Training

It is only natural that the ETSF is dedicated to spread the knowledge and experience gathered in its community, both from the view point of theoretical physics and the implementation in software packages. Training is provided in three ways:

Training in a glance

On-demand Training

A training project provides dedicated training on topics in theoretical spectroscopy (theory, methodology, software, etc.), delivered to individuals or groups, at the user's site, an ETSF research group, or at another suitable place. Proposals of scientists from any public- or private- sector organisation and from any country are welcome. They may be submitted at any time. Proposals are evaluated every two weeks.

guide for proposers
It is important that you read the Guide for Proposers before submitting a proposal.

Consultancy

First contact with the ETSF: You are not yet in contact with an ETSF scientist. You would like to know if the ETSF could implement a training project at your request.
ConsultancySubmission

To submit a Consultancy proposal, you will be asked to create a website account, enter your contact details and enter your question into an on-line form (800 characters max.).
An answer will be sent you by e-mail within 10 days.

Submitting a proposal

You have discussed with an ETSF scientist, you are ready to submit a proposal.

Proposals for training projects may be submitted at any time on this web site. Please use one of the templates provided below for the written part of the proposal.
ConsultancySubmission

To submit, you will be asked to log in, enter basic details into an on-line form, and upload a PDF document describing the proposal project itself, based on one of the templates provided below (1 or 2 pages).

Templates

Please user one of these forms to present your training needs: OpenOffice; MS Word; LaTeX.

Criteria and Evaluation Process

Evaluation criteria are (1) the scientific excellence and (2) the relevance with respect to theoretical spectroscopy. Proposals are evaluated every two weeks and those approved are then carried out by the ETSF.

  1. The Internal Training Panel evaluates the admissibility of the proposals, according to the evaluation criteria stated above. ETSF Beamline coordinators constitute the Internal Training Panel.
  2. The ETSF call coordinator verifies the ETSF capabilities in terms of resources for accepted proposals and informs proposers of the outcome of the call.

Step by step explanations may be found in the Guide for proposers.

Statistic reports about training proposals are done every six mouths, together with proposals for collaborative research.

Guide for proposers

The ETSF call itself is open all year round. Users may submit proposals at any time. All proposals must be submitted on-line.

Create user account

In order to submit a proposal a user account must be created on the ETSF website. After submitting the account information, an email containing the password will be send to the address provided, allowing users to access their account.

Screenshot of create an account page

When logged in, a bottom left menu appears “My Proposals”. Here, proposers can edit personal details and modify them, create a new proposal or, for Collaborative Research proposals only, edit an already existing proposal and modify it.

Submit Training

Submission process and proposal structure

All proposals must be submitted on-line. Completing the submission of a proposal will require the following information:

  1. the contact details of the main proposer;
  2. resource information (preferably provided together with an ETSF scientist).
  3. the training proposal as a PDF file (please use the templates)

Besides contact details of the proposer, the proposal is divided in two parts which are explained below.

Basic information

In this part, basic details about the project are described. In particular, the ETSF wishes to list on its website successful proposals titles and when projects are completed and published, a summary of the results. For this reason, the users' permission is asked for. Moreover, information about co-funding is required. If the proposer wishes to contribute with her/his own resources, the type of co-funding (money, CPU time, etc.) and the amount in euro should be described.

Screenshot of Basic Information page

Furthermore, the ETSF requires information about the resources which the proposed project will employ in order to estimate the character, the feasibility and the duration of a project. It is highly recommended that this part is discussed with an ETSF scientist beforehand. If you are not already in contact with an ETSF scientist, please submit a Consultancy proposal or contact one of the ETSF beamline coordinators.

Information required when submitting on-line:

  • the ETSF beamline(s) which are relevant to the project;
  • the name of the ETSF scientist with whom you have primarily discussed your project;
  • the type of training (by visiting an ETSF node, dedicated training at your institution or participation in a school) and the estimated duration (days, weeks or months).

Uploading the PDF file

In this part the planned training is described. Please use one of the templates.

The completed proposal is meant to be as short as possiblie. It must not exceed 2 A4 pages, when converted for submission into a "PDF" file and uploaded on-line.

Screenshot of upload written proposals page

Review and confirm information

The last step gives the proposer a chance to review all information and edit any part of the proposal in case changes are necessary.

The "human verification" tool is used to avoid automated spam submissions. Finally, when all the information is correct, click on "confirm" to submit.

After submitting, the proposer will be redirected to the "Proposal Control Panel" page, and a message will be displayed to notify the proposer that the proposal has been stored successfully: "Proposal has been successfully submitted". Your proposal is listed in the table "Your proposals".

Contacting the ETSF

If, having read this guide, you have questions or are having difficulty with submitting a proposal, please contact Olivia Pulci (olivia [dot] pulci [at] roma2 [dot] infn [dot] it).

Online Training Modules

A foreword

The ETSF on-line training modules introduce into the methods of theoretical spectroscopy, such are time dependent density functional theory (TDDFT) and the GW and BSE approaches of the many body perturbation theory. They also outline the density functional theory as the approach to the ground state on which in practice the investigation of the optical properties is frequently founded.

The on-line modules address the group of ETSF users, both theoreticians and experimentalists, and students, with interest to conduct theoretical projects on the excited state of matter. For readers with different background in theoretical physics this on-line training portal is ment to give first access to the theoretical methods. The training modules are organized in two levels, an introductory level and a tutorial level. The introductory level already outlines all the essential topics and the tutorial level complements more complex subjects and derivation. It is frequently embedded in the introductory level or appears as separate sections.

The on-line training portal starts with an introductory module to the subject. Modules on each of the methods will appear on-line subsequently in the near future.

Introductory Level

Training modules for software components of the ETSF code suite

Electrons and Nuclei

Molecules and solids consist of atoms that bind to each other due to the interaction of the positively charged nuclei with the electrons. In this interaction for instance covalent bonds are formed among pairs of atoms. The mutual interaction of nuclei and electrons is based on the Coulomb law, however, as a large number of particles are involved, the resulting bonding can be complex. Generally, one distinguishes different types of bonding

  • covalent bonding
  • metallic bonding
  • ionic bonding
  • van der Waals bonding

Any description of a molecule or solid from first principles starts with the full Hamiltonian $\hat{H}$ of the $N_\mathrm{k}$ nuclei and $N_\mathrm{e}$ electrons. The electrons have coordinates $\mathbf{r}_j$, with $j=1,\ldots,N_\mathrm{e}$ and mass $m_\mathrm{e}$. The nuclei posses the mass $M_{J}$, charge $Z_J\,e$, and coordinates $\mathbf{R}_J$, where $e$ is the elementary charge and $J=1,\ldots,N_\mathrm{k}$. The Hamiltonian $\hat{H}$ consists of the kinetic energy operators $\hat{T}_\mathrm{e}$ and $\hat{T}_\mathrm{k}$ of electrons and nuclei and the operators for the Coulomb interaction between the charged particles
\begin{equation}
\hat{H}= \hat{T}_\mathrm{e}+\hat{T}_\mathrm{k}+\hat{V}_{\mathrm{ee}}+
\hat{V}_{\mathrm{ek}}+\hat{V}_{\mathrm{kk}}\quad.
\end{equation}
The kinetic energy operators $\hat{T}_\mathrm{e}$ and $\hat{T}_\mathrm{k}$ are
\begin{equation}
\hat{T}_\mathrm{e}=-\sum_j\frac{\hbar^2}{2 m_\mathrm{e}}\frac{\partial^2}{\partial \mathbf{r}_j^2}
\qquad\mathrm{and}\qquad
\hat{T}_\mathrm{k}=-\sum_j\frac{\hbar^2}{2 M_J}\frac{\partial^2}{\partial \mathbf{R}_J^2}\quad,
\end{equation}
where $\frac{\partial}{\partial \mathbf{r}_j}$ and $\frac{\partial^2}{\partial \mathbf{r}_j^2}$ represents the gradient and Laplacian with respect to the coordinate $\mathbf{r}_j$. The mutual electron-electron and nuclei-nuclei interaction $\hat{V}_{\mathrm{ee}}$ and $\hat{V}_{\mathrm{kk}}$, and the interaction of the electrons with the nuclei read
\begin{align}
\hat{V}_\mathrm{ee}=\frac{1}{2}\sum_{i\neq j} \frac{e^2}{4\pi\varepsilon_0}
\frac{1}{|\mathbf{r}_i-\mathbf{r}_j|}\qquad&\qquad\qquad
\hat{V}_\mathrm{k\,k}=\frac{1}{2}\sum_{I\neq J} \frac{e^2}{4\pi\varepsilon_0}
\frac{Z_I Z_J}{|\mathbf{R}_I-\mathbf{R}_J|}\\
\hat{V}_\mathrm{ek}&=-\sum_{i J} \frac{e^2}{4\pi\varepsilon_0}
\frac{Z_J}{|\mathbf{r}_i-\mathbf{R}_J|}
\end{align}
The former two interactions are purely repulsive (the factor $\frac{1}{2}$ above accounts for double counting in the sums). Only the interaction between nuclei and electrons is attractive and thus is the source of bonding of molecules and solids.

In the absence of external potentials, the ground state properties and excited states are described by the Schrödinger equation
\begin{equation}
\hat{H}\,\Psi\left(\{\mathbf{r}_j\},\{\mathbf{R}_J\}\right)
=E\,\Psi\left(\{\mathbf{r}_j\},\{\mathbf{R}_J\}\right)\quad.
\end{equation}
The wave function $\Psi\left(\{\mathbf{r}_j\},\{\mathbf{R}_J\}\right)$ of the interacting electrons and nuclei is a function of $3\,(N_\mathrm{e}+N_\mathrm{k})$ degrees of freedom. This complexity of the Schrödinger does not allow for closed solutions. Physical approximations must be made in order to separate the problem into tractable parts. Such an approximation is the Born-Oppenheimer approximation or more correctly the adiabatic approximation. It allows to separates the nuclear motion from the electron-problem.

Before this, a note on units is due. It is common practice to drop the SI-units and to refer to atomic units, which greatly simplifies the equations. This means to set
\begin{equation}
\hbar \rightarrow 1\qquad\frac{e^{2}}{4 \pi\,\varepsilon_{0}} 1
\quad\mathrm{and}\quad m_{\mathrm{e}}\rightarrow 1\quad.
\end{equation}

Thereby the unit of length becomes Bohr=$\frac{4 \pi\,\varepsilon_{0}\,\hbar^{2}}{m_{\mathrm{e}}\,e^{2}}=0.529\,10^{-10}$m and the unit of energy is Hartree (Ha)=$\frac{m_{\mathrm{e}}\,e^{4}}{16 \pi^{2}\,\varepsilon_{0}\,\hbar^{2}}$ $=27.2116$eV.

Born-Oppenheimer or adiabatic approximation

The Rayleigh-Ritz principle

The Hartree-Fock approximation and beyond

The Hartree-Fock approximation and beyond

With the Born-Oppenheimer approximation the general problem of the electron and nuclei is separated. The first problem awaiting solution now is the Schrödinger equation of the interacting electrons moving in the potential of the momentary configuration of the nuclei
\begin{equation*}
\left(\hat{T}_{\mathrm{e}}+\hat{V}_{\mathrm{ee}}
+\hat{V}_{\mathrm{ek}}(\{\mathbf{R}_I\}_{N_\mathrm{k}})\right)
\,\Phi_n(\{\mathbf{r}_i\}_{N_\mathrm{e}};\{\mathbf{R}_I\}_{N_\mathrm{k}})
=E_n(\{\mathbf{R}_I\}_{N_\mathrm{k}})\,
\Phi_n(\{\mathbf{r}_i\}_{N_\mathrm{e}};\{\mathbf{R}_I\}_{N_\mathrm{k}})
\end{equation*}
The potential of the electron-nuclei interaction $\hat{V}_{\mathrm{ek}}$ can be cast into the form
$\hat{V}_{\mathrm{ek}}(\{\mathbf{r}_i\}_{N_\mathrm{e}},\{\mathbf{R}_I\}_{N_\mathrm{k}})=\sum_i V_\mathrm{ext}(\mathbf{r}_i)$, where the single particle potential $ V_\mathrm{ext}(\mathbf{r})$ in our context is the Coulomb potential of all nuclei with positions $\{\mathbf{R}_I\}$. In the following, however, $V_\mathrm{ext}(\mathbf{r})$ is not limited to this context and can be any single particle potential. Since no explicit reference to the coordinates of the nuclei will be made in the following, the above electron wave function $\Phi_n$ is written as a function of the electron coordinates only, i.e. $\Phi_n(\{\mathbf{r}_i\}_{N_\mathrm{e}})$.

Already for small atoms and molecules, not to mention solids, the Schrödinger equation cannot be solved, neither analytically nor numerically. For the purpose of assessing $\Psi_0$, therefore approximations to the wave function become necessary in one or the other form. The focus in the following will lie on the outline of the concepts for the discussion of the methods of theoretical spectroscopy, which also offer a complementary approach to this problem. The primary interest at the moment is to describe the electron ground state.

The idea of the Hartree-Fock method for this purpose is to construct an approximation of the many electron wave function from single particle wave functions. This approximate wave function must obey the Pauli principle. It is given in terms of the slater determinant
\begin{equation}
\Phi_{\mathrm{HF}}=\frac{1}{N_\mathrm{e}!}\det \bar{\phi}_{i,\sigma_j}(\mathbf{r}_j)
\qquad\mathrm{with}\quad j=1,\ldots,N_\mathrm{e}\quad.
\end{equation}
The single particle spin orbitals $\bar{\phi}_{i}(\mathbf{r})=\phi_i(\mathbf{r})\,\alpha_i(\sigma)$ are required to be orthonormal wave functions, i.e. $\langle\bar{\phi}_i(\mathbf{r})|\bar{\phi}_j(\mathbf{r})\rangle=\delta_{i j}$, such that $\Phi_{\mathrm{HF}}$ is normalized. Here $\alpha_i(\sigma)$ is the spin eigenfunction with eigenvalue $\bar{\sigma}_i$. The equation for the $\phi_i(\mathbf{r})$ is obtained from the Ritz principle. The variation is executed with respect to the complex conjugate of the independent single particle orbitals $\delta\phi^*_j(\mathbf{r})$ under the constraint $\langle \bar{\phi}_j(\mathbf{r})|\bar{\phi}_j(\mathbf{r})\rangle=1$. This leads to an equation of the form
\begin{equation}
\frac{\delta}{\delta\phi_j^*}\,\left({\langle\Phi_\mathrm{HF}|\hat{H}_\mathrm{e}|\Phi_\mathrm{HF}\rangle}
-\sum_i \varepsilon_i\langle\phi_i|\phi_i\rangle\right)=0\quad,
\end{equation}
where the $\varepsilon_i$ are the Lagrange multipliers to account for the constraint. In order to execute this variation, the expectation value of the Hamltonian $\hat{H}_\mathrm{e}$, namely the ground state energy $E_{\mathrm{HF}}$, has to be evaluated. It is given by
\begin{align}
E_\mathrm{HF}=\sum_{i}&\int d^3r \phi_i^*(\mathbf{r})\,
\left(-\frac{1}{2}\nabla^2+V_{\mathrm{ext}}(\mathbf{r})\right)\,
\phi_i(\mathbf{r}) \\
&+\frac{1}{2} \sum_{i\neq j}\iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_i(\mathbf{r})\,\phi_j(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\\
&-\frac{1}{2} \sum_{{i\neq j}} \delta_{\bar{\sigma}_j,\bar{\sigma}_j}\iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\quad,
\end{align}
where $\bar{\sigma}_i$ denotes the spin eigenvalue associated with the spin orbital $\alpha_i(\sigma)$. The above expression for the energy can be brought into a more conventional form. Firstly note that in the constraint $i\neq j$ in the summation can be dropped as the additional term is the same in both sums, therefore cancels out. The second simplification is accomplished then with the help of the electron density $n(\mathbf{r})$. It is given by
\begin{equation}
n(\mathbf{r})=\sum_i |\phi_i(\mathbf{r})|^2\quad.
\end{equation}
The second term, after dropping the constraint, now can be expressed with the help of $n(\mathbf{r})$. This energy term is the Hartree energy $E_\mathrm{H}$
\begin{equation}
E_\mathrm{H}=\frac{1}{2} \sum_{i,j}\iint d^3r\,d^3r'
\frac{|\phi_i(\mathbf{r})|^2\,|\phi_j(\mathbf{r}')|^2}{|\mathbf{r}-\mathbf{r}'|}=\frac{1}{2}\iint d^3r\,d^3r'\,
\frac{n(\mathbf{r})\,n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
\end{equation}
The Hartree energy thus corresponds to the classical electrostatic energy of the charge density $n(\mathbf{r})$. It contains a self-interaction of the charge density at $\mathbf{r}$ with itself, which is canceled by the following exchange energy term. This last sum in the above expression for $E_\mathrm{HF}$ solely arises from the Pauli principle as a consequence of the quantum mechanical nature of the indistinguishable electrons
\begin{equation}
E_{\mathrm{X}}=-\frac{1}{2}\sum_{i\,j}\delta_{\bar{\sigma}_1,\bar{\sigma}_j}
\iint d^3r\,d^3r'\,\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})
\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
\end{equation}

The Hartree-Fock equations for the orbitals $\phi_i$ are then obtained from the variation of $E_{\mathrm{HF}}$ respecting the constraint of orthonormality
\begin{equation}
\left(\vphantom{\int}-\frac{1}{2}\nabla^2+V_{\mathrm{ext}}(\mathbf{r})+V_\mathrm{H}(\mathbf{r})\right)\phi_i(\mathbf{r})
+\int d^3r'\,\frac{\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}=\varepsilon_i\,\phi_i(\mathbf{r})\quad,
\end{equation}
where the $\varepsilon_{i}$ are Lagrange multipliers to ensure the orthonormality constraint of the single particle orbitals. The Hartree potential $V_{\mathrm{H}}$ arises from the Hartree energy $E_{\mathrm{H}}$
\begin{equation}
V_\mathrm{H}(\mathbf{r})=\int d^3r' \frac{n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
\end{equation}
and although it is written here like a typical potential it depends on the electron density, which is calculated from all orbitals $\phi_i$. The last sum gives rise to the so called exchange potential. The exchange potential operator $\hat{V}_\mathrm{X}$ is an integral operator
\begin{equation}
\hat{V}_\mathrm{X}\,\phi_i(\mathbf{r})=-\sum_j \delta_{\bar{\sigma}_i,\bar{\sigma}_j}
\int d^3r' \frac{\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})}{|\mathbf{r}-\mathbf{r}'|}\,\phi_i(\mathbf{r}')
\end{equation}
$\hat{V}_\mathrm{X}$ is a non-local operator as $\phi_i$ appears under the integral together with $\phi_j^*$, whereas $\phi_j$ remains outside the intergration. Hence the exchange potential for each orbital $\phi_i$ is different. $V_\mathrm{H}$, in contrast, despite being an integral operator too, is the same for all orbitals and which is expressed by writing it in the form $V_\mathrm{H}(\mathbf{r})$.

Solving the Hartree-Fock equations is more complicated than solving the Schrödinger equation of a single-particle problem. The reason is its non-linearity in the orbitals $\phi_i$, or, in other words, the fact that the expressions for the potential operators $\hat{V}_{\mathrm{H}}$ and $\hat{V}_{\mathrm{X}}$ contain the orbitals $\phi_i$ themselves. A practicle scheme illustrated in Fig. 1.2 is the iterative solution of the equations for fixed approximations to the potentials $\hat{V}_{\mathrm{H}}$ and $\hat{V}_{\mathrm{X}}$ improved in each step. At the start of this iterative procedure an approximation of the initial density and orbitals $\phi^{\tau=0}$ is used for evaluating the first approximation $\hat{V}^{\tau=1}_{\mathrm{H}}$ and $\hat{V}^{\tau=1}_{\mathrm{X}}$, where $\tau$ is an iteration index. In the $\tau$-th. iteration one first solves the Hartree-Fock equations using the potential operators $\hat{V}^{\tau}_{\mathrm{H}}$ and $\hat{V}^{\tau}_{\mathrm{X}}$ of the previous iteration. From the new orbitals $\phi^{\tau}_i$ one evaluates the potential operators to generate a new approximation $\hat{V}^{\tau+1}_{\mathrm{H}}$ and $\hat{V}^{\tau+1}_{\mathrm{X}}$. The equations are solved when the sequence of potentials and orbitals (within a certain tolerance) have converged to final values.

Flowchart of the SCF-method

Fig. 1.2: Illustration of the method of the self-consistent field for solving iteratively the Hartree-Fock equations. The step highlighted in orange is the actual step being executed. Follow the scheme until a solution is obtained.

So far the $\varepsilon_i$ are mathematical objects only and were introduced as Lagrange multipliers to ensure the orthonormality of the orbital $\phi_i$. Allthough the similarity of the Hartree-Fock equation with a Schrödinger equation might suggest this, an interpretation in terms of a single particle eigenenergies is not permitted on a strict basis. The following considerations will illustrate this.

First of all, it is interesting to compare the Hartree-Fock groundstate energy $E_\mathrm{HF}$ with that of a single particle system with single particle eigenvalues $\varepsilon_i$. For this purpose $E_\mathrm{HF}$ is rewritten. First the expectation value $\langle\phi_i|\frac{1}{2}\nabla^2+V_\mathrm{ext}|\phi_i\rangle$ is expressed with help of the Hartree-Fock equation
\begin{align}
\int d^3r\
\phi_i^*(\mathbf{r})\left(\vphantom{\int}-\frac{1}{2}\nabla^2+V_\mathrm{ext}\right)
\phi_i(\mathbf{r}) =\varepsilon_i&-\iint
d^3r\,d^3r'\,\frac{|\phi_i(\mathbf{r})|^2\,n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
\\ &+\sum_j \delta_{\bar{\sigma}_i,\bar{\sigma}_j} \iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\quad.
\end{align}
Substituting this into the expression for $E_\mathrm{HF}$ yields
\begin{align}
E_\mathrm{HF}&=\sum_{i} \varepsilon_i
-\frac{1}{2} \sum_{i,j}\iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_i(\mathbf{r})\,\phi_j(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\\
&\hphantom{-\frac{1}{2} \sum_{{i,j}}}
+\frac{1}{2} \sum_{{i,j}} \delta_{\bar{\sigma}_j,\bar{\sigma}_j}\iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\\
&=\sum_i \varepsilon_i-E_\mathrm{H}-E_\mathrm{X}
\end{align}
The Hartree-Fock ground state energy unlike the ground state energy of a single particle system is not only given by the sum over the occupied single particle eigenvalues $\varepsilon_i$. In addition it contains two terms that arise from the mutual interaction of the electrons. Now consider the ionization of the $N_\mathrm{e}$-electron system by removing the particle with the highest value of $\varepsilon_i$. As a consequence the slater determinante will contain $N_\mathrm{e}-1$ single particle orbitals accept the orbital $\phi_i$. The ionization energy $I$ of the system is obtained from $E_\mathrm{HF}(N_\mathrm{e})-E_\mathrm{HF}(N_\mathrm{e}-1)$. Under the assumption that the number of electrons is large, one may approximate the orbitals of the ionized system by the remaining $\phi_j$ orbitals. With this approximation one obtains the ionization energy of an electron in the $i$th. quantum state by
\[
I=E_\mathrm{HF}(N_\mathrm{e}-1_i)-E_\mathrm{HF}(N_\mathrm{e})=-\varepsilon_i
\]
Thus, under the above assumption, the eigenvalue $\varepsilon_i$ can be identified with the ionization energy of the system. This statement is known as Kopmann's theorem.

Problems of the Hartree-Fock approximation

  • In metallic systems a logarithmic divergence of the
    density of states appears at the Fermi level.
  • In semiconductors or insulators the band gap turns out to be too large.
  • In molecules the HOMO-LUMO gap often is overestimated

Direct improvements of the Hartree-Fock approximation follow its spirit and address a refinement of the many-electron wave function $\Phi$. The central idea is to add to the solutions $\phi_i$, $i\leq N_\mathrm{e}$, of the Hartree-Fock equation orbitals $\phi_k$ with $k>N_\mathrm{e}$. These latter states are obtained by solving the Hartree-Fock equation with the density $n(\mathbf{r})$ given by the orbitals $\phi_i$ with $i\leq N_\mathrm{e}$ and the summation of the exchange term still restricted to the same set of orbitals. This set of orbitals forms a complete basis of the single particle Hilbert space.

With this complete set of single particle orbitals one expands $\Phi$
\begin{equation}
\Phi(\{\mathbf{r}_j\})=\sum_{\nu_1,\ldots\nu_{N_\mathrm{e}}} a_{\nu_1,\ldots,\nu_{N_\mathrm{e}}}
\phi_{\nu_1}(\mathbf{r}_1)\cdot\phi_{\nu_2}(\mathbf{r}_2)\cdots
\phi_{\nu_{N_\mathrm{e}-1}}(\mathbf{r}_{N_\mathrm{e}-1})\cdot\phi_{\nu_{N_\mathrm{e}}}(\mathbf{r}_{N_\mathrm{e}})
\end{equation}
here for each coordinate $\mathbf{r}_i$ the summation contains all possible orbitals $\phi_{\nu_i}(\mathbf{r}_i)$ with the quantum number $\nu_i$. The $a_{\nu_1,\ldots\nu_{N_\mathrm{e}}}$ are the expansion coefficients. They are determind such that the Pauli principle is fullfilled, i.e. that the sign of $\Phi(\{\mathbf{r}_j\})$ changes when the coordinates $\mathbf{r}_i$ and $\mathbf{r}_j$ are exchanged. This implies, that the coefficients with identical indices $\nu_i$ and $\nu_j$ must vanish. Therefore one can write $\Phi(\{\mathbf{r}_j\})$ as a linear combination of slater determinants
\begin{equation}
\Phi(\{\mathbf{r}_j\})=\sum_{\nu_1<\ldots<\nu_{N_\mathrm{e}}}
f_{\nu_1,\ldots\nu_{N_\mathrm{e}}} \frac{1}{\sqrt{N_\mathrm{e}!}}
\mathrm{det}\,\phi_{\nu_i}(\mathbf{r}_j)
\end{equation}
The summation thus contains the slater determinant of the ground state $\Phi_{\mathrm{HF}}$, determinants where one of the orbitals in $\Phi_{\mathrm{HF}}$ is replaced by an unoccupied orbitals, determinants where two of the orbitals $\Phi_{\mathrm{HF}}$ are replaced by unoccupied orbitals and so forth. Each of the unoccupied orbitals is associated with an $\varepsilon_{\nu_i}$ that is larger than any of the $\varepsilon_{\nu_i}$ of the orbitals $\phi_{\nu_j}$ included in $\Phi_{\mathrm{HF}}$. This can be rationalized as if one excites a number of electrons from the ground state into the unoccupied states as sketched in Fig. 1.3.

Expansion of $\Phi$ in terms of ground state and excited state slater determinants

Fig. 1.3: Schematic visualisation of the expansion of $\Phi(\{\mathbf{r}_{i}\})$ in terms of the Hartree-Fock ground state and slater determinats of excited states. Single particle orbitals corresponding to the lowest $\varepsilon_{i}$ (denoted by black lines) form the slater determinat of the ground state. In the second set of determinants one ground state orbital is replaced by an unoccupied orbital (unoccupied levels are represented by gray lines). The third set contains to unoccupied orbitals and so forth. The inclusion of the orbitals indicated by up and down arrows representing the electron spin.

The coefficients $f_{\nu_1,\ldots\nu_{N_\mathrm{e}}}$ are determined now by the application of the Ritz principle to the energy expectation value $\langle \Phi|\hat{H}_\mathrm{e}|\Phi\rangle$. This expectation value can be expressed in the form
\begin{equation}
\langle \Phi|\hat{H}_\mathrm{e}|\Phi\rangle=
\sum_{\nu_1<\ldots<\nu_{N_\mathrm{e}}}\sum_{\mu_1<\ldots<\mu_{N_\mathrm{e}}}f^{*}_{\nu_1,\ldots,\nu_{N_\mathrm{e}}}
f_{\mu_1,\ldots,\mu_{N_\mathrm{e}}}\frac{1}{N_\mathrm{e}!}
\langle \mathrm{det}\,
\phi_{\nu_i}(\mathbf{r}_j)|\hat{H}_\mathrm{e}|\mathrm{det}\,\phi_{\mu_i}(\mathbf{r}_j)\rangle
\end{equation}
The summation contains an infinite number of terms unless the basis set is finite. And even then the number of matrix elements one has to include is typically large. Thus in practise not all excited states can be considered and approximations have to be made. It is interesting to note, that the matrix elements among determinants corresponding to single excitations vanish, thus the smallest set of slater determinants contains the ground state and double excitations and the next larger set includes also the single excitations.

At least, in principle, one can describe the exact ground state with this procedure. In this ground state slater determinants of the Hartree-Fock ground state and excited states are correlated via the coefficients $f_{\nu_1,\ldots,\nu_N}$. Therefore the energy difference between the exact ground state energy $E_\mathrm{exact}$ and the Hartree-Fock ground state is called correlation energy
\begin{equation}
E_\mathrm{C}=E_\mathrm{exact}-E_\mathrm{HF}
\end{equation}
The concepts of exchange and correlation of identical particles is central to the physics of a many electron system and will be used in the subsequent discussion of methods of theoretical spectroscopy.

Back to Electrons and Nuclei

The Rayleigh-Ritz principle

The energy eigenvalue of each eigenstate $\Psi_n$ of the Hamiltonian $\hat{H}$ or more specifically $\hat{H}_\mathrm{e}$ or $\hat{H}_\mathrm{k}$ is equal to the expectation value
\begin{equation}
E_n=\frac{\langle\Psi_n|\hat{H}|\Psi_n\rangle}{\langle\Psi_n|\Psi_n\rangle}
\end{equation}
For any quantum state $\Psi$ of the system's Hilbert space, the expectation value of the energy $E$ is a functional of the wave function $\Psi$
\begin{equation}
E[\Psi]=\frac{\langle\Psi|\hat{H}|\Psi\rangle}{\langle\Psi|\Psi\rangle}
\end{equation}
The ground state $\Psi_0$ is associated per definition with the lowest energy, $E_0\lt E_{n\neq0}$. Therefore the variation of the functional with respect to $\Psi$ or its complex conjugate $\Psi^*$ has to be stationary for the ground state, i.e. $\delta E[\Psi]/\delta\Psi=0$ or $\delta E/\delta\Psi^*=0$. This variation yields the Schrödinger equation
\begin{equation}
\frac{\delta E[\Psi_0]}{\delta\Psi_0^*}=\frac{\hat{H}\Psi_0
-\langle \Psi_0|\hat{H}|\Psi_0\rangle \Psi_0}{\langle\Psi_0|\Psi_0\rangle}
=0\qquad\mathrm{and\ hence}\qquad \hat{H}\Psi_0=E_0\,\Psi_0\quad.
\end{equation}
Consider now a small deviation $\delta\Psi$ from the ground state wave function $\Psi_0$. Here $\delta\Psi$ -- without loss of generality -- shall be orthogonal to $\Psi_0$, i.e. $\langle\Psi_0|\delta\Psi\rangle=0$.
The energy expectation value of this state $\Psi_0+\delta\Psi$ shall be expanded in $\delta\Psi$
\begin{align}
E[\Psi_0+\delta\Psi]&=\frac{\langle\Psi_0+
\delta\Psi |\hat{H}|\Psi_0+\delta\Psi\rangle}{\langle\Psi_0+\delta\Psi|\Psi_0+\delta\Psi\rangle}\\
&=\frac{\langle\Psi_0|\hat{H}|\Psi_0\rangle+\langle\Psi_0|\hat{H}|\delta\Psi\rangle
+\langle\delta\Psi|\hat{H}|\Psi_0\rangle
+\langle\delta\Psi_0|\hat{H}|\delta\Psi_0\rangle}{\langle\Psi_0|\Psi_0\rangle+\langle\Psi_0|\delta\Psi\rangle
+\langle\delta\Psi|\Psi_0\rangle+\langle\delta\Psi|\delta\Psi\rangle}\\
&=\frac{E_0\,\langle\Psi_0|\Psi_0\rangle+\langle\delta\Psi |\hat{H}|\delta\Psi\rangle}{\langle\Psi_0|\Psi_0\rangle+\langle\delta\Psi|\delta\Psi\rangle}=E_0+o(\delta\Psi^2)
\end{align}
In the last line the fact that $\hat{H}\Psi_0=E_0\,\Psi_0$ and $\langle\Psi_0|\delta\Psi\rangle=0$ were used. Thus, for an approximation $\Psi_0+\delta\Psi$ to the ground state wave function $\Psi_0$ the error in the ground state energy will be quadratic in the deviation $\delta\Psi$. In particular, the approximation value $E[\Psi_0+\delta\Psi]$ for the ground state energy will always be larger than the true value $E_0$.

The Rayleigh-Ritz principle provides therefore a strategy to find approximations to the ground state wave function in terms of an energy minimization. Among two distinct approximations the one with the lower energy expectation value is the better approximation. The error in the approximation for the ground state energy converges with second order of the deviation $\delta\Psi$ from the ground state wave function.

For excited states the Functional $E[\Psi_n+\delta\Psi]$ is only stationary. Also the error is quadratic in $\delta\Psi$: Nevertheless the convergence of $E[\Psi_0+\delta\Psi]$ to $E_n$ is not monotonic as for the ground state.

One important application of this principle for the purpose of numerical calculations is the approximation of the infinite Hilbert space by a vector space spanned by a finite basis of normalized wave
functions $\{\bar{\Psi}_n\}$. The approximate wave function $\bar{\Psi}$ is thus expanded in terms of the basis functions and coefficients $a_n$
\begin{equation}
\bar{\Psi}=\sum_n a_n\,\bar{\Psi}_n
\end{equation}
Application of the Rayleigh-Ritz principle to the finite vector space yields the generalized eigenvalue problem
\begin{equation}
\sum_m \langle \bar{\Psi}_n|\hat{H}|\bar{\Psi}_m\rangle\,a_m
=E\,\sum_m \langle \bar{\Psi}_n|\bar{\Psi}_m\rangle\,a_m\quad.
\end{equation}

Another application is the simplification of the many particle wave function in terms of a generic form. The adiabatic approximation for the separation of electronic and nuclear degrees of freedom, for instance, can be seen in this light. Also Hartree-Fock theory and theories beyond it are based on approximations regarding the functional form of the wave function, namely they express $\Psi$ in terms of single particle wave functions as detailed in the following.

Back to Electrons and Nuclei

Tutorials and Hands-On

Due to the paramount importance of nanoscience and nanotechnology in industrialized countries, the Nanoquanta NoE and the ETSF developed an idea of training program in this field. Our aim is to provide training in ab initio techniques, for both ground- and excited-states properties.

Hands-On training schools are designed to train young researchers, both from the ETSF teams and outside, on Theory and Applications of Spectroscopic tools in Material Science and Nanoscience. These schools are characterized by full-time intensive programs, limited enrolment, highly available computer facilities, and informal environment; all elements that contribute to building the success of the events.

Very often some speakers are asked to deliver several hour-long lectures spread over the course of the schools and workshops. This format allowed the speakers to spend ample time educating students about their fields as well as to present their latest results.

As a byproduct, students especially enjoy the school's many opportunities for informal interaction with the lecturers. During informal late-afternoon "tutorials," small groups of students have ample time to ask individual lecturers deeper scientific or even technical questions.

We have established modular training units (training exercises, tutorial material), some of which are available to the etsf members and some others, through the Web. These modules form a starting point for more substantial web-based distance learning, which we envisage would extend to components of the proposed graduate programme.

Finally, worthwhile to mention the European Graduate Modules, in which ECTS credits will be awarded, in occasion of Hands-On or Schools, on the theoretical description of excitations.

An archive of training module is also available here.

Past Hands-On and Schools

Archive Training Material

Researchers from the ETSF are frequently giving lectures on both the theory of spectroscopy as well as software development. Collecting the nodes from these lectures we are creating an archive of training material.

Theoretical Spectroscopy Lectures

Software Development