Until quite recently (~15 years ago), usual wave-function methods (WF) were used to treat small or medium size objects while Density Functional Theory (DFT) approaches were capable to treat some larger systems but weakly correlated ones. Indeed, standard black box quantum chemistry methods are hardly able to treat large systems like nano-objects, biological molecules,... which are at the heart of modern chemistry developments.
Nowadays developments of linear scaling techniques, largely based on the use of local orbitals on one side, and modern functionals including dispersion effects on the other side have considerably broadened the field of application of quantum chemistry methods in general. At the same time, the border between WF and DFT has become less clear. State of the art methods are today capable to treat systems with thousands of electrons presenting strong correlation effects.

The aim of this tutorial is to give second year master and first years PhD students as well as young researchers an overview of modern quantum chemistry, (both wave-function and density functional) methods as applied to large molecular systems.
Indeed, theoretical chemistry students usually have a fairly good background in standard quantum chemistry methods but lack the knowledge of how to apply these on large systems.

This is particularly true nowadays due to the fact that, because of the lack of students, very often pure quantum chemistry courses have been reduced in many universities.

On an European scale, this has partly been overcome by the Theoretical Chemistry and Computational Modelling (TCCM) Erasmus Mundus Master. However, even in that framework very specific topics are only partly covered and required dedicated schools (see for instance the two 2013 EMTCCM schools sponsored by the CECAM on Molecular Excited States and Theoretical Solid State Chemistry).

Thus, it is important for future researchers in quantum chemistry to acquire a sound knowledge of state of the art techniques to address objects of this size and complexity.

This tutorial is a chance for students to complete their cursus in this field, which is crucial at a moment when requests from our experimentalist colleagues increase from day to day.

Nowadays,wave-function quantum chemistry methods are capable to treat molecules up to about one hundred atoms with reasonable accuracy
and computational efforts. For some properties like ground state geometries it is possible to overcome this limit. However, even the calculation of simple quantities,like harmonic frequencies, is a quite demanding task. In these cases, some other techniques have to be used. Some of these will be the subject of our Winter-School teachings.

1 - QM-MM:
For biological systems, as soon as some insights of the electronic structure are desired and excited states are involved, mixed Quantum Chemistry / Molecular Mechanics (QM/MM) approach is the method of choice. It is then possible to study very large molecular systems (up to many thousands of atoms) and concentrate the computational effort on a small part of the molecule where the electronic process takes place using high level quantum chemistry methods.

2 - Local orbitals:
The same philosophy can be applied to smaller systems by mixing different level of quantum chemistry leading to QM/QM' methods. To achieve this separation it is necessary to use molecular orbitals localization techniques. Indeed, these are the necessary step for all state-of-the-art linear scaling methods as density-fitting, Choleski decomposition, etc... Combined with Configuration-Interaction (CI) techniques, it is then possible to determine tiny energy differences of a few meV, like those that are typically interesting in poly-metallic magnetic complexes.

3 - Method of Increments:
Approximate wavefunction-based methods like truncated multireference CI with size-consistency corrections can be applied to large systems with potentially applications in nano-electronics using the method of increments. In the method of increments, the Hartree-Fock result for the energy of an extended system, EHF, is combined with results from electron correlation calculations for finite embedded fragments.
The correlation energy of the system is expressed as a many-body expansion in terms of one-body increments (representing the correlation energy from all electrons on one center), two-body increments (giving the non-additive correlation contributions from correlating electrons at two centers simultaneously), and so on.

4 - Applications at Nano-scale level:
Chemistry of nano-scale objects will be covered by three different situations: finite large metallic clusters, molecules on surfaces, and nano-particules for biomedical applications:
- Metallic clusters will be treated by an original use of the traditional plane wave approach, usually reserved to infinite periodic systems. In this way, it is possible to treat metallic clusters, in particular transition-metal ones, that are extremely promising in nanocatalysis.
- Plane wave techniques will also be covered to treat molecule-surface interactions, like the interaction of a molecule with graphene like or metallic surface. In these cases the most accurate approaches are based in the use of plane-wave and periodic conditions. In fact, for many applications in nanotechnology the molecules of interest are electron donors or acceptors and the substrate is generally a metallic surfaces. The main challenge in the theoretical treatment is to describe simultaneously the surface, in which electrons are delocalized, and the molecules, in which the electron are localized in bonds.
- Biological applications will be described through the DFT-B method. In recent years, this method has been widely used in the context of biological studies in particular with the refined formulation known as the Self Consistent Charge SCC-DFTB. It has been applied successfully to determine structures and dynamic behavior of many biomolecules such as polypeptides or nucleotides, whos computational complexity would be a challenge for more refined methods.

The proposed tutorial will contain both lectures and hands-on sessions. Typically, each topic will be covered by a lecture of around four hours, followed by practical sessions on computers. In this way, about seven hours will be globally devoted to each topic. In our opinion, this should be adequate to give students the opportunity to manage the basics of eventual new quantum chemistry codes and methods and to tackle the limitations of each approach during the lectures.

At the end of the tutorial, the students are expected to be able to apply these methods to their own problems and applications.

Finally, the students will be given the opportunity to present their own work during a poster session.

QM/MM Hybrid methods
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1) Hybrid Methods (QM/MM)
a. Interest.
b. Some recall on QM and MM methods.
c. QM/MM Interactions
d. Macromolecule treatment
2) Link atom methods
a. Mechanical Embedding
b. Electrostatic Embedding and beyond
3) No-extra atom methods
a. Connection atom methods (pseudo atoms)
b. Frozen orbitals methods
4) Applications
a. Electron capture by disulfide bond
b. Copper proteins. Redox and UV/Visible properties.
c. Electronic excited states of macromolecules.

Localized methods and Configuration Interactions for Large Systems
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0) Introduction to Configuration Interaction
a. Multiconfigurational wave functions
b. The configuration interaction matrix
c. Brillouin's theorem
d. Externally and internally contracted configuration interaction

1) Configuration Interaction Methods
a. Multi-reference configuration interaction
b. Configuration interaction by perturbation selected iterations (CIPSI)
c. Difference-dedicated configuration interaction (DDCI)

2) Localized Molecular Orbitals
a. Basis transformations
b. Localized Molecular Orbitals
c. Modern Approaches to Molecular Orbital Localization

The method of increments
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a wavefunction-based correlation method for solids and surface

Wavefunction-based correlation methods like the golden standard of quantum chemistry are widely used for electronic structure calculation in molecules. It is not straight-forward to apply these methods to extended systems like solids and surfaces. But in the last decade the idea of correlation methods relying not on canonical orbitals but on localized orbitals has attracted more and more interest.
One of these local correlation methods which can be applied to periodic systems in the method of increments developed in 1992 by Stoll [Stoll,92]. It determines the correlation energy in solids and is based on the expansion of the correlation energy in terms of localized orbital groups. Any size-extensive correlation method like coupled cluster can be used for the correlations, the Hartree-Fock treatment is performed for the extended systems. The method of increments can be routinely applied to insulators and semiconductors, for a review see [Paulus,2006]. In the second part of the lecture two extensions of the methods are presented, to metallic systems with multi-reference character [Paulus,2003] and to adsorption processes on surfaces [Müller, 2012].

Plane-wave DFT Methods Applied to Large Molecules and Nanoclusters in their electronic groundstate: a practical guide
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Plane-wave DFT methods within periodic boundary conditions are usually applied to bulk materials and surfaces, owing to some fundamental and computational reasons. It will be shown here that reliable results can be also obtained on organic, inorganic and metal finite systems. The practical interest of using such methods will be assessed on several cases. We will more particularly focus on metal nanocatalysts.

- Localized basis sets vs. plane waves ? Historical and Computational reasons.
- Periodic boundary conditions and the supercell approach.
- Efficiency of the DFT implementation. LDA, GGA and hybrid functionals.
- Analysis of the electronic structure ; semi-conductors vs. metals ; magnetic systems.
- Theory-experiments relationship: Geometry optimizations and X-Ray ; Vibrational normal modes and Raman or Infrared ; chemical shieldings or quadrupolar coupling constants and NMR.
- Multi-step chemical reactions and transition state search
- Examples

Theoretical treatment of molecules deposited on surfaces
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In these lessons we will present how we can treat organic molecules deposited on surfaces. For many applications in nanotechnology the molecules of interest are electron donors or acceptors and the substrate is generally a metallic surfaces. The main challenge in the theoretical treatment is to describe simultaneously the surface, in which electrons are delocalized, and the molecules, in which the electron are localized in bonds. In these cases the most accurate approaches are based in the use of plane-wave and periodic conditions. We will also show how to analyze charge transfers and will present the students with some examples in which molecules self organize on a surface.

The SCC-DFTB method and its biological applications
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Combining near quantum mechanics accuracy with solution speeds approaching those of classical atomistic methods, parameterized quantum methods offer an attractive alternative for studying large systems as they keep the quantum nature of the system, but at a much lower computational cost due to several approximations and the use of parametric functions avoiding integral calculations. This is the case of tight binding methods (TB) as the density functional based tight binding (DFTB) whose parameters are fitted from DFT calculations. In recent years, this method has been widely used in the context of biological studies in particular with the refined formulation known as the Self Consistent Charge SCC-DFTB. It has been applied successfully to determine structures and dynamic behaviour of many biomolecules such as polypeptides or nucleotides. In this lecture the theoretical basis of the method will be presented together with a review of its application to biological systems.