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the theory of quantum transport in nano - scale devices has evolved rapidly over the past decade , as advances in experimental techniques have made it possible to probe transport properties ( at different temperatures ) down to the single - molecule scale . furthermore simultaneous measurement of charge and heat transport through single molecules is now also possible @xcite . the development of accurate theoretical methods for the description of quantum transport at the single - molecule level is essential for continued progress in a number of areas including molecular electronics , spintronics , and thermoelectrics . one of the longstanding problems of quantum charge transport is the establishment of a theoretical framework which allows for quantitatively accurate predictions of conductance from first principles . the need for methods going beyond the standard approach based on density functional theory combined with landauer - like elastic scattering @xcite has been clear for a number of years . it is only recently that more advanced methods to treat electronic interaction have appeared , for example those based on the many - body @xmath0 approximation @xcite . alternative frameworks to deal with the steady - state or time - dependent transport are given by many - body perturbation theory based on the non - equilibrium ( ne ) green s function ( gf ) formalism : in these approaches , the interactions and ( initial ) correlations are taken into account by using conserving approximations for the many - body self - energy @xcite . other kinds of interactions , e.g. electron - vibron coupling , also play an important role in single - molecule quantum transport . inelastic tunneling spectroscopy constitutes an important basis for spectroscopy of molecular junctions , yielding insight into the vibrational modes and ultimately the atomic structure of the junction @xcite . there have been many theoretical investigations focusing on the effect of electron - vibron coupling in molecular and atomic scale wires @xcite . in all these studies , the interactions have always been considered to be present in the central region ( i.e. the molecule ) only , and the latter is connected to two non - interacting terminals . interactions are also assumed not to cross at the contracts between the central region and the leads . when electronic interactions are present throughout the system , as within density - functional theory calculations , they are treated at the mean - field level and do not allow for any inelastic scattering events . however , there are good reasons to believe that such approximations are only valid in a very limited number of practical cases . the interactions , in principle , exist throughout the entire system .    in a recent paper we derived a general expression for the current in nano - scale junctions with interaction present everywhere in the system @xcite . with such a formalism , we can calculate the transport properties in those systems where the interaction is present everywhere . the importance of extended interaction in nano - scale devices has also been addressed , for electron - electron interaction , in recently developed approaches such as refs .  [ ] .    in the present paper , we also consider interactions existing beyond the central region . we apply our recently developed formalism @xcite for fully interacting systems to a specific model of a single - molecule nanojunction . we focuss on a model system in the presence of electron - vibron interaction within the molecule and between the molecule and one of the leads . we show how the interaction crossing at one interface of the molecular nanojunctions affects the transport properties by renormalising the coupling at the interface in a bias - dependent manner . we also study the relationship between the non - equilibrium  charge susceptibility @xcite and the dynamical conductance for the present model of interaction crossing at the contacts . the paper is organised as follows : in sec . [ sec : transport ] , we briefly recall the main result of our current expression for fully interacting systems . in sec . [ sec : interac ] , we present the model hamiltonian for the system which include two kinds of electron - vibron interaction , an holstein - like hamiltonian combined with a su - schrieffer - heeger - like hamiltonian . in this section , we also describe how the corresponding self - energies are calculated and the implications of such approximations on the current expression at the left and right interfaces . in sec . [ sec : res ] , we show that our approximations are fully consistent with the constraint of current conservation . then the effects of the static non - equilibrium ( i.e. energy - independent but bias - dependent ) renormalisation of the coupling at the contact on both the current and the dynamical conductance are studied for a wide range of parameters . we also show that the ne charge susceptibility is still related to the dynamical conductance even in the presence of crossing interaction at the contact . we finally conclude and discuss extensions of the present work in sec . [ sec : ccl ] . we consider a two - terminal device , made of three regions left - central - right , in the steady - state regime . in such a device , labelled @xmath1 , the interaction  which we specifically leave undefined ( e.g. electron - electron or electron - phonon)is assumed to be well described in terms of the single - particle self - energy @xmath2 and spreads over the entire system .    we use a compact notation for the green s function @xmath3 and the self - energy @xmath4 matrix elements @xmath5 . they are annotated @xmath6 ( @xmath7 or @xmath8 ) for the elements in the central region @xmath9 ( left @xmath10 , right @xmath11 region respectively ) , and @xmath12 ( or @xmath13 ) and @xmath14 ( or @xmath15 ) for the elements between region @xmath9 and region @xmath10 or @xmath11 . there are no direct interactions between the two electrodes , i.e. @xmath16 .    in refs . [ ] , we showed that for a finite applied bias @xmath17 the steady - state current @xmath18 flowing through the left @xmath19 interface is given by : @xmath20 \\ + & { { \rm tr}}_{\{l\}}\left[\sigma^{{{\rm mb}},>}_l g^<_l - \sigma^{{{\rm mb}},<}_l g^>_l \right ] \end{split}\ ] ] where the @xmath21 quantities are @xmath22 by definition @xmath23 ( similarly for the @xmath24 components ) where @xmath25 are the nominal coupling matrix elements between the @xmath10 and @xmath9 regions . @xmath26 are the gf of the region @xmath10 renormalised by the interaction _ inside _ that region , where @xmath27 stands for the retarded , advanced and lesser gf components . for example , for the advanded and retarded components , we have @xmath28 where all quantities are defined only in the subspace @xmath10 .    the first line in the current equation eq . ( [ eq : ilfinal ] ) corresponds to a generalisation of the meir and wingreen @xcite result to the cases for which the interactions are present in the three @xmath29 regions as well as in between the @xmath30 and @xmath31 regions . the second trace in eq . ( [ eq : ilfinal ] ) corresponds to inelastic events induced by the interaction in the @xmath10 lead . when a local detailed balance equation helds , this terms vanishes since locally one has @xmath32 . ( [ eq : ilfinal ] ) bears some resemblance to the expression derived by meir and wingreen @xcite when written as : @xmath33 . \end{split}\ ] ] where we use the definitions @xmath34 and @xmath35 . hence @xmath36 becomes    hence one can see by comparing eq . ( [ eq : ilfinal ] ) and eq . ( [ eq : il_meirwingreen_bis ] ) that the quantities @xmath37 ( @xmath38 ) and @xmath39 are playing the role of the @xmath10 lead self - energy @xmath40 ( @xmath41 ) and @xmath42 respectively when the interactions cross at the @xmath19 interface . in meir and wingreen model , the leads are non - interacting , hence the second trace @xmath43 $ ] in eq . ( [ eq : ilfinal ] ) does not exist . we consider a single - molecule junction in the presence of electron - vibron interaction inside the central region and crossing at the contacts . using a model system to reduce these calculations to a tractable size , we concentrate on a single molecular level coupled to a single vibrational mode . a full description of our methodology , for the interaction inside the region @xmath9 , is provided in refs . [ ] . furthermore , we consider that the electron - vibron interaction exist also at one contact ( the left @xmath10 electrode for instance ) . this model typically corresponds to an experiment for a molecule chemisorbed onto a surface ( the left electrode ) with a tunneling barrier to the right @xmath11 lead .    in the following model , we consider two kinds of electron - vibron coupling : a local coupling in the sense of an holstein - like coupling of the electron charge density with a internal degree of freedom of vibration inside the central region , and an off - diagonal coupling in the sense of a su - schrieffer - heeger - like coupling @xcite to another vibration mode involving the hopping of an electron between the central @xmath9 region and the @xmath10 electrode . the hamiltonian for the region @xmath9 is @xmath44 where @xmath45 ( @xmath46 ) creates ( annihilates ) an electron in the molecular level @xmath47 . the electron charge density in the molecular level is coupled to the vibration mode of energy @xmath48 via the coupling constant @xmath49 , and @xmath50 ( @xmath51 ) creates ( annihilates ) a vibration quantum in the vibron mode @xmath48 . the central region @xmath9 is nominally connected to two ( left and right ) one - dimensional tight - binding chains via the hopping integral @xmath52 and @xmath53 . the corresponding electrode @xmath54 self - energy is @xmath55 with the dispersion relation @xmath56 where @xmath57 and @xmath58 are the tight - binding on - site and off - diagonal elements of the electrode chains .    to describe the electron - vibron interaction existing at the left contact , we consider that the hopping integral @xmath52 is actually dependent on some generalised coordinate @xmath59 . the latter represents either the displacement of the centre - of - mass of the molecule or of some chemical group at the end of the molecule link to the @xmath10 electrode . at the lowest order , the matrix element can be linearised as @xmath60 . hence the hopping of an electron from the @xmath9 region to the @xmath10 region ( and _ vice versa _ ) is coupled to a vibration mode ( of energy @xmath61 ) via the coupling constant @xmath62 ( itself related to @xmath63 ) . the corresponding hamiltonian is given by @xmath64 where @xmath65 ( @xmath66 ) creates ( annihilates ) a vibration quantum in the vibron mode @xmath61 , the generalised coordinate is @xmath67 , and @xmath68 ( @xmath69 ) creates ( annihilates ) an electron in the level @xmath70 of the @xmath10 electrode .    the hamiltonians eq .  ( [ eq : h_central ] ) and eq .  ( [ eq : vcrossing ] ) are used to calculate the corresponding electron self - energies at different orders of the interaction @xmath49 and @xmath62 using conventional non - equilibrium  diagrammatics techniques @xcite . furthermore , at equilibrium , the whole system has a single and well - defined fermi level @xmath71 . a finite bias @xmath17 , applied across the junction , lifts the fermi levels as @xmath72 . the fraction of potential drop @xcite at the left contact is @xmath73 and @xmath74 at the right contact , with @xmath75 and @xmath76 $ ] . the electron - vibron self - energies  in the central region @xmath9 are calculated within the born approximation . the details of the calculations are reported elsewhere @xcite so we briefly recall the different expressions for the self - energies   @xmath77 with @xmath78 with @xmath79 and @xmath80 and @xmath81 with the usual definitions for the bare vibron gf @xmath82 : @xmath83 \\ d_0^r(\omega ) & = \frac{1}{\omega - \omega_0 + i 0^+ }       - \frac{1}{\omega + \omega_0 + i 0^+ } \ ,     \end{split}\ ] ] where @xmath84 is the averaged number of excitations in the vibration mode of frequency @xmath48 given by the bose - einstein distribution at temperature @xmath85 . in the following , we work in the limit of low temperature for which @xmath86 .    as a first application of our transport formalism for crossing interactions , we consider a mean - field approximation for the electron - vibron coupling at the @xmath19 interface . this leads to the hartree - like expressions for the many - body self - energies at the @xmath19 interface : @xmath87 where @xmath88 similarly the self - energy @xmath89 is obtained from @xmath90 . one can see that the interaction crossing at the @xmath19 interface induces a static ( however bias - dependent ) renormalisation of the nominal coupling @xmath91 between the @xmath10 and @xmath9 regions . this non - equilibrium  renormalisation will induce , amongst other effects , a bias - dependent modification of the broadening of the spectral features of the @xmath9 region . since the renormalisation is static at the hartree level , we use a `` potential '' notation to represent the normalised coupling : @xmath92 with @xmath93 and similarly for @xmath94 . the static renormalisation of the nominal coupling @xmath52 is driven by the ratio @xmath95 . in the following numerical applications , we consider small to larger renormalisation effects , for which the ratio @xmath96 is ranging from @xmath97 to @xmath98 . one should note , however , that in all the calculations we have performed , the density matrix element @xmath99 is always of the order of @xmath100 . therefore the renormalisation effects are always smaller than the nominal coupling @xmath52 itself .    in order to get the renormalised couplings @xmath101 , we need the off - diagonal elements @xmath102 and @xmath103 . the closed expression for the @xmath102 gf matrix element is obtained from the corresponding dyson equation @xmath104^<_{lc}$ ] . after formal manipulation , we find that @xmath105 with @xmath106 is the renormalised advanced gf of the central region @xmath107^{-1 } -\sigma^a_c - v_{cr } g^a_r v_{rc } - \tilde v_{cl } g^a_l \tilde v_{lc } \right]^{-1 } , \ ] ] and the off - diagonal element @xmath108 is given by @xmath109^{-1 } - v_{cr } g^r_r v_{rc } \right]^{-1 } , \ ] ] where @xmath110 is the renormalised gf of the disconnected c region @xmath111^{-1}= [ g^r_c]^{-1 } - \sigma^r_c$ ] . after further manipulation , we get the following expression for @xmath108 by using the notation @xmath112 for the non - interacting @xmath11 lead self - energy : @xmath113^{-1 } -   \tilde v_{lc } \left [ [ \tilde g^r_c]^{-1 } - y_c^{r , r } \right]^{-1 } \tilde v_{cl } \right]^{-1 }   \tilde v_{lc } \left [ [ \tilde g^r_c]^{-1 } - y_c^{r , r } \right]^{-1 } .\ ] ]    similarily we can calculate the off - diagonal element @xmath103 from the corresponding dyson equations @xmath114^<_{cl}$ ] . we find @xmath115 where @xmath116 is the retarded version of eq . ( [ eq : gac ] ) and @xmath117^{-1 } - y_c^{r , a } \right]^{-1 }   \tilde v_{cl } g^a_l   .\ ] ]    as expected from the definition of the different gfs , we can see that indeed @xmath118 and @xmath119 . calculations are performed in a self - consistent manner . there are different ways to calculate the gf and the self - energies  in a self - consistent way : in the present work , we first perform self - consistent calculations for the central region @xmath9 . the new hopping matrix elements between the @xmath9 and @xmath10 regions given by eq . ( [ eq : vlc_renorm ] ) are then updated via the use of eqs . ( [ eq : glessercl ] ) and ( [ eq : glesserlc ] ) . the calculations are then re - iterated until full self - consistency for all self - energies and gf in the region @xmath9 and at the left contact @xmath120 is obtained .    for the model of crossing interaction we are considering here , further simplications can be introduced in the calculation of the current . since there is no other interaction inside the @xmath10 and @xmath11 regions , the current expression is given only by the first line of eq.([eq : ilfinal ] ) . furthermore , there are no lesser and greater components for the self - energy @xmath121 at the mean - field level . the current expression eq . ( [ eq : ilfinal ] ) reduces to : @xmath122 \ , \end{split}\ ] ] with the following simplified expressions for the @xmath123 quantities @xmath124 since @xmath125 and @xmath126 . furthermore , the current at the @xmath127 interface has a conventional meir and wingreen expression , since there are no crossing interactions : @xmath128 \ , \end{split}\ ] ] where we use the conventional definition for the @xmath11 lead self - energies : @xmath129 with @xmath130 . the main differences between the expression for the left and right currents @xmath131 and @xmath132 arise from the non - equilibrium static renormalisation of the coupling between the central region @xmath9 and the @xmath10 lead given in eq . ( [ eq : vlc_renorm ] ) . ( black line with dots ) and @xmath133 ( red line ) given respectively by eq . ( [ eq : il_hartree ] ) and eq . ( [ eq : ir_meirwingreen ] ) . the current is fully conserved as can be seen by the green dashed line which shows @xmath134 magnified by a factor of 2000 . the parameters are @xmath135 , @xmath136 , @xmath137 , @xmath138 , @xmath139 , @xmath140 , @xmath141.,width=272 ]    . current @xmath142 ( left panel ) , dynamical conductance @xmath143 in logarithmic scale ( top - right panel ) and the real part of non - equilibrium renormalized static coupling @xmath144 ( bottom - right panel ) for different values of the coupling @xmath62 strength at the left interface between @xmath145 and @xmath146 , corresponding to the follow ratio : @xmath147 . the ne renormalization of @xmath148 corresponds to a non - monotonic reduction of the nominal hopping integral @xmath52 with applied bias , which induces a diminution of the current , and modifications of the widths of the conductance peaks accompanied with a slight shift of the conductance peaks at high bias . see text for a complete discussion . the other parameters are @xmath136 , @xmath137 , @xmath138 , @xmath149 , @xmath141.,width=272 ] we have perfomed calculations for many different values of the parameters in the hamiltonians . we present below the most characteristic results for different transport regimes and for different coupling strengths @xmath62 , while the interaction in the region @xmath9 is taken to be in the intermediate coupling regime @xmath150 . the nominal couplings between the central region and the electrodes @xmath151 , before ne renormalisation , are not too large , so that we can discrimate clearly between the different vibron side - band peaks in the spectral functions . the values chosen for the parameters are typical values for realistic molecular junctions @xcite . the different transport regimes considered bellow are called the far off - resonant , the off - resonant , the intermediate regime and the quasi - resonant regime . they correspond to different position of the molecular level @xmath47 with respect to the fermi level @xmath71 at equilibrium . with the static renormalised molecular level @xmath152 , the far off - resonant regime corresponding to @xmath153 and @xmath154 several @xmath48 , the off - resonant regime corresponds to @xmath153 and @xmath155 one or two @xmath48 , the intermediate regime corresponds to @xmath156 and the quasi - resonant regime corresponds to @xmath157 .    in the following the current is given in units of charge per time , the conductance in unit of quantum of conductance @xmath158 and the bias @xmath17 and the normalised coupling @xmath94 in natural units of energy where @xmath159 and @xmath160 . one of the most important physical conditions that our formalism needs to fulfil is the constraint of current conservation . we use conserving approximations to calculate the interaction self - energies in the central region @xmath9 and for the crossing interaction at the left interface . however , since there is no interaction crossing at the @xmath127 interface while the interaction _ is _ crossing at the @xmath19 interface , we have to check that the current given by eq . ( [ eq : il_hartree ] ) for @xmath18 is equal to the meir and wingreen current given by eq . ( [ eq : ir_meirwingreen ] ) for @xmath133 , i.e. @xmath161 . figure [ fig : ex_iconserv ] shows that the condition of current conservation is indeed fulfilled , as expected . we have carefully checked that the current is conserved for all the calculations presented in the present paper , i.e. that @xmath162 .      in figures [ fig : eps090 ] to [ fig : eps020 ] , we show the dependence of the current @xmath142 , of the dynamical conductance @xmath143 and of the renormalised coupling @xmath148 on the applied bias @xmath17 , for different values of the interaction strength @xmath62 at the left contact . we consider different transport regimes ranging from the far off - resonant regime ( fig . [ fig : eps090 ] ) , the off - resonant regime ( fig .  [ fig : eps050 ] ) , the intermediate regime ( fig .  [ fig : eps030 ] ) and finally the quasi - resonant regime ( fig .  [ fig : eps020 ] ) .    the ne renormalisation of the coupling at the left contact given by eq . ( [ eq : vlc_renorm ] ) corresponds to an effective decrease of the hopping integral leading to a decrease of the current for increasing values of @xmath62 , as can be seen in the left panels of figs . [ fig : eps090 ] to [ fig : eps020 ] . the real part of @xmath148 shows a non - monotonic behaviour with the applied bias and presents features ( local dips ) at applied biases corresponding to peaks in the dynamical conductance ( bottom right panels of figs . [ fig : eps090 ] to [ fig : eps020 ] ) . therefore , the interaction crossing at the left contact not only decreases the value of the current but also affects the width of the peaks in the conductance , as can be seen in the top right panels of figs . [ fig : eps090 ] to [ fig : eps020 ] .    in all our calculations , it appears that each conductance peak now has an asymmetric shape , i.e. a different broadening on each side of the peak , which is due to the non - monotonic and asymmetric behaviour of @xmath94 versus applied bias . the detailed understanding of such a behaviour is rather complex . however , one can obtain a qualitative understanding of our results by considering the following analysis .    . the other parameters are the same as in figure [ fig : eps090 ] . see text for a complete discussion.,width=272 ]    . the other parameters are the same as in figure [ fig : eps090 ] . see text for a complete discussion.,width=272 ]    . same parameters as in figure [ fig : eps090 ] . see text for a complete discussion.,width=272 ]    we can use the current expressions for @xmath131 and @xmath163 given by eq .  ( [ eq : il_hartree ] ) and eq .  ( [ eq : ir_meirwingreen ] ) respectively to obtain a symmetrized current @xmath164 expression within a good approximation @xcite : @xmath165 where @xmath166 and @xmath167^{-1 } \ , \ ] ] with the self - energy in the central region @xmath168 given by the sum of eq . ( [ eq : sigmac_hartree ] ) and eq . ( [ eq : sigmacr_fock ] ) ; and we recall that @xmath169    in the off - resonant regime and for small bias @xmath170 , one can consider that @xmath171 ( see ref .  [ ] ) , hence @xmath172    we can now use eq . ( [ eq : isym_bis ] ) to understand the behaviour of the current . in figure [ fig : parametrici_vcl ] we show the parametric curves @xmath173 obtained for the far off - resonant transport regime ( shown in fig . [ fig : eps090 ] ) . figure [ fig : parametrici_vcl ] shows a complex dependence of the current versus the ne renormalised coupling @xmath94 . however at low bias , one can consider that @xmath174 and @xmath175 are independent of @xmath176 and hence take such quantities out of the integral in eq . ( [ eq : isym_bis ] ) . therefore we have @xmath177 and the current is quadratic in @xmath94 since @xmath178 . such a dependence can be clearly seen in figure [ fig : parametrici_vcl ] for the low bias regime where @xmath179 .    at larger bias , one can no longer neglect the @xmath176 dependence of @xmath174 ( and @xmath175 ) and more importantly the effects of the interaction in the central region , i.e. @xmath180 . hence the quadratic dependence of @xmath181 on @xmath182 is lost .     for the far off - resonant transport regime shown in fig . [ fig : eps090 ] ( using the same legend for the different @xmath62 values ) . the starting points of the parametric curves , i.e. @xmath183 , are at the bottom of the graph ( zero current values ) , and increasing @xmath17 corresponds to increasing values of the current . at low bias ( @xmath179 ) , @xmath131 is quadratic in @xmath94.,width=272 ]    we now turn onto the effect of the strength of the nominal hopping integral @xmath52 on the renormalised coupling @xmath148 at fixed values of the crossing interaction strength @xmath62 . figure [ fig : vcl_varyt0l ] shows the relative dependence of @xmath148 on the nominal coupling @xmath52 at the left interface versus applied bias . the figure shows that there is no simple relationship between @xmath148 and @xmath52 for the whole range of parameters explored . there is a small linear regime at low applied bias , otherwise the dependence of @xmath94 on both @xmath184 and @xmath17 is highly non - linear . there is a progressive washing - out of the features in @xmath148 for increasing values of @xmath184 , since a general increase of the coupling to the leads generates to a global broadening of the features in the spectral functions .      in a recent paper @xcite , we have developed the concept of the generalized susceptibilities for nonlinear systems @xcite and applied it to the charge transport properties in two - terminal nano - devices . we have introduced the non - equilibrium   charge susceptibility @xmath185 concept , defined by @xmath186 . we have shown that @xmath185 is related to the dynamical conductance @xmath187 . the relationship is formally different than the one obtained at equilibrium . in spectroscopic terms , both @xmath185 and @xmath187 contain features versus applied bias when charge fluctuation occurs in the corresponding electronic resonance . this relationship has been demonstrated for model calculations of interacting nanoscale devices but only when the interaction is present in the central region @xcite . we now check the validity of such a relationship between @xmath185 and @xmath187 when interaction crossing at the left contact is also taken into account . figure [ fig : nechigv ] shows the results we obtain for @xmath185 and @xmath187 for different sets of parameters corresponding to the off - resonant and quasi - resonant transport regimes . once more , we find that @xmath185 and @xmath187 present features at the same applied bias . such a behaviour validates the relationship between the non - equilibrium charge susceptibility and the dynamical conductance even when interactions cross at the contacts ( at least at the static mean - field level ) .     for the off - resonant transport regime ( @xmath188 ) and @xmath189 versus applied bias and for different values of the nominal hopping integrals @xmath184 ( @xmath54 ) . the relative dependence of @xmath94 , shown here for @xmath190 , is linear only for small bias . otherwise the dependence of @xmath94 on both @xmath184 and @xmath17 is highly non - linear . the other parameters are the same as in fig . [ fig : eps050].,width=272 ]     and dynamical conductance @xmath187 versus applied bias for different sets of parameters . @xmath185 is rescaled by a factor of 7 . ( a ) off - resonant regime @xmath191 and @xmath192 . ( b ) off - resonant regime @xmath191 and @xmath189 , ( c ) quasi - resonant regime @xmath193 and @xmath194 . @xmath185 and @xmath187 present features at the same applied bias . note the negative contribution to @xmath195 around @xmath196 . see main text for a complete discussion . the other parameters are the same as in fig . [ fig : eps050].,width=272 ]    finally , one should note the negative contribution to @xmath185 in panel ( c ) of figure [ fig : nechigv ] . such a behaviour originates from the properties of @xmath195 and not from the approximation used to calculate the crossing interaction @xcite . in fact , there are always two contributions to @xmath195 , one is positive and corresponds to electron fluctuation and the other is negative and corresponds to hole fluctuation . by electron ( hole ) fluctuations , we mean the variation of the occupancy ( versus applied bias ) of electronic resonances located nominally above ( below ) the fermi level at equilibrium ( or at small bias ) . therefore , for the off - resonant transport regime with @xmath197 , the features in @xmath185 corresponds to positive peaks ( as shown in panels ( a ) and ( b ) in figure [ fig : nechigv ] and in the different figures of ref . [ ] ) . for the off - resonant regime with @xmath198 , one would get negative peaks in @xmath185 . for the intermediate and quasi - resonant transport regime , one obtains both positive and negative contributions in @xmath185 as shown in panel ( c ) of figure [ fig : nechigv ] . the most peculiar case corresponds to a fully electron - hole symmetric system , i.e. the spectral function @xmath199 is fully symmetric around the equilibrium fermi level and for any applied bias @xmath17 with @xmath200 . in that case , one can easily show that @xmath201 is actually given by @xmath202 for symmetry reasons . hence @xmath203 is independent of the applied bias for conserving approximations for the self - energies , and consequently @xmath204 . such a behaviour can also be interpreted with the previous picture : for a fully electron - hole symmetric system , any contributions from electron fluctuation is exactly cancelled out by the opposite contribution from hole fluctuation , and @xmath185 is flat and equal to zero for each bias @xcite . we have studied the transport properties through a two - terminal nanoscale device with interactions present not only in the central region but also with interaction crossing at the interface between the left lead and the central region . to calculate the current for such an fully interacting system , we have used our recently developed quantum transport formula @xcite based on the negf formalism . as a first practical application , we have considered a prototypical single - molecule nanojunction with electron - vibron interaction . in terms of the electron density matrix , the interaction is diagonal in the central region for the first vibron mode and off - diagonal between the central region and the left electrode for the second vibron mode . the interaction self - energies are calculated in a self - consistent manner using the lowest order hartree - fock - like diagram in the central region and the hartree - like diagram for the crossing interaction . our calculations were performed for different transport regimes ranging from the far off - resonance to the quasi - resonant regime , and for a wide range of parameter values . they shown that , for this model , we obtain a non - equilibrium ( i.e. bias dependent ) static ( i.e. energy independent ) renormalisation of the nominal hopping matrix element between the left electrode and the central region . the renormalisation is such that the amplitude of the current is reduced in comparison with the current values obtained when the interaction is only present in the central region . such a result could provide an partial explanation for the fact in conventional density - functional based calculations , the values of the current are always much larger than in the corresponding experiments , since no non - equilibrium renormalisation of the contacts is taken into account in those calculations . however , even though it provides the right trends , the decrease in the current obtained by ne renormalisation of the coupling to the leads is not as important as the effects obtained from a proper calculation of the band - gap and band - alignment in realistic molecular system @xcite . the ne static renormalisation of the contact is highly non - linear and non - monotonic in function of the applied bias , and the larger effects occur at applied bias corresponding to resonance peaks in the dynamical conductance . the conductance is also affected by the ne renormalisation of the contact , showing asymmetric broadening around the resonance peaks and some slight displacement of the peaks at large bias in function of the coupling strengh @xmath62 . furthermore , we have also shown that , even in the presence of crossing interactions , the relationship between the ne charge susceptibility and dynamical conductance @xcite still holds for the different transport regimes considered here . extensions of the present study are now considered . one route is to develop more accurate ne renormalisation by considering , for example , a quasi - particle approach within a dynamical mean - field - like treatement of the crossing interaction self - energy @xcite . another route is to go beyond the hartree approximation for the crossing interaction self - energy using other many - body diagrams @xcite . this would lead to dynamical ne renormalisation of the contact involving inelastic scattering processes . finally , it should be noted that for the model system we considered here , one could also solve the problem by using more standard approaches ( typically the original meir and wingreen approach ) by extending the size of the central region to include all the interaction . the results then obtained should be strictly equivalent to our calculations . we have already commented in detail on this point ( in a formal and theoretical point of view ) in refs . [ ] . however our approach offers a more intuitive physical interpretation of the results , i.e. renormalisation of the contact ( in the present case in a static ne mean - field scheme ) . using an extended central region will not provide an easy physical interpretation of the results ; and as a matter of principle , it will not always be possible to increase at will the size of the central region , more especially when one considers future application of the method to much large molecular system with much more complex coupling to the leads .
we present an application of a new formalism to treat the quantum transport properties of fully interacting nanoscale junctions [ phys . 
 rev . 
 b * 84 * , 235428 ( 2011 ) ] . 
 we consider a model single - molecule nanojunction in the presence of two kinds of electron - vibron interactions . in terms of electron density matrix , 
 one interaction is diagonal in the central region and the second is off - diagonal in between the central region and the left electrode . 
 we use a non - equilibrium green s function technique to calculate the system s properties in a self - consistent manner . 
 the interaction self - energies are calculated at the hartree - fock level in the central region and at the hartree level for the crossing interaction . 
 our calculations are performed for different transport regimes ranging from the far off - resonance to the quasi - resonant regime , and for a wide range of parameters . 
 they show that a non - equilibrium ( i.e. bias dependent ) static ( i.e. energy independent ) renormalisation is obtained for the nominal hopping matrix element between the left electrode and the central region . 
 such a renormalisation is highly non - linear and non - monotonic with the applied bias , however it always lead to a reduction of the current , and also affects the resonances in the conductance . 
 furthermore , we show that the relationship between the non - equilibrium charge susceptibility and dynamical conductance still holds even in the presence of crossing interaction .