DFT

In Density Functional Theory (DFT) the electronic density ρ(x) is the key quantity. The "Hohenberg-Kohn theorem" establishes that the density is in a one-to-one correspondence with the external potential, v_ext (r), e.g. the potential determined by the positive ions which at the end is the only quantity that changes in the Hamiltonian when passing from a condensed matter system to another. Thanks to this theorem, all the ground-state observables can be expressed as unique functionals of the density, O[ρ(x)], without needing to resort to the complicated many-body wave function. In particular, the total energy of the system can also be expressed as an unique functional of the density, E[ρ(r)].

The Hohenberg-Kohn theorem further provides a variational principle which states that the exact ground state density of the system is that which minimizes the total energy, E₀ = min_ρ E[ρ(r)] . If we know the total energy functional expression in terms of the density, the solution to the problem can be found by numerically minimizing this functional.

The problem can, however, be reformulated in other terms: in parallel to the real electron-electron interacting system, one can introduce a ficticious non-interacting system, called the Kohn-Sham system, which features an effective external potential such that the electronic density of this system exactly coincides by construction with the electronic density of the real system. The calculation of the electronic density is hence simpler within this system than within the real system. One needs to solve a one-particle Schroedinger equation, with a Hamiltonian containing a kinetic and an effective external potential term. Then the states of the systems are filled with a Fermi-Dirac distribution until all the electrons are accounted for up to the Fermi level. The density is hence calculated by . This constitutes the Kohn-Sham set of equations and the scheme is known as the Kohn-Sham scheme. The only problem is now that we have to provide appropriate forms to the effective Kohn-Sham potential, which contains the real external potential, the Hartree classical repulsion term, and an unknown term, the exchange-correlation potential. This term need to be approximated. The most used approximations are the Local-Density Approximation (LDA), or the Generalized Gradient Approximation (GGA).

DFT is, in principle, an exact theory for predicting ground state observables, such as the ground state energy, the electronic density, the atomic structure (lattice parameters, atomic positions), but also (thanks to perturbation theory), elastic constants, bulk moduli, phonon and vibrational frequencies.

To access excited-state properties, one needs to introduce a complication into the theory, which is the time-dependence, thus passing to Time-Dependent Density-Functional Theory (TDDFT). The "Runge-Gross" extends the Hohenberg-Kohn theorem to time-dependent external potentials and densities, O[ρ(r,t)]. TDDFT can in principle access excited-state properties, in particular the neutral excitations (excitations in which the system does not undergo a change in the charge, the number of electrons being kept constant). These include: optical spectroscopies such as optical absorption, reflectivity, real and imaginary indexes of refraction, etc.; Dielectric spectroscopies, such as Electron Energy-Loss Spectroscopy (EELS), Inelastic X-Ray Scattering Spectroscopy (IXSS), and so on.

TDDFT is versatile and computationally efficient, but the accuracy of the result may be affected by the approximation that we always need to make for the exchange-correlation functional.

MBPT

Green's function theory, also called (improperly) Many-Body Perturbation Theory (MBPT), is a Quantum Field Theory based on a formalism of second quantization of operators. The fundamental degree of freedom is the Green's function or propagator, $G(r_1,t_1,r_2,t_2)$, which represents the probability amplitude for the propagation of an electron from $r_1,t_1$ to $r_2,t_2$. The main advantages of this theory are that:

1. it avoids having indices running over many particles;
2. fermionic antisymmetrization is automatically imposed;
3. systems with varying number of particles can be treated;
4. and most importantly, all the physics of the system is condensed inside the Green's function.

As in any other quantum field theory (for example QED), the many-body system can be expanded in perturbation theory, with the coupling being the many-body interaction term. The Green's function (as well as any other quantity of the theory, such as the self-energy or the polarization) can be calculated at a given order of perturbation theory. A Feynman diagrammatic analysis is hence possible. The theory at the first order is equivalent to Hartree-Fock theory.

However the coupling is not small (compare to, for example, the electron-ion interaction) and the expansion does not converge. The second order is not necessarily smaller than the first. Hence one needs to resort to more complicated methods to solve the theory, such as partial resummations of diagrams at all orders, or better, iterative methods.

In iterative schemes one introduces new quantities into the theory but relating them to the old, in the hope that at the end one can succeed in closing the equations. Indeed, MBPT can be solved thanks to a set of five integro-differential equations, called the Hedin equations, that have to be solved iteratively until self-consistency is achieved.

So far, nobody has solved the Hedin equations for a real system. Approximations are required to simplify the problem. Among the most widely used approximate schemes are the GW approximation for the self-energy and the Bethe-Salpeter Equation approach and its related approximations.

0D-Systems (biological and technological interest)

Towards biological molecules

Electronic excitations play a key role in biological processes and both the short-range effects around the excitation area, and the long-range ones must be described correctly in the complex biomolecules. We plan therefore technical developments (optimisation of existing codes to describe huge, partially empty supercells, including a better treatment of the continuum), theoretical issues (additional diagrams in the many-body approaches, a description of both short and long ranges in the response functions in time-dependent density functional theory TDDFT), as well as the development of a strategy for a systematic divide-and-conquer approach for excited states.

Developments are tested for the absorption spectra and the femtosecond dynamics of small inorganic molecules such as silanes and silicon clusters. Peculiar aspects like spin, Rydberg series or image states will be addressed. Organic molecules (e.g. acetylene and toluene), DNA bases, porphyrines and biomolecules in solution are our long-term objectives.

Photo-technology

Entire areas of today's technological developments, like opto-electronics or solar cells, are based on a controlled use of electronic excitations in nanostructures. Among the numerous very fundamental questions still open, we address the origin of the radiative decay channels and their relation to electronic states within crystallites or localized in the interfaces. Systems with several hundreds of atoms will be studied including many-body effects, in the framework of the combined solution of Dyson and Bethe-Salpeter equations. We are searching for a more efficient description of substrates and embeddings by using a basis of unperturbed bulk states instead of simple plane waves. We work in understanding the optical (emission) properties of Si and Ge quantum dots for various embeddings, alloying and structural/chemical properties of their interfaces. The oxidation and hydrogenation of nanocrystals will also help understanding nanostructured, porous, amorphous materials used in or proposed for solar cells.

Transport

In order to achieve the breakthrough in molecular electronics, a reliable theoretical description of transport is needed. Good candidates are the Landauer-Buttiker approach based on density-functional theory (LB-DFT), the non-equilibrium Green's function theory (NEGF), time-dependent density-functional theory (TDDFT), the Lang method or also the maximum-entropy approach.

Every direction contains obstacles: e.g new exchange and correlation functionals must be found in TDDFT to cope with the new dynamical correlations induced by the current flow; efficient implementations of NEGF taking into account out-of-equilibrium distributions and correlations. Even to specify the correct boundary conditions to calculate the current through a molecule has still to be settled. Applications will elucidate the I/V characteristic of molecules, the role of contacts, the Coulomb blockade, the Kondo effect, the current induced structural transformations, and the photoelectron injection in molecules.

Electron-ion dynamics

The idealized decoupling of electronic and ionic degrees of freedom can be far from reality, especially in localized structures. Understanding the transfer of energy between electrons and ions will help identifying the yet unknown mechanism(s) occurring during the fragmentation of a cluster excited by a fast and intense laser, or to predict the changes in the molecular conformation under excitation. For example we computed non-adiabatic photoemission spectra for fullerene C₆₀ and obtained striking agreement with experimentally measured spectra, previously not especially well understood.

We now are developing methods based on TDDFT and adiabatic as well as non-adiabatic dynamics, and also combine the ionic dynamics and the many-body framework (Bethe-Salpeter equation and/or newly developed TDDFT kernels). Our goal is to study the laser-induced chemical reactivity and isomerization of molecules: azobenzene ring, HCN-CNH, and then biological molecules like retinal.

1D-Systems

Phonons in 1D-structures

Electron-phonon coupling is at the core of many fundamental processes in material science, from superconductivity to material degradation. The combined description of electron-ion dynamics is of fundamental relevance to address the decaying mechanism and how those are affected by the 1D nature of the structures. Besides leading to temperature dependent broadening of optical spectral features this coupling can be use to control and monitor chemical reactions (and assembling) at surfaces

Magnetic effects in nanostructures

Low-dimensionality causes non-magnetic bulk material to exhibit magnetic response: this has important consequences for applications and up to now is poorly understood. The proper treatment requires the introduction of spin-orbit effects (magnetic anisotropy) and the existence of non-collinear magnetic structures. This task requires fundamental advances in the many-body (self-energy) description of electron correlations and, consequently, in the development of proper exchange-correlation functionals within TD-DFT. One natural application would be to study spintronics (spin-transport through molecular structures) and the behaviour of magnetic nanostructures supported on surfaces.

Nanowires, nanotubes, conjugated polymers including their composites and the environment (packing effects)

We have two main tasks:

• One, calculate the optical spectra of free-standing and interacting silicon wires as well as the effective electron-electron interaction influence of surface passivation on their structural properties. We will also study the electronic structure of metallic nanowires on semiconductor surfaces, as explained in C3, and the optical and EELS spectra of isolated carbon and boronnitride nanotubes. As a way towards complexity, we will tackle the analysis of response properties of more complicated structures, like multiwall nanotubes, the inclusion of other materials, and their 3D assemblies.
• Second, study the packing effects in the efficiency of polymer-based LEDs. We will also shed light on the nanotube/polymer precursor interaction, which is responsible of a change of the effective polymer conversion temperature thus influencing the efficiency of polymer-based LEDs. Finally, we will study the cohesion and resistance to light degradation of tube/polymer composites.

1D-Organic molecules on surfaces

We will study of the role played by defects, and of water, oxygen and organic groups attached to the 1D-structures in the electronic properties. We will also address the optimal control of laser pulses to achieve a desired binding or desorption of molecules. This works goes towards the development of chemical and biological sensors (e.g, nanotubes have been shown to exhibit important changes of conductivity upon absorption of different gases and enzymes, this, of course, needs to be quantified and controlled).

2D-Systems

Ground and excited states at surfaces and interface

Total Energy methods

Surface problems, such as the dynamics of molecules on a surface, including catalysed reactions) present a variety of bonding situations and energy barriers, and so test present density-functional theory (DFT) methods to beyond their limits of accuracy. We shall build on current collaborations to develop more accurate methods within DFT (including generalised- Kohn-Sham DFT) and many-body perturbation theory (MBPT).

York and S. Sebastian teams will contribute through developments of the MBPT-GW method; Berlin-FU and Louvain will contribute orbital-dependent functionals, combining the RPA with appropriate TDDFT-XC kernels within the adiabatic connection approach. Milan and Rome will also participate, using GW and Green's Functions Methods. The new methods will describe bonding situations ranging from the van der Waals limit to highly covalent bonding. Van der Waals bonded systems are notoriously difficult in traditional density functional theory.

Surface excited states

Improvements of present state-of-the-art methods for surface excited states:

• going beyond common approximations (e.g., extending the existing tools to spin-polarized systems);
• Developing numerically efficient schemes (i.e. with a favourable size-scaling), in order to allow for the study of large unit cells. Paris, S. Sebastian, Milan, Rome and Berlin-FU will collaborate on the development of new TDDFT Kernels; Rome and Jena will also contribute through the nondiagonal GW method.

Self-assembled structures

Formation of self-assembled structures on surfaces. We will use DFT together with thermodynamics, to investigate structures and the driving forces of the self-assembly. Electronic excitations will be treated using state-of-the-art methods, as well as the improved tools obtained within task C2. Rome, Paris and S. Sebastian will look at the problem of the optical properties, Jena will work on self-assembled nanowires of metal atoms on Si surfaces.

Organic biomolecules on surfaces

Interaction of organic biomolecules with surfaces, using parameter-free theoretical methods (DFT, TDDFT, MBPT). We plan to study atomic structures, electronic excitations, optical properties (RAS), Auger and vibrational spectra. Jena, Rome, York and Milan will study, e.g., DNA bases, molecules like porphyrines, and molecules having an amine group which can act as a hook for the attachment to a surface.

Reactions of small molecules on surfaces, in connection with the new experiments with short-intense lasers, which promote specific chemical reaction (e.g., isomerization, or bonding to specific sites), through transitions to excited-state Born-Oppenheimer surfaces. S. Sebastian, Milan and Berlin-FHI will collaborate through suitable developments of both Green-function techniques and TDDFT. Jena and York will study excited states of simple molecules starting with CO and NO on Si surfaces, and moving on to H2O, CH3Cl and unsaturated cyclic hydrocarbons. Lund will study simple chemisorbed molecules, their vibrational and photoelectron spectra, including shake-up of excitations in the substrate and vibronic shake-up.

Spectral features related to interfaces, in particular to grain boundaries. It requires participation of specialists in the latter field (present in the Paris node), of experts in complex band structure calculations (Louvain), and of specialists of surface optical properties and anisotropy (Rome and Milan). SiO2 and SiHfO4 interfaces with other materials will be studied.

Confinement effects

Goal: to understand the systematic changes in the electronic structure and electronic excitations (e.g. plasmons) when passing from a three-dimensional (bulk) material to a very thin layer. Confinement effects will be studied for selected materials, ranging from those used in semiconductor technology to catalytically active oxides. Multipole plasmons will be studied at Berlin-FHI; Berlin-FU will contribute using TDDFT, Rome will contribute through the GW method, and Milan with the embedding approach for a semi-infinite surface.

3D-Systems

Fundamental knowledge for advanced materials

Localized states

For many standard materials the commonly used approximation that the DFT wave functions are almost identical to the quasi-particle wave functions obtained from GW or other sophisticated methods is valid. However, special systems which attracted also scientific and technological interest in the last few years require a more advanced treatment.

As soon as localised states (defect or surface/interface states, molecular orbitals, but also shallow core states of transition metals and heavy main group elements) play an important role for electronic and optical properties this approximation often breaks down and one has to take into account more carefully interactions (correlations) and intermixing of different states. A simple but technologically important material system is for example InN where strong p-d repulsion effects require a full diagonalisation of the problem. Similarly other nitrides or oxides of heavy elements possessing shallow core states or of transition metals are further clear candidates. Other critical benchmarks shall be molecules and molecular crystals. In particular we plan applications for systems involving hydrogen bridge bonds as e.g. ice and organic molecules.

Full dynamical GW scheme

Dynamical effects are currently treated in a manner which allows an accurate treatment of lowenergy excitations typically occurring in bulk systems. Molecular systems (important for bioscience), however, raise new challenges since they require a proper treatment of dynamical effects for giant excitation energies which are not yet handled properly. Current work in the network points the way forward. Simple molecules shall act as benchmark systems.

Magnetic systems

Spin is an important degree of freedom for all magnetic but also various molecular systems. The lack of inclusion of the spin degree of freedom in existing codes and methods often prevents application of these powerful tools to magnetic or molecular systems where spin effects (magnetism) play an important role. It is therefore essential to improve the existing methods and codes in this direction. Benchmark applications will be transition metal oxides, molecular systems and magnetic semiconductors.

Optical absorption

TDDFT is a powerful tool for the calculation of optical properties. However, it turns out that the accuracy of TDDFT often breaks down for extended (bulk) systems. The basic reasons for this behaviour are already understood. It is now necessary to make use of these insights to improve the TDDFT codes to make them also applicable for bulk systems of technological relevance. Critical benchmarks shall be standard semiconductor materials and silicon dioxide, more advanced applications shall be amorphous Si, polymers, and organic crystals.

Non-linear effects

A comprehensive understanding of the nonlinear optical properties of solids is crucial for the improvement of the nonlinear materials and devices and provides an opportunity to search for new materials. Nonlinear optics also has a great potential as a characterization technique for materials, because of its sensitivity to symmetry. In order to extract the maximum amount of information from such measurements, a quantitative theoretical analysis is required.

One important nonlinear process is second harmonic generation. Since its discovery in 1961, many difficulties delayed the accurate calculation of the corresponding susceptibility for many years. Although several calculations have been performed for various semiconductors, the agreement with the experimental measurements is far from being satisfactory and a consistent picture could not emerge from these calculations.

We will develop theory and software to calculate nonlinear susceptibilities: in particular, we will generalize the developments made in the framework of linear response time-dependent density functional theory to the case of the nonlinear response, in order to get accurate values for the second and third order susceptibilities in an efficient way. Attention will be paid to excitonic and/or local-field effects

Theoretical developments

One of the greatest challenges of modern solid-state theory is to understand strongly correlated materials.

While the photo-electron spectrum of solids such as NiO or FeO can be interpreted to some extent on the basis of Hubbard-type model calculations, there exists, as yet, no parameter- free abinitio theory for these materials. This problem will be approached from two different angles: (i) with many-body perturbation theory by including suitably chosen vertex functions in the GW scheme; (ii) with a density-matrix functional approach. The latter has proven successful in the chemists’ paradigms of strongly correlated molecules [S. Goedecker, C.J. Umrigar, Phys. Rev. Lett. 81, 866 (1998); E.J. Baerends, Phys. Rev. Lett. 87, 133004 (2001); K. Yasuda, Phys. Rev. Lett. 88, 053001 (2002)] and the goal of this project is to generalise the method to periodic solids. Furthermore, the approach will be generalised to finite temperature to achieve an ab-initio description of the phase diagram of Mott insulators as well as strongly correlated superconductors.

Examples

A biological marker: the Green Fluorescent Protein

The study of the optical properties of biochromophores has developed into an important and active field of research. The rationale is clear, as the absorption and emission of light by biomolecules are at the center of crucial biophysical processes -- such as vision or photosynthesis, and has led to various important technological applications. Among photo-active proteins, and due to its unique photophysical properties, the family composed by the green fluorescent protein (GFP) and its mutants has attracted a considerable amount of attention during the last decade (read more).

New materials for solar cells

Solar (or photovoltaic) cells are the symbol of clean and safe energy production, as they convert into electric power the prime source of renewable energy, the sunlight. The present technology of solar cells is based on silicon. However, this solution is not satisfactory as it does not allow a good compromise between cost production and efficiency. New materials with interesting electronic properties are now under study in search for new candidates for future commercial cells. Simulation of the optical properties of promising materials for photovoltaic applications is work in progress in ETSF groups (read more).

Nanoelectronics

IOP publishing

The miniaturization process in the integration scale of modern Electronics according to Moore's law will attain in 2015 its ultimate limit: that one of Electronic Devices at the Atomic and Molecular scale of a Nanometer. The theory faces now the challenge to understand and model the Quantum Mechanisms governing the Electron Transport at such scale and, as a last issue, to predict from First Principles the I/V Electronic Characteristics of Nanodevices. The ETSF is actively involved into all these tasks (read more).

Optical data storage and rewritable DVDs

Phase change materials are a class of alloys characterized by a very fast structural change and a pronounced optical/electrical contrast between the crystalline and the amorphous phase. These materials are at the basis of the present technology of rewritable DVDs and are promising candidates for future non-volatile memories. To understand the microscopic mechanism that allows for such peculiar electronic properties is essential for a systematic material optimization of phase change alloys. Calculations performed by ETSF groups have made a link between the strong optical contrast and local structural changes (read more).

Surface Reconstruction Puzzle

The structure of the surfaces of Ge, Si or GaAs, with varying composition and reconstruction, can be established thanks to the comparison of measured spectra to high level calculations (read more).

Data Storage

Optical data storage and rewritable DVDs

Several Tellurides alloys containing Ge and Sb exhibit a pronounced contrast in the optical absorption of the crystalline and the amorphous state. This phenomenon is the basis for their application in optical data storage e.g. in rewritable DVDs. These two phases also present a profound change in electrical properties 'such as the resistivity change' which is one of the crucial features that would be used in phase change random access memories, a very promising candidate for future non-volatile memories.

Fig.1 Setup to measure electrical switching in PCRAM's

To understand the change in optical properties upon amorphization of the phase change materials (PCM), experimentalists in Germany [1] wanted to correlate a macroscopic measurement like the optical properties with microscopic details like the local atomic structure. Using ab initio calculations on structural models with different local order, theoreticians in Palaiseau node [2,3] proved that the optical contrast between the two phases is due to the change of the number of first neighbours of Ge atom and that it should be possible to adjust the optical contrast by changing the composition of the PCM. These results provide a fundamentally new insight in the physics of the optical absorption of amorphous materials and in addition they represent important contributions to a systematic material optimization of phase change alloys.

This example illustrates how first principles calculations can provide an ideal complementary tool in technological issues.

[1] A. Kolobov, P. Fons, A. Frenkel, A. Ankudinov, J. Tominaga, and T. Uruga, Nature Materials 3, 703 (2004).

[2] W. Welnic, A. Pamungkas, R. Detemple, C. Steimer, S. Blügel, and M. Wuttig, Nature Materials 5, 56 (2006).

[3] W. Welnic, S. Botti, L. Reining and M. Wuttig, Optical contrast in phase change materials, submitted.

Nano Electronics

Quantum Transport

Since the introduction of the integrated circuit or chip in the sixties, the power of microprocessors has always grown together with the integration scale, following what is called the Moore's law. If such a miniaturization rate will continue, modern electronics will attain its ultimate limit, the molecular and atomic scale, around the year 2015. As a consequence, Electronics at the Nanoscale, namely Nanoelectronics, represents the next years' technological challenge. A bottom-up process instead of a top-down as followed so far, will allow the realization of Electronic Devices made out of Molecules and Nanostructures. And this is boosted not only by the need for shorter integration scales, and device miniaturization, but also by the expectation that unusual quantum effects are going to be observed due to quantum phenomena effects.

Beside the experimental efforts to synthesize nanoelectronic devices, quantum transport theory has the formidable task to understand and to model the mechanisms behind these phenomena and to predict them from a first principles approach.

Several years ago, important progresses have been accomplished in the theory of quantum transport thank to the setup of two frameworks: the Landauer-Buttiker (LB) and Kubo-Greenwood formalisms. These two formalisms rely on theories able to provide the electronic structure of the nanodevices. And these can be the semi-empirical Tight-Binding (TB); or fully ab initio Density-Functional Theory (DFT).

The Landauer-Buttiker on the top of Density-Functional Theory, today to be considered the state-of-the-art, has demonstrated its ability to describe small bias coherent transport in nanojunctions. These approaches were successful in accounting for the contact resistance and conductance degrading mechanisms induced by impurities, defects and non-commensurability patterns in the conductor region.

The theoretical effort and trend today is to move toward theories able to account for non-coherent and dissipative effects due to electron-phonon and electron-electron scattering mechanisms inside the conductor and for non-linear response, far from equilibrium, finite-bias transport. Along these directions, the two major research lines are Time-Dependent Density-Functional Theory (TDDFT) [Runge, E.K.U. Gross, W. Kohn]; and Non-Equilibrium Green's Function (NEGF) theory [Schwinger, Baym, Kadanoff, Keldysh]. Both NEGF and also TDDFT [G. Stefanucci, C.-O. Almbladh] are in principle correct frameworks to address the above objections.

Researchers at the Rhone-Alpes associated ETSF node have demonstrated [Darancet et al], in the framework of NEGF, that a GW approximation on the Self-Energy can introduce diffusion and loss-of-coherence effects due to the electron-electron scattering and giving rise to reduction of conductance and appearance of resistance inside the conductor.

Gold Monoatomic Nanowire:
Differential conductance vs applied voltage

Together with the result of another theory group in Copenhaguen who introduced electron-phonon scattering effects through a self-consistent Born approximation (SCBA), their calculated conductance characteristics as a function of the applied voltage for a gold monoatomic nanowire fully explains all the features present in the experimental measures of Agrait et al.

P. Darancet, A. Ferretti, D. Mayou and V. Olevano, Phys. Rev. B (2007).
T. Frederiksen, M. Brandbyge, N. Lorente, and A.-P. Jauho, Phys. Rev. Lett., 93, 256601 (2004).
G. Stefanucci and C.-O. Almbladh, Europhysics Letters 67, 14 (2004).
N. Agrait, C. Untiedt, G. Rubio-Bollinger and S. Vieira, Phys. Rev. Lett., 88, 216803 (2002).

Surface Reconstruction Puzzle

We study the two lowest-energy isomers of the Ge(111)-(2x1) surface, by a state-of-the-art first-principles calculation of their optical spectra, including the electron-hole interaction effects. A comparison of our results with the available experimental data suggests that, at difference with the Silicon case, the stablest isomer differs from the standard buckled Pandey chains reconstruction. This conclusion is supported by accurate total-energy results.

Fig.1 Two possible reconstructions

Fig.2 Experiment compared with the BSE results of both reconstructions.

1. We have studied the two lowest-energy isomers of the Ge(111)-(2x1) surface, by a state-of-the-art parameter-free calculation of their total energy and optical spectra, including the electron-hole interaction effects.
2. The two isomers yield a surface geometry which differs only starting from the third atomic layer below the uppermost one, hence experimental probes like STM can hardly be employed to discriminate between the two isomers.
3. Optical properties deduced from electronic structure results obtained at the one-particle level - i.e. neglecting the electron-hole interaction effects - cannot be used to discriminate between the two isomers.
4. A comparison of our results with the available experimental data suggests that, at difference with the Silicon case, the stablest isomer differs from the standard buckled Pandey chains reconstruction.
5. The upper panel in Fig.2 is for the positively buckled (i.e., the traditional) Pandey chain while the lower panel is for the negatively buckled chain. In both panels, the full (dashed) curves include (neglect) excitonic effects.
6. In conclusion, the ground-state geometry of Ge(111)(2x1) is found to correspond to Pandey-like chains with a buckling angle in the opposite direction with respect to that of the commonly assumed geometry. This work has been published in Phys. Rev. Lett. 85, 5440 (2000)
7. Our conclusions have been later confirmed by low-temperature STM experiments: see R.M. Feeenstra, G. Meyer, F. Moresco and K.H. Rieder, Phys. Rev. B 64, 081306 (2001).