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silicon , one of the most relevant materials from the technological point of view , has excellent electronic properties as a semiconductor . however , in the crystalline form , it does not exhibit good optoelectronic properties because of the indirect gap , in the infrared range , between the valence and the conduction bands , and of the low probability of radiative electron - hole recombination . however it is possible to modify deeply such properties using a process of electrochemical etching  @xcite which produces a material known as light emitting porous silicon ( leps ) . this form , because of its luminescence properties , is presently the object of great scientific and technological attention  @xcite . in fact , inside this material one can observe some wire structures : the wires are strictly interconnected with each other , and have a mean diameter of a few tens of ngstrom . quantum confinement of charge carriers seems to be at the origin of the optical characteristics of the material , which are quite different from those of bulk silicon .    in the recent past many different theoretical studies  @xcite have been developed with the aim of analyzing the actual validity of the quantum confinement hypothesis in order to explain those new optical effects . according to the current experimental evidence , besides the presence of nanostructures of reduced dimensionality ( wires and/or dots ) , the key ingredients that are necessary to induce luminescence are crystallinity and surface passivation of dangling bonds . in the present work we concentrate on the quantum wire model proposed to explain the optical properties of leps . we use the _ ab initio _ molecular dynamics ( aimd ) method , as proposed by car and parrinello  @xcite , to study silicon quantum wires oriented along the ( 111 ) crystallographic direction . the aimd method has been successfully applied to a variety of different systems and problems  @xcite . porous silicon samples grown on a silicon substrate in the ( 111 ) lattice direction were characterized both in the morphology and in the optical properties  @xcite . the study presented here follows a similar work developed for quantum wires oriented along the ( 100 ) crystallographic direction  @xcite . this study showed the presence of a direct gap in the visible range and an important enhancement of the dipole matrix elements , which are directly related to the transition probability between the valence and the conduction band and , in consequence of that , to the absorption coefficient . this work , such as all the others quoted above , shows how the optical and electronic properties of silicon become deeply modified by quantum confinement . these predictions , with the limitations due to the local density approximation ( lda ) used in the aimd framework , support the thesis that the origin of porous silicon photoluminescence lies in the localization of charge carriers inside ordered structures of reduced dimensionality . the present work tries mainly to ascertain if the orientation may be relevant in determining the optical and electronic properties of the material since , _ a priori _ , it is not obvious that charge carriers confinement acts in a similar way along different symmetry planes . in section ii we describe the geometry of the model used and give some computational details . in section iii the results on the structural , vibrational and electronic properties of the model are discussed . the final section is devoted to the conclusions .    0.4truecm crystalline silicon is characterized by a diamond structure , in which atoms are distributed in a face centered cubic ( fcc ) lattice , each lattice site being occupied by a pair of particles whose chemical bond , about 2.35   long , lies in the ( 111 ) direction . the silicon lattice structure along the ( 111 ) direction can be characterized by three double layers periodically repeated with a periodicity @xmath1 , where @xmath2 is the lattice parameter of bulk crystalline silicon ( 5.43  ) . [ f : struc ] shows the smallest wire that one can build , still preserving the crystalline order along the ( 111 ) direction . the unit cell , indefinitely replied in space through periodic boundary conditions ( pbc ) , contains 26 silicon atoms , plus 30 hydrogen atoms which are necessary to saturate the dangling bonds at the surface of the wire . the geometry of the model is such that it exhibits a non uniform size along the longitudinal direction . in a way , this feature makes the model more similar to real samples of porous silicon . in fact , the diameter of the model has a maximum value of 7.67  , and a mean value of 5.51  . we have also studied a larger wire whose basic structure consists of 62 silicon atoms and 42 hydrogen atoms , with a maximum diameter of 11.72  , and a mean value of 10.37  . we note that a system consisting of 104 atoms is rather demanding on the computational side for _ ab - initio _ car - parrinello simulations . for this reason , we limited our calculations for the larger wire to those features which may confirm some specific properties of the thinner wire . in both cases the size of the supercell in the longitudinal direction corresponds exactly to the bulk silicon characteristic size of 9.41  along the ( 111 ) direction . the lateral size of the supercell is much larger than the diameter of the wire so as to avoid interactions of the particles with their images . the parameters needed in the aimd simulation were suggested by the previous experience on this material : we used a time step of 7 a.u . , an energy cutoff for the plane - wave expansion of 8 ry , an electronic mass of 300 a.u . , and pseudopotentials of the bachelet - hamann - schluter  @xcite type for describing the interaction between valence electrons and the nuclei with core electrons . as a first step , we wanted to verify the structural stability of the wire . to this aim , after the electronic ground - state relative to the initial ionic configuration was found , we let all the ionic degrees of freedom relax with a steepest descent technique . we notice a small increase of the distance of the most external atoms from the axis of the wire , of the order of 0.16  , analogously to what was observed in  @xcite . as expected , we observe no relevant variations of the characteristic distances , along the longitudinal direction . then , we let the system evolve dynamically for a time of about 2 ps , a considerable time in aimd , so that one may be confident that the system is reasonably close to thermal equilibrium . during this evolution the model did not exhibit any pathological behaviour : the system oscillates about the equilibrium configuration with no substantial atomic rearrangements . we observe that the total energy ( potential + kinetic ) fluctuates around an average value which remains constant , as should be in a microcanonical aimd simulation . the average temperature during this dynamical evolution is about 70 k.    we also computed the velocity autocorrelation function and , through its fourier transform , an indicative vibrational density of states ( vdos ) of the system . we find a good qualitative agreement with the typical experimental data of the vibrational modes of si - si and si - h bonds  @xcite . from a strictly quantitative point of view , we remark that the relative intensity of the peaks should not be taken too rigorously . however the positions of the peaks are in fair agreement with the experimental data . in particular we recognize in the spectra of fig . [ f : vibra ] the characteristic signal of the stretching of the si - h bond at about @xmath3 @xmath4 . the double peak has its origin in the presence of doubly hydrogenated silicon atoms , which give rise to symmetrical and antisymmetrical stretching of the bond at slightly different frequencies . the analysis of the oscillations of single and twin hydrogen particles shows that the higher - frequency peak can be ascribed to the symmetrical mode , while the lower - frequency signal arises from antisymmetrical stretching vibrations . we notice also , at lower frequencies , the typical signals of scissors mode ( @xmath5 @xmath4 ) of the twin hydrogen atoms bonded to one silicon atom , and of si - h bending ( around @xmath6 @xmath4 ) . the weak structure centered at about @xmath7 @xmath4 is likely associated with another bending vibrational mode of hydrogen atoms as it shows up in the motion of both single and twin passivating hydrogens . the low frequency noise produced by the finite time evolution renders more uncertain the identification of the modes below @xmath8 @xmath4 that are associated with the motion of silicon atoms . it is important to remark that such characteristic vibrational modes are observed , by infrared spectroscopy , also in real freshly prepared leps samples  @xcite .    for the larger size sample , we have also studied the relaxation towards the global energy minimum . it is noteworthy that in this case the mean radial relaxation , in the equilibrium configuration , is negligibly small if compared with that estimated for the thinner wire . this can be intuitively understood if one thinks that in the smaller - size model the ratio of surface to bulk atoms is definitely higher than in the larger wire . such a small relaxation originates from the competition between the doubly hydrogenated silicon atoms , which move slightly away from the axis of the wire , and the singly hydrogenated ones , which instead move inward .      once the minimum energy configuration has been determined , we computed the electronic band structure of the quantum wire along the one - dimensional brillouin zone ( bz ) . we computed the kohn - sham eigenvalues for six different k - points of the bz , from the center ( @xmath9 ) to the border ( which would correspond to l in the bulk ) . the resulting band structure is shown in fig . [ f : band ] for the ( 111)-oriented wire with a mean diameter of 5.51  . the main result is the prediction of a direct gap of 2.83 ev between the conduction and the valence band at the zone center . it is also interesting to notice that the second conduction state still shows a signature of the band structure of bulk silicon , i.e. a minimum located at about 0.2 . as to the gap , we recall that buda and co - workers found , for a ( 100)-oriented quantum wire with a diameter of 7.60  , a value of 2.33 ev  @xcite . the difference is to be ascribed , as we shall see later , to the different mean diameter of the two models , more than to the different symmetry of the wires . we recall that in bulk silicon the minimum of the conduction band along the @xmath9-l direction falls just at l , with a gap of 2.15 ev . for the larger wire we find an energy gap of 1.90 ev at the center of bz which , consistently with the quantum confinement hypothesis , is larger than that in bulk silicon , but smaller than the value for the thinner wire . the values of the two energy gaps , together with those of the ( 100 ) models and of bulk silicon as obtained in lda , are plotted in fig . [ f : egap ] . it is of striking evidence the linear behaviour of the energy gap as a function of the inverse size of the models . it is remarkable that both the ( 111 ) and ( 100 ) computed values lie on the same straight line , suggesting the independence of such property on the specific orientation of the wires . in the same figure we also show the theoretical prediction coming from effective mass arguments ( e@xmath10 ) , which should hold close to the bulk limit . it appears that the matching between these two different behaviours should occur in the region between 2 and 5 nm . the presence of a direct gap , and thus the possibility of zero - phonon transitions , is not a sufficient condition for ensuring good photoluminescence properties . the probability of an electronic transition as a result of the absorption of a photon is given by the fermi golden rule , which in the dipole approximation is @xmath11 where @xmath12 and @xmath13 are the conduction and valence states involved in the electronic transition , while * e * is the polarization vector and * p * the momentum operator . we determined all the matrix elements between kohn - sham states of the conduction and valence bands whose difference in energy falls in the visible range , or slightly above . table  [ t : small ] and table  [ t : big ] give the values of the most relevant of these dipole - matrix elements for the smaller and the larger ( 111)-oriented wires . we note the strong difference between the values relative to the longitudinal and transverse polarizations . moreover , the quantum confinement of the electrons mixes up the states , thus allowing for transitions that are forbidden in the bulk by selection rules . the dipole - matrix element between the top of the valence band and the bottom of the conduction band is not zero because of the mixing , even though not very large . we computed the imaginary part of the dielectric function , that is related to the absorption coefficient , through the formula @xmath14 where @xmath15 and @xmath16 are the length of the wire and the volume of the supercell , respectively . the results for the smaller model are shown in fig . [ f : eps2small ] . in the construction of this quantity we have considered only the dipole - matrix elements corresponding to energy differencies up to 4 ev . we observe that the component related to an external field parallel to the wire direction is much stronger than the perpendicular one , and this difference becomes larger and larger going to lower energies . probably , the most interesting result is the presence , in the parallel component , of a small signal at the band gap ( 2.83 ev ) which is completely absent in the perpendicular one . this is a clear effect of the mixing of the bulk states induced by the reduced dimensionality . this result seems to be peculiar of the ( 111 ) wire , since in the ( 100 ) models  @xcite the dipole - matrix element between the top of the valence band and the bottom of the conduction band was negligible in both the components .    the plot of the dielectric function for the larger - sized wire , that is shown in fig . [ f : eps2big ] , exhibits also a different behaviour in the parallel and the perpendicular components . we notice also in this case a clear difference in intensity between the parallel and the perpendicular components at higher energies . the most remarkable feature is the presence , below @xmath17 2.7 ev , of three peaks in the longitudinal component that are not obseved in the transverse one . in particular we observe again a significative signal in correspondence of the energy gap ( 1.90 ev ) , whose intensity is lowered with respect to the smaller wire . this is consistent with the fact that in the bulk limit this matrix element should go to zero .    0.4truecm we have presented an electronic structure calculation , within the lda , of two ( 111)-oriented silicon quantum wires of different size . the comparison of the present results with those obtained for the ( 001)-oriented models shows a substantial analogy of the electronic and optical properties of silicon quantum wires oriented along different crystallographic directions . in particular we find a direct energy gap at the zone center whose magnitude decreases for larger wires consistently with the quantum confinement hypothesis . quite surprisingly we find that the values of the gap for all the ( 111 ) and ( 001 ) wires studied are excellently fitted as a linear function of the inverse diameter . therefore this quantity seems to be independent on the orientation and geometrical details , and is affected only by the width of the wire . again , the present calculations support the idea that the photoluminescence properties of porous silicon can be ascribed to charge carriers confinement in nanostructures . we find , similarly to the ( 001 ) wires , a strong anisotropy in the imaginary part of the dielectric function . these results indicate that , if the leps structure consists of a network of randomly oriented wires  @xcite , all of them will contribute to the photoluminescence independently on their specific orientation . * acknowledgements * : this work was supported by the ministero delluniversita e della ricerca scientifica e tecnologica ( murst ) and by the istituto nazionale per la fisica della materia ( infm ) , italy . the authors thank the centro di calcolo elettronico delluniversita degli studi di messina for the allocation of computer time . a. m. s. would also like to acknowledge the hospitality of the ibm - ecsec centre in rome in the early stage of this work . very useful discussions with j. kohanoff , s. ossicini , and o. bisi are also acknowledged . @xmath18 & @xmath19 + 68 & 65 & 0.50@xmath20 & 0.11@xmath21 & 3.33 + 68 & 66 & 1.00@xmath20 & 0.14@xmath21 & 3.32 + 68 & 67 & 0.23@xmath22 & 0.86@xmath23 & 2.83 + 74 & 67 & 0.42@xmath20 & 0.17@xmath21 & 3.42 + 76 & 67 & 0.94@xmath24 & 0.45@xmath21 & 3.67 +       @xmath18 & @xmath19 + 147 & 138 & 0.17@xmath25 & 0.12@xmath26 & 2.84 + 148 & 139 & 0.18@xmath25 & 0.12@xmath26 & 2.84 + 146 & 145 & 0 & 0.41@xmath20 & 1.90 + 156 & 145 & 0.36@xmath22 & 0.14@xmath26 & 2.83 + 167 & 145 & 0.17@xmath25 & 0.57@xmath26 & 3.34 + 168 & 145 & 0.11@xmath24 & 0.23@xmath26 & 3.36 + 171 & 145 & 0.17@xmath26 & 0.11@xmath26 & 3.44 +
we analyze the influence of spatial orientation on the optical response of hydrogenated silicon quantum wires . 
 the results are relevant for the interpretation of the optical properties of light emitting porous silicon . 
 we study ( 111)-oriented wires and compare the present results with those previously obtained within the same theoretical framework for ( 001)-oriented wires [ f. buda , j. kohanoff and m. parrinello , _ phys . 
 rev . 
 lett . _ * 69 * , 1272 , ( 1992 ) ] . in analogy with the ( 001)-oriented wires and at variance with crystalline bulk silicon 
 , we find that the ( 111)-oriented wires exhibit a direct gap at @xmath0 whose value is largely enhanced with respect to that found in bulk silicon because of quantum confinement effects . the imaginary part of the dielectric function , for the external field polarized in the direction of the axis of the wires , shows features that , while being qualitatively similar to those observed for the ( 001 ) wires , are not present in the bulk . 
 the main conclusion which emerges from the present study is that , if wires a few nanometers large are present in the porous material , they are optically active independently of their specific orientation .    = 10000    0.4truecm