APPENDICES

APPENDIX 1

The generator matrix for pe

1. 0 0 0 0

1 . 0 0 0 0

1 . 0 0 0 0

. o o o o

. 0 0 0 0

. 0 0 0 0

. o o o o

. o o o o

. o o o o

. o o o o

. u u u u

. o o o o

. o o o o

. o o o o

. o o o o

APPENDIX 2

The generator matrix for ptfe

APPENDIX 3

The following IS the outputs of the correlation test and frequency test, done for

testing the randorn number generator used in the Monte Carlo Programs. The

outputs show that the random numbers generated are of almost uniform

distribution.

1. Output of frequency test

-- --

2. Output of correlation test

Appendices 194

APPENDIX 4

exact enurrieratlon of- unperturbed dimensions for ptfe & pe double precision R(0: 100,O: 100) ,BV(100,3), theta (100) ,phi (100) double precision TN(100,3,3) ,the(100) ,ph(100)

double precision te,nmrtr,dnmtr,thta,phii double precision etotl, etedsq,~, ss,ww, entd dimension l(100) n is no of bonds,te is temperatilre,ss is E-sigma,ww is E-omega bv is bond vector and rg is the gas constant tn is transformation matrix enpel isinput for pe

open(lO,file='enpe') open(8.file='enpe.out',status='old') READ(lO,*in, te,ss,ww,thta,phii close (10: write(8, +ln,te,ss,ww, thta,phii

rg = 4.2-1.98 pi = 4.*atan(l.1

generation of all configurations nn = 2*(3+*(n-3)) -1 nmrtr = 11.0 dnmtr = 0.0 do 800 kj = 0,nn etotl =0. etedsq = 0. w = 1.0 kf = k] do 530 1% = 2,n-1

ir = mod (kf,31 kf = kf/3 1 (ik) = ir c:ontinue do ik =2,n-1 if(l(ik1 .eq.O) phijik) =O. if (l(ik1 .eq.l) phi (ik) = phii if (l(ik1 .eq.2) phi (ik) = -(phii) enddo

do 25 ik = 1,n-1 theta(;k =tnta do 30 ik = 1 , n bv(ik,21 = 0. bv(ik,3! = 0. bv(ik,l) = 0.15:

hi (11 = 0. DO 45 NT =l,N-l the inti = theta (nt) *pi/l80. ph(nt! = phi (nt!*pi/l80. TN(NT,l,l) = COC,(?'HE(nt)) TN(NT, 1,2) = SIhi(?'HE(nt) ) TN(NT,1,3) = 0.0

TN(NT, 3 , 3 ) = -cos (:?hint) J 45 CONTINUE

call subham : s s , ww, kj,n,bv, tn,l,entd,etotl) ETEDSQ =entd W = DEXP(-ETOTL/RG/TE)

IF(KJ.GT. (3*+(li-3)-1)) W = 2.*W NMRTR = NMRTR +etedsq*w DNMTR = DNMTR + W

800 CONTINUE AVRSQ = NMRTR/DNMTR CHRTIO =AVRSQ/N

cn = chrtio/.153/.153 write I*, * ) 'avrsq=' ,avrsq

WRITE(8, *)dnrntr, 'r.umber of bonds =',n, ' cn= ' , cn, 'chrtio=' , CHRTIO

close (8, statusz'keep' ) STOP END SUBROUTINE RIK(n,bv,tn,entd)

c this rubroutine calculates rij,the distance between i-th6j-th at oms

double precision R(O:100,0:100) ,BV(100,3) double precisian TN(100,3,3) ,x, y, z,entd, K O ~ double precision P T ( 1 0 0 , 3 , 3 i , R l ( 0 : 1 0 0 , 3 ) , C ( 3 ) i = O j = n R1(0,11-0. RlIO,Zi=O. R1(0,3)=0. Rl(I+l,I) = BVl.i+l,l) Rl(I+1,2) = BV(F+1,2) R1 (Itl, 3) = BV(i+l,3) I1 = IT 1

C PRODUCT OF T(I+l) AND BV(It2) DO 15G 1: =:,3 C(IL) = 0.0 DO 130 KL = 1,3

130 C(IL) = ClIL) t TN(Il,IL,KL) *BVIi+Z,KL) 150 CONTINUE

Rl(I+Z,I) = Rl(I+l,l) +C(l) Rl(I+2,21 = R l ( I + 1 , 2 ) + C ( 2 ) R1(1+2,31 = RlII+1,3) +C(3)

C PRODUCT OF T(It1IAND T(I+ZI STORED AS PT(12)

. . DO 160 KT = 1,:

160 PT(I2,IT, J T ) =E'T(IZ,IT, JT) t TN(Il,IT,KT)* TN(I2,KT. JT) 170 CONTINUE C PRODUCT OF PT(I2) AND BV

DO 200 IL = 1,3 C(1L) = 0.0 DO 180 KL = 1.3

200 CONTINUE ~l(It3.1) = RlII+2,1) + C(l) Rl(I+3,21 = Rl(1+2,2) t C(2) ~l(1+3,3i = R1(1+2,3) + C(3i

C PRODUCT OF T S F'RCM I+1 TO J-1 C I1 IS I+L AND J1 IS J-1

. . DO 230 KT = 1,3

230 PT(N2+1, IT, JTI ' PT(NZ+l,IT, JT) + PT(N~,IT,KT) *TN(N~+~,KT, JT) 250 CONTINUE C PRODUCT OF PTS .W:3 BOND VECTOR

DO 265 ITL = 1,3 CIITL, - 0.0 DO 260 KTL =1,3

260 C(ITL) =C(ITL) t PT(N2+1, ITL,KTLi *BV(n2+2,KTL) 265 CONTINUE

Rl(N2~2,l) = Rl(NZ+l,li + Cili Rl(N2+2,2) - Rl(N2+1,2) +C(2) Rl(N2+2,3i = Rl (N2+1,3) tC(3)

270 CONTINUE x= RI (I,lj-nl(:r,l.i Y =RliI,2!-Rl(J,2) Z =Rl(I,3]-Rl(J,3) R(1, J) =x*x+y'y--z"z

c r(i:j, jj) are squares of rijs c rijjj =r(i, jiiz(!~,:j1*r(i, ]I

go to 276 276 contintie

entd = r(@,n) c310 CONTINUE

RETURN END subrouclne subham(ss, ww, kj , n , b v , tn,l,entd,etotll

c this s,dbroutine calls ell and rik for each configuration double precision bv(100,3) double precision tn(100,3,3i,ss,ww,entd,etotl,w

double precision r@n,el dimenslan 1 i 100)

nl=n-1 c K - KJ

w = 1. C EP IS THE ENEFIGI! OF THE PREVIOUS CHAIN C CALCULATION OF E- NL

call rik in, bv, tn, rOnI entd = rOn

c EL IS THE LOCAL ENERGY call e l l lr.,ss,w~.i,l,e:i E T W L = EL retur:: end

Appendices - 197

s u b r o u t i n e e l l ( n , s s , m r w , l , e l ) d i m e n s i o n 1 1 1 0 0 ) double p r e c i s i o n el., :;s, ww EL = 0 . 0 l ( n ) = 0 n l = n - 1 DO 69 I K = 2 , N l I F I L I I K ) . E Q . O ) GO TO 69 I F ( L I I K ) . G T . O ) EL = EL + s s I K 1 = I K + : I F ( L ( I K 1 ) . N E . O . A N I ) . I , ( I K l I . I I E . L ( I K ) ) EL = EL tww

6 9 CONTINUE r e t u r n end

a P W e s -- 198

APPENDIX 5

exact enumeration of- perturbed dimensions for ptfe & pe

adhing potentials-kihara potentials double precision ~(0:100,0:100),BV(100,3),theta(100],phi(100l double orecision TNi100,3,3), the(100) ,ph(100)

double preclslon te,nrnrtr,dnrntr,thta,phii,mirtrrl, double precision nnmrtr6,nmrtr8,nmrtrO double precision etotl,etedsq,w,ss,w,entd,sijnu,arsqlO double precision eted4,eted6,eted8,etedlO,arsq4,arsq6,arsqE double precision sqsij,snn,de, 10, sij dimension 1i100) n is no of bond:;,t.e is temperature,~~ is E--sigma,w is E-omega bv is bond vector and rg is the gas constant tn is transformation rnatrlx

openlltl,file='expt') openi8, flle='e:ipl:.out', status='old') READ ilC, *In, te, ss, ww, th:a,phli, de, rO close (101 write(8,') ri, te,ss,ww, thta,phii,de,rO

rg = 4.2*1.98 pi = 4.*atan(l.)

generation of all configurations nn = 2xi3ii(n-3)) -1 nmrtr = 0.0 dnmtr = 0.0 nmrtr4 = 0 .

nmrtr6=0. nmrtr8 = 0. nmrtr0 = 0. sijnu = 0. do 800 kj = 0,nn etotl =O. etedsq = 0. w = 1.0 kf = kj do 530 ik = 2,n-1

ir = rnod (kf, 3) kf = k f / 3 liik) = ir continue do ik =Z,n-1 if ( l ( r k ) . e q . C ) phi(ik) =0. ifil (lk! .eq.l) pk.ijik) = phii ifiliik) .eq.2) phi(ik) = -(phii) enddo

do 25 ik = I,n-l thetd(ikl =thta do 30 ik = l,n b v = 0. bv(ik,3) = 0.

Appendices - 199

30 b v ( i k , l ~ = 0 . 1 5 3 p h i ( 1 ) = 0 .

DO 4 5 NT = 1 , N - 1 t h e j n t ) = t h e t a : n t . i i p i / l H O . p h ( n t i = p h i ( n t i * g i / 1 8 0 . T N ( N T , 1 , 1 ) = C O S I T H E I n t ) ) TN(NT, 1 , Z j = S I N ( ? ' H E ( n t i I T N ( N T , 1 , 3 ) = O . C l T N ( N T , 2 , 1 ) = S I K I T H E I n t ) I * c o s ( p h ( n t ) ) TNINT,2,2I = -CC8S!THE!ntt ) * c o s ( p h ( n t ) ) TN(NT,:, 3 ) = s i n ( p h j n t ) 1 T N ( N T , 3 , 1 ) = s i n ( t h e ( n t 1 j * s i n ( p h [ n t ) ] TNINT, 3 ,Z l = - c o s ( t h e ( n t l i * s i n ( p h ( n t ) ) T N ( N T , 3 , 3 ) = - c o s i p h ( n t 1 )

4 5 CONTINUE c a l l s u b h a m (de,rO,ss,w,kj,n,bv,tn,l,entd,etotl,sij) ETEDSQ = e n t d

e t e d 4 = e t e d s q - e t e d s q e t e d b = e t e d 4 * e t e d s q e t e d 8 = e t e d 6 * e t e d s q e t e d l 0 = e t e d 8 + e t e t r l s g

w = D E X P ( -ETOTL/RG/TE) I F ( K J . G T . I 3 * * ( N - 3 1 - 1 ) ) W = 2.*W s i j n u = s i j n u t s i j * ~

NMRTR = 14MRTR t e t e d s q * ~ DNMTR = DNMTR + W

n r n r t r 4 = n m r t r 4 +et:ed4*w n m r t r 6 = n r n r t r b tet:ed6*w n r n r t r 8 = n r n r t r 8 t e t e d O * w n m r t r O = nmrtrO+etedl .O*w

800 CONTINUE sqs i j = s i j n u / d n m t r

a r s q 4 = n m r t r 4 / d n m t r : a r s q 6 = nrnr t r6 /dnmt . r a r s q 8 = nmrtrB/dnmt.r a r s q l 0 = n m r t r O / d n n i t r

AVRSQ = NMRTR/DNPITF. CHRTIG =AVRSQ/I.l

c n = c h r t i o / . 1 5 3 / . 1 5 3 s n n = s q s i j / n / . 1 5 3 / . 1 5 3 w r i t e ( * , * I ' a v r s q = ' , a v r s q w r i t e ( 8 , * I ' h i g h e r m o m e n t s ' w r i t e ( 8 , * ; a r s q 4 , a r s q 6 , a r s q 8 , a r s q l O

WRITEIB,*) 'n = ' , n , ' c n = ' , c n , ' c h r t i o = ' , C H R T I O w r i t e ( 8 , * 1 ' s i j s ' , s q s i j , s n n c l o s e i 8 , s t a t u s = ' k e e p ' i

STOP END SUBROUTINE RIK(de, rU,n,bv, t n , e n t d , e n l , s i j )

c t h i s r u b r o u t i n e c a l z u . l a t e s r i j , t h e d i s t a n c e b e t w e e n i - t h & j - t h a t oms

d o u b l e p r e c l s i o n R(O:100,1J:100),BV(100,3) d o u b l e p r e c i s i o n TN ( 1 0 0 , i, 3 ) ,x, y, z , e n t d , r O n d o u b l e p ~ e c i s l o n P T ( 1 0 0 , 3 , 3 ) , R 1 ( 0 : 1 0 0 , 3 ) , C ( 3 )

d o u b l e p r e c l s i o n s i j , e n l , r i j 3 , r i j 6 . r i j l Z , d e , r O i :; 0

Appendices -- .- 200

Rl(O,li=O. R1(0,21=0. R1 (0, 3)=C. Rl(It1,lI = BV(it1,l) Rl(It1,Zi = BV(i+l,2) Rl(I+1,3) = BV(i+l,3) I1 = I+ 1 PRODUCT OF T(I+l) .AND B'J(It2) DO 150 IL =1,3 C(IL) = 0.0 DO 130 KL = 1,3 C(1L) = C(1L) + TN(Il,IL,KL) *BV(itZ,KL) CONTINUE Rl(It2,l) = Rl(I+l,l) +C(l) R1(1+2,2) = Rl(I+1,2) tC(2) Rl(I+2,3j = Rl(It1,3) +C(3) PRODUCT OF T(I+l)AND T(It2) STORED AS PT(I2) I2 = It2 DO 170 IT = 1,3 DO 170 JT = 1,3 PT(IZ,IT,JTI = 0.0 DO 160 KT = 1,3 PT(I2,IT, JT) =PT (12,IT, JT) t TN(Il,IT,KT)* TN(IZ,KT, JT) CONTIN'JE PRODUCT OF PT (12) AND BV DO 200 IL = i ,3 C(IL; = U . 0 DO 180 KL = 1,3 C(1L) - C(IL) + PT(12,1L,KLlt BV(it3,KL) CONTINUi Rl(I+3,i; = Rl(It2,l) t C(1) Rl(It3,21 = Rl(Z+2,2) t C(2) R1(1+3,31 = Rl(l'+i:,3) + C(3) PRODUCT OF T S FROM 1+1 TO J-1 I1 IS I+1 AND J1. IS J-1

PT(N~+~,IT, J T ~ =: I'T(N2t1, IT,JT) t PT(NZ,IT,KT)*TN(N2tl,KT, JT) CONTINUE PRODUCT OF PTS I W I ) BOND VECTOR DO 265 ITL = 1,3 C(ITL) = 0.0 DO 260 KTL =1,3 C (ITLi =C (ITL! + P:?(N2+1, ITL, KTL)*BV(nZ+Z,K'I!L) CONTINUE Rl(N2+2,1) = R1 (N2+1,1l + C(li Rl(NZt2,Z) - Rl(N2t1,Zl rC(2) R1 (N2+2, 3) = R1 (N2+1, 3) rC(3) CONTINUL

do 305 i = 0,n do 305 j = 0 , ~ if (i.ge.]) gc to 305 X= Rl(I,11-Rl(J,ll Y =Rl(I,2l-RI(J,2I Z =R1(1,3)-Rl(J,31

Appendices - 20 1

RiI, J! =x*x+y'ytz*z sij = si] +r(i,]) if((j-il.lt.5) go to 305 if(r(i,ji.lt.(rO*r0/9.i) go to 310 sij = si] +r(i,j) rij3 = rO*rO/r!i, j l rij6 = r1]3*ri]3*rij3 rill2 = rij6*ri]6 en1 = enlr de*(rijl2-2,'rijb)

305 continue entd = r lO,ni

sij = si]/ lntli/(n+l) go to 315

310 en1 = C . entd = 0.

315 continue RETURIi END subroutirle subhanide, rO,ss, ww, kj,n,bv, tn,l,entd,etotl, sij)

c this subroutine calls ell and rik for each configuration double precision bv(100,31 double precision tn(100,3,3),ss,ww,entd,etotl,w

double precision rOn,el,enl,sij dimension l(100)

nl=n-1 c K = KJ

w = 1. C EP IS THZ ENERGY O F THE PREVIOUS CHAIN C CALCULATION OF E- UL

call rik(de,rO,n,bv, tn,rOn,enl,sij) entd = rOn

c EL IS THE LOCAL E:YERGY call ell (n, ss,ww, 1,el) ETOTL = enltEL return end

subroutine ell (n, s s , ~ ~ , 1, ell dimension l(1001 double precision el, .ss, ww EL = 0.0 l(n! = 0 nl = n-1 DO 69 IK = 2 , N I IF (L(IK! .EQ.C) GO TO 69 IF (LIIK) .ST. 0) EL = EL + s s IK1 = IK + 1 IF (L(IK1) .NE.O.AN:).:L(IKl) .IiE.L(IK)) EL = EL tww

6 9 CONTINUE return exd

Appendices 202

APPENDIX 6

average end to end distance b v matrix multi~lication method for pe and ptfe double precision Il!j, 15) ,g(15,15) ,gk(15,15) ,gs (15,15) ,un(15,15) double precision isuni (15.151 ,ux(l5,15),ts~15,15) .ult(15.151 . . . double precision e3 :li, 15j, lsi (15,15) ,lit (15, i5) , h t s (15.15) double prec~sion gn( : j , 15) , jkili,15), jksll5,15i, jksun(l5,15),gsqn(l5,15) double precision ue311!5,15),ue3~s(15,15) ,uls1(15,15),gsgnk(15,15) double precision g2(15,15) ,g1(15,15) ,e21st(15,15) ,elts (15,15) double precision 23 (15,15) ,e23ts(15,15) ,e211[15,15) ,glgn(l5,15) double precision jrlc.ng(15,15), t1(15,15) dimension n s 115) double precision t,pl,p2,p3,ls,pi,es,ew,tk,te,pll,pl2,pl3 double precision rg,s,w,z,gen,eted,cn,chrtio,tO real n open(9,file='pout') open(8,flle='peP) read(8,*Inns, ins(i),i = 1,nn.s) read(E,*)es,ew, tk, te read(8, *lpll,plZ,pl3 close(8) 15=. 153 do 24000 ; ~ n s = 1,nns n = ns (llns, n is no of bonds,t 1.3 theta,p 1 s ph1,s 1s sigma pi = 4.*atan(l.) p.L = pllip;i;dO. p2 = plZ'p1/18C. p3 = p13*pi/180. t = tkipi/18U. rg = 1.98'4.2 s = dexp (-rs/rg/te) w = dexpl-rw/rg/te) t is temp and rg is gas const

u the statistical wt matrix

u(l,li =l.O u(1,2)= s ~ ( 1 , 3 ) = s

~ ( 2 ~ 1 1 = 1 . : j ~ ( 2 , 2 1 = 5

u(2,3) = w's u(3,li = 1.,2 ui3,Z) = w - s u(3,3) = 5

to find u to the power of n-2 form prod~c; m a t r i x

call m u l t iu,u,un,3,3,.3) m := n-4

Appendices -- 203

find the f l n a l product ux do 20 1 = 1,m call mmult (un,u,ux, 3,3,3j do 15 i = 1,3 do 15 j = 1,3 un(i,j) = ux(i,j) continue

calulate z the part:ition fn jk(1,l) = I. jk(2,lj = 1. jk(3,lj = L. jks(1,l) =L. jks(l,2) =0. jks(l,3] = 0.

pr:e multiplication by jks call mmult (jks,ux,;k::un,1,3,3j

post multiplication by jk call mmult(jksun, jk, jsunj,1,3,1) z = jsunj [l,l)

to form the elements of the generator matrix G do 70 i = 1,15 do 70 j - l,l5 gli, jj = 0. glii,j) =O. tl(i,j) = 0. ts(i,jj =O. continue

g-genera to^ matrix ts-pseudodiagonal t matrix do 80 i = L,3 do 80 j = 1, 3 e3ii,,) =0. continue e3(1,1j =!. e3(2,2) = l . e3(3,3) =1.

Is1 is the column [ls, 0,O) and 1st the row lsl(1,l; = 1s lst(1,l; = 1s l~l(2,l) =O. lsl(3,l) =O. lst(l,2j = a . lst(1,3j =O.

1s is the bond length tl is the pseudo-diagonal t- matrix for the 1st bond t0 = tk*pi/l80. p10 = 0. p20 = 0. p30 = 0. call pstr(tO,plO,p20,p30, tlj

ts is the pseudo diagonal t-matrix

Appendices 204

call pstr(t,pl,p2,p3, tsi call kronek~u,lst,ult,3,3,1,3) (:all mmult lult, ts,ul.ts, 3, 9, 9) call kronek[e3,lst,e2lst,3,3,1,31 call mnult (ezlst, t.l,elts, 3,9,9) do 100 i =1,3 do 100 j = 1,9 k = j +3 gl(4, k) =elts (i, j) g(i,k) =ults (i, j ) do 110 i = 1,3 do 110 j = 1,3 gl(it12, jt12) = e3(i, j) g(i+12, j+12) = u(i, j) gl (i, ji =e3!i, j) g(i,j) = uii,~) call kronek(u,e3,ue3,3,3,3,3) call m u l t iue3, ts,ue3ts, 9,9,9) call kronekle3, e3, e23,3,3,3,3) call mmult (e23, tl,e23ts, 9, 9, 9) do 120 i =1,9 do 120 j = 1,9 il = i +3 j,l = j 7 3 gl(i1, jl) =e23ts (i,, j : ~ glil, jl) = ue3ts (i,, j:' do 125 i = 1,3 do 125 j = 1, 3 gli, j+12) - ~ ( 1 , ) ) sls'ls/2. gl!i,j+lZi = e3li,?)"ls*ls/:. continue call kronek(u,lsl,ulsl,3,3,3,1) call kronek(e3,lsl,e211,3,3,3,1) do 130 i =1, 9 do 130 1 = 1,3 il = i +3 jl = j +12

gl(i1, jll =eZll[i, ji g(i1, ]I)= ulsl(i, ji read gs the row and gk the column gs(1,li =I. do 140 i =2,15 gs(l,i) =O. gk(13,l) =I. gk(14,l) =1.0 gk(l5,l) = 1. do 150 i =1,12 gkli,li = 0. call prnt (gl, l5,l5)

to find y to the power of n-2 ca.11 mult(g,g,g2,l!j,:L5,15)

rng = n-4 do 160 i = 1,mg call mmult (cj,gZ,gn, L 5 . 1 5 , l i ) do 155 3 = 1 .15 do 155 k = 1, 15 g2:j.k) = g r ~ ( j , k )

Appendiccss - 205

160 continue

c premultiplication by gl and postmultiplication by gn call mmult (g1,gn,glgn,15,15,15) call mmult(glgn,g1,glgng, 15,15,15)

c to find the end to end distance c pre multiplication with gs

call m u l t (ys, glgng,gsgn, 1,15,15)

c post multiplicaion with gk call mmult (ysgn,gk,gsgnk, 1,15,1) gen = gsgnk(1,l) eted = 2.*gen/z cn = eted/r; chrtio = sr:/ls/ls write(*;) ' n= ' ,n, 'eted=',eted

10002 format (3fB.Z,e23.81 25002 continue 25001 continue 25000 continue 24000 continue

stop end subroutine rnmult (a,b, c,l,n,rnl double precision a(l5,151,bi15,15),~(15,15) do 100 i = 1 , l

do 101 j = l,m c(i,i) = 0. do 162 k = l,n

c(i,j) = cii,j) + a(i,k)*b(k,j) 102 continue 101 continue 100 continue

return end subroutine kronek(a, b,c,nl,n2,n3,n4) double precision a (15,15) ,bi15, I S ) , c[15,15) do 10 k =l,n2 do 10 1 =l,nl do 10 i = l,n3 ii =(l-1; *r.3+i d3 10 1 = l , n 4 jj = (k-l:'n4+,

10 c(ii, jj) = a(l,kl+>(~, jl return end subroutine prnt (a,.n,~nI double precision a ( 1 5 , 1 5 )

return end

subroutine pstr lt,plrp2,p3, rr)

Appendices --- 206

d o u b l e p r r c i s ~ o n t r ( 1 5 , 1 5 i double p r e c i s i o n t . , p l , p 2 , p 3 do i = 1,5 do j = l , Y t r ( i , j j = V . enddo enddo t r ( 1 , l ) = d c o s l t l t r I 1 , Z ) = d s i n l t ) t ~ I l , 3 ! = 0 . t r ( 2 , l ) = d s i n ( t ) * d c o s ( p l ) t r ( 2 , Z ) = d c o s ( t j i d c o s l p l 1 t r ( 2 , 3 ) = d s i n ( p l j t r ( 3 , l ) = d s i n ( t ) * d s i n ( p l ) t r ( 3 , Z ) = - d c o s ( t ) ' d s i n l p l i t r ( 3 , 3 ) =-dcos l p l ) t r ( 4 , 4 ) = d c o s ( t 1 t r ( 4 , 5 ) = d s i n ( t ) t r ( 4 , 6 1 = G . t r ( 5 , 4 ) = d s i n l t ) * d c ~ s ( p 2 ) t r ( 5 . 5 ) = -dcos ( t ) 'dcos ( p 2 ) t r ( 5 , 6 ) = d s i n ( p 2 ) t r ( 6 , 4 ) = 3 ~ 1 n ( t ) * d s : i n ( p Z ) t r ( 6 . 5 ) = - d c o s ( t ) " d s i n ( p Z ) t r ( 6 , 6 ) = -dcos (p2 i t r ( 7 . 7 ) = d c o s It) t r ( 7 . 8 1 = d s i n ( t )

t r ( 9 , 7 ) = d s i n i t ) * c i s i . n ( p 3 ) t r ( 9 , 8 ) = -dcos i t ) +.ds:in ( p 3 ) t r ( 9 , 9 ) = - d c o s (p3) r e t u r n end

Appendices - -- 207

APPENDIX 7

Monte Carlo proyram for pe and ptfe with L.J. potential dimension 1(100) double precision R(O:100,0:100),BV(3) double precision T(O:3,3,3),TN(100,3.31

double precision RS(0:100,0:100) double precision dnmtr,w,pw,etotl,ep,enl,el,enlt,wl

double precision nrnrtr, nrnrtr4, nmrtr6,nmrtr8, nmrtrO double precision etedsq,eted4,eted6,eted8,eted1O0ent double precision dee, te, rg,psq,prsq, chrtio,avrsq, arsq4, arsq6 double precision arsqlO,arsq8,de,rO,entd,ran,seed,ttl,tt2,tt3 double precision cnn,snn,sij,psij,sijnu,sqsij

THE PROGRAMME FOR INTERDEPENDENT <R**2> WITH EXCLUDED VOLUME BV IS BOND VECTOR, L ( 1 ) IS 0 FOR TRRNS,l FOR G+,2 FOR G-,N IS NO OF BONDS RG IS GAS CONSTANT,TE IS TEMP

open(ll,file='ptf') open (12, file= 'mc.out') READ ill,') nli,N,NCl,I!JC,TE,de, rO,TH,PH,B,SS,!W close (11) write (12, * J N, NC1, N'J, TE, de, r0, TH, PH, B, SS,WW PI = 4.* ATAN(l.1 THETA = TH*PI/180.3 B'ri(1) = B BV(2) = 0.0 BV(3) = 0.0 RI; = 1.38'4.2 n l = n - 1

GENERATION OF' (THE ALL TRANS CHAIN) xxxxxx do 25 ik = 1 , n l l(ik) = 0

continue call altra:ls In, theta, bv,de, rO,psq,ent, sij) p:;ij = sil pxsq = psq enlt = ent do 45 lj =3,2 if(lj.eq.31 ph: = 0 . 1 e . 1 phl = phi.pi/180. if(lj.eq.2 phi = -ph*pi/lBfI. t l j l 1 = cos (theta) tilj,l,Z) = sinlthrta) t l j , l , 3 = 0. t l 2 1 = sln(theta)*cos(phi) tilj,2,) = -coslthet.a)*cos!phi) tllj,2,3) = slniphi) t 3 1 = sin(tht:ta)*sin(phi) tilj,3,2) = -cos (theta) *sin(phi) t(lj,3,3) = -cos [phi)

continue

GENERATION OF RA&DC'M CONFIGURATIONS

NMRTR = i. 0 nmrtr4 = 0. mrtr6 = 0. nrnrtr8 = 0. nmrtr0 = 0. DNMTR = 0.0 sijnu = 0. EP = 0.0 PW = 1.0 DO 100 KJ = 1,NC if (kj.eq.nc1) then nmrtr = 0. dnrntr = 0. nmrtr4 = 0. nmrtr6 = 1;.

rimrtr8 = U. nmrtr0 = il. sijnu = 3. go to 100 else go to 50 endi f

lf ik~!100UO0'100000.eq.kj) 1 write(+,*l 'Current configuration number =',kj

to choose a bond between 2 and n-1 at random ttl = raI,.(r~.ra) ik = 0 lkls = nl -~ 1 nttl = ttl * ~ k l s + 2 i:< = nttl ipc = 0 if (k.ne.11 lpc = ~k

to change 1 (lkl randomly tt2 = ran(nra) if(ttZ.lt.0.5) l(1k) = l(ik) + 1 if (ttZ.ge.0.5) l(ik! =l(ik) - 1 if(l(ik) .gt.2) liik) = 0 if(l(1k) .lt.0) l(ik) = 2 call subharn (theta,enlt,ss,bw, kj,n,bv, t,de, rO,l,entd,etotl,sij) if( (entd.eq.O.).ana.. (etotl.eq.0.)) go to 100

calculating probability IF' (EP.GT.ETOTL) GO TO 72

DEe = ETOTL - EP TT3 = F L W I N R K ) W 1 = DEXP (-DEe/RG/TE) IF (TT3.LE.VIl) GO T3 1 2 GO TO 78

CONTINUE ZALCULRTIO?I OF AVIVPRtGE END TO END DISTANCE

ETEDSQ = E~ITLI ps. i j = sij PRSQ = ETEDSQ

t i = EXP I-ETOTL/RG,'TE) PW = W CONTINirE

GO TO 79 ETEDSQ = r ' ! iSQ

W =: PW sij = psi]

7 9 CONTINUE EP = ETOTL if((kj.gt.nii).and.(kj.lt.(nii+20))) go to 21111 go to 21112

21111 write(*,*) kj,etotl,w 21112 continue

eted4 = etedsq*etedsq eted6 = eted4*etedsq eted8 = eted4*eted4 etedlO = etedatetedsq NMRTR = NMRTR +etedsq*w nmrtr4 = nnrtr4 + eted4fw nmrtr6 = nmrtr6 +eted6*w nmrtr8 = nrnrtr8 +eted8*w nmrtrO = nmrtrO + etedlO*w DNMTR = DNMTR + W sijnu = sijnutsij'w

100 CONTINUE AVRSQ = NMRTR/DNMTR CHRTIO =AVRSQ/N sqsij =sijnu /dnmtr arsq4 = nmrtr4/dnn.tr cnn = chrtio/b/b

snn = sqsij/n/b/b arsq6 = nmrtr6/dnrrLtr arsq8 = nrnrtr8/dnnttr arsql0 = nrnrtrO/dnmtr var = arsq4-avrsqi*Z write(+,*) 'var=',var WRITE(*,+) n, te, 'avrsq=',AVRSQ, 'chrtio=',CHRTIO,cnn

c write(l2,*) 'arsq4=',arsq4,'arsq6=',arsq6 c write(l2,'1'arsq8=',arsq8,'arsqlO=',arsqlO c write(l2,*l'sqsij=',sqsij,snn

STOP END subroutine altrans (n, theta,bv,de, rO,prsq, enlt, sij) double precision bv(3) ,tn(100,3,3) ,r(O: 100,O: 100) double precision rs(0:100,0:100) double precision de, rO,prsq, enlt, en1,entd

C GENERATION OF THE ALL TRANS CHAIN nl = n-1 DO 32 NT =l,Nl TN INT, 1, 1, = COS (THE,TA)

. . ?'N(NT,Z,ll = SIN(THE:TA) TN (NT, 2,21 = C O S (THETA) TN(NT,2,3] = 0.C ?'N(NT,3.1) = O . C ?'N(NT,3,?) = O . C TN(NT,1,3! = -1.0

32 CONTINUE C CALCULATION OF THE: NON LOCAL ENERGY OF THE ALL TRANS CHAIN

E:NLT =O. 0 call rlk(ue, rU,n,bv,tn,enl,entd,sij)

C ENLT IS THE NON LOCAL ENERGY OF ALL TRANS CHAIN ENLT = ENLT + en1 PRSQ = entd ' entd return

Appendices --A

2;o

end subroutine ell(n,ss,ww,l,el) dimension l(100) double precision el ES = S S Ejy = WW EL = 0.0 l(n) = 0 nl = n-1 D'2 69 IK = 2 ,EJ l IF (LIIK! .EQ.O) GO TO 69 IF (L(IK! .GT.O! EL = EL + ES

. . 69 CONTINUE

return end FUNCTION KAN ( fJXA)

c REAL A double precision ran,seed,a,y y = 1. A =(SEED +4.*datan(yl)**5 data seed /4.0/ SEED = A - AINT (A) PAN = SEED RETURN END SUBROUTINE RIK(de, rO,n, bv, En, enl, entd,sij) dimension 1(100! double precision R10:100,0:100),BV(3) double precislon T t0:3,3,3) ,TN(100,3,3)

double precislon E3T(100,3, 3) ,R1(0:100,3),C(3) double precision sij,enl,rij3,rij6,rij12,x,y,z,entd,de,rO en1 = 0. s i j = 0. i = O j = n Rl(0,l)l.O. Rl(0,Zl-0. R1 (0,3)=0.

R1(1+1,3) - BV(3) I1 = I+ 1

C PRODUCT OF T(I+1) AND BV(I+2) DO 150 IL 1 , 3 C(IL) = 0.0 DO 130 KL - L , 3

130 CIIL) = CiILI + TN(Il,IL,KL) *BV(KLl 150 CONTINUE

Rl(It2,l) = Rl(I+l,l! tC(l! Rl(I+2,2) = Rl(It1,Z) +C(2! R1(1+2,3! = Rl(I+1,3) +C(31

C PRODUCT OF T (Itl).?.ND T (I+2! STORED AS PT(I2) I2 = 1+2 DO 170 IT = ?,3 DO 170 JT = 1,3 PT(IZ,IT,JT) = 0.0 DO 160 KT = 1,3

Appendices 211

160 PT(IZ,IT, JT) =P'T(I2,IT, JT) t TN(Il,IT,KT)* TN(I2,KT, JTI 170 CONTINUE C PRODUCT OF PT (12) AND BV

DO 200 IL = 1,3 CIIL) = 0.0 DO 180 KL = 1,3

180 CiIL) = CIIL) + PT(I2,IL,KLI * BV(KL) 200 CONTINUE

Kl(It3,l) = RlII+2,1) + C i l ) R1(1+3,2) = RliI+2,2) + C(2) Kl(It3,3) = Rl(It2.3) t Cl3)

C PRODUCT OF T S FROM It1 TO J-1. C I1 IS It1 AND J1 IS J-1

ir2 = J - 2 DO 270 N2 = 12,JZ I)O 250 IT =1,3 DO 250 JT =I, 3 E'T(N2+1,IT,JT) = 0 . 0 DO 230 KT = 1,3

230 PT(N2+1,IT, JT) =: E'T(N2+1,IT, JT) + PT(N2,1TrKT)*TN(N2t1,KT, JT) 250 CONTINUE C PRODUCT OF PTS PND BOND VECTOR

DO 265 ITL = 1,3 C(1TL) = 0.0 DO 260 KTL =1,3

260 C(1TL) =CiITL) + PT(N2+1,1TL,KTL) *BV(KTLl 265 CONTINUE

Rl(N2+2,lr = KlIN2+1,1I t C(I1 Rl(N2t2,ZI = Rl(N2+1,2) tC(2) Rl(N2+2,3i = RliN2+1,31 +Ci31

270 CONTINUE DO 305 IJ = 0 , N DO 305 JJ = O , N IF(IJ.GE. JJi GO TO 305 X= R1(IJ,1I-R1(JJ,ll Y =Rl(IJ,2)-RlIJJ,Z) Z =Rl(IJ,3)-RliJJ,3) R(I3, JJ1 =(X*X+Y*YtZkZ) sij = si]trjlj, jj) if ((jj-ijl .lt.5) go to 305 if(r(ij,]jl .le. (rO'r0/9.)) go to 308 rij3=(rO*rO/r~~j, 13)) rij6 = ~ij3~rij3*rij3 rijl2 = rij6*rij6 en1 = en1 + dec(rij12 - 2.*rij6)

305 CONTINUE entd = r (O,n) slj = sl]/ (ntli / {n--1) go to 310

308 eritd = 0. en1 = 0.

310 CONTINUE c: writei*, *)en1

RE:TURN END subroutine

subham(th,enlt,ss,ww, kj,r,bv,t,de, rO,l,entd,etotl,sij) double prsclslon bvi31 ,t(0:3,3,3) ,enlt,etedsq,enl,el,sij double precision tn il00,3,3i ,de, r0, entd,etotl,rOn

dirnenslon l(1001 nl=n-1 K = KJ ETEDSQ = 0.0

EP IS THE ENERGY OF THE PREVIOUS CHAIN t of the first bond

tnll,l,l)= coslth) tn11,1,21= sinlth) tn(1,1,3) = 0. tni1,2,11 = sinith) tn(1,Z.Z) - -cos(th) tni1,2,3i = 0. tn(l,3,1)=0. tn(1,3,2) -0. tn(1,3,3) = -1. DO 57 IK = 2,Nl DO 57 NI = 1,3 DO 57 NJ = 1, 3 TN(IK,NI,NJ) =T(L(IK) ,NI,NJ)

CONTINUE CALCULATION OF E- .NL

call rik lde, 10, n,bv, tn, enl, rOn, s i j ) entd = rOn if((enl.eq.O.).and.(entd.eq.O.l) go to 100 ENL = ENL - ENLT

CALCULATION OF THE LOCAL ENERGIES FOR THE CHAIN EL IS THE LOCAL ENERGY

call ell:n,ss,ww,l,el) ETOTL = EL + ENL 93 to 110 etotl = 0. continue return end