38 centered on te orgn, and for ostve ntegers K let P K () denote te set of lattce vectors n suc tat n=k P. Gven N onts r 1 ;::: ;r N n te unt cell U,

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1 37 Aendx D Ewald Summaton D.1 Introducton From macromolecular structure, to aqueous bologcal systems, accurate comutaton of electrostatc and van der Waals nteractons s te most dcult task n comuter modelng. Smulatons of etdes and membranes as well as of ons n aqueous solutons ave rovded clear-cut evdence of artfactual beavor due to te use of cutos. Works done by Yor 1 ave sowed usng current force eld wtout truncaton of Coulombc nteractons do not exbt smlar artfactual beavor. Te general aroac to ts roblem s te Ewald metod. D. Lattce Sums For Inverse Power Of Dstance Consder crystal lattce arwse sums of te tye E n = 1 C n q j q k r n j6=k jk (D.1) were n = 1 for Coulomb nteracton and n = for selded Coulomb nteracton, wle n =6ste London dserson term, or te van der Waals attracton sum. We'll derve te lattce summaton formulas used n our rogram for te case of dserson nteractons as well as for Coulombc nteractons. Tese lattce sums do not n general converge absolutely, so we need to secfy te asymtotc order of summaton, corresondng to te asymtotc sae of te nte crystal made u of te unon of lattce translatons of unt cell U. Let denote te set of all lattce vectors n = n 1 a 1 + n a + n 3 a 3. In order to descrbe te order of summaton n R 3, we ntroduce a closed, bounded regon P,

2 38 centered on te orgn, and for ostve ntegers K let P K () denote te set of lattce vectors n suc tat n=k P. Gven N onts r 1 ;::: ;r N n te unt cell U, and real constants C j, we consder 1 E (r 1 ;::: ;r N )= lm K!1 0 np K () ;j C j jr, r j + nj (D.) were te rme denotes tat terms wt = j and n = 0 are omtted. Let's start wt some denttes for te nverse owers 1=jrj, > 0, were r s any nonzero vector n R 3. Te followng two formulas are used,(x) = 1 1 t x,1 e,t dt = z t x,1 e,t dt 0 0 (D.3) and e,a w = a 1 0 e, u a e,wu du (D.4) were,(x) s te Euler gamma functon. Gven a 3-dmenson vector r, substtute = jrj = r and z = =. For arbtrary ostve number, we ten ave,( ) 1 = t r,1 e,rt dt + t,1 e,rt dt (D.5) 0 In te second term, f we substtute t by s, wt r t = s, we ave 1 t,1 e,rt dt = 1 1 r r s,1 e,s ds (D.6) For te rst term, we wrte r = x + y + z and aly Equaton (D.4) n all tree dmenson; and we ave 0 t,1 e,rt dt = 3 0 t, 5 R3 e, u t e,ur d 3 udt (D.7)

3 39 Integrate over t rst and substtutng t wt s, were u = ts,weave 3 0 t, 5 e, u t dt = 3,3 ( u 1 ),3 s, e,s ds u (D.8) Consder te recrocal unt cell U made u of te onts u n R 3 suc tat, 1 a u 1 and te fact tat R 3 can be decomosed as te unon of te onts sets U + m, over all recrocal vectors m, we ave 1 r = 3,3 m were we ave dened jv + mj f ( )e,(v+m)r d 3 v + g (r) U r (D.9) 1 f (x) = x,3,( ) s, e,s ds x (D.10) and g (x) = 1,( ) s,1 e,s ds x (D.11) also noted lm( 1 r!0 r, g (r) )= 1 r,( ) t,1 dt = 0,( ) (D.1) For v U we wrte v = w 1 a 1 + w a + w 3 a 3 were w k = v a k, k =1;;3. For r U and any lattce vector n = n 1 a 1 + n a + n 3 a 3, suc tat r + n 6= 0,we extend Equaton (D.9); cangng varables n te ntegral over U, we ave 1 jr + nj = 3,3 V m e,mr 1, 1 1, 1 1, 1 ;m;n (w)e,wn d 3 w + g (jr + nj) jr + nj (D.13)

4 40 were wt v = w a ;m;n (w) =f ( jv+mj )e,vr (D.14) Alyng te above formula to Equaton (D.), and usng te fact tat te sum of te Fourer coecents of te smoot, bounded functon ;m;r, m 6= 0 converges to ;m;r (0) = f ( jmj ), we can wrte E (r 1 ;::: ;r N ) = 1 0 n j,,( ) C C j g (jr, r j + nj) jr, r j + nj + 3,3 V m f ( jmj ) j C j e,m(r,r j ) (D.15) were te last term s te correcton term (self-energy) for r =0. D.3 Coulomb Sums D.3.1 Energy, Force, Stress Wen = 1 wc s Coulomb nteracton, we ave f 1 (x) = e,x ; g x 1(x) =erfc(x) (D.16) Ten we can wrte E 1 (r 1 ;::: ;r N ) = 1 0 n j, q q j erfc(jr, r j + nj) jr, r j + nj q + 1 lm K!1 np K () + 1 V U j m6=0 e, m m q q j e, m v S(m)S(,m) e,v(r,r j ) e,vn d 3 v (D.17)

5 41 were S(m) = q e,mr (D.18) s te Coulomb structure factor. Te last term dverges, but wen we aly a secondorder Taylor seres exanson to te functon e, m e,v(r,r j), exandng about v = 0. Te zerot and rst-order terms, wc account for te sngularty n te ntegral, are cancelled by te double summaton over and j for neutral unt cell. Te remander term, wc s of order tree, can be wrtten as J(D) = lm K!1 np K () U (v D) were D = P q r s te unt cell dole moment. v e,vn d 3 v (D.19) Followng te above argument, f we neglect te unt cell dole moment contrbuton (wc deends on te surface boundary of te bulk materal) and relace by =1=, weave E Coulomb = 1 L;;j Q j erfc(a) a + 0 S()S(,), e,b, 1 Q (D.0) were te rme ndcates tat te term at te orgn s excluded, a = jr, r j, R L j ; ~ =~m = H ~,1 ~n (D.1) wt matrx H contans te real sace unt cell vectors n Cartesan coordnates, and b = ; = det H (D.)

6 S() s te structure factor S() =C j q j e,r j (D.3) It was roved tat S()S(,) =C 1 ( q cos( r ) + ) q sn( r ) (D.4) Snce " 1 (r) = 1 1 r r r e,t t 1 dt # =, 1 e, r r r ( r, r r )=,e r (D.5) and S(,)S() r ; ( = C 1 q, sn( r ) q cos( r ) + cos( r ) = C 1 q q sn[ (r, r )] q sn( r ) ) (D.6) Te force on eac atom F ; =, E 1 r ;ala = C 1 erfc(a ) q q (r ;, r ;, R L; ) 3 L; + 4C 1 0 a 3 + e,a a e,b q q sn [ (r, r )] (D.7) were a = j~r, ~r, ~ R L j (D.8)

7 43 Te nternal stress can be calculated as were =, E 1 = 1, L;;j " # erfc(a) Q j + e,a a a 0 = L;;j 0 e,b S()S(,) " erfc(a) Q j + e,a a 3 a a ( 1 ), 1 # (e,b ) (r, r j, R L ) (r, r j, R L ) S()S(,) e,b ", 1+b # (D.9) r =~ H,1 n H s ; =n s (D.30) s ndeendent of H matrx, or, and a = 1 a = 1 a a (s, s j, S L ) G (s, s j, S L ) = 1 a (r, r j, R L ) (r, r j, R L ) (D.31) = 4 (n G,1 n ) G,1 = 4 n n G =,4 n G,1 G,1 n =, (D.3)

8 44 were A,1 =,A,1 B A B A,1 (D.33) was used. 1 =, 1 G G =, 1 G G,1 =, 1 (D.34) D.3. Accuracy Seced Cutos Wt seced tere s stll an nnte number of terms n te sums over te real sace and recrocal sace lattces, and an accuracy crtera s used to secfy lmts on tese sums. Ts s aceved by secfyng a tolerance and carryng out te sums untl te neglected terms ave a total contrbuton smaller tan. For structure otmzaton, te energy based cutos s desred, wle for dynamcs smulaton, te force based cutos s more arorate. We'll look at te two cases searately. Energy Based Cutos Usng a cuto dstance R cut ntroduces an error n te total energy for te real sace sum of E real = 1 L;;j were (R jl, R cut )steste functon. Q j erfc( RjL ) R jl (R jl, R cut ) (D.35) To estmate ts error, we relace te dscrete sum by a contnous ntegral. Den-

9 45 P ng te average nteracton as <q >= q =N, we ave By usng te nequalty E real ' N <q > 1 4R erfc( R R cut R ) dr (D.36) we obtan erfc( R )= t,1 e,t dt 1 R 1 1 R R e,t dt = 1 R e, R (D.37) E real N <q > 1 4 1, e R <q > dr = N erfc( R cut R cut ) (D.38) Usng a cuto H cut n te recrocal sace sum ntroduces an error n te total energy for te recrocal sace sum of E rec = 0 S()S(,) e,b (, H cut) (D.39) Relacng te sum by anntergral and relacng S()S(,) byn <q >,weobtan E rec ' N <q > Force Based Cutos H cut 4 e,( ) d = N <q > erfc( H cut 1 ) (D.40) For te real sace sum, te error ntroduced n force of atom by R cut can be wrtten as jf ;real j = 1 3 L; " erfc(a ) jq jr L + e,a #(R a 3 a L, R cut ) (D.41)

10 46 by usng average nteracton and relacng te sum wt ntegral, we obtan " # erfc(a) jf real j' N<q > 1 4R R 3 R cut a 3 + e,a a dr (D.4) furter usng Equaton (D.37), jf real j ' 4N <q > 1 Rcut 4N <q > 4N <q > = 4N <q > 1 Rcut 1 Rcut erfc(a)+ ae,a da 1 a e,a + ae,a da e,a + Rcut erfc( R cut )+e, ae,a da R cut (D.43) For recrocal sace sum, te error ntroduced n force of atom by R cut s jf ;rec j = 4 0 e,b Q sn ( r ) (, H cut) (D.44) Relacng te sum by ntegral and relacng P Q sn ( r ) by N < q >, we obtan jf rec j ' 4 N <q > 1 e,b 8 3 H cut d = N <q > 1 H cut e,b d = N <q > e, 1 4 H cut (D.45) For a gven, by usng Equaton (D.43) and Equaton (D.45) we can evaluate te cuto dstances R cut and H cut to obtan a gven accuracy Q. Because of te neutralty of te cell under wc Ewald calculaton s carred out, tere wll be a great deal of cancellaton wen we use te average nteracton <q >. Consequently, Equaton (D.43) and Equaton (D.45) overestmate te errors.

11 D.4 Dserson Sums 47 D.4.1 Energy, Force, Stress For = 6, wc s te London dserson nteracton, noted f 6 (x) = 1 3 (1, x )e,x +x 3 erfc(x) (D.46) and g 6 (x) = (1 + x + 1 x4 )e,x (D.47) we can wrte energy sum as E London = 1 C j (a,6 + a,4 +1a, )e,a 6 L;;j ;j C j cos[ (r, r j )] 3 " C j, ;j C 1 erfc(b)+ 1 b 3,1 b were a, b, and are dened n te revous secton. If we assume e,b # (D.48),C j = C C jj (D.49) we ave C j cos[ (r, r j )] =, jc j cos( r ), jc j sn( r ) ;j (D.50)

12 48 Te force on atom s F ; =, E London r ; = 1 8 L C (r, r, R L ) (6a,8 +6a,6 +3a,4 +a, )e,a 0 erfc(b)+( 1 C sn[ (r, r )] 3 b,1 3 b )e,b (D.51) Te stress = 1 8 L;;j C j (6a,8 +6a,6 +3a,4 0 ;j 0 ;j +a, )e,a (r,r,r L ) (r,r,r L ) " # 1 C j cos[ (r, r j )] 3 1 erfc(b)+( b,1 3 b )e,b C j cos[ (r, r j )]3 1 e erfc(b),,b + 3 b 6 3 ;j C j (D.5) D.4. Accuracy Seced Cutos Followng secton of Coulombnteracton, we'll dscuss te energy based accuracy and force based accuracy for London dserson nteracton. Energy Based Cutos Usng a cuto dstance R cut ntroduces an error n te total energy for te real sace sum of E real = 1 B 6 j (a,6 + a,4 + 1 a, )e,a (R Lj, R cut ) L;;j (D.53)

13 49 By usng average nteracton strengt and relacng te sum wt ntegral, we obtan E real ' N 1 6 Rcut N ( Rcut 4 Rcut (a,6 + a,4 + 1 a, )e,a R dr + 1 ) 1 Rcut e,a da = 3 N ( Rcut 4 Rcut )erfc(r cut ) (D.54) Usng a cuto dstance H cut ntroduces an error n te total energy for te recrocal sace sum of E rec = 3 4 " 0 erfc(b)+ 1 B j cos ( r j ) 3 b,1 e #(,H,b cut ) 3 b j (D.55) Relacng te sum wt ntegral, and relacng P j B j cos ( r j ) wt N , we obtan E rec ' 3 N H cut 3 " erfc(b)+ 1 = N 1 " b 3 5 erfc(b)+ 6 1 Hcut By usng erfc(b) e,b =b, we ave b 3,1 b e,b #d 1 b,b 4 e,b #db (D.56) E rec ' N = N Hcut Hcut b e,b db e, 1 4 H cut + erfc( 1 H cut) (D.57)

14 Force Based Cutos 50 Usng a cuto dstance R cut ntroduces an error n te force on atom for te real sace sum of jf ;real j' 1 8 L B r L (6a,8 +6a,6 +3a,4 +a, )e,a (r L, R cut ) (D.58) Relacng te sum wt ntegral and usng te average nteracton , we ave jf real j ' N 1 R (6a,8 +6a,6 +3a,4 +a, )e,a dr 8 R cut = N 1 (6a,6 +6a,4 +3a, +1)e,a da 5 Rcut N ( ) 5 Rcut 6 Rcut 4 Rcut 1 Rcut e,a da = 3 N ( )erfc( R cut 5 Rcut 6 Rcut 4 Rcut ) (D.59) Usng a cuto dstance H cut ntroduces an error n te force on atom for te recrocal sace sum of jf ;rec j' " erfc(b)+ 1 B sn ( r ) 3 b,1 e #(,H,b cut ) 3 b (D.60) Relacng te sum wt ntegral and usng P B sn ( r ) ' N, we obtan jf rec j ' 3 N 1 erfc(b)+( H cut b,1 3 b )e,b d N b 3 e,b d H cut = 4N 1 3 b e,b db 6 1 Hcut = N 3 H cut e, Hcut + erfc( 1 H cut) (D.61)

15 51 D.5 Partcle-Mes Ewald Sum Snce n cut = 1 H cut, te cost for recrocal sum s roortonal to 4 3 n3 cut N N N for conventon comutaton, were 4 3 n3 cut s te volume n sace, wle N s te cost of structure factor comutaton. For gven accuracy, otmze eta arameter so tat te comutaton cost mnmzed, we can get a scalng of N 3 = NN were te N and N s te cost n real sace sum and recrocal sace sum. It's not ractcal to erform smulaton wt N 1000; 000. To mrove seed, Lee Pedersen et al. roosed te so-called artclemes Ewald (PME) metod,4 wc sannlog N metod for te recrocal sace sum. D.5.1 Teory Te artcle-mes Ewald metod nvolves coosng sucently large tat atom ars for wc r j exceeds a seced cuto are neglgble n te drect sace sum wc reduces te real sace sum to order N. Te recrocal sace sum s ten aroxmated by multdmensonal ecewse-nterolaton. Te aroxmate recrocal energy and forces are exressed as convolutons and tus can be evaluated quckly usng 3D fast fourer transforms (FFTs). Te resultng algortm s of order N ln N. Let's look at te second term n Equaton (D.15) E rec = 3,3 V 0 m f ( jmj ) j C j e,m(r,r j ) (D.6) Dene te recrocal lattce vector m by m = m x a x + m y a y + m z a z wt m x,m y,m z ntegers not all zero, and te structure factor S(m) by ~S(m) = = N j=1 N j=1 q j e mr q j ex (m x s xj + m y s yj + m z s zj ) (D.63)

16 5 were s j ; = x; y; z are te fractonal coordnates of atom j. In order to aroxmate te above dened structure factor (Coulomb, or London), we'll nterolate te comlex exonentals aearng n te above equaton. Gven ostve ntegers K x ; K y ; K z and a ont r n te unt cell, denote ts fractonal coordnates by u x ; u y ; u z,.e., u = K a r, for = x; y; z. Due to erodc boundary condtons, we may assume tat 0 u K. Ten ex(m r) =ex( m xu x K x ) ex( m yu y ) ex( m zu z ) (D.64) K y K z Tere are several ways of nterolatng te above exonental. Lagrangan nterolaton and Cardnal B-slnes are te two wc get te most attenton. Lagrangan wegt functons are contnuous and terefore gve rse to aroxmate unt cell energes wc are contnuous as functons of artcle ostons. But tey are only ecewse derentable, so te aroxmate recrocal energy cannot be derentated to arrve at forces. Te forces and stresses ave to be nterolated as well. Wle by usng te Euler exonental slne wc nterolate exonentals wt te Cardnal B-slnes, we can derentate te energy to get forces and stresses, due to several nce roertes of te Cardnal B-slnes. For any real number u, let M (u) denote te lnear at functon gven by M (u) = 8 < : 1,ju,1j 0u 0 oterwse (D.65) For n greater tan, dene M n (u) byte recurson It can be roven tat M n (u) = u n,1 M n,1(u)+ n,u n,1 M n,1(u,1) (D.66) d du M n(u) =M n,1 (u),m n,1 (u,1) (D.67)

17 53 Clearly, for n >, M n (u) s n, tmes contnously derentable. It's also roved wen n s even we can wrte ex( m K u ) ' b(m ) 1 k=,1 M n (u, k)ex( m K k) (D.68) were agan = x; y; z, and b(m )= ex (n, 1) m K P n, k=0 M n(k +1)ex( m K k) (D.69) Proceedng as above, we can ten aroxmate te structure factor by ~S(m) =b(m x )b(m y )b(m z ) 1 k x;k y;k z=,1 Q(k x ;k y ;k z )ex( m x k x )ex( m y k y )ex( m z k z ) K x K y K z (D.70) were Q(k x ;k y ;k z )= N j=1 q j M n (u j x, k x, n x K x )M n (u j y, k y, n y K y )M n (u j z, k z, n z K z ) n x;n y;n z (D.71) Dene B(m x ;m y ;m z )=jb(m x )j jb(m y )j jb(m z )j (D.7) and F (Q)(m x ;m y ;m z )= 1 k x;k y;k z=,1 Q(k x ;k y ;k z )ex( m x k x )ex( m y k y )ex( m z k z ) K x K y K z (D.73)

18 54 Te aroxmate recrocal energy s now gven by E rec = 3,3 = 1 V 0 m x;m y;m z K x,1 K y,1 K z,1 k x=0 k y=0 f ( jmj )B(m x;m y ;m z )F(Q)(m x ;m y ;m z )F(Q)(,m x ;,m y ;,m z ) k z=0 Q(k x ;k y ;k z )( rec Q)(k x ;k y ;k z ) (D.74) were rec Q s te convoluton of rec and Q, and rec = F 3,3 f ( jmj V )B(m x;m y ;m z ) We ave used te followng roertes of dscrete fourer transform A B = F F,1 (A B) = F F,1 (A)F,1 (B) (D.75) (D.76) F,1 (A)(m x ;m y ;m z )=F(A)(,m x ;,m y ;,m z ) (D.77) m F (A)(m) B(m) = m A(m)F(B)(m) (D.78) Snce rec does not deend on artcle ostons, we get E rec r = K x,1 k x=0 K y,1 k y=0 K z,1 k z=0 Q r (k x ;k y ;k z )( rec Q)(k x ;k y ;k z ) (D.79) Also, snce m r does not deend on te unt cell arameters, we can comute recrocal contrbuton of stress as te followng two terms: 1; = E rec (D.80)

19 55 wc orgnate from and wt ; = 1 = F K x,1 k x=0 K y,1 k y=0 K z,1 k z=0 Q(k x ;k y ;k z )( Q)(k x ;k y ;k z ) 3,3 f ( jmj V m m )B(m x;m y ;m z ) (D.81) (D.8) For te Coulomb case, wc s =1,weave 1 Q rec = F V e, m m B(m x ;m y ;m z ) (D.83) For van der Waals attracton, or London dserson nteracton, we ave = 6, wc gves us L rec = F " 3 3 (1, 6V D.6 References m )e, + 5 m 3 3 # erfc( m ) B(m x ;m y ;m z ) (D.84) 1. D.M. York, T.A. Darden, and L.G. Pedersen, \Te eect of long-range electrostatc nteractons n smulatons of macromolecular crystals: A comarson of te Ewald and truncated lst metods," J. Cem. Pys. 99(10), 1993, U. Essmann, L. Perera, M.L. Berkowtz, T. Darden, H. Lee, and L.G. Pedersen, \A smoot artcle mes Ewald metod," J. Cem. Pys. 103(19), 1995, H.G. Petersen, \Accuracy and ecency of te artcle mes Ewald metod," J. Cem. Pys. 103(9), 1995, T. Darden, D. York, and L. Pedersen, \Partcle mes Ewald: An N log(n)

20 56 metod for Ewald sums n large systems," J. Cem. Pys. 98(1), 1993,

Not-for-Publication Appendix to Optimal Asymptotic Least Aquares Estimation in a Singular Set-up

Not-for-Publication Appendix to Optimal Asymptotic Least Aquares Estimation in a Singular Set-up Not-for-Publcaton Aendx to Otmal Asymtotc Least Aquares Estmaton n a Sngular Set-u Antono Dez de los Ros Bank of Canada dezbankofcanada.ca December 214 A Proof of Proostons A.1 Proof of Prooston 1 Ts roof