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the measurement of the temperature dependence of the rate constant is one of the important tools of chemical kinetics in determining mechanisms of chemical reactions.@xcite significant deviations from a simple exponential behavior can be evidence of tunneling and of other nuclear quantum effects.@xcite these effects are particularly strong for hydrogen transfer reactions with a high activation barrier or at low temperatures . recently , however , quantum effects have been observed also in many enzymatic reactions at physiological temperatures.@xcite it therefore becomes more and more important to have accurate theoretical methods for computing the temperature dependence of the rate constant.@xcite    probably the oldest yet still the best known expression for the thermal rate constant @xmath6 at temperature @xmath7 is the empirical arrhenius law,@xcite@xmath8 here @xmath9 is the boltzmann constant , @xmath10 the activation energy , and the temperature dependence is purely exponential . an improvement over the arrhenius law was provided by the transition state theory ( tst),@xcite in which @xmath11 where @xmath12 is the planck s constant , @xmath13 and @xmath14 are the partition functions of the transition state and the reactants , respectively , and @xmath15 is the barrier height for the reaction . here the temperature dependence includes a fractional power @xmath16 in addition to the exponential . nevertheless , both arrhenius law and tst are basically purely classical , so they can not take into account tunneling and other nuclear quantum effects . although the simplest quantum effects _ can _ be taken into account within the tst in an _ ad hoc _ fashion , by replacing the partition functions by their quantum analogs for the simple harmonic oscillator ( which takes into account the zero - point energy , the wigner tunneling correction,@xcite and approximate quantization of the vibrational motion ) , a more systematic approach requires quantum treatment of the nuclear motion . this is of course extremely difficult , and therefore various approximate yet accurate methods have been developed . these include , e.g. , the semiclassical methods@xcite or the so - called quantum transition state theories.@xcite    in this paper , we evaluate the temperature dependence of the rate constant starting from the quantum instanton ( qi ) approximation.@xcite this quantum transition state theory has been shown to describe correctly not only all of the above - mentioned quantum effects , but also corner - cutting , coupling of vibrational and rotational motions , multiple tunneling paths , etc . as we evaluate the temperature dependence of the rate constant directly , we can speed up the qi  calculation significantly by avoiding the tedious umbrella sampling necessary for computing the rate constant itself . furthermore , if it is only the temperature dependence of the rate constant that is needed , we can increase the accuracy of the qi approximation for the rate constant by canceling some small remaining systematic errors . this can be useful , e.g. , if we know @xmath17 at a temperature @xmath18 very accurately and would like to know @xmath6 at other temperatures . the temperature dependence of the rate constant is computed via a thermodynamic integration @xcite with respect to the inverse temperature . a similar thermodynamic integration in the framework of the qi model was used by ceotto and miller to compute the rate constant for several one and two - dimensional systems using a discrete variable representation.@xcite below , we develop a general thermodynamic integration procedure based on the path integral implementation , which is suitable for many - dimensional systems . the method is tested on two simple systems for which exact quantum dynamical calculations are feasible : the eckart barrier and the full nine - dimensional h @xmath0h@xmath1  h@xmath2 h reaction . while the former system is the simplest one - dimensional model of a bimolecular reaction , the latter is the simplest bimolecular chemical reaction with an energy barrier . as such , it has been widely studied and attained the status of a benchmark reaction.@xcite despite the apparent simplicity , this reaction remains a challenging test for new approximations . this is due to the presence of strong quantum effects , as the lightest atoms are involved in both bond breaking and bond formation . this reaction was investigated not only for the temperature dependence of its rate constant , but also for the kinetic isotope effect,@xcite the presence of the geometric phase effect,@xcite etc . the remainder of this paper is organized as follows : section ii describes the methodology , i.e. , the qi approximation , the thermodynamic integration with respect to the inverse temperature , the path integral formalism , and the relevant estimators . computational details and , in particular , the analysis of various errors are presented in sec . section iv contains the results . first , it is shown how the low and high temperature limits are obtained . the numerical results for the eckart barrier and the h@xmath19h@xmath20 reaction are then presented . the results are compared with the arrhenius law , tst , tst with the wigner tunneling correction , and the exact quantum calculation . finally , we discuss how the efficiency is related to the dependence of the statistical error on the number of imaginary time slices in the path integral , and how this error , in turn , depends on the system under study . section v concludes the paper . the qi approximation for thermal rate constants was introduced in ref . . the most direct derivation @xcite starts from the exact miller - schwartz - tromp formula for the rate constant,@xcite@xmath21 where @xmath22 is the symmetrized flux - flux correlation function,@xmath23 with hamiltonian operator @xmath24 , inverse temperature @xmath25 , time @xmath26 , and @xmath27 the flux operator through the dividing surface @xmath28 . a stationary - phase approximation applied to eq . ( [ miller_schwartz_tromp ] ) yields the qi approximation for the rate constant,@xcite @xmath29 where @xmath30 is a specific type of energy variance,@xcite @xmath31   ^{1/2}. \label{dh}\ ] ] the delta - delta correlation function used above is defined as @xmath32 where the generalized delta function operator is given by @xmath33   \equiv\delta\left [   \xi\left (   \mathbf{\hat{r}}\right ) -\xi_{\gamma}\right ]   m^{-1/2}\left\vert \nabla\xi\right\vert \label{delta_op}\ ] ] and @xmath34 is the reaction coordinate such that @xmath35 at the transition state . similarly , @xmath36 defines the position of the dividing surface @xmath37 . we have used mass - scaled coordinates in which all degrees of freedom have the same mass @xmath38 . in practice , the exact delta function constraint is approximated by a gaussian constraint corresponding to a harmonic constraint potential @xmath39,@xcite@xmath40   &   \approx   \sqrt{\frac{\beta}{2\pi \sigma^2 } } e^{-\beta v_{\text{constr}}(\mathbf{r})},\label{gauss}\\ v_{\text{constr}}(\mathbf{r } )   &   = \frac{1}{2}\left (   \frac{\xi(\mathbf{r})-\xi_{\gamma}}{\sigma}\right )   ^{2}. \label{v_constr}\ ] ]    the accuracy of the qi approximation has been already verified in numerous applications.@xcite the main shortcoming of the qi method is the neglect of recrossing which is however , neglected in any quantum or classical transition state theories . the recrossing effects on the quantum instanton rate constant have been quantified for several collinear reactions by ceotto and miller.@xcite fortunately , the recrossing effects become generally less important in higher dimensions .      the goal of this paper is to compute the temperature dependence of the rate constant , i.e. , the ratio @xmath41 . within the qi approximation , this ratio is given by @xmath42 where we multiplied and divided the numerator and denominator by @xmath43 . in this expression , quantities @xmath30 and @xmath44 can be computed directly by the metropolis monte - carlo procedure because they are thermodynamic averages . on the other hand , the ratios @xmath45 and @xmath46 can not be computed this way since they involve ratios of quantities at different temperatures . these ratios can , however , be calculated by the method of _ thermodynamic integration _ @xcite with respect to the inverse temperature @xmath47,@xmath48   , \label{ti_qr}\\ \frac{c_{\text{dd}}\left (   \beta_{2},0\right )   } { c_{\text{dd}}\left (   \beta _ { 1},0\right )   }   &   = \exp\left [   -\int_{\beta_{1}}^{\beta_{2}}e^{\ddag}(\beta)d\beta\right ]   , \label{ti_cdd}\ ] ] where @xmath49 and @xmath50 are the thermal energies of the reactants and of the transition state , respectively . a similar thermodynamic integration was used within a discrete variable representation of the qi approximation to compute the rate constant for several collinear triatomic reactions.@xcite unlike @xmath51 and @xmath52 , the energies are normalized quantities because they can be written as logarithmic derivatives:@xmath53 hence they can be computed directly by a monte carlo procedure . quantum thermodynamic effects can be treated rigorously using the imaginary time path integral ( pi).@xcite let @xmath54 be the number of degrees of freedom ( @xmath55 for the eckart barrier and @xmath56 for the h@xmath57 potential ) and @xmath58 the number of imaginary time slices in the pi . the pi representations of the partition function@xcite and the delta - delta correlation function@xcite are@xmath59   , \label{pi_qr}\\ c_{\text{dd}}^{p}\left(\beta ,   0\right )    &   = c\int d\mathbf{r}^{\left (   1\right ) } \cdots\int d\mathbf{r}^{\left (   p\right )   } \delta\left [   \xi_{a}\left ( \mathbf{r}^{\left (   0\right )   } \right )   \right ] \nonumber \\   & \times \delta\left [   \xi_{b}\left ( \mathbf{r}^{\left (   p/2\right )   } \right )   \right ]   \exp\left [   -\beta \phi\left (   \left\ {   \mathbf{r}^{\left (   s\right )   } \right\ }   \right ) \right ]   , \label{pi_cdd}\ ] ] where @xmath60^{dp/2}$ ] , @xmath61 is a @xmath54-dimensional vector representing the @xmath62th time slice , and the effective potential @xmath63 is given by@xmath64 for @xmath65 , the above expressions reproduce classical statistical mechanics , while exact quantum statistics is reached in the limit @xmath66 .    in practice , there are two main strategies for evaluating thermodynamic averages using the pi : the pi  molecular dynamics ( pimd)@xcite or pi monte carlo ( pimc).@xcite we use the pimc procedure together with the metropolis algorithm . the basic idea is to sample the pi  configuration space according to an appropriate weight @xmath67 , which is , e.g. , for @xmath52 given by @xmath68   \delta\left [   \xi_{b}\left (   \mathbf{r}^{\left (   p/2\right ) } \right )   \right ]   \exp\left [   -\beta\phi\left (   \left\ {   \mathbf{r}^{\left ( s\right )   } \right\ }   \right )   \right ]   , \label{rho_ts}\ ] ] and then , at each sampled configuration , to evaluate the so - called estimator @xmath69 of the relevant physical quantity @xmath70 . the final estimate of @xmath70 is given by the average @xmath71 along the pimc trajectory . using the pi representation of @xmath51 and @xmath52 , one can obtain  estimators for all quantities needed in eq . ( [ qi_tdep ] ) , i.e. , @xmath72 , @xmath73 , and the logarithmic derivatives of @xmath74 , @xmath52 ( i.e. , the energies @xmath49 , @xmath50 ) . those for @xmath72 and @xmath73 are listed in ref . . the simplest estimator for the energy @xmath49 , the so - called barker or thermodynamic estimator ( te),@xcite can be derived directly from eq . ( [ er_logder ] ) and the pi expression ( [ pi_qr ] ) , giving@xmath75 as observed by herman _ et al._,@xcite the te can have a large statistical error , which can be avoided with the so - called virial ( ve)@xcite or centroid virial ( cve),@xcite estimators . invoking the virial theorem , the kinetic energy in these two estimators is replaced by an expression involving the gradient of the potential energy.@xcite this is convenient in the pimd implementations since the gradient is already available . in pimc simulations , however , only the potential is needed for the random walk , and in order to avoid computing the gradients , alternative approaches have been proposed . one can , e.g. , employ the centroid thermodynamic estimator@xcite or more generally , use a procedure based on rescaling coordinates@xcite in which the gradients of the potential are replaced by a single derivative that can be evaluated by finite difference.@xcite variants of the latter approach have been applied successfully to compute thermal energies and heat capacities,@xcite kinetic isotope effects@xcite , equilibrium isotope effects@xcite , or the derivatives of the flux - flux correlation function@xcite needed in the generalized qi model.@xcite    the ve for @xmath49 can be derived most directly by the change of coordinates @xmath76 in the pi ( [ pi_qr ] ) , yielding@xmath77}{d\delta\beta }   \right\ } \nonumber \\   & = \frac{1}{p } \sum_{s=1}^{p } \left\ { v(\mathbf{r}^{(s ) } ) + \frac{dv[(1+q)^{1/2}\mathbf{r}^{(s)}]}{dq } \right\ } ,   \label{ev}\ ] ] where @xmath78 is a small dimensionless parameter and the @xmath78-derivative is evaluated by finite difference at @xmath79 . similarly , the cve can be obtained by the change of variables @xmath80 , where one first subtracts the so - called centroid coordinate @xmath81 . the resulting estimator is@xmath82}{dq } \right\}. \nonumber \ ] ]      in the case of constrained simulations near the transition state , the constrained weight function ( [ rho_ts ] ) can be approximated by using the gaussian approximation of the delta function from eqs . ( [ gauss])-([v_constr ] ) . besides a prefactor , this amounts to adding a constraint potential@xmath83 to the effective potential @xmath63 . assuming that @xmath84 is independent of temperature and following a derivation similar to that for estimators of @xmath49 , one obtains the thermodynamic , virial , and centroid virial estimators for @xmath85@xmath86 } { dq},\label{e_ts_ve_rig}\\ e_{\text{cve}}^{\ddag , p }   &   = e_{r,\text{cve}}^{p}-\frac{1}{\beta}+\phi_{\text{constr}}(\{\mathbf{r}^{(s)}\ } ) \nonumber \\   &   + \frac{d\phi_{\text{constr}}[\{\mathbf{r}^{(c)}+(1+q)^{1/2}(\mathbf{r}^{(s)}-\mathbf{r}^{(c)})\}]}{dq}. \label{e_ts_cve_rig}\ ] ]    although the above estimators converge to the exact results , we found that the statistical errors can be decreased slightly by employing an alternative set of estimators , derived using an exact relation@xmath87 which is valid for a harmonic constraint potential for any value of @xmath58 . the new estimators are@xmath88}{dq } , \label{e_ts_ve_adh}\\ e_{\text{cve}}^{\ddag , p }   &   = e_{r,\text{cve}}^{p}+ \frac{d\phi _ { \text{constr}}[\{\mathbf{r}^{(c)}+(1+q)^{1/2}(\mathbf{r}^{(s)}-\mathbf{r}^{(c)})\}]}{dq}. \label{e_ts_cve_adh}\ ] ] estimators ( [ e_ts_te_adh])-([e_ts_cve_adh ] ) are in a way more intuitive than estimators ( [ e_ts_te_rig])-([e_ts_cve_rig ] ) : in the limit of a sharp constraint , the constrained energy should be independent of the type of constraint .    it should be stressed that the last terms in the ve and cve in eqs . ( [ e_ts_ve_rig ] ) , ( [ e_ts_cve_rig ] ) , ( [ e_ts_ve_adh ] ) , and ( [ e_ts_cve_adh ] ) are important ; without them the agreement among the te , ve , and cve is lost . in other words , an intuitive guess such as @xmath89 would not give a correct answer for the constrained energy . finally , we also tested a constraint potential that is proportional to temperature , i.e.,@xmath90 where @xmath91 is a harmonic potential independent of temperature . as a result , the constraint ( [ gauss ] ) itself is actually independent of temperature . following again a derivation similar to that for estimators of @xmath49 , one obtains another set of the te , ve , and cve for @xmath85 that look exactly like the estimators ( [ e_ts_te_adh])-([e_ts_cve_adh ] ) for @xmath84 independent of temperature . the only difference is that the random walk is done with a different constraint . all calculations were performed with a pimc code implemented in fortran 90 . sampling of the configurational space in the pimc simulation was done using three types of moves . staging algorithm @xcite was employed to move all unconstrained beads . constrained beads , i.e. , beads @xmath92 and @xmath93 which feel the constraint potential @xmath84 , were sampled with the free particle single slice algorithm.@xcite finally , whole chain moves@xcite were used to speed up sampling of the potential energy surface . the gaussian constraint potential must be strong enough in order to exert the constraining effect on the system . when this condition was satisfied , the converged results were independent of the constraint . however , the statistical root mean square error ( rmse ) of the transition state energy @xmath50 increases with the strength of the constraint because sampling of the configuration space becomes more difficult . therefore the selected strength of the constraint should take into account these two effects . we have used @xmath94 a.u . in both systems . all quantities needed in the ratio ( [ qi_tdep ] ) were evaluated using the above mentioned estimators . the thermodynamic integrations ( [ ti_qr ] ) and ( [ ti_cdd ] ) were evaluated with the simpson rule using 25 values of @xmath47 between @xmath95 and @xmath96 with the reference temperature @xmath97 and the minimum temperature @xmath98 . the number of beads was chosen to be inversely proportional to the temperature , with the maximal number of beads ( used for @xmath98 ) being @xmath99 for the eckart barrier and @xmath100 for the h@xmath19h@xmath20 reaction . the error of the final result consists of four main error contributions : a ) the statistical error due to the monte carlo simulation , b ) the error due to the discretization of the ti , c ) the error due to the discretization of the pi ( i.e. , the finite @xmath58 error ) , and d ) the actual error of the qi approximation . we have carefully separated these four contributions and attempted to make the first three contributions small in comparison with the error of the qi . in more complicated systems , this may not be possible and especially the final statistical error may be comparable to or larger than the error of the qi . because the exponentiation of the ti is quite sensitive to various errors , a detailed analysis of errors was carried out for the ratio @xmath101 , i.e. , over the largest temperature range , where the first three types of errors are the greatest . the ti was evaluated by four different numerical methods , namely the trapezoidal , simpson , simpson @xmath102 , and boole methods.@xcite    comparing the analytical bounds on the _ discretization errors _ of the ti integrals using a numerical estimate of a higher order derivative,@xcite one can conclude that both the simpson and simpson 3/8 methods were much better than the trapezoidal rule and that the boole method did not provide any further improvement . specifically , for the eckart barrier the error of the ratio due to the discretization of the ti was @xmath103 , @xmath104 , @xmath105 , or @xmath106 for the trapezoidal , simpson , simpson @xmath102 , or boole methods , respectively . for the h+h@xmath20 reaction , the discretization error of the final ratio was @xmath107 , @xmath108 , @xmath109 , or @xmath108 , in the same order . it should be emphasized that these error estimates are very conservative , as the actual difference between the final ratios based on different methods was an order of magnitude smaller than what one would expect from the error estimates . the final results displayed in the plots used the simpson method . the _ statistical _ rmses were estimated with the block averaging method using a variable block size @xcite to remove correlation of the pimc data . the statistical error of the ti was evaluated using an appropriate formula for each integration method and assuming that the statistical errors of energies at different temperatures were uncorrelated . as expected , the statistical error did not depend much on the integration method , and was always close to the statistical error for the simpson method . the statistical error of the final ratio was @xmath108 for the eckart barrier and @xmath110 for the h+h@xmath20 reaction . the _ finite _ @xmath58 _ error _ , i.e. , the error due to the discretization of the feynman pi , was obtained by repeating calculations of all quantities at all temperatures with twice smaller numbers of beads ( @xmath111 ) and then extrapolating each quantity to @xmath66 , assuming @xmath112 convergence . we emphasize that we used the extrapolated results only for estimating the finite @xmath58 error of the computed ratio and _ not _ for estimating the ratio itself , which could be dangerous . the finite @xmath58 error of the ratio was @xmath113 for the eckart barrier and @xmath114 for the h+h@xmath20 reaction . [ we note that for h+h@xmath20 one of the temperatures ( @xmath115 , a temperature in the vicinity of which a sharp bend in the @xmath116 dependence occurs ) required a five times longer simulation to reduce the ti discretization errors . this was because a small statistical error had a huge effect on the estimate of the fourth derivative and hence on the analytical estimate of the discretization error . ]    to sum up , in both systems the ti discretization error was negligible to the statistical and finite @xmath58 errors , which , in turn , were small in comparison to the error of the qi approximation . at high temperatures , the rate constant is expected to behave classically and follow the arrhenius law or the more accurate tst result . whereas the arrhenius law ( [ k_arrh ] ) predicts the rate constant ratio to be a simple exponential function of the inverse temperature , @xmath117 tst ( [ k_tst ] ) gives the ratio of rate constants @xmath118 in particular , assuming the partition functions @xmath119 and @xmath51 to be separable into products of classical rotational and vibrational partition functions , the temperature dependence ( [ tdep_tst ] )  of tst rate constant includes an additional fractional power law besides the exponential dependence in the arrhenius law ( [ tdep_arrh ] ) .    at somewhat lower temperatures , when quantum effects start to play a role , the basic tst expression ( [ tdep_tst ] ) can be improved in several ways : first , classical partition functions @xmath51 and @xmath120  can be replaced by their exact quantum analogs for a harmonic potential . second , quantum tunneling can be included approximately via the wigner tunneling correction.@xcite this method corrects the rate constant with a multiplicative factor@xmath121 where @xmath122 is the imaginary frequency of the asymmetric stretch along the reaction coordinate . the correction can be derived by treating the motion through the transition state as a vibration on an upside down potential and expanding the quantum partition function to second order in @xmath47 . although an improvement over tst , the wigner tunneling correction can not describe multidimensional tunneling .      a simple model of an activated chemical reaction is provided by the eckart barrier , a one - dimensional system described by the potential @xmath123   ^{-2}.\ ] ] we use standard parameter values @xmath124 , @xmath125 , mass @xmath126 and reaction coordinate @xmath127 . the exact quantum rate constant @xmath128 for this reaction can be obtained by integrating the exact quantum mechanical cumulative reaction probability , which is known analytically.@xcite    figure [ fig : eckart ] ( a ) compares the qi results with the exact qm results , tst ( which is equal to the arrhenius law here ) , and the tst including the wigner tunneling correction . the reference temperature is @xmath129 and the plot shows ratios for temperatures down to @xmath130 . since classical recrossing does not occur for the eckart barrier , all tsts should converge to the correct quantum results at high temperatures . the figure confirms that this is indeed the case :  note that all curves are tangent at the high temperature limit . at low temperatures , one reaches the quantum regime where tunneling is important and consequently the arrhenius plot of the exact qm result has a large curvature . while tst has a huge error , the qi approximation agrees very well with the qm result . note that the wigner tunneling correction improves over the tst and captures the tunneling effect partially but still fails to recover the curvature of the exact result .    . ]    figure [ fig : eckart ] ( b ) shows the relative error of the rate constant ratio for the different methods . whereas both tst and tst with the wigner tunneling correction deteriorate rapidly with decreasing temperature , the qi method has an error below @xmath131 for all temperatures above @xmath132 . the qi approximation has a significant error ( @xmath133 ) only at very low temperatures , below @xmath134 . however , this error was well understood already in the original paper by miller et al . @xcite and can be remedied by considering two separate dividing surfaces at very low temperatures . ( here we have used a single dividing surface at all temperatures for simplicity . )    the temperature dependence of the reactant and transition state energies is shown in fig . [ fig : eckart_ti ] . while both curves are quite smooth , small discretization errors in the integrals can have large effects on the exponentiated result . by a detailed error analysis described in sec . iii , we found that the simpson method was sufficient for the ti over the whole temperature range . note that the ve for @xmath135 gives zero , but can be easily corrected with an analytical correction @xmath136 . the three different estimators for the constrained energy @xmath50 at @xmath137 are compared in figs . [ fig : eckart_evsp_adhoc ] and [ fig : eckart_evsp_rigorous ] . panel ( a ) of fig . [ fig : eckart_evsp_adhoc ] , which uses the simpler estimators ( [ e_ts_te_adh])-([e_ts_cve_adh ] ) , shows that the te , ve , and cve agree for all examined values of @xmath58 and , in particular , converge to the same value for @xmath66 . the three estimators , however , differ in their statistical convergence . unlike for the unconstrained result , where the cve is the optimal estimator , for the constrained energy , the optimal estimator is the ve . this can be clearly seen in fig . [ fig : eckart_evsp_adhoc ] ( b ) which shows the rmses of the different estimators for different values of @xmath58 . while the rmse of the te increases with @xmath58 , the rmses of the ve and cve remain approximately constant as a function of @xmath58 , with the ve having a much smaller statistical error . assuming that the desired convergence is achieved for @xmath138 , the speedup factor achieved by using the ve compared to the te and cve is approximately @xmath139 and @xmath140 respectively . it is clear from the figure that both the speedup factor and the best estimator depend on @xmath58 and hence on the temperature .    . the constraint potential is independent of temperature and estimators ( [ e_ts_te_adh])-([e_ts_cve_adh ] ) are used . ] figure [ fig : eckart_evsp_rigorous ] shows the same results , but computed with the estimators ( [ e_ts_te_rig])-([e_ts_cve_rig ] ) . the statistical errors are very similar , although for the ve slightly larger than those in fig . [ fig : eckart_evsp_adhoc ] . the temperature dependence of the rate constant of the @xmath141 reaction was studied on the boothroyd - keogh - martin - peterson ( bkmp2 ) reactive potential energy surface.@xcite the classical transition state of this system has a collinear configuration with equal bond lengths @xmath142 . a suitable reaction coordinate is therefore given by the difference of the bond lengths , @xmath143    figure [ fig : h3 ] ( a ) shows the temperature dependence of the rate constant in the range from @xmath144 to @xmath4 . the exact qm  results are from ref . . at high temperatures the tst curve is tangent to the exact qm curve , but at low temperatures , there is a significant discrepancy even for the tst with the wigner tunneling correction . on the other hand , the qi approximation agrees very well with the exact qm result all the way to @xmath144 . the relative error of the rate constant ratio is shown in fig . [ fig : h3 ] ( b ) which confirms that the error of the qi approach is within @xmath5 in the full temperature range whereas all other approximations have huge errors already for temperatures as high as @xmath145 . h@xmath146h@xmath147h reaction . ( a ) temperature dependence of the rate constant . ( b ) temperature dependence of the relative error of the ratio @xmath148 . ]    in case of the @xmath141 reaction , the ve had to be corrected for both @xmath135 and @xmath50 calculations . the reason is that the virial theorem only holds for bound systems . the transition state of the @xmath149 system can translate freely as a whole and the three translational degrees of freedom yield a correction of @xmath150 to the ve . in the reactant region , both the @xmath151 atom and @xmath152 molecule can move freely and the six translational degrees of freedom give a correction of @xmath153 to the ve . the temperature dependence of the reactant and transition state energies is shown in fig . [ fig : h3_ti ] . while both curves are quite smooth , small discretization errors in the integrals can have large effects on the exponentiated result . by an error analysis described in sec . iii , we found that the simpson method was sufficient for the ti over the whole temperature range . h@xmath146h@xmath147h reaction . temperature dependence of the reactant ( a ) and transition state ( b ) energies . ]    figures [ fig : h3_evsp_adhoc ] and [ fig : h3_evsp_vconstr_propto_t ] show how the constraint energy @xmath50 and the rmse of @xmath50 depend on @xmath58 for @xmath137 . panel ( a ) of fig . [ fig : h3_evsp_adhoc ] , which uses the estimators ( [ e_ts_te_adh])-([e_ts_cve_adh ] ) and a constraint potential independent of temperature , shows again that the te , corrected ve , and cve give approximately the same results for all values of @xmath58 and , within a statistical error , converge to the same limiting value for @xmath66 . panel ( b ) of fig . [ fig : h3_evsp_adhoc ] shows that while the statistical error of the cve is approximately constant as a function of @xmath58 , the rmses of the te  and ve grow with @xmath58 . however , in this case , the results are quite well converged for @xmath154 and at this point the rmse of the cve is still larger than the rmse of the te , although it is already smaller than the rmse of the ve . while for lower temperatures where larger values of @xmath58 are needed , cve would eventually become the optimal estimator , it is not so for @xmath137 . the growth of the rmse of the ve with @xmath58 is due to the fact that unlike for the eckart barrier , the transition state of the h@xmath57 system can move freely as a whole . h@xmath146h@xmath147h reaction at @xmath137 . the constraint potential is independent of temperature and estimators ( [ e_ts_te_adh])-([e_ts_cve_adh ] ) are used . ]    finally , fig . [ fig : h3_evsp_vconstr_propto_t ] shows analogous results , still using estimators ( [ e_ts_te_adh])-([e_ts_cve_adh ] ) , but obtained with a constraint potential ( [ v_constr_propto_t ] ) proportional to @xmath7 ( chosen such that the two types of constraints coincide for @xmath137 ) . as expected , the statistical errors of the te , ve , and cve are similar to those in fig . [ fig : h3_evsp_adhoc ] [ obtained with the constraint potential ( [ v_constr ] ) independent of @xmath7 ] . h@xmath156h@xmath147h reaction at @xmath137 . the constraint potential is proportional to temperature and estimators ( [ e_ts_te_adh])-([e_ts_cve_adh ] ) are used . ] a general method for the direct evaluation of the temperature dependence of the quantum rate constant was presented . the main advantage of this method is the increased efficiency : evaluating the temperature dependence directly , without computing the rate constant at any given temperature , allows us to avoid a tedious umbrella sampling procedure . besides efficiency , the direct calculation of the temperature dependence of the rate constant can also improve the accuracy : our ratios @xmath157 for both the eckart barrier and the @xmath158 reaction have somewhat smaller relative errors than the errors obtained for the absolute qi rate constants in previous studies of these systems.@xcite the smaller relative error in the ratio of rate constants is due to a favorable cancellation of various systematic errors , such as the systematic error of about @xmath159 of the qi model at high temperatures ( that can also be removed by an ad hoc correction of @xmath72)@xcite and small recrossing effects in the @xmath141 reaction , also at high temperatures . it is noteworthy that for both reactions , the rmses of transition state energies depend on the strength of the constraint . weakening the constraint facilitates sampling of the configuration space and the error of the cve decreases , approaching the well - known unconstrained situation where the cve is typically the optimal estimator . however , at the same time the constraint must be strong enough to exert the constraining effect and describe the situation near the transition state properly . as a result , `` ranking '' of the estimators is not universal but can change with the potential used as a constraint and is in general different from the ranking for unconstrained simulations . the dependence of the error of the ve on @xmath58 is best understood in terms of the ring polymer interpretation@xcite of the discretized pi : the quantum thermodynamics of the original system can be interpreted as the classical thermodynamics of the ring polymer . the constrained pi simulation for the one - dimensional eckart barrier is completely bound , resulting in the rmse independent of @xmath58 . in the full - dimensional hydrogen exchange reaction , on the other hand , even the constrained simulation allows the system to move as a whole . this is exactly where the ve is known to have a rmse increasing with @xmath58 . the cve estimator is usually the optimal estimator in unconstrained systems with some translational ( i.e. , free - particle ) degrees of freedom . in a system in which only two slices are bound ( in our case , slices @xmath160 and @xmath58 ) , the symmetry between different slices is lost and so is , to some extent , the advantage of subtracting the centroid . this explains why the rmse of the ve can sometimes be smaller than the rmse of the cve for all values of @xmath58 , which we observed in figs . [ fig : eckart_evsp_adhoc ] and [ fig : eckart_evsp_rigorous ] .    to sum up , while in generic systems at very low temperatures , the cve is expected to be the optimal estimator for energy , at finite temperatures in constrained simulations , the ve or even the te can have the smallest rmse . the results obtained in this paper can serve as a guide for choosing the best estimator for a given system . however , since the additional cost of evaluating all three estimators is negligible in comparison to the cost of the pimc random walk or pimd simulation , we recommend computing all three estimators , evaluating their rmses , and using the one with the smallest rmse in a given situation . this research was supported by the swiss nsf ( grant no . @xmath161 ) and by the epfl . authors thank tom zimmermann for helpful discussions .
a general method for the direct evaluation of the temperature dependence of the quantum - mechanical reaction rate constant in many - dimensional systems is described . 
 the method is based on the quantum instanton approximation for the rate constant , thermodynamic integration with respect to the inverse temperature , and the path integral monte carlo evaluation . 
 it can describe deviations from the arrhenius law due to the coupling of rotations and vibrations , zero - point energy , tunneling , corner - cutting , and other nuclear quantum effects . 
 the method is tested on the eckart barrier and the full - dimensional h @xmath0h@xmath1  h@xmath2 h reaction . in the temperature range from @xmath3 to @xmath4 , 
 the error of the present method remains within @xmath5 despite the very large deviations from the arrhenius law . 
 the direct approach makes the calculations much more efficient , and the efficiency is increased even further ( by up to two orders of magnitude in the studied reactions ) by using optimal estimators for reactant and transition state thermal energies . 
 which of the estimators is optimal , however , depends on the system and the strength of constraint in a constrained simulation .