document: polymers with contour length @xmath1 much larger than the persistence length @xmath2 , which is the correlation length for the tangent - tangent correlation function along the polymer and is a quantitative measure of the polymer stiffness , are flexible and are described by using the tools of quantum mechanics and quantum field theory @xcite-@xcite . 
   if the chain length decreases , the chain stiffness becomes an important factor . 
   many polymer molecules have internal stiffness and can not be modeled by the model of flexible polymers developed by edwards @xcite . 
   the standard coarse - graining model of a wormlike polymer was proposed by kratky and porod @xcite . 
   the essential ingredients of this model are the penalty for the bending energy and the local inextensibility . 
   the latter makes the treatment of the model much more difficult . 
   there have been a substantial number of studies of the kratky - porod model in the last half century @xcite-@xcite ( and citations therein ) . in recent years 
   there has been increasing interest in the theoretical description of semiflexible polymers @xcite-@xcite . 
   the reason for this interest is due to potential applications in biology allemand05 ( and citations therein ) and in research on semicrystalline polymers @xcite . 
   it was found in the recent numerical work  by lattanzi et al . 
   lattanzi04 , and studied analytically in @xcite within the effective medium approach , that the transverse distribution function of a polymer embedded in two - dimensional space possesses a bimodal shape for short polymers , which is considered to be a manifestation of the semiflexibility . the bimodal shape for the related distribution function of the 2d polymer was also found in recent exact calculations by spakowitz and wang @xcite . 
   in this paper we study the transverse distribution function @xmath3 of the three dimensional wormlike chain with a fixed orientation @xmath4 of one polymer end using the exact representation of the distribution function in terms of the matrix element of the green s function of the quantum rigid rotator in a homogeneous external field @xcite . 
   the exact solution of the green s function made it possible to compute the quantities such as the structure factor , the end - to - end distribution function , etc . 
   practically exact in the definite range of parameters @xcite , @xcite . 
   our practically exact calculations of the transverse distribution function of the 3d wormlike chain demonstrate that it possesses the bimodal shape in the intermediate range of the chain lengths ( @xmath0 ) . 
   in addition , we present analytical results for short and long wormlike chain based on the exact formula ( [ gtkp ] ) , which are in complete agreement with the previous results obtained in different ways @xcite ( wkb method for short polymer ) , @xcite ( perturbation theory for large chain ) .    the paper is organized as follows . 
   section [ sect1 ] introduces to the formalism and to analytical considerations for short and large polymers . 
   section [ numer ] contains results of the numerical computation of the distribution function for polymers with different number of monomers . 
   the fourier - laplace transform of the distribution function of the free end of the wormlike chain with a fixed orientation @xmath5 @xmath6 of the second end is expressed , according to @xcite , in a compact form through the matrix elements of the green s function of the quantum rigid rotator in a homogeneous external field @xmath7 as @xmath8where @xmath9 , and @xmath7 is defined by @xmath10with @xmath11 and @xmath12 being the infinite order square matrices given by @xmath13and @xmath14 . 
   the matrix @xmath11 is related to the energy eigenvalues of the free rigid rotator , while @xmath12 gives the matrix elements of the homogeneous external field . 
   since @xmath7 is the infinite order matrix , a truncation is necessary in the performing calculations . 
   the truncation of the infinite order matrix of the green s function by the @xmath15-order matrix contains all moments of the end - to - end chain distance , and describes the first @xmath16 moments exactly . 
   the transverse distribution function we consider , @xmath3 , is obtained from @xmath17 , which is determined by eqs . 
   ( [ gtkp])-([d ] ) , integrating it over the @xmath18-coordinate , and imposing the condition that the free end of the chain stays in the @xmath19 plane . as a result 
   we obtain @xmath20 is the bessel function of the first kind abramowitzstegun . taking the @xmath18-axis to be in the direction of @xmath21 yields @xmath22 , so that the arguments of the legendre polynomials in eq . 
   ( [ gtkp ] ) become zero , and consequently only even @xmath23 will contribute to the distribution function ( [ gyn ] ) . 
   we now will consider the expansion of ( [ gtkp ] ) around the rod limit @xmath24 , which corresponds to the expansion of @xmath25 in inverse powers of @xmath26 . to derive such an expansion 
   , we write @xmath11 in the equivalent form as@xmath27with @xmath28 and @xmath29 . 
   further we introduce the notation @xmath30 with @xmath31 and @xmath32 defined by@xmath33the iteration of @xmath11 and @xmath34 results in the desired expansion of @xmath32 and consequently of @xmath35 in inverse powers of @xmath26 , which corresponds to an expansion of @xmath36 in powers of @xmath37 . 
   the leading order term in the short chain expansion is obtained by replacing @xmath11 by @xmath38 in eq . 
   ( [ gtkp ] ) as @xmath39 _ { 0l}\sqrt{2l+1}p_{l}(\mathbf{t}_{0}\mathbf{n } ) . \label{gtkp0}\]]the latter coincides with the expansion of the plane wave landau - lifshitz3@xmath40where @xmath41 is the angle between the tangent @xmath4 and the wave vector @xmath42 . 
   the connection of @xmath43 with the plane wave expansion is due to the fact that the kratky - porod chain becomes a stiff rod in the limit of small @xmath37 . 
   we have checked the equivalency between the plane wave expansion ( [ plw ] ) and the distribution function ( [ gtkp0 ] ) term by term expanding ( [ gtkp0 ] ) in series in powers of the wave vector @xmath44 . 
   the arc length @xmath37 is equivalent for stiff rod in units under consideration to the chain end - to - end distance @xmath45 . in the @xmath45 space 
   the plane wave ( [ plw ] ) corresponds to the stiff rod distribution function @xmath46    the iteration of @xmath11 in ( [ dit ] ) and @xmath34 in ( [ y ] ) generates an expansion of @xmath35 in inverse powers of @xmath26 . 
   the corrections to the plane wave to order @xmath47 are obtained as@xmath48 _ { 0l}\sqrt{2l+1}% p_{l}(\mathbf{t}_{0}\mathbf{n})+ .... \end{aligned}\ ] ]    the above procedure yields for @xmath49 the short chain expansion of the distribution function of the free kratky - porod chain , which was studied recently in @xcite . 
   unfortunately , we did not succeed yet in analytical evaluation of @xmath50 . 
   such computation would be an interesting alternative to the treatment of the short limit of the wormlike chain by yamakawa and fujii @xcite within the wkb method . nevertheless , following the consideration in stepanow05 we succeeded in computing the anisotropic moments @xmath51 for small @xmath37@xmath52the first - order correction coincide with that obtained in @xcite using the wkb method , while the second - order correction is to our knowledge new . 
   the higher - order terms in ( [ rt_0 ] ) can be established in a straightforward way using the present method . 
   note that the computation of @xmath53 does not require the knowledge of the full distribution function @xmath54 .      in studying the end - to - end distribution function for large @xmath37 
   we utilize the following procedure . we expand first the expression @xmath55 in powers of @xmath56 . 
   the structure of this expansion for @xmath49 is presented in table [ table1 ] . 
   the subseries in powers of @xmath56 in the @xmath57th column are denoted by @xmath58 . 
   thus we have @xmath59the series @xmath58 with small values @xmath23 and @xmath57 possess a simple structure and can be summed up . 
   for example @xmath60 and @xmath61 are given by @xmath62@xmath63while @xmath61 corresponds to the distribution function of the gaussian chain , @xmath64 give the @xmath57th correction to the gaussian distribution . 
   the inspection of the series for @xmath65 and @xmath66 shows that they are expressed by @xmath61 and @xmath67 as@xmath68however , it seems that there is no general recursion relation for @xmath69 . 
   the results of computations of @xmath70 for @xmath71 by taking into account @xmath72 ( @xmath49 ) and @xmath73 ( @xmath74 ) are summarized in table [ table2 ] .    . 
   [ cols="^,<",options="header " , ]     the inverse laplace - fourier transform of @xmath70 given in the table yields the expansion of the end - to - end distribution function @xmath75 to order @xmath76 as @xmath77 ,   \label{grt}\end{aligned}\]]where @xmath78 , @xmath79 , and @xmath80 is the angle between @xmath81 and @xmath82 . 
   the latter is in accordance with the result by gobush et . 
   @xcite derived in a different way . 
   the expansion of @xmath83 for large @xmath37 can be extended in a straightforward way to include higher - order corrections . 
   the computation of the distribution function of the polymer with the fixed orientation of one end is performed by truncating the infinite order matrices in ( [ gtkp ] ) with the finite ones , and by taking into account the finite number of terms in the summation over @xmath23 . 
   the inverse laplace transform of ( [ gtkp ] ) is carried out with maple . 
   the results of the calculation of the distribution function @xmath3 ,     for various chain lengths , computed with @xmath84 matrices . 
   the insets show the distribution function at the onset of bimodality , and in the region of its disappearance . ]    using the truncations with @xmath84 order matrices and restricting the summation over the quantum number @xmath23 at @xmath85 , are given in fig . 
   fig - doublepeak . 
   the results show that the distribution function possesses the bimodal shape at intermediate chain lengths within the interval @xmath86 . 
   we also find that the distribution function becomes gaussian for very short and very long chains . at the onset 
   there is a tiny maximum at @xmath87 , which we interpret as remnant of the gaussian behavior of short chains . 
   the maximum at @xmath87 for 3d chain is a rather small effect , which is difficult to be explained in a qualitative way . 
   we now will discuss qualitatively the origin of the bimodal behavior of the projected distribution function of the free end of the wormlike chain . 
   the very short wormlike chain behaves similar to a weakly bending stiff rod , so that the distribution function of the free end is gaussian with the maximum at @xmath87 . 
   the typical conformation of the chain in this regime looks like a bending rod with constant sign of curvature along the chain . for larger contour lengths 
   the curvature fluctuations are small and are still controlled by the bending energy , however with varying sign of curvatures along the chain . 
   the typical conformation of the chain can be imagined as undulations along the average conformation of the polymer . 
   the projected distribution function of the free end in this regime is expected to be roughly uniform within some range of @xmath88 . 
   we expect that the inhomogeneities of curvature fluctuations in the vicinity of the clamped end are the reason for the maximum at @xmath89 . 
   the larger curvatures in the vicinity of the fixed end result in larger displacements @xmath88 of the free end , and therefore contribute preferentially to the maximum at @xmath89 . with further increase of @xmath37 
   the conformations correspond to undulations around the average conformation of the chain , which is now a meandering line . 
   fluctuations become now less controlled by bending energy , which results in weakening of the bimodality . since the difference between 2d and 3d chain is assumed to be marginal for short chains on the projected distribution function , we will compare the onset of the bimodality in both cases . 
   because the transversal displacement is measured in both cases in units of the contour length , we have to recompute the number of segments for 3d and 2d chain at the onset according to @xmath90using the dependence of the persistence length on dimensionality benetatos05 , @xmath91 , we obtain @xmath92 . according to @xcite @xmath93 at the onset , 
   hence we obtain @xmath94 , which is not far from our numerical result , @xmath95 ( see fig . 
   [ fig - doublepeak ] ) .     for chain length @xmath96 . 
   squares : truncation with @xmath97 matrices ; continuous line : truncation with @xmath84 matrices . ] 
   we now will address the issues of accuracy of the calculations , which depends on the size of the matrices @xmath11 and @xmath12 , and the maximal @xmath23 at which the summation over @xmath23 is stopped . in order to check our computation 
   we verified that at large @xmath37 ( @xmath98 ) the numerical evaluation of ( gyn ) gives with very high accuracy the gaussian distribution @xmath99 . 
   the general tendency is such that the sufficient level of matrix truncations and the number of terms in the expansion over the legendre polynomials increase with decreasing @xmath37 . in the limit @xmath24 
   the whole series over @xmath23 should be taken into account . 
   the accuracy of the computations is demonstrated in fig . 
   [ fig - n0 - 5 ] showing the computation of the distribution function for @xmath96 . 
   the squares and the continuous curve correspond to the truncations by @xmath97 matrices and @xmath84 matrices , respectively . in both cases the summation was stopped at @xmath85 . 
   the corrections due to higher @xmath23-s are negligibly small . 
   for example , the corrections associated with @xmath100 and @xmath101 contribute only in 3rd and 5th decimal digits , respectively . 
   thus , the computations depicted in figs . [ fig - doublepeak ] , [ fig - n0 - 5 ] can be considered as exact .    figure [ figure3 ] shows the results of the computation of the 3d distribution functions @xmath102 of the free polymer obtained by performing the inverse laplace - fourier transform of the term @xmath49 in eq . 
   ( [ gtkp ] ) for different chain lengths , and its comparison with the monte carlo simulations @xcite .    , @xmath103 , @xmath104 , @xmath105 , @xmath106 ( from left to right ) computed with @xmath107 matrices . 
   the symbols are the monte carlo simulation data extracted from fig . 1 in @xcite . 
   ]    our results are in excellent agreement with the numerical data . 
   to conclude , we have studied the transverse distribution function of the free end of the three dimensional wormlike chain with fixed orientation and position of the second end using the exact solution for the green s function of the wormlike chain . within the procedure of truncations of the exact formula with finite order matrices 
   we find that the distribution function @xmath108 for intermediate chain lengths , belonging to the interval @xmath0 , possesses the bimodal shape with the maxima at a finite value of the transverse displacement , which is consistent with the recent studies @xcite and @xcite for the two dimensional chain . 
   in contrast to the 2d wormlike chain , the transverse 1d distribution function of the 3d chain shows only a tiny peak at @xmath87 in the vicinity of the onset of bimodality , which however disappears for larger @xmath37 . 
   we present also results of analytical considerations for short and large polymers which are in complete agreement with the classical works @xcite where these limits were investigated using different methods . 
   the computation of the three dimensional distribution function of a free polymer is in excellent agreement with the monte carlo simulations @xcite .
summary: we study the distribution function of the three dimensional wormlike chain with a fixed orientation of one chain end using the exact representation of the distribution function in terms of the green s function of the quantum rigid rotator in a homogeneous external field . 
   the transverse 1d distribution function of the free chain end displays a bimodal shape in the intermediate range of the chain lengths ( @xmath0 ) . 
   we present also analytical results for short and long chains , which are in complete agreement with the results of previous studies obtained using different methods .