Academic literature on the topic 'Ensemble density-functional theory'

Ensemble density functional theory extends the usual Kohn-Sham machinery to quantum state ensembles involving ground- and excited states. Recent work by the authors [Phys. Rev. Lett. 119, 243001 (2017); 123, 016401 (2019)] has shown that both the Hartree-exchange and correlation energies can attain unusual features in ensembles. Density-driven (DD) correlations – which account for the fact that pure-state densities in Kohn-Sham ensembles do not necessarily reproduce those of interacting pure states – are one such feature. Here we study atoms (specifically S–P and S–S transitions) and show that the magnitude and behaviour of DD correlations can vary greatly with the variation of the orbital angular momentum of the involved states. Such estimations are obtained through an approximation for DD correlations built from relevant exact conditions, Kohn-Sham inversion, and plausible assumptions for weakly correlated systems.

Cette thèse traite du développement et de l’implémentation de nouvelles méthodes visant à décrire la corrélation électronique forte dans les molécules et les solides. Après avoir introduit l’état de l’art des méthodes utilisées en chimie quantique et en physique de la matière condensée, une nouvelle méthode hybride combinant théorie de la fonction d’onde et théorie de la fonctionnelle de la densité (DFT) est présentée et s’intitule “site-occupation embedding theory” (SOET). Celle-ci est appliquée au modèle de Hubbard à une dimension. Ensuite, le problème du gap fondamental est revisité en DFT pour les ensembles, où la dérivée discontinue est réécrite comme une fonctionnelle de la densité de l'état fondamental. Enfin, une extension à la chimie quantique est proposée, basée sur une fonction d’onde de séniorité zéro complémentée par une fonctionnelle de la matrice densité, et exprimée dans la base des orbitales naturelles<br>The thesis deals with the development and implementation of new methods for the description of strong electron correlation effects in molecules and solids. After introducing the state of the art in quantum chemistry and in condensed matter physics, a new hybrid method so-called ``site-occupation embedding theory'' (SOET) is presented and is based on the merging of wavefunction theory and density functional theory (DFT). Different formulations of this theory are described and applied to the one-dimensional Hubbard model. In addition, a novel ensemble density functional theory approach has been derived to extract the fundamental gap exactly. In the latter approach, the infamous derivative discontinuity is reformulated as a derivative of a weight-dependent exchange-correlation functional. Finally, a quantum chemical extension of SOET is proposed and based on a seniority-zero wavefunction, completed by a functional of the density matrix and expressed in the natural orbital basis

Le fer, en tant que brique élémentaire de la Terre, a reçu beaucoup d’attentions. Des efforts considérables ont été mis en œuvre pour déterminer ses propriétés thermodynamiques et thermophysiques à des conditions atteignant celles du noyau terrestre. Cependant, ses propriétés physiques dans le domaine des faibles densités sont moins explorées, et il manque en particulier la position de la courbe d’équilibre liquide-gaz et du point critique. Les informations manquantes entravent le développement d’une équation d’états complète qui couvrirait l’état de détente post onde de choc, et donc empêchent la caractérisation des grands impacts planétaires. Cette étude vise à réduire le fossé de connaissances sur l’équilibre liquide-gaz du fer. Pour cela nous utilisons la Dynamique Moléculaire et la méthode Monte Carlo dans lesquelles les énergies et les forces sont estimées à partir de la théorie de la fonctionnelle densité. Nous utilisons ensuite des méthodes statistiques et thermodynamiques pour construire la position du point critique, le dôme liquide-gaz, et caractériser les propriétés physiques du fer à l’état de fluide.Tout d’abord nous avons déterminé la position du point critique à partir de simulations de dynamique moléculaire ab initio selon plusieurs isothermes. Les résultats des simulations nous ont donné la position du spinodal liquide au-dessus de 3000 K, et du spinodal gazeux à proximité du point critique. La position du point critique est estimée entre 9000-9350 K et 1.85-2.4 g/cm3, ce qui correspond à 4-7 kbars. Nous avons également caractérisé la structure et les propriétés de transport du fer fluide pour une large gamme de densités et températures, avec une attention particulière sur l’état supercritique.Ensuite nous avons calculé deux courbes Hugoniot à partir de deux conditions initiales réalistes. En comparant les valeurs d’entropie calculées le long de ces courbes à celle du point d’ébullition, nous avons trouvé que la pression requise pour atteindre le seuil de vaporisation est significativement plus basse que précédemment estimée. Cela suggère que les simulations hydrodynamiques précédentes sous-estiment la production de vapeur de fer, et que le noyau de Théïa aurait subi une vaporisation partielle lors de l’impact géant. De même nous avons trouvé qu’une grande fraction des planétésimaux ayant frappé la Terre lors du vernis tardif ont dû voir leur noyau vaporisé partiellement. La facilité avec laquelle les noyaux se vaporisent devrait améliorer l’équilibration fer-silicate, ce qui permettrait d’expliquer les observations géochimiques.Enfin, nous avons déterminé l’équilibre liquide-gaz du fer. Pour cela nous avons amélioré et implémenté la méthode Monte Carlo dans l’ensemble de Gibbs couplée avec la théorie de la fonctionnelle densité en températures finies. Le premier test de référence avec le sodium nous as donné un bon accord avec les résultats expérimentaux. Nous avons donc appliqué cette technique au fer et calculé sa densité liquide à l’équilibre avec la phase vapeur. Nous avons également montré que l’importance du magnétisme diminue à l’approche du point critique<br>Iron as a building block material of the Earth naturally received significant attention. Considerable efforts have been made to determine its thermodynamic and thermophysical properties up to the Earth’s inner core’s conditions. However, its physical properties in the low-density regime are less explored, and notably the position of the liquid-vapor equilibrium line and of the critical point are lacking. The missing information inhibits developing a complete equation of state that covers the released state after shock waves, and thus hinders the characterization of large planetary impacts.The present study aims at closing the knowledge gap on the liquid-vapor equilibrium dome of iron. For this we exploit molecular dynamics and Monte Carlo methods where the energy and the forces are estimated by the density functional theory. We then employ statistical and thermodynamics methods to construct the position of the critical point, build the liquid-vapor dome, and characterize the physical properties of the fluid iron.First we determine the position of the critical point from ab initio molecular dynamics simulations along several isotherms. The simulation results provide the position of the liquid spinodal above 3000 K, and the gas spinodal close to the critical point. We bracket the position of the critical point in the 9000-9350 K temperature range, and 1.85-2.40 g/cm3,density range, corresponding to 4-7 kbars pressure range. Additionally, we characterize the structure and the transport properties of the fluid iron over a wide density and temperature range, with a particular focus on the supercritical state.Then we compute two Hugoniot lines starting with two realistic initial conditions. By comparing the entropy values calculated along these Hugoniot lines to that at the boiling point, we find that the pressure required to reach the onset vaporization is significantly lower than previous estimates. It suggests that previous hydrodynamic simulations underestimate the iron vapor production, and that the core of Theia underwent partial vaporization during the giant impact. Similarly, we find that a large fraction of the planetesimals falling on Earth during the late veneer must have had their cores undergoing partial vaporization. The readily achieved partialcore vaporization would enhance the iron-silicates equilibration, which helps explain geochemical observations.At last, we determine the liquid-vapor equilibrium line of iron. For this, we have extended and implemented the Gibbs ensemble Monte Carlo method coupled with the finite-temperature density functional theory. The first benchmark test to sodium shows a good agreement with available experimental results. We then apply this technique to iron and calculate its liquid density in equilibrium with the vapor phase. We also show the importance of magnetism diminishes as approaching the critical point

Au cours des dernières décennies, la théorie de la fonctionnelle de la densité (DFT) s'est imposée comme une approche rigoureuse pour la description de l'état fondamental des systèmes électroniques. Grâce à son faible coût computationnel et à l'élaboration d'approximations sophistiquées pour la fonctionnelle d'échange-corrélation (xc-DFA), la DFT est devenue la méthode de choix pour le calcul de structure électronique. Néanmoins, il subsiste nombre de défis que la DFT ne parvient pas à surmonter. En réalité, ces carences ne sont pas le fruit de la théorie elle-même mais plutôt du fait de défauts intrinsèques des approximations utilisées. Il existe une formulation plus générale de la DFT pour les nombres fractionnaires d'occupation qui permet la description de systèmes avec nombre fractionnaire d'électrons, la PPLB-DFT. Cette formulation grand canonique de la DFT peut être mise en place à l'aide d'un formalisme d'ensemble et permet une extraction directe d'énergies d'excitation chargée et d'autres propriétés à partir d'un seul calcul de type DFT. Malheureusement, l'incapacité des DFAs à reproduire la fameuse dérivée discontinue (DD) s'est avérée être particulièrement préjudiciable pour la prédiction d'énergies d'excitation chargée, telles que les potentiels d'ionisation et les affinités électroniques, donnant lieu à des erreurs conséquentes, et connue comme le problème du gap fondamental. Dans ce contexte, la DFT d'ensemble (eDFT) offre une alternative très attrayante du fait de sa capacité à user de DFAs dépendantes du poids de l'ensemble pour reproduire la DD via leur dérivée. La DFT est connue pour montrer des limites vis-à-vis du calcul d'énergies d'excitation chargée et neutre. La procédure standard pour accéder aux états excités neutralement dans le cadre de la DFT est à travers son extension dépendante du temps, la TD-DFT. En effet, l'usage est de recourir à la TD-DFT pour obtenir des prédictions acceptables pour les énergies de transition des niveaux excités les plus bas, cela avec un coût computationnel relativement modéré. Bien que la TD-DFT se soit avérée incroyablement fructueuse pour accéder aux énergies d'excitation neutre, elle a également montré certaines limites lors de la description de certains phénomènes et propriétés physiques. En cela, l'eDFT constitue une alternative prometteuse à la TD-DFT pour le calcul des énergies d'excitation électroniques. En eDFT, il est possible d'extraire n'importe quelle énergie d'excitation neutre d'un système électronique en un seul calcul à l'aide d'un ensemble Gross-Oliveira-Kohn (GOK), et cela avec un coût computationnel et un niveau d'approximation pour la fonctionnelle d'xc, similaires à ceux de la DFT standard. La GOK-DFT est une alternative moins connue mais tout autant rigoureuse que la TD-DFT, où le large choix de poids de l'ensemble et la dépendance en poids de la fonctionnelle xc peuvent significativement influer sur la qualité des énergies calculées. En temps normal, accéder aux énergies d'excitation chargée nécessite de faire varier le nombre d'électrons du système, ce qui peut s'avérer problématique dans certains cas. Très récemment, un nouveau formalisme canonique a été développé, l'eDFT N-centrée, rendant possible l'extraction d'énergies d'excitation chargée sans altération du nombre d'électrons. Le comportement des DFAs standard dans le cadre de l'eDFT peut offrir une compréhension plus poussée de la nature intrinsèque des erreurs systématiques dont elles souffrent, telles que la violation des conditions exactes de linéarité par morceaux et de constance de l'énergie. En outre, la mauvaise description des systèmes avec charge et spin fractionnaires a prouvé avoir un impact majeur dans la description des systèmes fortement corrélés ainsi que dans les processus de dissociation et la prédiction de gaps d'énergie. Tout cela pourrait donner un nouvel essor au développement futur de la DFT et à des applications émergentes jusqu'alors inaccessibles<br>Over the last few decades, density-functional theory (DFT) has proved to be a rigorous approach for describing the ground-state of any electronic system. Due to a relatively low computational cost and the elaboration of sophisticated density-functional approximations (DFAs), DFT became the prevailing method used in electronic-structure calculations. Still, there remain numerous challenges that standard DFAs fail to overcome. These limitations are not attributed to failures of the theory itself but are rather due to deficiencies of the currently used approximate exchange-correlation (xc) functionals. There exists a generalization of ground-state DFT to fractional occupation numbers which allows for the description of systems with fractional number of electrons, PPLB-DFT. Such grand canonical extension of DFT can be achieved through the use of the ensemble formalism and enables direct extraction of charged excitation energies and other properties from a single DFT-like calculation. Unfortunately, the inability of commonly used exchange-correlation DFAs to mimic the infamous derivative discontinuity (DD) has proved to be highly detrimental to the prediction of charged excitations such as ionization potentials and electron affinities, yielding substantial errors, and known as the fundamental-gap problem. Regarding this matter, ensemble DFT (eDFT) offers a very appealing alternative benefiting from the possibility for explicitly weight-dependent xc-functionals to mimic the infamous DD through their derivatives with respect to the ensemble weights. DFT is known to possess deficiencies when it comes to computing charged and neutral excitations. The most popular way to access neutrally excited states within the scope of DFT is through its time-dependent extension, TD-DFT. Indeed, one would usually turn to TD-DFT to get accurate transition energies for low-lying excited-states with a relatively moderate computational cost. Although TD-DFT has been incredibly successful to access neutral excitation energies, it still suffers from some limitations and fails to provide accurate descriptions of some phenomena and properties. eDFT constitutes a promising alternative to TD-DFT for computing electronic excitation energies. In eDFT, it is possible to extract any neutral excitation energies of a N-electron system from a single calculation through the use of a Gross-Oliveira-Kohn (GOK) ensemble, with a similar computational cost and level of approximation for the xc-functional than in an usual DFT calculation. GOK-DFT is a less well-known but comparably rigorous alternative to TD-DFT where the large choice of ensemble weights and the weight-dependence of DFAs can significantly impact the accuracy of the energies. In DFT, it is well-known that the HOMO-LUMO gap can be a very poor estimation of the fundamental gap of the system, whereas eDFT may provide better predictions. Nevertheless, accessing charged excitations usually require to vary the number of electrons of the system, which can be problematic for some systems. Very recently, a new canonical eDFT formalism has been developed, the N-centered formalism, which allows for the extraction of charged excitation energies without any alteration of the number of electrons of the system. The behaviour of standard approximations in the scope of eDFT may provide additional insight into the intrinsic systematic errors of DFAs, such as the violation of the piecewise-linearity and constancy-condition exact properties. Indeed, poor descriptions of systems with fractional charges and fractional spins have shown to have major implications on the description of strongly correlated systems, which are known to suffer from large static-correlation errors, as well as on the prediction of asymptotic integer dissociations and band-gap predictions. These considerations may lead the way to further development and refinement of the DFT scheme towards both current and emerging applications

L'adsorption de CO dans la faujasite échangée au CuI et au Na+ a été modélisée à l'aide des approches quantiques (DFT) et classiques (Monte Carlo). Grâce à l'approche DFT, la surface d'énergie potentielle de la faujasite a été explorée. Différents types d'interactions de CO avec les cations ont été identifiés, pour chacune les effets induits par l'adsorption de CO aux niveaux structural et énergétique ont été analysés, et le calcul de la fréquence de vibration de CO a été réalisé. Grâce aux valeurs obtenues, une nouvelle attribution des spectres d'adsorption de CO dans CuY et NaY a été établie. D'un autre côté, grâce aux simulations Monte Carlo dans l'ensemble Grand Canonique, les propriétés d'adsorption (isothermes et enthalpies) de la faujasite vis-à-vis de CO ont été modélisées, et le mécanisme microscopique d'adsorption de CO a été établi. La mise en œuvre de ces simulations a nécessité de paramétrer un nouveau champ de force destiné à décrire les interactions CO/faujasite et CO/CO<br>CO adsorption in CuI and Na+ exchanged faujasite has been modeled by mean of quantum (DFT) and classical (Monte Carlo) approaches. By mean of the DFT calculations, faujasite potential energy surface has been explored. Different types of CO interactions with the cations have been highlighted, for each one of them CO adsorption effects on the structural and energetic parameters have been analyzed, and calculations of the CO stretching frequency have been performed. Thanks to our calculated values, a new attribution of CO adsorption spectra in CuY and NaY has been established. On another side, by mean of Monte Carlo simulations in the Grand Canonical ensemble, faujasite adsorption properties regarding CO (isotherms and enthalpies) have been modeled, and the CO adsorption mechanism has been established at the microscopic level. The implementation of these simulations has required the derivation of a new force field describing the CO/faujasite and CO/CO interactions

Les travaux présentés dans ce manuscrit de thèse peuvent être divisés en deux parties. Dans une première partie, nous nous sommes intéressé à une extension multiconfigurationnelle de la théorie de la fonctionnelle de la densité (DFT) par l'intermédiaire d'une séparation de portée permettant un traitement hybride entre DFT et fonction d'onde multiconfigurationnelle « state-averaged ». Ainsi, nous récupérons en même temps la corrélation dynamique et la corrélation statique. De plus, cette étude est réalisée en considérant la DFT pour les ensembles afin de considérer une alternative à la méthode usuelle utilisée (DFT dépendante du temps) pour la détermination des états excités d'une molécule, évitant ainsi certains problèmes théoriques rencontrés avec cette approche. En particulier, les intersections coniques entre états excités nous intéressent particulièrement car il s'agit de cas pour lesquels une approche multiconfigurationnelle est primordiale. Dans une seconde partie, le développement de nouvelles fonctionnelles est réalisé sur le dimère de Hubbard asymétrique afin de tester de nouvelles approximations et d'étudier plus en détail les processus auto-cohérents. De plus, des couplages non-adiabatiques sont calculés en utilisant des énergies déterminées dans le cadre de la DFT pour les ensembles ayant la particularité de ne pas être dépendant du temps<br>This thesis manuscript can be divided in two parts. In the first one, we are interested in a multiconfigurational extension for the density functional theory (DFT) including a range separation to deal with a hybrid theory between DFT and state-averaged wave function theory. In this case, we recover, at the same time, the dynamical correlation and the static correlation. Moreover, this study is performed considering the ensemble DFT to use an alternative to the usual method (time-dependent DFT) to describe the excited states of a molecule, avoiding some theoretical problems known with this approach. Particularly, conical intersections between excited states are interesting because a multiconfigurational approach is necessary. In the second part, new functionals development are performed and applied on the non-symmetric Hubbard dimer in order to test new approximations and to study more in detail self-consistency processes. In addition, non-adiabatic couplings are calculated using energies from ensemble DFT framework without time-dependence

Submitted to the Faculty of Science Wits University May 29, 2018<br>Density Functional Theory (DFT) is an elegant reformulation of quantum mechanics in which the density distribution is the variable that formally contains all the information about a system. It was placed on a formally sound theoretical footing by Hohenberg and Kohn [1] in 1964 and an implementation for determining the ground state density and energy was proposed by Kohn and Sham the following year [2]. Despite more than fty years since Hohenberg and Kohn showed that the density can be used as the controlling variable, there is no known exact way to implement DFT. Nevertheless, DFT has been successfully applied using approximations and has become the standard approach for investigating structural properties of solids and molecules. In this project we examine properties of DFT functionals for a nite single band Hubbard chain. The advantage of using a Hubbard model is that for short chains exact solutions can be found numerically and for a uniform in nite chain an analytic solution is available. The exact solutions can be used as a reference for approximate implementations of DFT. We explore DFT on a lattice in an ensemble formulation which allows a formal implementation of DFT for fractional particle numbers. We show that even for a simple uniform density approximation the resulting functional derivatives have a spatially independent discontinuity as a function of particle numbers at integer particle number, as the required by the exact formalism. An approximate exact implementation of Kohn-Sham DFT with the neglect of the DFT correlation energy can be implemented exactly and results show that it can compare very well with the exact solution, but that the success of the approximation is not consistent under all circumstances. Finally we show that it is possible to achieve the original goal of Kohn-Sham Density Functional Theory which was to nd the ground state density and energy of an interacting system while all calculations are performed for a ctitious independent particle model. We introduce a mapping of the ground state wavefunction basis function expansion coe cients of a single band Kohn-Sham Hubbard model onto the coe cients of the interacting Hubbard model and derive a set of exact self-consistent equations that can be solved within an ctitious Kohn-Sham framework to nd the interacting ground state density and energy.<br>MT 2018

In this work, experimental and theoretical approaches are applied to the study of chemical reaction dynamics. In Chapter 2, two applications of transition state theory are presented: (1) Application of microcanonical transition state theory to determine the rate constant of dissociation of C2F3I after π∗ ← π excitation. It was found that this reaction has a very fast rate constant and thus is a promising system for testing the statistical assumption of molecular reaction dynamics. (2) A general rate constant expression for the reaction of atoms and molecules at surfaces was derived within the statistical framework of flexible transition state theory. In Chapter 4, a computationally efficient TDDFT approach was found to produce useful potential energy surface landscapes for application to non-adiabatic predissociative dynamics of the molecule CS2 after excitation from the ground state to the singlet C-state. In Chapter 5, ultrafast experimental results of excitation of CS2 to the predissociative neutral singlet C-state is presented. The bandwidth of the excitation laser was carefully tuned to span a two-component scattering resonance with each component differently evolving electronically with respect to excited state character during the quasi-bound oscillation. Scalar time-resolved photoelectron spectra (TRPES) and vector time-resolved photoelectron angular distribution (TRPAD) observables were recorded during the predissociation. The TRPES yield of photoelectrons was found to oscillate with a quantum beat pattern for the photoelectrons corresponding to ionization to the vibrationless cation ground state; this beat pattern was obscured for photoelectron energies corresponding to ionization from the vibrationally excited CS2 cation. The TRPAD data was recorded for two general molecular ensemble cases: with and without a pre-excitation alignment laser pulse. It was found that in the case of ensemble alignment (Chapter 6), the “molecular frame” TRPAD (i.e. TRMFPAD) was able to image the purely valence electronic dynamics of the evolving CS2 C-state. The unaligned ensemble TRPAD observable suffers from excessive orientational averaging and was unable to observe the quantum beat. Engineering efforts were also undertaken to eliminate scattered light background signal (Chapter 7, Appendix A) and improve laser stability as a function of ambient pressure (Appendix B) for TRMFPAD experiments.<br>Thesis (Ph.D, Chemistry) -- Queen's University, 2012-09-11 22:18:20.89

Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed

In this chapter, several examples of NN fitting of databases obtained using either ab initio electronic structure methods or an empirical potential will be discussed. The objective of this presentation is not to provide a complete and comprehensive review of the field nor is it to acquaint the reader with the details of the reaction dynamics of the particular systems employed as examples. It is rather to provide a clear picture of the power and limitations of NN methods for the investigation of reaction dynamics. We begin with a brief overview of the literature in the field. Neural networks provide a powerful method to effect the fitting of an ensemble of potential energy points in a database. In 1993, Blank et al. employed an NN to fit data derived from an empirical potential model for CO chemisorbed on a Ni(111) surface. Two years later, these same investigators also examined the interaction potential of H2 on a Si(100)-2 × 1 surface using a data set comprising 750 energies computed using local density functional theory. To the best of our knowledge, these were the first two examples in which NNs were employed to provide the PES for a dynamics study. Hobday et al. have investigated the energies of C-H systems by using a Tersoff potential form in which the three-body term is replaced by an NN comprising five input nodes, one hidden layer with six nodes, and an output layer. In this work, the five input elements are computed by consideration of the bond type, i.e., C-C or C-H, the three-body bond angle θ, which is input to the NN in the form (1 + cos θ)2, the connectivity of the local environment, and the second neighbor information. The method was applied to carbon clusters and a wide variety of alkanes, alkenes, alkynes, aromatics, and radicals. Comparison of the atomization energies obtained using the NN potential surfaces with experimental values showed the errors for 12 alkanes, 13 alkenes, 4 alkynes, 7 aromatics, and 12 radicals to lie in the ranges zero to 0.3 eV (alkanes), 0.1 to 1.5 eV (alkenes), 0 to 0.5 eV (alkynes), zero to 1.0 eV (aromatics), and zero to 2.8 eV (radicals).

An ab initio molecular dynamics study of femtosecond laser processing of germanium is presented in this paper. The method based on the finite temperature density functional theory is adopted to probe the nanostructure change, thermal motion of the atoms, dynamic property of the velocity autocorrelation, and the vibrational density of states. Starting from a cubic system at room temperature (300 K) containing 64 germanium atoms with an ordered arrangement of 1.132 nm in each dimension, the femtosecond laser processing is simulated by imposing the Nose Hoover thermostat to the electron subsystem lasting for ∼100 fs and continuing with microcanonical ensemble simulation of ∼200 fs. The simulation results show solid, liquid and gas phases of germanium under adjusted intensities of the femtosecond laser irradiation. We find the irradiated germanium distinguishes from the usual germanium crystal by analyzing their melting and dynamic properties.