-7de602469d01cell_id$6f347a85-610d-4ea4-98ef-7de602469d01codemd"# Addendum"metadatashow_logsèdisabled®skip_as_script«code_folded$d26799e2-bd9a-4ff5-89b2-7d4a8cc61b74cell_id$d26799e2-bd9a-4ff5-89b2-7d4a8cc61b74codeTableOfContents()metadatashow_logsèdisabled®skip_as_script«code_folded$150891dc-51ab-4f29-9962-c8eb6e6f87ffcell_id$150891dc-51ab-4f29-9962-c8eb6e6f87ffcode_md""" - For the time-dependence we obtain the ODE ```math \begin{align} c'(t) &= - i E c(t) \\ \Rightarrow c(t) &= c(0) \exp(- i Et) \end{align} ``` - The space dependence yields the time-independent Schrödinger equation (TISE) ```math \mathcal{H} \varphi(x) = E \varphi(x) \tag{TISE} ``` which is an **eigenvalue problem**. """metadatashow_logsèdisabled®skip_as_script«code_folded$8993ae47-1173-46c1-8f1e-55412215b61fcell_id$8993ae47-1173-46c1-8f1e-55412215b61fcodeHif show_outline RobustLocalResource("https://teaching.matmat.org/math-materials/sidebar.md", "sidebar.md") Sidebar(toc, ypos) = @htl("""""") Sidebar(Markdown.parse(read("sidebar.md", String)), 265) endmetadatashow_logsèdisabled®skip_as_script«code_folded$17dd79e4-c998-48cc-854e-34e20f3338f9cell_id$17dd79e4-c998-48cc-854e-34e20f3338f9codeOmd""" To fully translate the problem to the eigenfunction setting, we need to consider the initial condition $u_0.$ - On the one hand, in the eigenbasis, ```math u_0 = \sum_{i=1}^∞ \underbrace{\langle \phi_i, u_0 \rangle}_{u_{0,i}} \phi_i. ``` - On the other hand, ```math c_i(t) = α_i \exp(- λ_i t). ``` which yields ```math u(x,t) = \sum_{i=1}^∞ α_i \exp(- λ_i t) \phi_i(x). ``` - Taking $t → 0$, we obtain ```math \sum_{i=1}^\infty u_{0,i} \phi_i(x) = u_0(x) = u(x,0) = \sum_{i=1}^\infty \alpha_i \phi_i(x). ``` - Since $\phi_i$ is a basis, we deduce $\alpha_i = u_{0,i}$ and finally obtain ```math u(x,t) = \sum_{i=1}^\infty u_{0,i} \exp(- \lambda_i t) \phi_i(x) ``` We observe the following: - The solution decays to $u_\infty = 0$ as $t \to \infty$. - The decay rate depends on $\lambda_i$. """metadatashow_logsèdisabled®skip_as_script«code_folded$e1132fa4-c9ff-487e-b716-0b3157c2cdd0cell_id$e1132fa4-c9ff-487e-b716-0b3157c2cdd0code^md""" In analogy to classical mechanics, one may obtain the dynamics of the particle by investigating its total energy (considering the mass of the particle to be 1). - The total energy is given by ```math \begin{align} E(t) &= \text{kinetic energy} + \text{potential energy} \\ &= \int_{\mathbb R^3} \frac{|p|^2}{2} | \hat \Psi (p,t) |^2 dp + \int_{\mathbb R^3} V(x) | \Psi (x,t) |^2 dx \end{align} ``` where $V : \mathbb R^3 \to \mathbb R$ is the potential inducing the particle dynamics. - From the Fourier identity $p \hat \Psi (p) = -i \widehat{(\nabla \Psi)}(p)$, we can develop the first term into ```math \begin{align} \int_{\mathbb R^3} |p|^2 | \hat \Psi (p,t) |^2 dp &= \int_{\mathbb R^3} |p \hat \Psi (p,t)|^2 dp \\ &= \int_{\mathbb R^3} \lvert -i \widehat{(\nabla \Psi)}(p,t) \rvert^2 dp \\ &= \int_{\mathbb R^3} \lvert \widehat{(\nabla \Psi)}(p,t) \rvert^2 dp \\ &\hspace{-0.9em} \stackrel{\text{Parseval}}{=} \int_{\mathbb R^3} | \nabla \Psi(x,t) |^2 dx \end{align} ``` where we used Parseval's theorem to go from the third to the fourth line. """metadatashow_logsèdisabled®skip_as_script«code_folded$b944f68e-85a6-4613-8c5e-0fe3d47ff0d4cell_id$b944f68e-85a6-4613-8c5e-0fe3d47ff0d4codemd""" # Introduction """metadatashow_logsèdisabled®skip_as_script«code_folded$d5581136-7b5e-4987-a67d-241286d424c8cell_id$d5581136-7b5e-4987-a67d-241286d424c8codemd""" - Consider the time-dependent part ($c$) ```math \begin{align} c'(t) &= μ c(t) \\ \Rightarrow c(t) &= α \exp(μ t). \end{align} ``` - Separately, consider the space-dependent part ($v$) ```math \begin{align} \Delta v &= μ v && \text{in } \Omega \\ v &= 0 && \text{on } ∂\Omega \end{align} ``` *which is the Dirichlet-Laplace eigenvalue problem*! """metadatashow_logsèdisabled®skip_as_script«code_folded$bd045f8e-7baa-4fbf-94bc-22e39cac9a0dcell_id$bd045f8e-7baa-4fbf-94bc-22e39cac9a0dcodemd""" The goal of this lecture is to discuss mathematical techniques which aid with the estimation of error in density-functional theory and related methods. Within the lecture we will distinguish various forms of error and consider (some) of them in separate discussions throughout the semester. For example we will distinguish the **model error**, i.e. the error introduced by the selected physical model compared to the ground truth, as well as the **numerical error**, i.e. all additional error from the numerical procedure employed for solving the chosen physical model. Of particular focus in the lecture will be understanding the various aspects of the numerical error, that is in particular the errors due to the chosen **discretisation basis**, the **algorithms** to solve the numerical problem. Additional effects (such as the errors due to the chosen **floating-point arithmetic**) will be mentioned on the aside. Overall our goals in this class are to introduce density-functional theory and its mathematical structure, discuss numerical techniques for solving it, analyse their convergence and provide computable estimates for the basis set error. Towards the end we will further look into data-driven techniques to complement or substitute DFT simulations, moving towards the data-driven, physics-informed modelling pipelines that are nowadays commonly employed when modelling materials in practice. """metadatashow_logsèdisabled®skip_as_script«code_folded$7229e2b6-cf77-4b32-b52d-1fc7d218b007cell_id$7229e2b6-cf77-4b32-b52d-1fc7d218b007codemd""" Which physical effects can be neglected and which numerical techniques are most appropriate is often application-specific and **many alternatives exist**. Understanding the **dominant sources of error** in a simulation is thus a crucial component in understanding whether each respective **approximation choice** has been appropriate and --- most importantly --- which of them **needs to be refined** in order to improve the result. """metadatashow_logsèdisabled®skip_as_script«code_folded$94c2368c-243e-4e29-ba9b-a6c9253fa74fcell_id$94c2368c-243e-4e29-ba9b-a6c9253fa74fcodemd""" - Under suitable assumptions for the potential $V$ *(which we will discuss later)*, one can show that the operator $\mathcal{H}$ on the unbounded domain $\mathbb{R}^3$ admits an infinite sequence of eigenvalues $E_0 \leq E_1 \leq \ldots \leq E_n \leq \ldots $ as well as associated $L^2(\mathbb{R}^3)$-orthonormal eigenfunctions $\phi_0, \phi_1, \ldots, \phi_n, \ldots$. I.e. functions with ```math \mathcal{H} \phi_n = E_n \phi_n ``` with ```math \langle \phi_n, \phi_m \rangle = ∫_{\mathbb{R}^3} \overline{\phi_n(x)} \phi_m(x) dx = δ_{mn}. ``` - These form a complete basis of $H^1(\mathbb{R}^3)$, the relevant Hilbert space for this problem. This justifies the ansatz ```math \Psi(x,t) = \sum_{n=0}^∞ c_n(t) \phi_n(x) ``` we made before, where $\phi_n$ are now eigenfunctions of the Hamiltonian operator. - Combining with our solution of the time dependence, we obtain ```math \Psi(x,t) = \sum_{n =0}^\infty c_n e^{- i E_n t} \phi_n(x) ``` where the coefficients $c_n = \langle \phi_n , \Psi(\cdot, 0) \rangle$ are fixed by the initial condition. - We observe the following: - The solution oscillates with different modes. - The oscillation frequency of each mode is given by the eigenvalue $E_n$. - Much of the physical and chemical properties of materials are usually dominated by the eigenpair corresponding to the smallest eigenvalue $E_0$ of $\mathcal{H}$, called the *ground state*. In particular knowing the ground state energy $E_0$, the ground state wave function $\Psi_0$ and how this object changes as we perturb (alter) the Hamiltonian $\mathcal{H}$ is exactly what gives access to prediction of material properties. - Computing the ground state will be central to this course. """metadatashow_logsèdisabled®skip_as_script«code_foldedënotebook_id$36fafd3c-d814-11f1-84c9-b73e0d74e5b1bondscell_results$7142e1bd-200f-4395-94bf-0f0b3bb7ecd0queued¤logsrunning¦outputbody

This motivates a separation of variables ansatz:

$$\Psi(x,t) = \sum_{n = 0}^{\infty} c_n(t) \varphi_n(x).$$

We will see later that this postulate is justified.

To start, let's just consider one term of the sum, i.e. $\Psi(x,t) = c(t) \varphi(x)$.

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The dynamics of the system are now given by the time-dependent Schrödinger equation (TDSE)

$$i \frac{\partial \Psi}{\partial t} = \mathcal{H} \Psi \tag{TDSE}$$

In general, solving this equation is quite complicated. However, since it is linear, we know that superpositions of multiple solutions are also a solution.

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The Schrödinger equation and quantum mechanics

At the microscopic level, the physics and chemistry of materials is governed by the interaction of electrons and nuclei. At this scale, the regime of quantum mechanics applies.

In quantum mechanics the state of a system is described by the complex-valued square-integrable wave function $\Psi : (x,t) ↦ \Psi(x,t) \in \mathbb C$, where $x$ corresponds to the degrees of freedom (e.g. position or spin) of the particles (e.g. electrons) in the system. Without going into too many details we want to briefly discuss the key ideas how quantum mechanics describes physical systems.

For simplicity, take $x \in \mathbb R^3$, i.e. one particle in 3D. Part of the meaning of $\Psi$ appears through its modulus squared. Indeed, for all $t$, $|\Psi(x,t)|^2$ corresponds to the probability distribution of finding the particle at position $x$ at time $t$.

Further information appears through its Fourier transform, as $|\widehat \Psi (p,t)|^2$ corresponds to the probability distribution of finding the particle at momentum $p$ at time $t$, where the Fourier transform is given by

$$\widehat \Psi (p,t) = \frac{1}{(2\pi)^{3/2}} \int_{\mathbb R^3} \Psi (x,t) e^{- i \ p \cdot x}dx.$$

For those already familiar with quantum mechanics, note that we set $\hbar = 1$.

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To solve this problem, we perform separation of variables

$$u(x,t) = \sum_{n = 1}^{\infty} c_n(t) v_n(x).$$

We will see later that this postulate is justified.

To start, let's just consider one term of the sum, i.e. $u(x,t) = c(t) v(x)$.

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Show outline

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Opening remarks

Slides of the presentation on moodle: These summarise the following paragraphs and the content of the course in a short presentation.

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This justifies the ansatz

$$u(x,t) = \sum_{n=1}^∞ c_n(t) \phi_n(x) $$

we made before, where $\phi_n$ are now eigenfunctions of the Laplace operator.

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The connection between linear and eigenvalue problems: the heat equation

Let $\Omega \subset \mathbb R^n$ with a Lipschitz boundary. We seek $u : \Omega \times \mathbb R^+ → \mathbb R$ such that

$$\begin{align} \frac{∂ u}{∂ t} - \Delta u &= 0 && \text{in } \Omega \times \mathbb R^+ \tag{Heat equation} \\ u(\cdot,0) &= u_0 && \text{in } \Omega \tag{Initial conditions} \\ u &= 0 && \text{on } ∂ \Omega \times \mathbb R^+ \tag{Dirichlet b.c.} \end{align}$$

Where the initial conditions $u_0 : \Omega → \mathbb R$ are of appropriate regularity.

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Less importantly for this course the above argument also shows that eigenvalue problems are closely linked to the dynamics (or time propagation) of differential operators. Understanding eigenvalue problems and the spectral properties of operators therefore also gives us understanding about PDE problems in general.

Another example that drives this message home in a clearer fashion is the classic heat equation, sketched below.

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For many technologically relevant applications, the available materials are what keeps engineered solutions from becoming faster, more efficient or less costly. In particular questions related to the energetic transition (better solar cells or battery devices), communication (data transmission and storage) or computing (quantum materials, energy efficiency or power of CPUs) are examples that come to mind.

Nowadays the discovery of new materials with better-suited physical or chemical properties features simulations at center stage to predict properties of fictitious materials without the need to make them in the lab –- which would be more costly and time-intensive.

Given the scale at which such materials simulations are performed (thousands to millions) and the need to employ complex quantum-mechanical models to properly describe important physical effects, obtaining a balance between physical soundness, computational cost and predictive power needs to be found.

In practice this means that for simulating "real" materials of practical interest the full problem of quantum mechanics cannot be treated, as we will see. Instead one needs to rely on more approximate, thus simpler, physical models, but also employing numerical instead of analytical techniques for solving the underlying equations. In this course we will focus primarily on an extremely popular model of the field, namely density-functional theory. An approximate (mean-field) version of the full Schrödinger equation of quantum mechanics.

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Addendum

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To fully translate the problem to the eigenfunction setting, we need to consider the initial condition $u_0.$

We observe the following:

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In analogy to classical mechanics, one may obtain the dynamics of the particle by investigating its total energy (considering the mass of the particle to be 1).

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Introduction

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The goal of this lecture is to discuss mathematical techniques which aid with the estimation of error in density-functional theory and related methods. Within the lecture we will distinguish various forms of error and consider (some) of them in separate discussions throughout the semester. For example we will distinguish the model error, i.e. the error introduced by the selected physical model compared to the ground truth, as well as the numerical error, i.e. all additional error from the numerical procedure employed for solving the chosen physical model. Of particular focus in the lecture will be understanding the various aspects of the numerical error, that is in particular the errors due to the chosen discretisation basis, the algorithms to solve the numerical problem. Additional effects (such as the errors due to the chosen floating-point arithmetic) will be mentioned on the aside.

Overall our goals in this class are to introduce density-functional theory and its mathematical structure, discuss numerical techniques for solving it, analyse their convergence and provide computable estimates for the basis set error. Towards the end we will further look into data-driven techniques to complement or substitute DFT simulations, moving towards the data-driven, physics-informed modelling pipelines that are nowadays commonly employed when modelling materials in practice.

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Which physical effects can be neglected and which numerical techniques are most appropriate is often application-specific and many alternatives exist. Understanding the dominant sources of error in a simulation is thus a crucial component in understanding whether each respective approximation choice has been appropriate and –- most importantly –- which of them needs to be refined in order to improve the result.

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