development of linear scaling methods , or order - n methods , is of primary importance in quantum chemistry . the former linearizes every step of hartree - fock ( hf ) or kohn - sham ( ks ) schemes . nowadays , the construction of hf and ks matrices is linear [ 18 ] . to reach this goal and to solve hf / ks equations several techniques are applied , for example , prescreening techniques , the continuous fast multipole method [ 1 , 1014 ] , order - n exchange , near - field exchange , tree - code approaches [ 14 , 17 ] , density fitting , density matrix minimization techniques [ 1821 ] , linear scaling quadrature methods [ 22 , 23 ] , sparse matrix algebra [ 24 , 25 ] , code parallelization , and gpu accelerators . it assumes that the electronic structure of a given molecular fragment is predominantly determined by its nearest neighborhood and the influence of very distant fragments is negligible . they differ in conditions imposed on mutual interactions among fragments . in the most advanced fragment molecular orbital ( fmo ) method [ 3034 ] and elongation ( elg ) method [ 3541 ] all interactions are rigorously taken into account . the opposite situation appears when fragments are treated completely independently [ 42 , 43 ] . however , one should remember that by simplifying the interactions and narrowing the variation space a systematic error is introduced . in this paper we describe a simple way to improve the accuracy of the fragmentation methods . the self consistent field ( scf ) calculations for selected molecular fragments are performed in the electrostatic field created by the point - charge distribution of the fragment s complementary part . the field diminishes in each step of the elongation procedure and finally disappears after the electronic structure of the whole system is obtained . this intermediate electrostatic field introduces the long - distance polarization into elg calculations and is a step beyond the nearsightedness approximation . there have been some attempts to include the polarization effects in charged polymers , however , they limited the fragment s complementary part to charged groups only . here , we propose a uniform treatment of the fragment s complementary part . point charge distribution was also used by exner and mezey   to diminish the distance criterion in the adjustable density matrix assembler method . first , we briefly describe the elg methods and charge sensitivity analysis ( csa ) used to derive the point - charge distributions . the electronic structure of the whole molecular system ( m ) is synthesized by enlarging a so called starting cluster ( m1 ) . next , the size of the system is enlarged by adding a new molecular fragment c1 . the localized molecular orbitals ( lmos ) assigned to fragment a1 , which is far away from the chain propagation center , the lmos assigned to ai are excluded from variation space ( they are frozen in the elg process ) . the lmos assigned to bi together with mos of ci constitute the active space ( si ) . the whole elg procedure can be summarized as follows:1a\documentclass[12pt]{minimal } 
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				 \begin{document}$$ { m}_1\overset{\mathrm{lmo}}{=}\left({a}_1\big|{b}_1\right ) $ $ \end{document}m1=lmoa1|b11b\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \left\{\begin{array}{l}{m}_2=\left({a}_1\big|{b}_1+{c}_1\right)\overset{\mathrm{scf}}{\equiv}\left({a}_1\big|{s}_1\right)\overset{\mathrm{lmo}}{=}\left({a}_1+{a}_2\right|{b}_2\left)\equiv \right({a}^2\left|{b}_2\right)\\ { } { m}_3=\left({a}^2\big|{b}_2+{c}_2\right)\overset{\mathrm{scf}}{\equiv}\left({a}^2\big|{s}_2\right)\overset{\mathrm{lmo}}{=}\left({a}^2+{a}_3\right|{b}_3\left)\equiv \right({a}^3\left|{b}_3\right)\\ { } \kern2em \vdots \\ { } { m}_{n-1}=\left({a}^{n-2}\big|{b}_{n-2}+{c}_{n-2}\right)\overset{\mathrm{scf}}{\equiv}\left({a}^{n-2}\big|{s}_{n-2}\right)\overset{\mathrm{lmo}}{=}\left({a}^{n-2}+{a}_{n-1}\right|{b}_{n-1}\left)\equiv \right({a}^{n-1}\left|{b}_{n-1}\right)\end{array}\right . $ $ \end{document}m2=a1|b1+c1scf(a1|s1)=lmo(a1+a2|b2)(a2|b2)m3=a2|b2+c2scf(a2|s2)=lmo(a2+a3|b3)(a3|b3)mn1=an2|bn2+cn2scf(an2|sn2)=lmo(an2+an1|bn1)(an1|bn1)1c\documentclass[12pt]{minimal } 
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				 \begin{document}$$ m={m}_n=\left({a}^{n-1}\big|{b}_{n-1}+{c}_{n-1}\right)\overset{\mathrm{scf}}{\equiv}\left({a}^{n-1}\big|{s}_{n-1}\right ) $ $ \end{document}m = mn = an1|bn1+cn1scfan1|sn1 equations  1a , 1b , and 1c correspond to initialization , chain propagation , and chain termination , respectively . notice that there is no need to localize mo in the final step ( eq . one can eventually take advantage of the sparseness in the lmo representation [ 39 , 40 , 52 , 53 ] . in the limit of perfect localization , lmos assigned to the frozen fragment ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \mathbf{c}}_{a^i } $ $ \end{document}cai ) have no tails in the active fragment and vice versa , lmos assigned to the active fragment ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \mathbf{c}}_{s_i } $ $ \end{document}csi ) have no tails in the frozen one . therefore , instead of constructing the fock or kohn - sham matrix of mi-1 ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \mathbf{f}}_{m_{i-1}{m}_{i-1}}^{\mathrm{ao } } $ $ \end{document}fmi1mi1ao ) , the \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \mathbf{f}}_{s_i{s}_i}^{\mathrm{ao } } $ $ \end{document}fsisiao block is built . such elongation cutoff scheme ( elg / c ) operates within a low - dimensional subspace,2\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \begin{array}{l}{\boldsymbol{c}}_{s_i}^{\dagger } { \boldsymbol{f}}_{i+1}{\boldsymbol{c}}_{s_i}=\kern1em \left(\begin{array}{cc}\hfill { \mathbf{0}}_{a^i{s}_i}^{{}^{\dagger } } \hfill & \hfill { \mathbf{c}}_{s_i{s}_i}^{\dagger}\hfill \end{array}\right)\kern0.5em \left(\begin{array}{cc}\hfill { \mathbf{f}}_{a^i{a}^i}^{\mathrm{ao}}\hfill & \hfill { \mathbf{f}}_{a^i{s}_i}^{\mathrm{ao}}\hfill \\ { } \hfill { \mathbf{f}}_{s_i{a}^i}^{\mathrm{ao}}\hfill & \hfill { \mathbf{f}}_{s_i{s}_i}^{\mathrm{ao}}\hfill \end{array}\right)\kern0.5em \left(\begin{array}{c}\hfill { \mathbf{0}}_{a^i{s}_i}\hfill \\ { } \hfill { \mathbf{c}}_{s_i{s}_i}\hfill \end{array}\right)\\ { } \kern5em \equiv { \mathbf{f}}_{s_i{s}_i}^{\mathrm{mo}}={\mathbf{c}}_{s_i{s}_i}^{\dagger } { \mathbf{f}}_{s_i{s}_i}^{\mathrm{ao}}{\mathbf{c}}_{s_i{s}_i}\end{array } $ $ \end{document}csifi+1csi=0aisicsisifaiaiaofaisiaofsiaiaofsisiao0aisicsisifsisimo = csisifsisiaocsisiand avoids the known bottleneck of the scf calculations , i.e. , diagonalization . the lower indices in eq . ( 2 ) are introduced to indicate dimensions and the block \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \mathbf{0}}_{a^i{s}_i } $ $ \end{document}0aisi is filled with zeros . in addition , the number of two - electron repulsion integrals is substantially reduced as long as the total energy of mi is not needed . it was demonstrated that elg / c scheme is linear in cpu time at hf   and ks   levels of theory for a linear or quasi - linear polymer . in such a generalized elg scheme the frozen lmos of a given fragment can reenter the variation space [ 5456 ] . in the elg ( elg / c ) scheme the molecular fragments are not treated equivalently . namely , the starting cluster does not know its  future  while sn-1 possesses the whole knowledge of m. such way of building m is consistent with the nearsightedness approximation   which is a common assumption in the fragmentation methods . the elg ( elg / c ) process can be performed in an approximate field arising from complementary part of mi , for example its point charge distribution . the ith subsystem s complementary part is denoted by \documentclass[12pt]{minimal } 
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				 \begin{document}$$ \overline{m_i } $ $ \end{document}mi. by enlarging the system size ( i    n ) the complementary part becomes smaller and finally ( i  =  n ) disappears ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$ \overline{m_n } $ $ \end{document}mn is a zero set ) . this electrostatic field is an intermediate construction , therefore , the final energy of m should be greater than the reference hf ( ks ) one . csa is based on second - order taylor expansion of the system s energy em with respect to atomic charges . the csa formalism in global resolution ( without constraints on charge flow ) can be summarized in a single matrix equation : 3\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \left(\begin{array}{cc}\hfill 0\hfill & \hfill \begin{array}{cccc}\hfill 1\kern1em \hfill & \hfill 1\kern1em \hfill & \hfill \cdots \kern0.62em \hfill & \hfill 1\hfill \end{array}\hfill \\ { } \hfill \begin{array}{c}\hfill 1\hfill \\ { } \hfill 1\hfill \\ { } \hfill \vdots \hfill \\ { } \hfill 1\hfill \end{array}\hfill & \hfill \begin{array}{cccc}\hfill { \eta}_{11}\hfill & \hfill { \eta}_{12}\hfill & \hfill \cdots \hfill & \hfill { \eta}_{1n}\hfill \\ { } \hfill { \eta}_{21}\hfill & \hfill { \eta}_{22}\hfill & \hfill \cdots \hfill & \hfill { \eta}_{2n}\hfill \\ { } \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ { } \hfill { \eta}_{n1}\hfill & \hfill { \eta}_{n2}\hfill & \hfill \cdots \hfill & \hfill { \eta}_{nn}\hfill \end{array}\hfill \end{array}\right)\kern0.5em \left(\begin{array}{c}\hfill -\chi \hfill \\ { } \hfill \begin{array}{c}\hfill { q}_1\hfill \\ { } \hfill { q}_2\hfill \\ { } \hfill \vdots \hfill \\ { } \hfill { q}_n\hfill \end{array}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill q\hfill \\ { } \hfill \begin{array}{c}\hfill -{\chi}_1^{*}\hfill \\ { } \hfill -{\chi}_2^{*}\hfill \\ { } \hfill \vdots \hfill \\ { } \hfill -{\chi}_n^{*}\hfill \end{array}\hfill \end{array}\right ) $ $ \end{document}011111111121n21222nn1n2nnq1q2qn = q1*2*n*where   =  { ij  =  em/qiqj } is the hardness matrix , q is the total charge and   =  em/q is the global electronegativity [ 59 , 60 ] of a molecular system m composed of n atoms . vectors q  =  ( q1 ,  q2 ,    qn ) and   =  ( 1 * ,  2 * , the first equation in ( 3 ) is a closure relation:4\documentclass[12pt]{minimal } 
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				 \begin{document}$$ q={\displaystyle \sum_{i=1}^n{q}_i } $ $ \end{document}q=i=1nqi the remaining equations5\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \chi = { \chi}_i={\chi}_i^{*}+{\displaystyle \sum_{j=1}^n{\eta}_{ij}{q}_j},\kern1em j=1,\kern0.5em 2,\kern1em , \kern0.5em n $ $ \end{document}=i=i*+j=1nijqj , j=1,2,,nare the electronegativity equalization equations ( 1  =  2  =    =  n  =   ) . ( 3):6\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \kern0.5em \left(\begin{array}{c}\hfill -\chi \hfill \\ { } \hfill \begin{array}{c}\hfill { q}_1\hfill \\ { } \hfill { q}_2\hfill \\ { } \hfill \vdots \hfill \\ { } \hfill { q}_n\hfill \end{array}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill -\eta \hfill & \hfill \begin{array}{cccc}\hfill { f}_1\kern1em \hfill & \hfill { f}_2\hfill & \hfill \kern1.5em \cdots \hfill & \hfill { f}_n\hfill \end{array}\hfill \\ { } \hfill \begin{array}{c}\hfill { f}_1\hfill \\ { } \hfill { f}_2\hfill \\ { } \hfill \vdots \hfill \\ { } \hfill { f}_n\hfill \end{array}\hfill & \hfill \begin{array}{cccc}\hfill -{\beta}_{11}\hfill & \hfill -{\beta}_{12}\hfill & \hfill \cdots \hfill & \hfill -{\beta}_{1n}\hfill \\ { } \hfill -{\beta}_{21}\hfill & \hfill -{\beta}_{22}\hfill & \hfill \cdots \hfill & \hfill -{\beta}_{2n}\hfill \\ { } \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ { } \hfill -{\beta}_{n1}\hfill & \hfill -{\beta}_{n2}\hfill & \hfill \cdots \hfill & \hfill -{\beta}_{nn}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill q\hfill \\ { } \hfill \begin{array}{c}\hfill -{\chi}_1^{*}\hfill \\ { } \hfill -{\chi}_2^{*}\hfill \\ { } \hfill \vdots \hfill \\ { } \hfill -{\chi}_n^{*}\hfill \end{array}\hfill \end{array}\right ) $ $ \end{document}q1q2qn=f1f2fnf1f2fn11121n21222nn1n2nnq1*2*n * here ,   =  { ij  =  em/vivj  =    qj/vi  =    qi/vj } is the polarization matrix ( linear response matrix ) . the remaining undefined quantities are the global hardness [ 62 , 63 ]   =  em/q and the fukui function ( ff )   vector f  =  { fi  =  ( qi/q)v  =    ( /vi)q}. the hardness matrix  and the vector of atomic electronegativities  are the base parameters of csa . in the force - field atoms resolution they depend on the atomic number , hybridization and local chemical environment of atoms constituting the system . we have recently derived these parameters for different population analyses [ 64 , 65 ] . all elg ( elg / c ) calculations were performed at restricted open - shell hartree - fock ( rohf ) level of theory using gamess package [ 66 , 67 ] . three different basis sets , namely : sto-3 g , 6 - 31 g , and 6 - 31g(d ) , were applied . the starting cluster was built of 15 amino acid units . in each step of the elongation eight units were frozen and another eight units were added to the system . this elg propagation scheme is denoted as 15/8 . the elongation process was terminated for 55 glycine units . in the case of elg / c calculations , the cutoff procedure was initialized for a system made up of 31 units ( 15/8:31 ) . we have chosen this conformer since , for a given partitioning scheme , the error in its total energy was bigger than for other conformers ( c5 , c7 , and 310-helix ) . five different charge distributions were used , namely , bader ( b ) , hirshfeld ( h ) , mlliken ( m ) , natural ( n ) , and voronoi ( v )   population analyses . the errors in total ( tot ) , kinetic ( kin ) , and potential ( pot ) energies were defined with respect to the conventional ( supermolecule ) rohf energies:7\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \vardelta { e}_x={e}_x^{\mathrm{elg}}-{e}_x^{\mathrm{rohf}},\kern1.75em x=\left(\mathrm{tot},\mathrm{kin},\mathrm{pot}\right ) $ $ \end{document}ex = exelgexrohf , x = tot , kin , pot by definition em  =  etot is greater than zero since elg ( elg / c ) is a variational method . there are no such restrictions on its components , therefore , ekin ( epot ) may be either negative or positive . we have chosen rohf calculations since the propagation at rhf level of theory would require saturation of  broken  bonds . this means that each intermediate subsystem ( 15 , 23 , 31 , 39 , and 47 units ) would have to be saturated by a hydrogen atom . to avoid perturbation created by this artificial hydrogen atom rohf scheme one can expect that the intermediate electrostatic field should stabilize the system and should improve the scf convergence . the proposed modification of elg ( elg / c ) scheme has no influence on cpu time since one - electron integrals are computed only once , before the scf process . the errors introduced by elg and modified elg schemes are plotted in fig .  1 . parts a , b , and c correspond to sto-3 g , 6 - 31 g , and 6 - 31g(d ) basis sets , respectively . the error in the total energy the elg and modified elg methods are variational , therefore the calculated energies are higher than the reference rohf energies . the intermediate energies can not be directly compared with the reference rohf energies since they have quite a different meaning . the first error bar in each figure corresponds to standard elg calculations with \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \mathbf{q}}_{\overline{m_i}}=0 $ $ \end{document}qmi=0 ( no field ) . the remaining error bars correspond to different population analyses employed in the modified elg procedure . the error in elg calculation depends on the basis set and increases with its size . for 6 - 31g(d ) basis set it is about two - orders of magnitude greater than for the minimal basis set . the presence of the intermediate field arising from charges from every population analysis reduces the error in the total energy . the reproduction of conventional rohf energies is improved by about one order of magnitude for sto-3 g ( fig . the improvement for 6 - 31g(d ) basis is the most pronounced ( fig . the error is reduced by three orders of magnitude for b , m , and n population analyses . for all basis sets the modified elg scheme with v and h charges works slightly worse than with b , m , and n charges . the changeability in total energy with the basis set size is the smallest for b and n charges . the errors in total energies are equal to 0.01 , 0.02 , and 0.02  kcal mol for sto , 6 - 31 g , and 6 - 31g(d ) basis sets , respectively.fig . 1the error [ a.u . ] in the total energies introduced by 15/8 elg and modified elg schemes [ with bader ( b ) , hirshfeld ( h ) , mlliken ( m ) , natural ( n ) , and voronoi ( v ) charge distributions ] with respect to conventional rohf calculations for polyglycine chain made up of 55 units . parts ( a ) , ( b ) , and ( c ) correspond to sto-3 g , 6 - 31 g , and 6 - 31g(d ) basis sets the error [ a.u . ] in the total energies introduced by 15/8 elg and modified elg schemes [ with bader ( b ) , hirshfeld ( h ) , mlliken ( m ) , natural ( n ) , and voronoi ( v ) charge distributions ] with respect to conventional rohf calculations for polyglycine chain made up of 55 units . parts ( a ) , ( b ) , and ( c ) correspond to sto-3 g , 6 - 31 g , and 6 - 31g(d ) basis sets therefore , by taking the long - distance polarization into account , even in the simplified way , the reproduction of the reference rohf energy is much better and the error does not exceed 1  kcal mol . it should also be mentioned that , although fig .  1 corresponds to elg and modified elg schemes , they present the errors in the elg / c and modified elg / c schemes . the reason for that is that the cutoff error ( etotelg    etotelg / c ) is at least two orders of magnitude smaller than the elongation error ( etotelg    etotrohf ) . to understand the improvement in reproduction of the system s total energy we have decomposed etotelg into its potential ( epotelg ) and kinetic ( ekinelg ) components . the errors with respect to conventional potential ( epotrohf ) and kinetic ( ekinrohf ) energies are illustrated in fig . as it was mentioned in computational details , the errors in potential and kinetic energies may be either positive or negative . depending on the population analysis , the error in kinetic energy is positive ( b and n ) or negative ( h , m and v ) . regardless of the basis set and the population analysis , the error in potential energy always has the opposite sign to ekin . this error cancelation causes the accuracy in the total energy to be one order of magnitude greater than in its potential and kinetic components . however , one should remember that the virial theorem is exact for the true ground - state wave function . the approximate wave function fulfils it only approximately . for more extended basis sets [ 6 - 31 g and 6 - 31g(d ) ] , epot for modified elg scheme is negative and ekin is positive . in the case of standard elg scheme , epot is negative for 6 - 31 g and positive for 6 - 31g(d ) basis sets , respectively.fig . 2the error [ a.u . ] in the potential ( white bars ) and kinetic ( black bars ) energies introduced by 15/8 elg and modified elg schemes [ with bader ( b ) , hirshfeld ( h ) , mlliken ( m ) , natural ( n ) , and voronoi ( v ) charge distribution ] with respect to conventional rohf calculations for polyglycine chain made up of 55 units . parts ( a ) , ( b ) , and ( c ) correspond to sto-3 g , 6 - 31 g , and 6 - 31g(d ) basis sets the error [ a.u . ] in the potential ( white bars ) and kinetic ( black bars ) energies introduced by 15/8 elg and modified elg schemes [ with bader ( b ) , hirshfeld ( h ) , mlliken ( m ) , natural ( n ) , and voronoi ( v ) charge distribution ] with respect to conventional rohf calculations for polyglycine chain made up of 55 units . parts ( a ) , ( b ) , and ( c ) correspond to sto-3 g , 6 - 31 g , and 6 - 31g(d ) basis sets it is clear from the figures that the intermediate electrostatic field improves the reproduction of kinetic and potential energies . one should expect such behavior since slightly populated polarization functions on heavy atoms should be sensitive to the intermediate field used in the modified elg calculations . all the results of modified elg calculations at 6 - 31g(d ) basis set show that the kinetic energy is overestimated due to the influence of point charges . finally , let us analyze the scf convergence during elg calculations . all intermediate subsystems are radicals ( dublets ) . only the whole system is closed - shell . 3 . parts a , b , and c correspond to sto-3 g , 6 - 31 g , and 6 - 31g(d ) bases , respectively . one can observe that , except for b , m , and n charges at sto-3 g basis set calculations , the field stabilizes the radicals . it means that fewer scf cycles are required to reach the same assumed accuracy in comparison to standard elg scheme . the differences between elg and modified elg schemes are more pronounced for the starting cluster . for larger basis in general , the natural population analysis ( n ) gives the best performance of calculations both in accuracy and convergence . at the end of the elongation process the system is then closed - shell and the number of iterations in the scf cycle is small.fig . 3the scf convergence during elg and the modified elg [ with bader ( b ) , hirshfeld ( h ) , mlliken ( m ) , natural ( n ) , and voronoi ( v ) charge distribution ] calculations . parts ( a ) , ( b ) , and ( c ) correspond to sto-3 g , 6 - 31 g , and 6 - 31g(d ) basis sets the scf convergence during elg and the modified elg [ with bader ( b ) , hirshfeld ( h ) , mlliken ( m ) , natural ( n ) , and voronoi ( v ) charge distribution ] calculations . parts ( a ) , ( b ) , and ( c ) correspond to sto-3 g , 6 - 31 g , and 6 - 31g(d ) basis sets the field is exerted by distributed monopoles located in the positions of atoms of the system s complementary part and it disappears in the final stage of the elongation calculations . the modified elg scheme was tested for alpha - helix of polyglycine chain ( 55 glycine units ) . it was shown that the long - distance polarization , introduced by the field , improved the performance of the elg method . the errors in the total , kinetic , and potential energies were reduced by at least one order of magnitude . the natural , bader , and mlliken population analyses gave the best agreement with the reference , conventional rohf energies for the largest basis set . the modified elg method improved the convergence of the scf process at rohf level of theory . the temporary electrostatic field stabilized the intermediate radical , therefore fewer cycles were involved during the scf step .
a simple way to improve the accuracy of the fragmentation methods is proposed . 
 the formalism was applied to the elongation ( elg ) method at restricted open - shell hartree - fock ( rohf ) level of theory . 
 the -helix conformer of polyglycine was taken as a model system . 
 the modified elg method includes a simplified electrostatic field resulting from point - charge distribution of the system s environment . in this way 
 the long - distance polarization is approximately taken into account . 
 the field attenuates during the elg process to eventually disappear when the final structure is reached . 
 the point - charge distributions for each elg step are obtained from charge sensitivity analysis ( csa ) in force - field atoms resolution . 
 the presence of the intermediate field improves the accuracy of elg calculations . 
 the errors in total energy and its kinetic and potential contributions are reduced by at least one - order of magnitude . 
 in addition the scf convergence of rohf scheme is improved .