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this is the fifth in a series of papers@xcite investigating the properties of strongly correlating liquids,@xcite i.e. , liquids that have strong correlations between their constant - volume equilibrium fluctuations of potential energy , @xmath4 , and virial@xcite @xmath5 where @xmath6 is the position of particle @xmath7 at time @xmath8 . as is well known , the average virial @xmath0 gives the configurational contribution to the pressure : @xmath9 letting @xmath10 denote instantaneous deviations from equilibrium mean values , the @xmath3 correlation is quantified by the correlation coefficient @xmath11 ( with @xmath12 denoting equilibrium average ) : @xmath13 perfect correlation gives @xmath14 . as a pragmatic definition we have chosen `` strongly correlating liquids '' to designate liquids that have @xmath15 in the @xmath2 ensemble ( constant volume , @xmath16 , and temperature , @xmath17 ) . strongly correlating liquids have simpler physics than liquids in general , an observation that has particular significance for the highly viscous phase.@xcite thus it has been shown that strongly correlating viscous liquids to a good approximation have all eight frequency - dependent thermoviscoelastic response functions@xcite given in terms of just one@xcite ( i.e. , are single - parameter liquids in the sense of having dynamic prigogine - defay ratio@xcite close to unity@xcite ) . strongly correlating viscous liquids moreover obey density scaling@xcite to a good approximation , i.e. , their dimensionless relaxation time @xmath18 ( where @xmath19 is the average particle mass ) depends on density @xmath20 and temperature as @xmath21.@xcite    paper i@xcite presented computer simulations of 13 different systems , showing that van der waals type liquids are strongly correlating , whereas hydrogen - bonding liquids like methanol or water are not . strongly correlating liquids include@xcite , for instance , the standard lennard - jones ( lj ) liquid , the kob - andersen binary lj ( kablj ) mixture , an asymmetric rigid - bond dumbbell model , a seven - site united - atom toluene model , and the lewis - wahnstrm otp model . paper ii@xcite analyzed the cause of @xmath3 correlations with a focus on the lj potential . the strong correlations were related to the well - known fact that an inverse power - law ( ipl ) pair potential , @xmath22 where @xmath23 is the distance between two particles , @xcite implies perfect @xmath3 correlation@xcite ,    @xmath24    with @xmath25 . around the potential energy minimum , the lj potential is well described by an `` extended '' inverse power - law potential ( eipl),@xcite @xmath26 . at constant volume the linear term contributes little to the virial and potential - energy fluctuations : when one nearest - neighbor interatomic distance increases , others decrease in such a way that the sum is almost constant . thus systems interacting via the lj potential inherit strong @xmath3 correlations from an underlying inverse power - law - they have a `` hidden scale invariance''.@xcite    paper iii@xcite gave further numerical evidence for the explanation for strong @xmath3 correlations presented in paper ii , and theoretical results were given on the statistical mechanics and thermodynamics of the hidden scale invariance that characterizes strongly correlating liquids . it was also shown that strong virial - potential energy correlations are present even in out - of - equilibrium situations - the hidden scale invariance is a property of the potential energy surface , not just of the equilibrium states . paper iv@xcite introduced the concept of `` isomorphs '' in the phase diagram of a strongly correlating liquid . starting from a single assumption a number of isomorph invariants were derived . in particular , structure and dynamics were shown to be invariant on isomorphs when reduced units are used .    in the present paper further simulation results supporting the isomorph predictions are presented using systems interacting with the multicomponent generalized lj potential : @xmath27 where @xmath28 is an ipl potential acting between the two particles @xmath7 and @xmath29 : @xmath30 for systems interacting via a generalized lj potential , a prediction for the shape of the isomorphs in the @xmath3 phase diagram is derived in section iii and demonstrated to fit well to simulation results.@xcite interestingly , the isomorph shape depends only on the exponents @xmath19 and @xmath31 . thus , e.g. , all 12 - 6 lj systems have the same isomorphs in the @xmath3 phase diagram . finally we briefly present two applications of the theory . one tests the paper iv prediction that solid - liquid coexistence lines are isomorphs . the second application gives an approximate equation of state for systems interacting via generalized lj potentials ; this is arrived at by combining the present theory with rosenfeld and tarazona s expression for the isochoric temperature dependence of the potential energy . we term a microscopic configuration `` physically relevant '' if its influence on the thermodynamics and dynamics of the sustem is not _ a priori _ negligible ( paper iv ) . for instance , any configuration with very strong particle overlap is physically irrelevant ; note , however , that even unlikely configurations like transition states are relevant . two state points ( 1 ) and ( 2 ) with temperatures @xmath32 and @xmath33 and densities @xmath34 and @xmath35 , respectively , are defined to be _ isomorphic _ ( paper iv ) if they obey the following : any two physically relevant configurations of state points ( 1 ) and ( 2 ) , @xmath36 and @xmath37 , which trivially scale into one another , @xmath38 have proportional configurational boltzmann statistical weights : @xmath39 here @xmath40 is the potential energy function and it is understood that the constant @xmath41 depends only on the state points ( 1 ) and ( 2 ) . the property of being isomorphic defines a mathematical equivalence relation on the set of state points . the corresponding equivalence classes are smooth curves in the phase diagram termed isomorphs or isomorphic curves . equation ( [ eq : isomorphdef ] ) implies identity of the normalized boltzmann probabilities for scaled relevant configurations of isomorphic state points . as detailed in paper iv this identity implies that several quantities are invariant along an isomorph . examples include the configurational entropy , the isochoric specific heat , n - particle distribution functions in reduced units ( in particular , the radial distribution function(s ) ) , reduced - unit dynamics ( both for newtonian and stochastic dynamics ) , normalized reduced - time autocorrelation functions , reduced - unit transport coefficients , etc . it was further shown in paper iv that a jump from an equilibrium state to an isomorphic state point brings the system instantaneously to equilibrium ( see also ref . ) . ipl potentials are euler homogeneous functions , i.e. , obey @xmath42 . such purely repulsive potentials  which do not describe real intermolecular interactions  are the only potentials that have 100% correlation between virial and potential energy fluctuations ; this reflects the well - known ipl virial - theorem identity @xmath43 . likewise , only ipl liquids have exact isomorphs ; they satisfy eq . ( [ eq : isomorphdef ] ) with @xmath44 for all configurations of two state points obeying @xmath45 where @xmath25 . appendix a of paper iv showed that strongly correlating liquids generally have isomorphs to a good approximation . consequently , to a good approximation these liquids inherit a number of the exact invariants that ipl liquids have along their curves given by @xmath46 , namely all the ipl invariants that do not rely on the ipl identity @xmath44 . examples include the above mentioned thermodynamic , static , and dynamic isomorph invariants . ipl invariants that are not inherited by strongly correlating liquids in general include , e.g. , the helmholtz free energy over temperature , the potential energy over temperature , and the reduced - unit compressibility .    the fact that strongly correlating liquids have isomorphs to a good approximation should be understood as follows . any liquid has curves in its phase diagram of constant excess entropy , curves of constant isochoric specific heat , curves of constant reduced relaxation time , curves of constant reduced - unit transport coefficients , etc . for a strongly correlating liquid these curves are almost identical , and they define its ( approximate ) isomorphs . points on these isomorphs have approximately identical structure and approximately identical dynamics as probed , e.g. , by normalized reduced - time - autocorrelation functions .    for clarity of presentation below we shall not repeatedly mention that the existence of isomorphs is not exact for generalized lj systems . accordingly , we shall speak of isomorphs as a unique , well - defined concept . in other words : all properties of isomorphs derived below in generalized lj systems would be exact if these systems did have exact isomorphs , e.g. , fulfilled eq . ( [ eq : isomorphdef ] ) ; since this is not the case , the properties are in principle  as well as practice  approximate . the structure in reduced units ( @xmath47 ) is predicted to be invariant along an isomorph ( paper iv ) . the excess entropy , @xmath48 , depends only on structure and was shown in paper iv also to be invariant along an isomorph ( here and henceforth `` excess '' refers to the quantity in question in excess to that of an ideal gas with same density and temperature ) . in the below simulations we generate an isomorph as a set of state points in the phase diagram with constant @xmath49 . to change density and temperature keeping @xmath49 constant , the following identity is used ( paper iv ) : @xmath50 where @xmath51 and a maxwell relation was applied . utilizing the standard fluctuation formulae this leads ( paper iv ) to :    @xmath52    the procedure applied to generate isomorphs numerically can be summarized as : 1 ) an equilibrium @xmath2 simulation was performed at a given state point ; 2 ) @xmath53 was calculated from the fluctuations using eq . ( [ eq : gammasex ] ) ; 3 ) the density @xmath54 was changed slightly ( of order 1% ) , and @xmath53 was used to calculate the corresponding change in temperature in order to keep @xmath55 constant ( eq . ( [ eq : gammasex ] ) ) ; 4 ) a simulation at the new state point was performed , and the procedure was repeated . figure  [ fig : structure](a ) demonstrates the isomorph invariance of the radial distribution function for the large ( a ) particles of the kob - andersen binary lj mixture @xcite ( kablj ) . for comparison , [ fig : structure](b ) shows the same on an isotherm with a similar ( but smaller ) density change . clearly , structure is to a good approximation invariant on the isomorph , whereas this is not the case on the isotherm .    ) of the kablj mixture ; time is given in reduced units : @xmath56 and the q - vector is kept constant in reduced units . b ) same as in a ) , but along an isotherm ( density variation @xmath57).,title="fig:",scaledwidth=45.0% ] ) of the kablj mixture ; time is given in reduced units : @xmath56 and the q - vector is kept constant in reduced units . b ) same as in a ) , but along an isotherm ( density variation @xmath57).,title="fig:",scaledwidth=45.0% ]    taking the logarithm of eq . ( [ eq : isomorphdef ] ) shows that the potential energy surface in reduced units ( @xmath58 ) is the same for two isomorphic state points , except for an additive constant . this constant does not influence the dynamics , and consequently the dynamics in reduced units is the same for the two isomorphic state points . figure  [ fig : dynamics](a ) demonstrates the isomorph invariance of the incoherent intermediate scattering function of the a particles of the kablj mixture . for comparison , [ fig : dynamics](b ) shows the same property on the @xmath59 isotherm . like the structure , the dynamics is also invariant to a good approximation on the isomorph - both regarding average relaxation time and the shape of the relaxation function ; this is far from the case on the corresponding isotherm .     calculated from equilibrium fluctuations via eq . ( [ eq : gammasex ] ) for the kablj system . red squares : state points belonging to the isomorph presented in figs . [ fig : structure ] and [ fig : dynamics ] . black circles : isochoric state points . straight lines are linear regression fits . standard errors on slopes are 0.06 ( isomorph ) and 0.04 ( isochore ) . , scaledwidth=45.0% ]    the exponent @xmath53 calculated from eq . ( [ eq : gammasex ] ) generally depends on the state point . figure  [ fig : gammasex ] shows ( as red squares ) @xmath53 along the isomorph of figs . 1(a ) and 2(a).temperature changes by almost a factor of two , but @xmath53 changes less than 5% . paper iv argued that for a system with good isomorphs @xmath53 depends only on density . this is supported by fig . [ fig : gammasex ] in which @xmath53 on an isochore is shown to vary only @xmath60% when temperature is changed by a factor of two ( black circles ) .    , @xmath61 , eq .  ( [ eq : genlj ] ) ) and parameters chosen to place the minimum of the potential at the same distance and depth as in the 12 - 6 kablj system . likewise , 18 - 9 kablj is using the exponents @xmath62 and @xmath63 . finally 12 - 6 wblj is the wahnstrm 50:50 mixture@xcite using the standard 12 - 6 lj potential . straight lines are linear regression fits . the inset shows a zoom on the data for the two 12 - 6 lj systems . for each system the simulated temperatures ranges from where non - exponential relaxation sets in ( @xmath64 ) to a viscous state with two - step relaxation ( @xmath65 ) . , scaledwidth=45.0% ]    it follows from eq . ( [ eq : gammasex1 ] ) that the slope of isochores in the @xmath3 phase diagram is given by @xmath53 : @xmath66 the insignificant change of @xmath53 along isochores ( black circles in fig . [ fig : gammasex ] ) means that these to a good approximation are straight lines in the @xmath3 phase diagram . this is illustrated in fig . [ fig : uwischores ] for four different binary generalized lj mixtures . what is the shape of an isomorph in the @xmath3 phase diagram ? this section answers this question in generality for the multi - component generalized lj potential . recall that this potential ( eq .  ( [ eq : genlj ] ) ) is the sum of two ipl s . correspondingly , the potential energy and virial can be expressed as sums of two ipl terms : @xmath67 for pair interactions the virial is given by@xcite @xmath68 where the prime denotes the derivative with respect to @xmath69 . from this we get @xmath70 for any point in the @xmath3 phase diagram we can solve eqs . ( [ eq : u ] ) and  ( [ eq : w ] ) for @xmath71 : @xmath72     phase diagram for systems interacting via the standard 12 - 6 lj potential . included in this figure are also two theoretical isomorphs , plotted using eq . ( [ eq : parabola ] ) . the accessible points are those above both dashed lines ( see text ) . the part of the isomorphs between the dashed lines in the @xmath73 quadrant are thus not accessible to the standard 12 - 6 lj ( they are accessible to the purely repulsive 12 - 6 lj potential with @xmath74.),scaledwidth=45.0% ]    these equations allow one to determine which regions of the @xmath3 phase diagram are accessible to the system . adopting the convention @xmath75 , we restrict ourselves to the case @xmath76 in order to have a repulsive core for all interactions . this implies @xmath77 for all configurations , and therefore also for all thermodynamic averages . combining this with eq . ( [ eq : um ] ) implies @xmath78 . if all interactions have an attractive part ( @xmath79 ) we have @xmath80 , which in combinations with eq . ( [ eq : un ] ) implies @xmath81 . as an example , the region of the @xmath3 phase diagram accessible to the standard 12 - 6 lj potential is given by @xmath82 and @xmath83 ( the later inequality being relevant when @xmath84 ) . see fig . [ fig : genericwu ] for an illustration .      along an isomorph the structure is invariant in reduced units ( fig . [ fig : structure ] ) . from this it follows that @xmath85 is invariant for any @xmath86 along an isomorph ( @xmath47 ) : @xmath87 if we let `` * '' denote a reference point , @xmath88 , we can use eqs . ( [ eq : um ] ) and ( [ eq : un ] ) to get @xmath89 and find for other state points on the same isomorph ( where @xmath90 ) :    @xmath91    combining eq .  ( [ eq : scaled_u ] ) with eqs . ( [ eq : u ] ) and ( [ eq : w ] ) we obtain a parametric description of an isomorph in the @xmath3 phase diagram : @xmath92 once the numbers @xmath93 and @xmath94 have been determined from a reference state point using eqs .  ( [ eq : um ] ) and ( [ eq : un ] ) , the entire isomorph to which this state point belongs is thus traced out in the wu phase diagram using eqs . ( [ eq : isomorphu ] ) and ( [ eq : isomorphw ] ) . alternatively , given a collection of isomorphic state point generated , e.g. , as described in section iic , the predicted relation between @xmath1 , @xmath0 , and @xmath95 can be tested by linear regression : @xmath96 this is done in fig . [ fig : isomorphuwtest ] for the 12 - 6 kablj isomorph presented in figs . [ fig : structure ] and [ fig : dynamics ] . the potential energies were fitted by linear regression by means of eq . ( [ eq : isomorphu_lin ] ) . subsequently , the virials were compared to eq . ( [ eq : isomorphw_lin ] ) using the parameters estimated from the potential energies ( eq .  ( [ eq : isomorphu_lin ] ) ) , i.e. , without performing a new fit .     and [ fig : dynamics ] ) . top panel : u fitted to eq . ( [ eq : isomorphu_lin ] ) using @xmath97 ( i.e. , @xmath98 ) . bottom panel : w compared to eq . ( [ eq : isomorphw_lin ] ) , using the parameters @xmath93 and @xmath94 found from @xmath1 ( top panel).,scaledwidth=45.0% ]     phase diagram . the points give simulation results , the full curves the predictions of eqs . ( [ eq : isomorphu ] ) and ( [ eq : isomorphw ] ) . two isomorphs belong to the kablj system , the third result from simulations of wahnstrm binary lj liquid@xcite ( wblj ) , and the fourth ( sclj ) is the single component lj liquid . , scaledwidth=43.0% ]     by scaling with @xmath99 defined as the virial on each isomorph when @xmath100 . the points give simulation results , the dashed curve the prediction of eq . ( [ eq : parabola ] ) . , scaledwidth=45.0% ]    ) and  ( [ eq : isomorphw ] ) ) . for reference , also the isomorph prediction for 12 - 6 lj systems is included . , scaledwidth=45.0% ]    we note here two important consequences of the above . 1 ) if the reference state point @xmath101 generates the isomorph @xmath102 , then the reference state point @xmath103 generates the isomorph @xmath104 . this means that all isomorphs for a given system have the same shape in the @xmath105 diagram and can be scaled onto one another . 2 ) the shape of the isomorphs depends only on the exponents , @xmath19 and @xmath31 . this means that all 12 - 6 lj systems have the same isomorphs in the @xmath3 phase diagram they may be scaled onto a single `` master isomorph '' . figure  [ fig : isomorph ] shows four isomorphs in the @xmath3 phase diagram for three different 12 - 6 lj systems.@xcite figure  [ fig : figure10 ] shows the same isomorphs scaled onto the 12 - 6 master isomorph , figure  [ fig : isomorph_18_6 ] shows an isomorph for the 18 - 6 kablj system and compares it to a 12 - 6 lj isomorph . overall , the agreement between simulation results and predicted shapes is quite good .      combining eq . ( [ eq : scaled_u ] ) with eqs . ( [ eq : um ] ) and ( [ eq : un ] ) we get @xmath106 eliminating density and rearranging , we find an invariant for the isomorph : @xmath107 choosing the reference point with zero potential energy @xmath108 ( this reference point is guaranteed to exist if all interactions have attraction ) , we get : @xmath109 @xmath110 is a unique number identifying the isomorph . equation  ( [ eq : invariant ] ) implies that the isomorph is the solution to @xmath111 here it can be seen directly that - as argued above - for a given set of exponents @xmath112 all isomorphs have the same shape , suggesting the name `` master isomorph '' ; @xmath110 is the parameter that determines the scale of each isomorph . for @xmath113 ( e.g. , the standard 12 - 6 lj potential ) the solution to this ( quadratic ) equation is : @xmath114 this has real solution(s ) whenever @xmath115 , where equality corresponds to @xmath116 . figure [ fig : genericwu ] plots eq . ( [ eq : parabola ] ) with @xmath61 for two values of @xmath110 . ) . left panel : eq . ( [ eq : isomorphu_lin ] ) , right panel : eq . ( [ eq : isomorphw_lin ] ) ; data points are simulation results from ahmed and sadus , @xcite straight lines are linear regression fits . the lower panel plots the fit - parameters from eqs . ( [ eq : isomorphu_lin ] ) and ( [ eq : isomorphw_lin ] ) against each other , demonstrating that the fits to energies and virials , respectively , are consistent . , title="fig:",scaledwidth=23.5% ] ) . left panel : eq . ( [ eq : isomorphu_lin ] ) , right panel : eq . ( [ eq : isomorphw_lin ] ) ; data points are simulation results from ahmed and sadus , @xcite straight lines are linear regression fits . the lower panel plots the fit - parameters from eqs . ( [ eq : isomorphu_lin ] ) and ( [ eq : isomorphw_lin ] ) against each other , demonstrating that the fits to energies and virials , respectively , are consistent . , title="fig:",scaledwidth=23.5% ] ) . left panel : eq . ( [ eq : isomorphu_lin ] ) , right panel : eq . ( [ eq : isomorphw_lin ] ) ; data points are simulation results from ahmed and sadus , @xcite straight lines are linear regression fits . the lower panel plots the fit - parameters from eqs . ( [ eq : isomorphu_lin ] ) and ( [ eq : isomorphw_lin ] ) against each other , demonstrating that the fits to energies and virials , respectively , are consistent . , title="fig:",scaledwidth=30.0% ]    , plotting potential energy versus density ( left panel ) and virial versus density ( right panel ) . dashed lines : isomorphic predictions ( eqs . ( [ eq : isomorphu ] ) and ( [ eq : isomorphw ] ) ) , using the parameters determined in fig . [ fig : uwcoexist_lin ] . , title="fig:",scaledwidth=23.5% ] , plotting potential energy versus density ( left panel ) and virial versus density ( right panel ) . dashed lines : isomorphic predictions ( eqs . ( [ eq : isomorphu ] ) and ( [ eq : isomorphw ] ) ) , using the parameters determined in fig . [ fig : uwcoexist_lin ] . , title="fig:",scaledwidth=23.5% ]    ) ) with @xmath117 and @xmath118 respectively . open circles indicate results from simulations in the solid and liquid phase respectively . both phases are ( very ) strongly correlating with @xmath119 . inset shows the solid - liquid phase behavior in a @xmath120-diagram with logarithmic axes . dashed red lines are the isomorph prediction @xmath121 using @xmath53 found using eq . ( [ eq : gammasex ] ) from the simulations of the two phases ( open circles ) giving @xmath122 and 5.18 respectively . full black lines are free fits giving @xmath123 and 6.49 respectively . , scaledwidth=45.0% ]      in paper iv it was argued briefly that eq . ( [ eq : isomorphdef ] ) predicts that solid - liquid coexistence lines are isomorphs . in more detail the argument goes as follows . first one notes that an isomorph can not cross a solid- liquid coexistence line : recall that the isomorph concept relates to equilibrium ensemble probabilities . for a pair of isomorphic state points all pairs of microscopic configurations related by eq . ( [ scaledef ] ) have identical boltzmann probabilities according to eq . ( [ eq : isomorphdef ] ) . consequently , the two isomorphic state points must also have the same macroscopic phase behavior . in other words , if an isomorph were to cross the coexistence line , part of the isomorph would have liquid microstates as the most likely and part of it would have mixed crystalline - liquid microstates as the most likely . this would violate the proportionality of boltzmann statistical weights for scaled states , the property that defines an isomorph . given that an isomorph can not cross the solid - liquid coexistence line , the next point is to note that because all state points belong to some isomorph  a curve infinitesimally close to the coexistence line must be an isomorph . this argument applies in the @xmath124 phase diagram , as well as in the density - temperature or the pressure - temperature phase diagrams . in the first two cases there is a multitude of isomorphs in the coexistence region ; for instance one particular isomorph will be characterized by 53% liquid and 47% crystal , etc . in the pressure - temperature phase diagram these `` coexistence '' isomorphs collapse into one . the prediction that the melting line , both slightly to the liquid side and slightly to the solid side , are isomorphs has a number of consequences and it sheds new light on well - known phenomenological melting rules ( as well as exceptions to these rules for non - strongly correlating liquids , paper iv ) . thus , for instance the observation that the lindemann melting criterion is pressure independent for a given liquid follows from the isomorph property of the melting line ( whereas the existence of a universal lindemann criterion does not follow ) . indeed , as is easy to show from paper iv the isomorph theory implies ross `` generalized lindemann melting law '' from 1969 @xcite , according to which the reduced configurational partition function of the crystalline phase @xmath125 $ ] is invariant along the melting line ( here @xmath1 is the potential energy , @xmath126 is the potential energy of the crystal with all atoms located exactly at their lattice sites , and the tilde denotes reduced coordinates . further well - known melting rules , which the isomorph theory explains , include ( paper iv ) : the reduced viscosity , the reduced surface tension , the reduced diffusion constant , and the reduced heat conductivity are all invariant in the liquid phase along the melting line . likewise , the reduced - unit static structure factor is invariant along the melting curve . these rules have been confirmed for several liquids in both experiment and simulation ( see references of paper v ) .    like other isomorph predictions following from eq . ( [ eq : isomorphdef ] ) , the prediction that solid - liquid coexistence curves are isomorphs is only expected to be exact for systems that obey eq . ( [ eq : isomorphdef ] ) exactly , i.e. , perfectly correlating systems . indeed , it straightforward to show that in ipl systems the phase behavior depends only on @xmath127 , and thus the co - existence line is an ( exact ) isomorph . the interesting question now is : how well do strongly correlating systems , e.g. , generalized lj systems , follow the prediction ? note that we do not expect the liquid - gas co - existence lines to be isomorphs ( and indeed they are not ) ; the gas - phase is not strongly correlating and thus eq . ( [ eq : isomorphdef ] ) can not be fulfilled in the relevant part of configuration space . in contrast , at the solid - liquid coexistence both phases involved are strongly correlating for lj systems ( paper i ) , and we thus in the following test to what degree the isomorph prediction is fulfilled . ahmed and sadus used in a recent paper @xcite a novel numerical method for determining solid - liquid coexistence . they reported results for the generalized lj potential with the repulsive exponent @xmath19 varying from 12 to 7 and fixed attractive exponent @xmath61 ( ahmed and sadus termed the repulsive exponent @xmath31 , not @xmath19 ; we keep , however , the above notation ) . figure [ fig : uwcoexist_lin ] shows the state points reported by ahmed and sadus for the liquid side of the solid - liquid coexistence curve compared to the isomorphic predictions as expressed in eqs . ( [ eq : isomorphu_lin ] ) and ( [ eq : isomorphw_lin ] ) . the lower panel shows that the parameters found by fitting to eqs . ( [ eq : isomorphu_lin ] ) and ( [ eq : isomorphw_lin ] ) respectively are consistent . figure  [ fig : uwcoexist ] shows the same data plotting , respectively , potential energy ( left panel ) and virial ( right panel ) versus density . figure [ fig : uwcoexistphase ] shows the 12 - 6 lj solid - liqud phase behavior in the wu - diagram and in the @xmath128-diagram ( inset ) , including here also data from mastny and de pablo.@xcite    the prediction that solid - liquid coexistence lines are isomorphs agrees well on the liquid side with the simulation results of ahmed and sadus@xcite as well as with the results for 12 - 6 lj from mastny and de pablo@xcite . on the solid side the agreement is less convincing ( see fig .  [ fig : uwcoexistphase ] ) ; in particular the logarithmic derivative in the @xmath128 diagram ( slope in the inset ) differs by 25% . at present we do not have an explanation for this . it might be related to inter - particle distances longer than the first peak of the radial distribution function playing a bigger role for the phase behavior on the solid side@xcite than on the liquid side .    ) ) plotting potential energy versus @xmath129 on isochores for the systems investigated in fig . [ fig : uwischores ] . , scaledwidth=45.0% ]     phase diagram using @xmath130 as reference density . curves and lines : theoretical predictions ( eqs . ( [ eq : ustar ] ) to ( [ eq : isomorph_scaling_w ] ) ) for three isochores , an isomorph , and an isotherm . data points : simulations of the 12 - 6 kablj mixture . the parameters of the equation of state were found from the two state points indicated by circles , giving @xmath131 . by comparison , the corresponding parameters found by fitting to the @xmath130 isochore in figs . [ fig : uwischores ] and [ fig : urosenfeld ] are @xmath132 , illustrating the robustness of the fitted equation of state . applying the fitting procedure to the simulated state points on the @xmath133 isotherm gives the parameters ( 38.5 , 5.19 , -8.63 , 2.61 ) . lower panel : same data as above in a @xmath134 diagram . , title="fig:",scaledwidth=45.0% ]   phase diagram using @xmath130 as reference density . curves and lines : theoretical predictions ( eqs . ( [ eq : ustar ] ) to ( [ eq : isomorph_scaling_w ] ) ) for three isochores , an isomorph , and an isotherm . data points : simulations of the 12 - 6 kablj mixture . the parameters of the equation of state were found from the two state points indicated by circles , giving @xmath131 . by comparison , the corresponding parameters found by fitting to the @xmath130 isochore in figs .  [ fig : uwischores ] and [ fig : urosenfeld ] are @xmath132 , illustrating the robustness of the fitted equation of state . applying the fitting procedure to the simulated state points on the @xmath133 isotherm gives the parameters ( 38.5 , 5.19 , -8.63 , 2.61 ) . lower panel : same data as above in a @xmath134 diagram . , title="fig:",scaledwidth=45.0% ]      equations  ( [ eq : isomorphu ] ) and ( [ eq : isomorphw ] ) give the potential energy and the virial as functions of density along an isomorph . the temperature variation along the isomorph can be found by integrating eq . ( [ eq : gammasex ] ) . since @xmath53 generally changes only little ( fig . [ fig : gammasex ] ) , in many situations it is a good approximation to assume it constant , leading to the relation @xmath135 along an isomorph , where @xmath136 is the temperature at the reference point . taken together eqs . ( [ eq : isomorphu ] ) , ( [ eq : isomorphw ] ) , and ( [ eq : gammasex ] ) thus imply that from a single reference point @xmath137 we have a prediction for @xmath138 on the isomorph to which the reference point belongs . figure [ fig : uwischores ] demonstrated that in the @xmath3 phase diagram , isochores to a good approximation are straight lines with slope @xmath53 , i.e. , that one can write @xmath139 . the only ingredient missing to generate a full equation of state expressed as @xmath140 and @xmath141 is a relation between temperature and potential energy or virial along an isochore . rosenfeld and tarazona @xcite derived from density functional theory an expression for the potential energy on an isochore : @xmath142 rosenfeld and tarazona noted that the expression `` ... provides a good estimate of the thermal energy ( ... ) near freezing densities only for predominant repulsive interactions '' , and confirmed their expression by simulations of ipl potentials with different exponents . equation  ( [ eq : urosenfeld ] ) has since been shown , however , to be an excellent approximation also for several models with attraction , including the kablj mixture.@xcite this was regarded as a bit of a mystery , but can now be understood as a consequence of the fact that dynamics and heat capacity of strongly correlating liquids are well reproduced by a corresponding ipl system@xcite  a consequence of the hidden scale invariance discussed briefly in the introduction ( see papers i - iii ) . figure  [ fig : urosenfeld ] tests the rosenfeld - tarazona prediction for the systems investigated in fig . [ fig : uwischores ] . the prediction works very well , but less so for the supercooled wahnstrm blj system , for which the formation of extended `` frank - kasper '' clusters at low temperatures affects the temperature dependence of the potential energy . @xcite    combining eq . ( [ eq : urosenfeld ] ) with our results on isomorphs and isochores we can construct an equation of state for generalized lj systems . the idea is to map any state point @xmath138 to the corresponding isomorphic state point @xmath137 on a reference isochore ( where it is implecitely understood that the parameters ( @xmath143 ) are evaluated at the reference isochore @xmath144 ) : @xmath145 from ( @xmath146 ) we can calculate the contribution to the potential energy from the two ipl terms of the potential ( compare eqs . ( [ eq : um ] ) and([eq : un ] ) ) : @xmath147 finally , we use isomorphic scaling ( eqs . ( [ eq : isomorphu ] ) and ( [ eq : isomorphw ] ) ) to go back to the @xmath138 state point : @xmath148 the new equation of state given by eqs . ( [ eq : umseos ] ) to ( [ eq : isomorph_scaling_w ] ) contains four parameters ( @xmath143 ) evaluated at a reference density @xmath144 . the parameters can be calculated from @xmath149 , and @xmath150 at a single state point . a more convenient way of estimating the parameters might be : for a collection of @xmath138 state points , at least two of which are non - isomorphic , use isomorphic scaling ( eqs . ( [ eq : isomorphu])and  ( [ eq : isomorphw ] ) ) to get the corresponding @xmath101 at the chosen reference density , and fit them to eq . ( [ eq : wisochor ] ) to determine @xmath151 and @xmath53 . next , using the fitted value for @xmath53 , fit to eq . ( [ eq : ustar ] ) in order to find @xmath152 and @xmath153 .    to put the new equation of state to a test , the procedure described above was used to determine the four parameters ( @xmath143 ) for the 12 - 6 kablj system using merely two state points . the resulting equation of state is shown as full lines in fig . [ fig : uwstate ] , where it is compared to simulation results shown as data points . the agreement is good . the largest deviations are seen when changing density away from the reference density , most evident for the isotherm in the @xmath3 diagram and the isomorph in the @xmath134 diagram . better agreement is expected if the density dependence of @xmath53 is taken into account , but such a correction will not be attempted here . to summarize , combining the isomorph theory with two further approximations , a state - point independent exponent @xmath53 and the rosenfeld - tarazona expression , a realistic equation of state with four parameters is arrived at . paper iv demonstrated that strongly correlating liquids have isomorphs , i.e. , curves in the phase diagram along which structure and dynamics to a good approximation are invariant in reduced units . the main new results in this paper are predictions specific to generalized lennard - jones systems and the supporting numerical evidence .    starting from the invariance of structure along isomorphs , a prediction for the shape of these in the @xmath3 phase diagram was presented and shown to agree well with simulation results . it was shown that for a given system , the isomorphs all have the same shape in the @xmath3 phase diagram ; they are simply scaled versions of each other . furthermore , the isomorph shape depends only on the exponents @xmath19 and @xmath31 , i.e. , all systems with , say , 12 - 6 lj interactions have same isomorphs in the @xmath3 phase diagram . what differs from system to system is the values of the density and temperature along an isomorph .    throughout papers i - v the @xmath3 phase diagram was referred to extensively . this particular phase diagram has not been used much before , but it is evident that for strongly correlating liquids several features stand out when presented in the @xmath3 phase diagram . prominent examples are that isochores are straight lines ( with slope @xmath53 ) and the existence of the `` master '' isomorph discussed above for generalized lj systems . the class of strongly correlating liquids includes , we believe , all van der waals liquids and metallic liquids ( but excludes hydrogen - bonding , covalent , as well as strongly ionic liquids ) . more simulations , as well as experiments , are needed to substantiate that the class of strongly correlating liquids is this large , but preliminary simulations of simple molecular models like@xcite the asymmetric dumbbell and the lewis - wahnstrom otp three - site model are encouraging . in particular , the isomorph properties of more complex molecule models need to be simulated , as well as theoretically contemplated ; the scaling properties of molecules with fixed bond lengths are not trivial . it should be noted that even a quite complex system like a phospholipid membrane can be strongly correlating with respect to its slow degrees of freedom,@xcite the interactions of which are dominated by van der waals forces . another interesting question is to study in detail the properties of liquids where the exponent @xmath53 varies significantly throughout the phase diagram , for instance the weeks - chandler - andersen cutoff variant of the generalized binary lj liquid that was recently investigated om detail by coslovich and roland , @xcite as well as berthier and tarjus.@xcite another system that deserves further investigation is the 200 - 100 lj system@xcite which we find to be an exception to the general observation that generalized lj systems are strongly correlating  presumably as a consequence of its extremely narrow range of interactions . this series of papers investigated in depth the properties of strongly correlating liquids . we believe to have demonstrated that strongly correlating liquids are simpler than liquids in general . hopefully this insight will turn out to be useful in furthering the understanding of the properties of liquids in general . acknowledgments + urp is supported by the danish council for independent research in natural sciences . the center for viscous liquid dynamics `` glass and time '' is sponsored by the danish national research foundation ( dnrf ) . unless stated otherwise , simulations are performed using a newly developed molecular dynamics code optimized for nvidia graphics cards . the code is available as open source at http://rumd.org . potentials were cut and shifted at @xmath154 . reported potentials and virials do not include contributions beyond the cut - off . simulations reported in figs . 7 and 8 were performed using gromacs ; h. j. c. berendsen , d , van der spoel , and r. van drunen , comp . comm . * 91 * , 43 ( 1995 ) ; e. lindahl , b. hess , and d. van der spoel , j. mol . mod . * 7 * , 306 ( 2001 ) .
this series of papers is devoted to identifying and explaining the properties of strongly correlating liquids , i.e. , liquids with more than 90% correlation between their virial @xmath0 and potential energy @xmath1 fluctuations in the @xmath2 ensemble . 
 paper iv [ n. gnan _ 
 et al . 
 _ , j. chem . phys . * 131 * , 234504 ( 2009 ) ] showed that strongly correlating liquids have `` isomorphs '' , which are curves in the phase diagram along which structure , dynamics , and some thermodynamic properties are invariant in reduced units . in the present paper , using the fact that reduced - unit radial distribution functions are isomorph invariant , we derive an expression for the shapes of isomorphs in the @xmath3 phase diagram of generalized lennard - jones systems of one or more types of particles . 
 the isomorph shape depends only on the lennard - jones exponents ; thus all isomorphs of standard lennard - jones systems ( with exponents 12 and 6 ) can be scaled onto to a single curve . 
 two applications are given . 
 one is testing the prediction that the solid - liquid coexistence curve follows an isomorph by comparing to recent simulations by ahmed and sadus [ j. chem . 
 phys . 
 * 131 * , 174504 ( 2009 ) ] . 
 excellent agreement is found on the liquid side of the coexistence , whereas the agreement is worse on the solid side . 
 a second application is the derivation of an approximate equation of state for generalized lennard - jones systems by combining the isomorph theory with the rosenfeld - tarazona expression for the temperature dependence of potential energy on isochores . 
 it is shown that the new equation of state agrees well with simulations .