The MICMAC team has been created jointly by the Ecole Nationale des Ponts et Chaussées (ENPC) and the INRIA in October 2002.

It is hosted in the CERMICS laboratory (Centre d'Enseignement et de Recherches en Mathématiques, Informatique et Calcul Scientifique) at ENPC. The scientific focus of the team is to analyze and improve the numerical schemes used in the simulations of computational chemistry at the microscopic level, and in the simulations coupling this microscopic scale with larger, meso or macroscopic, scales.

Quantum Chemistry aims at understanding the properties of matter through the modeling of its behavior at a subatomic scale, where matter is described as an assembly of nuclei and electrons.

At this scale, the equation that rules the interactions between these constitutive elements is the Schrödinger equation. It can be considered (except in few special cases notably those involving relativistic phenomena or nuclear reactions) as a universal model for at least three reasons. First it contains all the physical information of the system under consideration so that any of the properties of this system can be deduced in theory from the Schrödinger equation associated to it. Second, the Schrödinger equation does not involve any empirical parameter, except some fundamental constants of Physics (the Planck constant, the mass and charge of the electron, ...); it can thus be written for any kind of molecular system provided its chemical composition, in terms of natures of nuclei and number of electrons, is known. Third, this model enjoys remarkable predictive capabilities, as confirmed by comparisons with a large amount of experimental data of various types.

On the other hand, using this high quality model requires working with
space and time scales which are both very
tiny: the typical size of the electronic cloud of an isolated atom is
the Angström (

It is certainly not true that all macroscopic properties can be
simply upscaled from the consideration of the short time behavior of a
tiny sample of matter. Many of them proceed (also) from ensemble or bulk
effects, that are far from being easy to understand and to model.
Striking examples are found in the solid state or biological
systems. Cleavage, the ability minerals have to naturally split along
crystal surfaces (e.g. mica yields to thin flakes) is an ensemble
effect. Protein folding is
also an ensemble effect which originates in the presence of the
surrounding medium; it is responsible of peculiar properties
(e.g. unexpected acidity of some reactive site enhanced by special
interactions) on which rely vital processes.

However, it is undoubtedly true that on the other hand
many macroscopic phenomena originate from
elementary processes which take place at the atomic scale. Let us
mention for instance the fact that
the elastic constants of a perfect crystal or the color of a chemical
compound (which is related to the wavelengths
absorbed or emitted during optic transitions between electronic
levels) can be evaluated by atomic scale calculations. In the same
fashion, the lubrifying properties of graphite are essentially due to a
phenomenon which can be entirely modelled at the atomic scale.

It is therefore founded to simulate the behavior of matter at the
atomic scale in order to understand what is going on at the
macroscopic one.
The journey is however a long one. Starting from the basic
principles of Quantum Mechanics to model the matter at the subatomic
scale,
one finally uses statistical mechanics to reach the macroscopic
scale. It is often necessary to rely on intermediate steps to deal with
phenomena which take place on various mesoscales.
Possibly, one couples one approach to the others within the so-called
multiscale models.
In the following we shall indicate how this journey can be done,
focusing rather on the first scale (the subatomic one), than on the
latter ones.

It has already been mentioned that at the subatomic scale, the behavior of nuclei and electrons is governed by the Schrödinger equation, either in its time dependent form or in its time independent form. Let us only mention at this point that

both equations involve the quantum Hamiltonian of the molecular system under consideration; from a mathematical viewpoint, it is a self-adjoint operator on some Hilbert space;

boththe Hilbert space and the Hamiltonian operator depend on the nature of the systemalso present into these equations is the wavefunction of the system; it completely describes its state; its

${L}^{2}$ norm is set to one.

The time dependent equation is a first order linear evolution equation, whereas the time-independent equation is a linear eigenvalue equation.

For the reader more familiar with numerical analysis
than with quantum mechanics, the linear nature of the problems stated
above may look auspicious. What makes in fact extremely difficult the
numerical simulation of these equations
is essentially the huge size of the Hilbert
space: indeed, this space is roughly some
symmetry constrained subspace of

Clever approximations of the Schrödinger problems
are therefore needed. The main two approximation
strategies, namely the Born-Oppenheimer-Hartree-Fock and the
Born-Oppenheimer-Kohn-Sham strategies, end up with
large systems of coupled nonlinear partial differential equations,
each
of these equations being posed on

As the size of the systems one wants to study increases, more efficient
numerical techniques need to be resorted to. In computational chemistry,
the typical scaling law for the complexity of computations with respect
to the size of the system under study is

how to improve the nonlinear iterations that are the basis of any ab initio models for computational chemistry

how to more efficiently solve the inner loop which most often consists in the solution procedure for the linear problem (with frozen nonlinearity)

how to design a small enough variational space, whose dimension is kept limited while the size of the system increases.

An alternative strategy to diminishing the complexity of ab initio computations, is to try and couple different models at different scales. Such a mixed strategy can be either a sequential one or a parallel one, in the sense that

in the former the results of the model at the lower scale is simply used to evaluate some parameters that are inserted in the model for the larger scale: one example is the parameterized classical molecular dynamics, which makes use of force fields that are fitted on calculations at the quantum level;

while in the latter, the model at the lower scale is concurrently coupled to the model at the larger scale: an instance of such a strategy is the so called QM/MM coupling (standing for Quantum Mechanics/Molecular Mechanics coupling) where some part of the system (typically the reactive site of a protein) is modeled with quantum models, that therefore account for the change in the electronic structure, and therefore for the modification of chemical bonds, while the rest of the system (typically the inert part of a protein) is coarse grained, and more crudely modeled by classical mechanics.

The coupling of different scales can even go up to the macroscopic scale, with methods that couple a microscopic description of matter, or at least a mesoscopic one, with the equations of continuum mechanics at the macroscopic level.

The laser control of chemical reactions is today an experimental reality. Experiments, carried out by many groups of researchers and in many different contexts and settings have demonstrated the feasibility of controlling the evolution of a quantum system using a laser field. All these experiments exploit the remarkable properties of quantum interactions (interferences) between one, or more, external interactions (e.g. lasers) and the sample of matter under study. In order to create the ad hoc interferences that will drive the system to the desired goal, one can either play with the dephasing between two beams, or conveniently choose the frequencies of the beams, or also make use of the two aspects mixed together, which amounts to allowing for ``all'' possible laser fields as in optimal control schemes.

Whatever the strategy, the success of these numerous experiments not only validate the idea of manipulating and controlling quantum systems with lasers, but also motivate the need for further theoretical studies in this direction, in order to even improve the results and the range of their applicability; interest in this research area has also been increasing in more applied communities. The standard modeling for the problem of the laser control of a molecular system involves the time-dependent Schrödinger equation which rules the evolution of the wavefunction describing the state of the system. On the basis of the Schrodinger equation, one then state a control problem, either in the framework of exact control or in the framework of optimal control.

The first fact to underline as a crucial features of the
problem of laser control is the orders of magnitude in time and space
that are typically encountered here.
The space scale is indeed that of an atom, say

A second point is to remark the way in which the control enters the problem: the control multiplies the state. Theoretically and numerically, this bilinear nature causes difficulties. Finally, we deal here with open-loop control, at least for two reasons: first, the timescale on which the phenomenon goes is too short for the current capabilities of electronic devices, which prevents closing the loop within one experiment; but secondly, feedback control means measuring something, which in a quantum framework means interacting with and thus perturbing the system itself. These two bottlenecks might be overcome in the future, but this will undoubtedly require a lot of theoretical and technical work.

A third peculiarity regards the choice of admissible laser fields as control :
what types of

A final key issue is robustness. It is of course a standard fact in every control problem that the control obtained needs to be robust, for obvious practical reasons. The somewhat unusual feature in the present setting is that the experiments show that they are surprisingly robust with respect to all kinds of perturbations (noise, uncertainties in the measures, ...). Clearly, there is here something to be understood on the theoretical level, e.g. by envisioning new modeling strategies that incorporate undesirable perturbations.

We have continued our studies on the algorithms used for electronic structure calculations. The different tracks followed are

improvements of the so-called SCF approaches, in collaboration with the group of Gustavo Scuseria at Rice

acceleration techniques for the linear subproblem based in particular on deflation techniques

introduction of reduced basis approaches in the framework of quantum chemistry; some exploratory works have been conducted in close collaboration with the group of Antony Patera at MIT.

Apart from this main stream, two numerical works, of different nature have been conducted.

First, we have developed in

Second, numerical schemes have been developed for the
simulation of a nonlinear Schrödinger
equation close to the one encountered
in time dependent electronic structure calculations and modeling
Bose-Einstein condensates

All of these new developments and methods are meant to be inserted in the long term inside the software platform that we are currently developing (in collaboration with Yves Achdou, University Paris 7). The current status of this software allows us to perform electronic structure calculations on simple systems, within a C++ environment, and relying on rigorous numerical analysis tools.

In parallel with these numerical works, we have pursued in our enterprise
to put the models on a safe mathematical ground. We have studied the
optimized effective potential approach from the theoretical viewpoint in

A survey of all these new developments together with a precise and
comprehensive state of the art of the mathematical knowledge in the
field of quantum computational chemistry has been published in a
book

Molecular dynamics is often used in statistical physics as an
alternative to Monte Carlo methods for computing ensemble averages. Based
on a theoretical result concerning the convergence of the numerical
averages toward the exact result, we were able to propose in

Following a series of works devoted to the optimal control
of the alignment of a molecule by laser field and population inversion
algorithms

We also intend to address in the near future and in collaboration with
Pierre Rouchon (Ecole des Mines, Paris) questions related to the
inversion paradigm: use the laser field as a tool to obtain
additional information on the molecular system. A related topic
was already the subject of a recent work

In addition to this, and on the level of numerical analysis, we have published some mathematical results on what
are the most commonly used optimization algorithms in the field of
optimal control for this type of laser-matter interaction problems,
namely the genetic algorithms. In

This research program divides into a theoretical part and a more numerical one.

On the theoretical side, following a series of works, in collaboration with Xavier Blanc
(Laboratoire Jacques-Louis Lions, Paris), Isabelle Catto (University
Paris 9), and Pierre-Louis Lions Collège de France, Paris), we have
addressed the question of how to define
ground state energies for systems composed of an infinite number of
particles.
The framework is that of quantum chemistry, where the state of matter is
modelled though variational problems that couple a classical description
of the nuclei with a quantum description of the electrons. Starting from
a model for the molecule (finite number

In addition, we have continued our program consisting in passing from
the microscale to the macroscale on the basis of quantum models at the
microscale. This program has been initiated in

On a more numerical level, Claude Le Bris and Frédéric Legoll, in collaboration with Xavier Blanc, have begun the mathematical and numerical analysis of models used in simulations for material science where an atomistic description of the sample is coupled with a macroscopic continuum description.

The subject of this activity covers two different applications and settings.

The first one is that of the modeling of polymeric fluid flows. We have
continued our endeavor to mathematically study the models commonly used
in numerical simulations by experts at nonnewtonian fluid dynamics

The second topic of interest deals with concentrated
suspensions, and is done in collaboration with Isabelle Catto
(University Paris 9). So far, the study is mainly of theoretical nature, with a view to
tackle in a near future more numerical and practical aspects. We have
established the well-posedness of the Cauchy problem for the equations
modeling the suspension

In collaboration with Jean-Frédéric Gerbeau (Inria, BANG), and in
association with
Aluminium Péchiney, we have pursued our efforts for the numerical
simulation of electrolytic cells for the industrial production of
Aluminium. The results obtained so far have been detailed in

In the context of a collaboration with A.Danchin (Pasteur Institute) and
Ng. Tuen (Hong-Kong University) we studied the impact of a biological
hypothesis on the SARS virus propagation data

Many research activities of the team are indeed conducted in close collaboration with private companies: Pechiney for the modeling of electrolytic cells, Electricité de France for computational chemistry, molecular dynamics and multiscale simulation of solids, and companies from the elastomer industry for the multiscale simulation of rubber-like materials.

The team is shared between INRIA and Ecole Nationale des Ponts et Chaussées.

The team is part of the research action GDR Density Functional Theory whose scientific leader is H. Dreysse (Physique, Strasbourg), devoted to the development of DFT methods for the simulation of materials and complex systems. It is also a part of the research action GDR Interaction de particules, whose scientific leader is Th. Goudon, on questions related to the modeling of many particles systems.

Some members of the team participate into the european project IHP ``HYKE'' ( Hyperbolic and kinetic equations: Asymptotis, numerics, Applications) whose Scientist in charge is Benoît Perthame (BANG), on theoretical aspects related to the resolution of the Schrödinger equation for large systems and long times.

Continuous and permanent cooperations have been established with the group of Gustavo Scuseria at Rice University on questions related to electronic structure calculations for large systems, that of Herschel Rabitz at Princeton University and that of André Bandrauk at Unversity of Sherbrooke (Canada), respectively on questions related to laser control and to the solution of the Schrödinger equation for a large number of degrees of freedom.

Claude Le Bris is VP of the SMAI (french SIAM), more particularly in charge of relation with the industrial companies.

Simulation moléculaire: aspects théoriques et numériques, cours de DEA, université Paris 6 (C. LeBris).

Simulation moléculaire, cours de DEA, université Paris 9 (E. Cancès).

Systèmes multiéchelles, cours de la majeure SEISM, Ecole Polytechnique (C. LeBris).

Calcul scientifique et Analyse, cours à l'Ecole Nationale des Ponts et Chaussées, (E. Cancès).

Analyse en fréquences, cours à l'Ecole Nationale des Ponts et Chaussées, (E. Cancès).

Modéliser, Programmer, Simuler, cours à l'Ecole Nationale des Ponts et Chaussées, (C. LeBris).

Members of the team have delivered lectures in the following seminars, workshops and international conferences:

AMAM conference, Nice, February 2003 (M. Barrault, E. Cancès, C. LeBris, T. Lelièvre, G. Turinici)

IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Sevilla, April 2003 (C. LeBris, G.Turinici)

``Spitalfields Day'' of the London Mathematical Society and the Isaac Newton Institute, Cambridge, May 2003 (C. LeBris)

Rigorous Ab-Initio Studies of Periodic Systems: Approaches to Electron Correlation, CECAM Workshop, May 2003, Lyon (E. Cancès, C. LeBris)

XIIIth International Workshop on Numerical Methods for non-Newtonian Flows, June 2003, Lausanne (C. LeBris)

PCAM Seminar of the Applied Mathematics department, Princeton University, June 2003 (C. LeBris)

Calais, June 2003 (M. Barrault)

Bonn University Mathematics department seminar, June 2003, (C. LeBris)

Gerone, July 2003 (F. Lodier)

SIAM Annual meeting, Montréal, June 2003 (E. Cancès, C. LeBris)

International congress of quantum chemistry, Bonn, july 2003 (E. Cancès)

London, September 2003 (T. Lelièvre)

Oberwolfach Workshop PDE and Materials, September 2003, (C. LeBris)

Warwick University Mathematics department seminar, October 2003, (C. LeBris)

Oberwolfach Workshop Classical and Quantum Mechanical Models of Many-Particle Systems, November 2003 (C. LeBris)

Workshop Discrete atomistic models and their continuum limit, Berlin, December 2003 (C. LeBris)

Workshop Multiscale problems in quantum mechanics and averaging techniques, Leipzig, December 2003 (C. LeBris)

PRESTISSIMO workshop, Institut Henri Poincaré, December 2003 (E. Cancès)