{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 270 65 "WATER- Molecular Dynamics Integration for the internal molecular " }}{PARA 0 "" 0 "" {TEXT 271 45 "vibrational motion using the Verlet algorithm" }{MPLTEXT 1 0 1 "\n " }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 261 21 "Goals and Objective s:" }{TEXT 262 0 "" }{TEXT -1 59 "To understand molecular vibrations i n polyatomic molecules " }}{PARA 0 "" 0 "" {TEXT -1 99 "by direct simu lation of those vibrations using molecular dynamics (Verlet algorithm) to numerically" }}{PARA 0 "" 0 "" {TEXT -1 87 "integrate Newton's equ ations of motion assuming a simple force field holding the atoms " }} {PARA 0 "" 0 "" {TEXT -1 93 "together inside the molecule. As a subsid iary objective, a viscerally intelligible example of" }}{PARA 0 "" 0 " " {TEXT -1 95 "using the Fourier Transform is part of the analysis of \+ the simulation; it is hoped that it will" }}{PARA 0 "" 0 "" {TEXT -1 93 "provide enough transparency that readers/users of this code will a ppreciate how the Transform" }}{PARA 0 "" 0 "" {TEXT -1 93 "provides i nformation in its output which relates to measureables in the system u nder study.\n\n" }{TEXT 263 13 "Introduction:" }{TEXT -1 76 " In order to carry out molecular dynamics one needs two things, a forcefield" } }{PARA 0 "" 0 "" {TEXT -1 91 "and an integration algorithm for numeric ally integrating Newton's equations of motion. The " }}{PARA 0 "" 0 " " {TEXT -1 93 "force field may be derived from a potential energy func tion, as in the case considered here. " }}{PARA 0 "" 0 "" {TEXT -1 88 "Although modern molecular dynamics is usually concerned with aggregat es of molecules our" }}{PARA 0 "" 0 "" {TEXT -1 91 "example here is su fficiently complex as to illustrate the method and allow for expansion to" }}{PARA 0 "" 0 "" {TEXT -1 87 "more complex systems. The integrat ion scheme used here is the simplest of an ascending " }}{PARA 0 "" 0 "" {TEXT -1 331 "family of more and more complex schemes.\n\nThis work sheet addresses the problem of simulating the motion of nuclei inside \+ a \nmolecule subject to a potential energy field binding the nuclei in to a molecule.\nSince the simulation is specific to water, H-O-H, wher e the H atoms are protons,\nwe start with the potential energy functio n:\n" }{OLE 1 34314 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]: :yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::fyyyyyI>nYnyyyYE:G:I:K:M:O:c:S:UJ:n;v;;JBB:]:_:a:C:e:g:i:k:m:o :q:s:u:w:y:;[:nYnYV>^>f>n>v>>?F?N?V?^?f?n?v?>`:B:];_;a;c;e;g;i;k;m;o;q ;s;u;wK:vA>B::::::::Nygx;>:::::::=Jyyy;d:yayAbs^L::A;UTT AeVYuVYeScEB[DTYEWYEBEDSMEWYeV;sFWCF;CNMuRcUW_USt:VdJ?gik\\LpnJ:l::JQH ]nIalHH>::::=J:D:<::::::::::yayY:;J<>:=Z:vYxY;J:JdQ\\:Vt=Jbq=<:::::::: ::::::::::::::::::::::::::::FZ;FZ;::B:=R:@:B:R::::<:@::::::::::Z:Z;:<: @:;:=:::B::;j:R:::::::::::J:F:j:>:FZ::R:B:R:J:j:: :::::::::J:j:J:j::::>:F:::::::::::>:F:>:F:>:F:::J::;j::j:::::::::::<:@ ::::::>:B:=Z;>Z;j:<:@Z::R:B:R:J:j:::::::::::Z:Z;B:>:@j::::;:=::::::::: ::<:@Z:J:R:=J:j::::;:<:@J:@Z:j:@::::::::::Z:Z;:Z;:F:B:R:J:j:::::::::::Z:Z;>Z:>Z:R:F:::J:@:=:::::::::::<:@J::<:=R::;:=:;B:j:::::;:FZ:F:@:>:F::;:=B:;R:F:::::B:<:@Z:Z: >:@Z;Z:F:@:<:@:::::::::::B:R:::::<:@:::::::::::B:R::J:j:::::Z:>Z;j::Z:J:@Z;F::::>Z:J:@Z;F:;:=Z:Z;:J:j:<:@:::::::::<:@:<:@:::::::::::J: j:::>:F:Z:Z;:::::::::::>:F::>:F::<:@:::::;:=Z:Z;>:F::::<:@J:j:B:R:;Z:j :@:J::F:::::::::>:F:B:R::::::::::::;:=::J:j::<:@:::::::::::J:j::Z:Z;:B :R:::::>:F:<:@J:j::::B:R:;:=Z::;B:;j:R::;R::=::R:F::::::B:=Z;:<:@J ::F::::;:=J:j:J:j:::<:Z;::::::::J:j::<:@Z:>:;R:F:;J: F:=J:J:<:=j:::::;j::=j:>:>Z:j:F::::J:FZ:F :@:J:@:=::B:R:Z:Z;::::B:R:>Z:>Z:Z;F:=:::J:B:R:F:>Z;j:::<:;Z;F::::>Z:B: ::::=:=Z;:<:@J:Z:F:@:Z:>Z::@:=Z:R:F::>Z;j:::;:=Z:>Z:>:j:R:;Z:::F:@Z:F:@: ;:=B:R:B:;J::=Z;:;R:FZ:Z;:;:=::B::R:F::::?Jq@jLB:;J\\uZ:>f:B :ry>:<:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::?:;:::: :cpcIj:@j:@::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::; 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:>Z;j:::>:F:Z:Z;>Z:B:F:=:::::::B:>:@Z;:>Z;j:::;:=:B:R:::::<:@:;:=J:j:: ::>:F:;:=:;:=:J:<:@:;R::=:::::::::;:=:<:@:::::::::::J:j:::>:F:Z:Z;:::: :::::::>:F::B:R::<:@:::::;:=Z:Z;>:F::::<:@J:j:B:J::F:::;:=:B:R::::::::::::;:=::;:=:Z:Z;::::J :j:B:R:;:=:::Z:Z;>:F:<:@J:B:FZ;>Z:Z;J:@:=B:;:=::::::F:<:@:;B:R:Z;B:>Z;R:=::::J:@::F::: ::::::::B:R:;B:;B:@:=:::>Z;j:::::::::::Z:Z;>Z:B:@:=J:@:=:::::;Z:j:@:J: j:J:<:F::::J::=B:=Z;:;:=J::F::::::: ::::B:R:<:;Z;F:::J:j:::::::::::Z:Z;B:>:@j:>:F:::J:Z:Z;>Z;B:FZ;:::::::: ::B:R:;B:R::::::;R:J:@j::=J:j:<:Z;Z:Z;:;:=:::::::::::;:=:;:=:::J:j:::: :::::::J:j:J:j:J:j::::;:>:=::=::::::::::B:R:;B:R:::::::F:;:=B :;B:;j:R:B:=Z;:<:@::::::::::Z:Z;:<:@:::Z:Z;::::::::::B:R:Z:Z;J:j:::::=Z;:::::::::B:=:R:;B:=Z;:::::B:<:@::F:;:=:::B:;:@::=Z;:::::B:=B:@Z;:;:j:>::=Z:Z;::::::::::::::;:=::::::::::::::::>:F:; :=:::B::=j::::::::::::;B:R::::::>Z;>:=j:Z:>Z:Z;B:B:;Z;R:::=J:j::::>:;R:=j:Z:>Z;j::::::::::::;: @::::::;R:;B:;B:=Z;:::Z:R:@:::::::::::::Z:>:R::::::::::::::::B:Z;J:@:= :::J:>::F::::::::::::::::::::: ::::::::::Z:Z;:::J:j::::::::::::::;B:R::::::B:=Z;::::J:j:::::::::::::: ::::::::::::::::::<:@::::<:@:::::::::::::>Z:Z;:::::Z:F:@:::::<:@:::::: :::::::::::::::::::::::::>:F::::B:R:::::::::::::J:<:@::::::;:@:::::>Z; j:::::::::::::::::::::::::::::::B:R::::J:<:@:::::::::::::>:R:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::>:R:::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::>:C:IS:yyyA:;:::Ja@Na`>;B :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::1:" }{TEXT -1 128 "\nwhich will be employed. There are \+ two force constants, one for bond length deformation, \nthe other for \+ bond angle deformation.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 0 "" }{TEXT 265 29 "The simulation's Maple code: " }{TEXT -1 52 "To be gin the simulation, we clear the Maple system, " }}{PARA 0 "" 0 "" {TEXT -1 91 "and load the package that will contain the Fourier Transf orm (to be employed at the end).\n\n" }{MPLTEXT 1 0 94 "restart;\nwith (inttrans):#in order to use the Fourier Transform\nwith(linalg):\nprin tlevel := 0;\n" }{TEXT -1 212 "\nWe define the initial displacements o f both bond length and bond angle that we will allow.\nThe units are c m for bond lengths and bond length deformations, and degrees for bond \nangles and bond angle deformations." }{MPLTEXT 1 0 78 "\n\ndelta_r : = 0.1e-8;#initial displacement\ndelta_theta := 10;# initial degrees\n " }{TEXT -1 44 "\nWe create a time step (h), in seconds (!). " }{TEXT 259 34 "We shall use cgs units throughout." }{MPLTEXT 1 0 1 "\n" } {TEXT -1 300 "The \"trick\" that we will use is to alter this h-value \+ so that the number of steps used \nin the simulation works out to be a n integral number of cycles of the vibration\nunder study. Thus, we wi ll \"fiddle\" with this value so that we do not actually have to\nloca te the maximum in the Fourier Transform (" }{TEXT 257 10 "vide infra" }{TEXT -1 2 ")." }{MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 84 "\nh := 0.965e-16;\n#(this is the time step in seconds)\n#RESONABLY GO OD VALUE for m=10\n" }{TEXT -1 202 "\nNow, the problem specific materi als, which could have been done earlier. We\ndefine the equilibrium O- H bond length (0.9584 Angtrom), and then the H-O-H \nequilibrium angle (104.5 degrees):\nRemember that" }{OLE 1 16904 1 "[xm]Br=WfoRrB:::wk; nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::fyyyyyy:nYnyyyYE:G:wAwAMJ:N;V;f:[Z:F=F=N=V=^=f=n=v=nYvY:::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::nwhGYX>efI=?R:E:yay=;Z:::: :::^<>:F:AlqfG[maNFO=;::::::::_:C:yay=;Z::::::j:>:e=;:wA?:Jyky;::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::>:<::Jj`NFJ:::::::F:wyyAbfccWflwgmwg`_hZ>dbwgnwgZfb^ Ggnwgl?^mn_j>^JGg]_hoOh_c=jc:CJ:F:Z;F:=::::::J:j::::Z:>Z;j:Z:B:<:@Z;>Z;j:>Z::=::::::B:=Z; ::::::B:R:::::::<:@:::::::Z:Z;:::::::J:j:::::::>:F:::::::;:=:::::::J:j ::::::::B:R:::::::;B:j:::::::;:F:::::::Z:F:@:::::::Z:>:R:::::::::::::: :::::::Z::@::::::::::::::::::::::::::::::::::::::::::::::::::::Z:F:@:: ::::::>:F:::::::Z:B:=R:@::::::::::::::;j:F :::::::::::::>:;j::::::::::J?N:yyyxI:;Z::::::j;>:c:;:?ja:[Lsf FaMR>@>Z::::::::kZ:vYxI:;Z::::::J><:ODAB::::::::::::jysy:>:<:::::::::: :::::::::vYxI:;Z::::::::jysyA:C:=B:;jysy?Z:>:fPNS>[Vj:F;HJ:ek:gx>S`:V=e?;jG:;JN>:eG:V=;B:=J>J:>gMJ\\u:C@;R:< J:<:jy;;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::N:eoGJ:^R>f:B:ryZ::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::J;>Z:>:::::fpLNs:=j::::::::;J:F:=:::::::::::::;J:F:=::::::: Z:B:@Z;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::B:;R:=j:::::::::::::::F: :::::::B:R::::::>:F:::::::::::::::B:R::::::::J:j::::::>:F::J::=j:: :Z:B:;:@:=:;:;j:@Z;:::;J:j::;R:F:J:Z;j::>Z:FZ:FZ;Z:J:R:@::::B:@:=Z:B:<:@Z;:B:=Z;:<:@:J::F:J: j:Z::R:;B:;J:j:F::;R:;R:=j::J:Z:>Z::@:=:; 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The last two lines are literature values \nof the force constants noted above. They can be changed at the discretio n \nof the user to obtain better or worse fits to experimental data." }{MPLTEXT 1 0 329 "\n\nm_H_1 := 1.0078/(6.023e23);#grams/atom\nm_H := \+ m_H_1;# in case you choose m_H_2 for deuterium\nm_O := 16/(6.023e23);# again, one could choose an isotope\nr_e := 0.9584e-8;#in cm\ntheta_e : = evalf((104.5)*Pi/180);\ntheta_e_delta := evalf(delta_theta*Pi/180); \nkOH := 7.76e5;#dynes/cm, pg 218 Barrow\nk_theta := 0.699e5*r_e^2;#dy nes/rad\n" }{TEXT -1 82 "\nTo fix the number of simulation points that will be employed, we choose the power" }}{PARA 0 "" 0 "" {TEXT -1 100 "of 2 that we require. 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:::>:F::B:R:<:@:::::::::::;:=:B:R::Z:Z;:<:@::;:=:>:F::B:R:<:@::::::::: :Z:>:F:Z:F:@::B:R:J:<:@::;R:F:B:;:=:::F:::Z :Z;:::B:R:<:@::::::::::::Z:Z;::>:F:J:j:>:F:<:@:B:R::B:R:<:@::::::::::: :Z:Z;::>Z:Z;::;R:F:Z:J:F:=::Z:J:R:@:::;R:F:<:@:::::::::::::;:=: ::::::::Z:Z;Z:Z;::::::::::::Z:Z;:::::::::B:R:B:R:::::::::::::B:R:::::: ::::<:@:<:@:::::::::::::>:F::Z:Z;:::<:@:::;:=:<:@:::::::::::::>:F:::;: =:::>:F::Z:Z;:<:@:::::::::::::B:R:::F:::J:j:::;:=:B:R::::::::::::::<:@::<:@:::B:R::Z:Z;:B:R ::::::::::::::<:@::::::::B:R::<:@::::::::::::::::::::::::B:R:::::::::: ::::::::::::::Z:^:v^;vyyY:>::::WTJWTL>Z::::::::::::::::::::::::::::::: ::::::::::::::::4:" }{TEXT -1 82 "holds for any two vectors. These def initions are\nexploited in the following code:\n" }{MPLTEXT 1 0 426 " \n#calculate the derivatives of the potential with respect to the 9 co ordinates.\n t1 := [x11,x12,x13]-[O1,O2,O3]:\n t2 := [x21,x22,x23]-[ O1,O2,O3]:\n t1_mag := sqrt(t1[1]^2+t1[2]^2+t1[3]^2):\n t2_mag := sq rt(t2[1]^2+t2[2]^2+t2[3]^2):\n #theta_mag := arccos((t1[1]*t2[1]+t1[2 ]*t2[2]+t1[3]*t2[3])/(t1_mag*t2_mag)):\n theta_mag := angle(t1,t2): \+ \n t3 := (kOH/2)*((t1_mag-r_e)^2+(t2_mag-r_e)^2)+(k_theta/2)*(theta_m ag-theta_e)^2:\n" }{TEXT -1 90 "\nIf you change the colon to a semi-co lon at the end of the last line of code, you will see" }}{PARA 0 "" 0 "" {TEXT -1 56 "that the term \"t3\" is written in Cartestian Coordina tes!" }}{PARA 0 "" 0 "" {TEXT -1 22 "\nHaving converted the " }{TEXT 260 12 "displacement" }{TEXT -1 265 " coordinates into Cartesian form, in t1, t2, and obtained \ntheir magnirtudes, we symbolically obtain e xpressions for t3 which is the recasting of the potential\nenergy term into Cartestian form. Then, and only then can we obtain the appropria te partial derivatives: " }{MPLTEXT 1 0 207 "\n\n d11 := diff(t3,x11) :\n d12 := diff(t3,x12):\n d13 := diff(t3,x13):\n d21 := diff(t3,x2 1):\n d22 := diff(t3,x22):\n d23 := diff(t3,x23):\n dO1 := diff(t3, O1):\n dO2 := diff(t3,O2):\n dO3 := diff(t3,O3):\n\n" }{TEXT -1 162 "which we have obtained as functions.\n\nWe define a function for the \+ potential energy for future usage (the code is virtually\nidentical to the code used (above)):\n\n" }{MPLTEXT 1 0 174 "V := proc(r_H_1,r_H_2 ,r_O);\nreturn(evalf(subs(x11=r_H_1[1],x12=r_H_1[2],x13=r_H_1[3], x21= r_H_2[1],x22=r_H_2[2],x23=r_H_2[3],\nO1=r_O[1],O2=r_O[2],O3=r_O[3],\nt 3))):\nend proc;\n" }{TEXT -1 110 "\nLastly, we choose which normal mo de, with the choice of 0 being no normal mode, just arbitrary\ndisplac ements." }{MPLTEXT 1 0 1260 "\n\nnorm_mode := 2; #choose your normal m ode.\nr_O := [0,0,0];#set this in common, but change later\nif norm_mo de = 0 then \n r_H_1 := [0.9e-8,0.5e-8,0]:\n r_H_2 := [-0.7e-8,0.6e- 8,0]:\nelif norm_mode = 2 then\n#WAGGING\nprint(` theta varying normal mode`);\n x_H_1 := r_e*cos(theta_e/2+theta_e_delta);\n y_H_1 := r_e *sin(theta_e/2+theta_e_delta);\nprint (`x,y = `,x_H_1, y_H_1);\n x_H_ 2 := x_H_1;\n y_H_2 := - y_H_1;\n r_H_1 := [x_H_1,y_H_1,0];\n r_H_ 2 := [x_H_2,y_H_2,0];\nprint (`h-h dist = `,dist(r_H_1,r_H_2));\nelif \+ norm_mode = 1 then\n#SYMMETRIC STRETCH\nprint(` symmetric stretch norm al mode`);\n x_H_1 := (r_e+delta_r)*cos(theta_e/2);\n y_H_1 := (r_e+ delta_r)*sin(theta_e/2);\n x_H_2 := x_H_1;\n y_H_2 := -y_H_1;\n r_H _1 := [x_H_1,y_H_1,0];\n r_H_2 := [x_H_2,y_H_2,0];\nprint (`h-h dist \+ = `,dist(r_H_1,r_H_2));\nelif norm_mode = 3 then\n#ANTISYMMETRIC STRET CH\nprint(` ANTI symmetric stretch normal mode`);\n x_H_1 := (r_e+0.2 e-8)*cos(theta_e/2);\n y_H_1 := (r_e+0.2e-8)*sin(theta_e/2);\n x_H_2 := x_H_1;\n y_H_2 := -y_H_1;\n r_H_1 := [x_H_1,y_H_1,0];\n r_H_2 : = [x_H_2,y_H_2,0];\nr_O := [0.1e-8,0,0];\n\nprint (`h-h dist = `,dist( r_H_1,r_H_2));\n\nend if;\nprint (`starting H1 coords = `,r_H_1);\npri nt (`starting H2 coords = `,r_H_2);\nprint (`starting O coords = `,r_O );\n" }{TEXT -1 165 "\nNow, we zero the velocities (this is arbitrary, and we can change any of these, including\ninducing a superimposed ro tation by just changing one velocity component)." }{MPLTEXT 1 0 112 " \n\nv_H_1 := [0,0,0]:\nv_H_2 := [0,0,0]:\nv_H_1_p := [0,0,0]:\nv_H_2_p := [0,0,0]:\nv_O := [0,0,0]:\nv_O_p := [0,0,0]:\n" }{TEXT -1 83 "\nNo w we zero the forces, initially, something again which is completely a rbitrary.\n" }{MPLTEXT 1 0 111 "\nF_H_1 := [0,0,0]:\nF_H_2 := [0,0,0]: \nF_O := [0,0,0]:\nF_H_1_p := [0,0,0]:\nF_H_2_p := [0,0,0]:\nF_O_p := \+ [0,0,0]:\n" }{TEXT -1 238 "\nThe new (p stands for prime, i.e., F_H_1_ p = F' for proton 1) forces which will be the ones\nfound at the end o f each time step, and substituted back into the original forces as the \nstarting forces for the next time step, are also zero'd." }{MPLTEXT 1 0 81 "\n\nV_start :=V(r_H_1,r_H_2,r_O):\nprint (`starting potential \+ energy = `,V_start);\n\n" }{TEXT -1 89 "Now, we set the number of time steps so that we get the most efficient Fourier Transform." } {MPLTEXT 1 0 17 "\n\nn_stop := 2^m;\n" }{TEXT -1 152 "\nWe need two ve ctors, one for the imaginary and one for the real component of the dis placement\nthat we are going to analyze using the Fourier Transform." }{MPLTEXT 1 0 48 "\n\nl := array(1..n_stop);\ny := array(1..n_stop);\n " }{TEXT -1 94 "\nAnd finally, here is the molecular dynamics simulati on, one time step (h seconds) at \na time." }{MPLTEXT 1 0 196 "\n\nfor i from 1 by 1 to n_stop do\nV_1 :=V(r_H_1,r_H_2,r_O):\nKE_1 := 1/2*( m_H*v_H_1_mag^2+m_H*v_H_2_mag^2+m_O*v_O_mag^2):\nE_1 := KE_1+V_1:\npri ntlevel := 0;\nif (i = 1) then printlevel := 3; end if;\n" }{TEXT -1 229 "\nWe substitute the current coordinates into the previously prepa red appropriate partial derivative\nhere. Since the coordinates are ch anging, and the function isn't, we needed to have pre-prepared\nthe fu nction, which we did above." }{MPLTEXT 1 0 27 "\n\n\nF_H_1[1] := mYsub (d11):\n" }{TEXT -1 132 "\nAnd now we have the x-component of force on proton #1\nWe repeat this 8 more times, for each component of each at om in the molecule." }{MPLTEXT 1 0 191 "\n\n\nF_H_1[2] := mYsub(d12): \nF_H_1[3] := mYsub(d13): \nF_H_2[1] := mYsub(d21):\nF_H_2[2] := mYsu b(d22):\nF_H_2[3] := mYsub(d23):\nF_O[1] := mYsub(dO1):\nF_O[2] := mYs ub(dO2):\nF_O[3] := mYsub(dO3):\n" }{TEXT -1 250 "\n(There may be a mo re sophisticated method for coding this, but what is presented\nhere w orks, and seems educationally valid!)We've completed the evaluation of the \ninstantaneous force vector each nucleus experiences.Now, we use the Verlet algorithm:\n" }{OLE 1 24073 1 "[xm]Br=WfoRrB:::wk;nyyI;G:; :j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::fyyyyyyN>V>^>f>n>v>>?F?N?V?^?f?nYvY:::::: ::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::: :::::gjwhGYX>efI::::::::N:GECB:VB:E`:B:::::::JFNZ;f:vYxI>:<::::::?J:^<>:F:; 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j:Z:>:R:F::;B:R:>:R:;R::=:<:@:J:j:::::::J:j:>::=:;:=Z:Z;::>Z:>:;B:=R:@ :::::;J:<:F:@:<:Z:>:F:J:@:=::::::: ;:=Z:B:@Z;Z:>:R::F::::::Z:B:@Z;:<:@:J:j:::::::Z: Z;B:::=j:@::::::::::::::::;R:F::::::Z:B:@Z;:>:F:J:j::::: ::Z:Z;B::Z:j:@:::::::::::::::::::::::;:=::::::J:<:F:;j:Z:>:= Z;:;R:F:;:=::<:@Z:Z;:::::::::::::::::::::::;:=::::::J:B:FZ;>:FZ:J:R:=: B:R:;:=::;:=:;:=::::::::::::::::::::::::::::Z;j:J:@:j:<:J:R:=::::: :<:@::::::::::::::::::::::::::::>:F:Z:J:R:=B:R:;Z:j:@::::::<:@:::::::: ::::::::::::::::::::B:R::;B:j:Z;:F::::::::::::::::::::::::::::::::::::::J::F:: ::::::::::::::::::::::::::::::::::::Z:Z;>:F::::::::::::::::::::::::::: :::::::::::B:R:;:=::::::::::::::::::::::::::::::::::::::;B:;B:R:F::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::C:IS:yyyA:;:::Ja@Na`>;B::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::6:" }{MPLTEXT 1 0 1 "\n" }{TEXT -1 76 "with a similar equation for velocity (to be used \+ further down in the code):\n" }{OLE 1 4105 1 "[xm]Br=WfoRrB:::wk;nyyI; G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::fyyyyyqyyyYJ ^:fBWMtNHm=;:::::::n:;`:Z@[::JjTTd\\sJIS]Aj;JZTZ:B:fB ]mtFFcmnvGWMJnC==nHE=;:::::JJNZ:vyyuy:>:<::::::j^K:j:vCSmlJ::::::::::O J;Zy=J:B::::::^:;JDJ:J;vCJbNHEms>@[C:>Z::::::::kJ;@J;B:=J:vYxY:B:::::: f:;B:sH;B:F:N:;j;JyKyK=j=J>j>J?>:Q:S:UJ:n;v;;JBB:]:_:a:c:e:gJ:v<>=;Jyk y;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::J HJ;>g:UR:B:KHx_jEIMbXq;@j:HJ?dj:Dk;Ua=v[;r<=b>AfUDk;Q`LVjrc>AFWFZ;N\\=pBA:JqjlJvJt:JCHR]:jqJpJlZ:N[;NZBP @_rZUM:VZ;j:Djv:K:FZ:b:;B:<:::VZ:JB>Z:ftBnKJBeME>^:vi=[::yayY:^:;j:jysy ?:;:[c>fb>JGM:=j>r:Ud:VZ:NF;B:;jDZ:B:E:?R:=:E:=b:yyyyI:E:WS:k:E:Qb: v=>jJN\\:B:;xyyqVyyyyYZ<;mv:V[:>:G;Sj`@Pt\\Pd `QrP`:>r:>;;JwE:[Z::CZ:f_;j:ab:DJTqJHB:qi:EbyAa:>:[V:>Z<> Z<>ZJVdsgg\\wgfkPqJ?LWRaYC>q^XxqGr^Xxq?^jfxqg]HnfWa:f?=J>JSdJiDjqqj:JN `QHB:qAJ:>Lb:DZJ:Y=og=Ord[Wx_:FR>>:jPNZ:^:>x;FZ::_cL=J:^ZcTTPpsq^lZKJ::Uk:^:>X?B:K:_KjDjbxj:Jt`Q>J S^k=fgDF:fgs?;N@UU:eu<=J:nbs?;N@flAF:;J]:_KjvZ>F:nhs?;N@^@=:gMHjw?^yM: <:[>;J<::::=@;;:f??Jl;>Z:b:;b:QFR>>:jPF:C:[Q;> ;N`D>f:Q;N@FvZ:VY;><:[>< JQFR>>:jPN:C:[q:>;N@SnMaj:Z_OF;B:G;mo`@@RSeTOUG:AR:=:bWH:h:t@ P@\\ALA]J?RIDk;UQ::rUBUBS:h\\=Z;NZ:>:sZ:VY[j=J:^q :OB:=K?A:X?B:AJ:aQYfDYJ;NjJyK:B::nYv:ym:<:K]:>:>=>:>=vyy?Z:n I;B::4:" }{MPLTEXT 1 0 1 "\n" }{TEXT -1 70 "Here, we assign the \"next \" value as prime, and handle the coordinates " }{MPLTEXT 1 0 151 "\n \nr_H_1_p := r_H_1 + h*v_H_1 + ((h^2)/(2*m_H))*F_H_1;\nr_H_2_p := r_H_ 2 + h*v_H_2 + ((h^2)/(2*m_H))*F_H_2;\nr_O_p := r_O + h*v_O + ((h^2)/(2 *m_O))*F_O;\n\n" }{TEXT -1 157 "Now we evaluate the new (updated) forc es, using the new (updated) coordinates. We\nneed these (updated) forc es for the velocity part of the Verlet algorithm.\n" }{MPLTEXT 1 0 242 " \nF_H_1_p[1] := mYsubP(d11):\nF_H_1_p[2] := mYsubP(d12):\nF_H_1_ p[3] := mYsubP(d13):\nF_H_2_p[1] := mYsubP(d21):\nF_H_2_p[2] := mYsubP (d22):\nF_H_2_p[3] := mYsubP(d23):\nF_O_p[1] := mYsubP(dO1):\nF_O_p[2] := mYsubP(dO2):\nF_O_p[3] := mYsubP(dO3):\n\n\n\n" }{TEXT -1 73 "Here we update the velocities using the Verlet algorithm, as noted above: " }{MPLTEXT 1 0 146 "\n\nv_H_1_p := v_H_1 + (h/(2*m_H))*(F_H_1_p + F_H _1):\nv_H_2_p := v_H_2 + (h/(2*m_H))*(F_H_2_p + F_H_2):\n\nv_O_p := v_ O + (h/(2*m_O))*(F_O_p + F_O):\n\n" }{TEXT -1 83 "Next, we make the \" old\" coordinates the new ones in preparation for the next cycle:" } {MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 139 "\nr_H_1 := r_H_1 _p:#update coordinates\nr_H_2 := r_H_2_p:\nr_O := r_O_p:\n\nv_H_1 := v _H_1_p:#update velocities\nv_H_2 := v_H_2_p:\nv_O := v_O_p:\n" }{TEXT -1 68 "\nNext, we update some quantities we might want to investigate \+ later:" }{MPLTEXT 1 0 173 " \n\nv_H_1_mag := sqrt(v_H_1[1]^2+v_H_1[2]^ 2+v_H_1[3]^2);#for kinetic energy\nv_H_2_mag := sqrt(v_H_2[1]^2+v_H_2[ 2]^2+v_H_2[3]^2);\nv_O_mag := sqrt(v_O[1]^2+v_O[2]^2+v_O[3]^2);\n" } {TEXT -1 95 "\nIn preparation for the Fourier Transform, we zero the i maginary part of the incoming function." }{MPLTEXT 1 0 322 "\n\ny[i] : = 0;#imaginary part\n \nKE(i) := 1/2*(m_H*v_H_1_mag^2+m_H*v_H_2_mag^2 +m_O*v_O_mag^2):#\nPE(i) := V(r_H_1,r_H_2,r_O):\n\nE_total(i) := evalf ((KE(i)+PE(i)+E_1)/2):#averaging the starting and ending energy\n# \+ in a cycle\nE_totaloverV[i] := (E_total(i)/V_start)*100;\nE_totalo verV_plot(i) := (E_total(i)/V_start)*100;\n" }{TEXT -1 111 "\nFor the \+ Fourier Transform, and also for plotting purposes, we assign l[i] in A ngstrom rather than centimeters." }{MPLTEXT 1 0 228 "\n\nl[i] := dist( r_H_1,r_H_2)*1e8:#convert to angstrom\nr_plot(i) := l[i]:\ntt1 := r_H_ 1-r_O;\ntt2 := r_H_2-r_O;\ntheta(i) := evalf(angle(tt1,tt2)*180/Pi); \+ \nif (normal_mode = 2) then l[i] := theta(i); end if;\nprintlevel := 0 ;\nend do;\n" }{TEXT -1 65 "\nAnd we are done with the main loop. Afte r the loop has executed " }{OLE 1 7177 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;: j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::fyyyyyq:nY^:f:v:nY>;F;N;V;^;nYvY::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq ^H;C:ELq^H_mvJ::::::::gjjJJ:j[vCWLdfBW`: B:::::::JFNZ;f:vYxI>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N<^:vYxI>:<:::: ::c:e=;:wA?:AJ:^:;j<>:G:IZ:>;;j>J?j?J@j@>:W:YJ:>\\:B:]:_J:VV>^>f>n>v>>?F?N?V?^?f_:JQjQJRB:];_;a;c;e;g;i;k ;m;o;q;s;u;wK:vA>Bi;^;>:::::::?jFv;>:Ch=:;:::::ef:UV:=R:=::::::::::B:@:j:@j::::@j:::;Z::>:=R:=:::F:::>Z:>Z:j:R:::::::::::C:=B:;B:yayQ:>:fL;j@A:Cd:<::::::gJ:^<>:N:Y<>D_mlVH[ KRJ:<:::::::>=:<::::::i:Gf<>Z:::::::::::::yay=J:B:::::::::::::: :::::jysy:>:<::::::::EZ:V[Z:>Z:vYJ:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::C:cc:V:;jTf:vZ;V:;JtE:At=;K`kAF=;B:;jA:;JN>:m:YJ:F:K:; JLA:C@C@;R::@:=:: <:R::R:B:=Z;::B:;:@:>Z;j::::R: :;R:F::B:;:@:J:@Z::>Z;R:=B:<:@Z:F:@J:B::FZ;B:;:@Z:F:Z;>:<:R:=B:;:@:;:F :;j:<:@:=B:::::WTJWTLB::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::2:" }{TEXT -1 34 "it's time to analyze \nour results." }{MPLTEXT 1 0 35 "\n\ni := 'i':#reset the variable \"i\"\n" }{TEXT -1 41 "\nWe're \+ ready to do the Fourier Transform:" }{MPLTEXT 1 0 15 "\n\nFFT(m,l,y); \n\n" }{TEXT -1 91 "All that is left to do, after the simulation has r un, is to plot and interpret the results." }{MPLTEXT 1 0 1228 "\n\npri nt(` r_plot(i):`);\nPLOT(POINTS(seq([i,r_plot(i)],i=1..n_stop),\nSYMBO L(DIAMOND),LEGEND(`r versus t`)));\n\n# i := 'i':#reset the variable \+ \"i\"\n\nprint(` theta_plot(i):`);\nPLOT(POINTS(seq([i,theta(i)],i=1.. n_stop),\nSYMBOL(DIAMOND),LEGEND(`theta versus t`)));\n\ni := 'i':#res et the variable \"i\"\n\nFreqSpectrum := [seq([(i-1),(2*mag(l[i],y[i]) /(2^m))],i=1..floor(((2^m)/2)))]:\nplot([seq(FreqSpectrum[j],j=2..floo r(((2^m)/2)))],title=\"Fourier Transform\");\n#plot([seq(FreqSpectrum[ j],j=2..40)],title=\"Fourier Transform\");\n#use the line above to see more detail, changing 40 to some low number in order to\n#discern whe re the maximum is.\n\nfrequency := number_of_cycles/(h*n_stop);\nprint (`sec^-1 = `,frequency,` times # of peaks`);\nprint(`cm^-1 = `,evalf(f requency/3e10),` times # of peaks`);\n\n\nprint(` KE(i):`);\nPLOT(POIN TS(seq([i,KE(i)],i=1..n_stop), SYMBOL(CROSS),LEGEND(`kinetic energy ve rsus t`)));\nprint(` PE(i):`);\n\nPLOT(POINTS(seq([i,PE(i)],i=1..n_sto p), SYMBOL(CIRCLE),LEGEND(`potential energy versus t`)));\nprint(` E_t otal(i):`);\n\nl_plot := [[ n, E_totaloverV_plot(n)] $n=1..n_stop]:\np lot(l_plot, n=1..n_stop, style=line,symbol=circle,labels=[`time step`, `% deviation of total energy`],labeldirections=[HORIZONTAL, VERTICAL]) ;\n\n" }{TEXT -1 1 "\n" }{TEXT 269 21 "Analysis of the Data:" }{TEXT -1 119 " (1)We can adjust the time step (h) in the code, to get as clo se as \npossible to an integral number of cycles with the " }{OLE 1 7176 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyyq :nY^:f:v:nY>;F;N;V;^;nYvY::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj::::::::NZ:j[:fBjcnDaKR:nYN:V:;J<>:EJ:n:v:>: M:OJ:V;^;f;;JAjA>:[B::a:wAe:Jyyyy?=F=N=V=^=f=n=v=>>AFANAVA^AfAnA;jYJZB:=:<::JrDLFj:Z;FZ;Z:FZ;R:B:R:::N:mjA J:^rHB:;]:Zy=<:::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :?:;:::::ef:UV:=R:=::::::::::B:;R:;:F:=:B::F:@:B:=:R:;Z::@j:: ;:=B:>::@j:::;Z::>:=R:=:::F:::> Z:>Z:j:R::::::::::Z:F:@:::::yyyyyyAJ:^:FZ:>Z:vYxY;J:jD?:UV:^bDj:F;HJKPZ:F:ci:>Z:VZ:J IF:YLpfF;B:;:::::::::N;?jyyiy=J:B::::::n<;JDJ:J;vCJbNHEms>@[;;B::::::: JFB:yay=J:B::::::v=Eb:;jus:@J::ek:ci:VbI>>S@YjF>Z:>:Y:>:KK:F=v;;j:>;>:CW:^R^R>Z;B :;B::yA>:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::j:@Z;:::Z::@Z:J:R:F::B:Z;B:=Z;>:F ::B:;:@:;R:F::::R:J:@:=::B:;:@:B:=Z;::Z;j::Z:>:R ::;R:<:J:@Z;FZ:B:R::R::R:>:j:>:=B:R:FZ:B :@Z;:Z:R:;B:j:FZ:B:R::::B:Z;:::Z:JZ;j:::::::::::::::::::::::::::::::::::::::::: ::B>N:F:nyyyyy]: :yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::fyyyyyQ=nYnyyyYE:G:I:K:M:O:c:S:UJ:n;v;;JBB:]:_:a:C:e:g:i:k:m:w AwAs:u:w:y:;[:F>N>V>^>f>n>v>>?F?N?V?^?f?n?v?>`:B:];_;a;wAyA::::::::::: :::::::::::::::::::::::::::::::::::::::_lqvGcMJ:::::::JEf: yyyxIF:>bs^lKnupvsKflxBjKGJMStB::>d[GeRHnMpr;V:>Z;Z:j[vCWLdfBW`:B:::::::JFNZ;f:vYxI>:<:: ::::JDJ:j:VBYmp>HYLkNG>::::::::N<^:vYxI>:<::::::=J:fH>:nYN::wAyA:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::J:B::::::::::F:wyyAbfccWflwgmwg`_hZ>dbwgnwgZfb^Ggnw gl?^mn_j>^JGg]_hoOh_c=jc:CJ:F:[>Jf=]:>::::::::::::::::::::::::::::::::::::j: @j:@:::::::J:@j:<:=R::::::B::R:;R:F:;B:;B:j:R:::::::::::::::::::::::::::::::::::::::::;R:FZ:Z ;::::::::::::::::::::::::::::::::::::::::::B:R:;:=:::::::::::::::: ::::::::::::::::::::::::Z;j::>Z;:F::R:F::::::::::::::::::::::::::: :::::::::::::Z:F:;Z;F:::::::::::::::::::::::::::::::::::::::::::>:;j:F :::::::::::::::::::::::::::::::::::::::::::B:;R:F::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::;:@::::::::>:R::::::::::::::J:Z; 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jR:_;sF=AZ:Fapf`r>;N@^aFF:n?K:_;su<=:[cGK:_KqFhKF:f`p>;N@F\\KF:n?JSjb< k:jV>;N@SNMTj:jiTM><:_KqfaAF:;jT:_;uh;=:W;N@f]@F:FAs:qQJZ:JBA:DJ:DJ:q^lZKJ::Uk:^:>X?B:K:_cJS>VIU>=J:f?sZ:V Y;><:[>DZjqjFlJ?dJnPXC:=@;;JX=J>JSdJjDJtmk:jXTM>JSj_Xk:JTU M>JS^KEr>=B:^PJSJwuj:<:c?N@;MUdj:jX:_K@_]>F:^PJSnGuI;=:qgGK:_Kjnv;F:;j X^=>Z\\X?J>JSJ?DL;jkpp>JS^Kcs;CJ:Vhnwgj? ;N@^^?F:ngn?;N@N`=F:>gn_=VYZ:JBK:DJ ;N`D>Fwyy=J::sg:B:=J;Dlc`qsLqlp`\\P_KnF;B :OjJNk;Z:FZ:J:B>>:uyyyQyyI:v Y::::::::::::::3:" }{TEXT -1 42 " \nwhen we use \"m\" large enough to \+ collect " }{OLE 1 7176 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyy y]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::fyyyyyq:nY^:f:v:nY>;F;N;V;^;nYvY::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ:: ::::::gj::::::::NZ:j[:fBjcnDaKR:nYN:V :;J<>:EJ:n:v:>:MJ:N;V;^;f;;JAjA>:[B::a:wAe:Jyyyy?=F=N=V=^=f=n =v=>>AFANAVA^AfAnA; jYJZB:=:<::JFDpEj:Z;FZ; Z:FZ;R:B:R:::N:mjAJ:^rHB:;]:Zy=<:::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::?:;:::::ef:UV:=R:=::::::::::B:;R:;:F:=:B:: F:@:B:=:R:;Z::@j::;:=B:>::@j:::;Z::>:=R:=:::F:::>Z:>Z:j:R::::::::::Z:F:@:::::yyyyyyAJ:^:FZ:>Z:vYxY;J:jD?:UV:^bDj:F; HJKPZ:F:ci:>Z:VZ:JIF:YLpfF;B:;:::::::::N;?jyyiy=J:B::::::n<;JDJ:J;vCJb NHEms>@[;;B:::::::JF:<:::::: :::::::::::::vYxI:;Z::::::::j:ia=R:;B:;B:yA>:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::J?;jFj A>:=J>J:^n;J\\K\\K:@Z:>Z::vYJ::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::=R:@::::<:R :<:;Z;j::Z::@Z:F:@J:j::Z:>:R:>Z;j:::B:;:@::R:Z:F :@::B:;:@:J:@:=::Z;B::;R:@j::<::=R:Z::=j:<::::::::::::::::::::::::::::::::::J;>Z:>:::::v_U>[>FZ;FZ;:::::: ::::::::::::::::::::::::::::5:" }{TEXT -1 230 "=2048 points. Our value for h is approximate.\n\n(2)Tinkering with the force constants would \+ then allow getting the closest fit of simulation frequencies \nto expe rimental frequencies. One can think of this as a form of data fitting. " }}{PARA 0 "" 0 "" {TEXT -1 112 "\n(3)Then, with just a slight altera tion in the coding, one could, for instance, predict the spectrum\n of HDO or " }{OLE 1 7688 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyy y]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::fyyyyyy:nY^:f:n:>;nYF;;J?j?J@j@>:wAyA:::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_m vJ::::::::gj::::::::N:F:AlqfG[maNFO=;::::::::_:C:yay=;Z::::::jFjtJ: JyK;j;>:CJ:f:;J=j=B:K:M:O:Q:S:U:W:YJ:>\\:B:]:_J:V<^:wAo:Jyy yyg=n=v=>>AFANAVA^A fAnA;jYJZB:=:<::JvUwEFZ;FZ;::Z:FZ;R:::J;F?]J:>:Ch=:F::B:R:<:@::<:@:;:=:Z:Z;J:j::B:R:B :R::>:F:<:@::;:=:;:=:Z:Z;J:j::>:F:B:R::<:@:<:;B:R:=Z:F:@:<:@::;:=Z:J:R :j:>:F:>:F::<:@:;:=Z:Z;B:R:>:F::<:@Z:>Z;j:J:@:=Z:Z;>:F::Z;j:::::;R:F::::::J:B:FZ;:::::J::C:=B:r:GX;m=Z:VZ:JIf:?Eb:vXHZ;>Z:>Z:vY<:;:::::: :::::::::::::::::::j:vCSmlJ::::::::::OJ;vyyuy:>:<::::::qZ:^<@[C:>Z::::::::kJ;:<::::::s:Wx:<::::::: ::::::::::::vYxI:;Z::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::^:^`:kv:VbI>>S@]B:MK:?;jNjB>Z:F:K:;JpE:C@C@;R:Z;j:Z::@:;R:F:;B:=Z;Z:F:@:;:=B:R:B:;B:R:B:=Z;J:j:B:R:<:@:;:=Z: >:R:B:=Z;J:B:F:R:<:@:<:@Z:>:R:B:=Z;Z:J:<:@j:B:=Z;Z:Z;B:;:@:>Z;j:B:R::< :@:>:FZ:>:R:J:@:=Z:Z;:J:j:J:j::FZ:>:R:Z:F:@J:j:::;:=B:R:::::WTJWTLB:=Z;::::::::B:=Z;::::::::::::::::::::::::J: B:FZ;:::::Z:B:=R:@:::::::::::1:" }{TEXT -1 52 ". Changing to Tritium i sotopes is equally as easy.\n\n" }{TEXT 266 0 "" }{TEXT 267 33 "Sugges tions for Alternative Uses:" }{TEXT 268 0 "" }{TEXT -1 611 " (1) Given the above code, it is easy to add a fourth\n atom, say another oxygen , and treat hydorgen peroxide, or say a proton, and transmutate \nthe \+ oxygen into nitrogen and simulate ammonia (or, if you're wedded to the oxygen, \nperhaps the hydronium cation). In any case, adding to the c ode is straightforward, \nbut one should be warned that this is the sl owest method extant for doing molecular dynamics,\nand given enough nu cleii, the number of cycles of vibration will rapidly become too\nsmal l to allow analysis.\n\n(2)It is possible to obtain a quantity which i s rarely (if ever) presented in the literature of" }}{PARA 0 "" 0 "" {TEXT -1 100 "vibrational spectroscopy, viz., the sensitivity of the v ibrational frequency to the force \nconstant," }{OLE 1 10760 1 "[xm]Br =WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::fyyyyyI;;JyK J?JyKyyyy_;f;;JAjA>:[Z:F^:fBWMtNHm=;:::::::nwhGYX>efI=?R:E:yay=;Z::::::: ^<>:F:AlqfG[maNFO=;::::::::_:C:yay=;Z::::::JQjtJ:JyK;j;>:CB:;j<>:G:IZ: >;F;N;V;^;f;;JAjAJB:i:kJ:F=N=V=^=f=n=v=>>:<::JnUmEJ:::::::N:gkLJ:^rHB:;]:Zy=<::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::?J:>:::::fl?Vt=FZ;FZ;::::::::::::::::: :::Z;j:Z::;R:F::::F:: >:F:::<:@::<:@:;:=::;:=B:;:=:Z:F:@::<:@:;:=:>Z:>:;R:FZ:R:=j::F:B:J:j:>Z;j:<:@:<:@::<:@:;:=:;B:R:;R:F:;:=B:R:B:R::B:R:>:F:>:FZ :Z;J::F:B:R::B:R:>:F:>:FZ:Z;J:B:FZ;:<:@::<:@:;:=:;:j:J:>:=j::<:@::;B:; :=J:@:=::F:::::B:R::: J:j:::::Z:Z;:::;:=:::::<:@:::>:F:::::B:R:Z:>Z:>:j:>Z;R:=Z:>:;:::F:@Z:F :@:::>:F:::::B:R:::J:j:::::Z:Z;:::;:=:::::<:@:::>:F:::::B:R:::J:j::Z;j:>:=Z;:B:=Z;:::;:=:>:B:R::F::B:R:<:@:Z:Z;:J:<:@:;R:F: :B:R:;:=:Z:Z;:J:<:@J:@J:F:=:B:;B:<:@J:@J::;R:=j::;: :;j:F:B:;:=B:=::F:B::@J::;j:F:B:R::B:;R:F: ;R:F:J:Z;Z:F:@:B:=Z;:::;B:;B:=Z;::::;Z:F:@:::::::::J:@J:F::F:::: Z:Z;Z::;R:F:;R::<::::::[[:^<>:N:Y<>D_mlVH[KRJ:<:::::::>=?B:yay=J :B::::::V;NyJJ:::::::::::::yay=J:B:::::::::::::::::::jysy:>:<::::::::= B:;b:Z:vYxY;J:jD?:ax:>qJ j:@Z;:::;R:=B:FZ;:::::::::::::::::::Z:B:=R:@::::::::::::::::::::::B:;: @::::::::::::::::::::vyyyyyY:QF:MZ=^eBj:n_@Z:VZ:B:w j:e[:j=@j;>:Wc;VbI>>S@EKU>Z:>:E;>:KK::=J>J:>]B>Z:^R^R>Z;B:;B::y A>:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::j:@Z;::::::::J:F:@::J:>Z:F:@::::>:Z:>:F:;R:B:=Z;::J:F:=:<:@ J:<:@:B:=J:R:j:Z:Z;>:FZ:B:@Z;:B:R:;B:R:Z:F:J:R:=:<:@J:j::=j:Z:>:FZ:F:B:@Z;:B:R:;B:R:Z:F:@J:>:=j::;:<:@J:@J:F:=:B: R:;B:R:J:Z;>Z;>:=j::Z;>Z:B:=R:@:Z:Z;::>Z;j::Z:Z;B:R::<:@J:<:@::J:@:=:>:B:R::j:>Z;R:=Z:>:;:::F:@Z:F:@J::F:;R:F:B:;:@: >:=Z;:B:=Z;>Z:Z;::F:>::=:;J:@j:F:Z:Z;>Z:Z;::F:>:FZ:Z;J:B:F Z;:<:@J:<:@:B:=Z;J:j:J:j:<:@:;:j:Z:Z;>Z:Z;::F:>Z:Z;>Z;j:>:FZ:Z;Z :Z;>Z:Z;::F:B:J:j:>Z;j:<:@:<:@J:<:@:B:=Z;J:j::;B:;J:j:Z:Z;::F::>:FZ:>:F::Z:Z;Z:B: R:>Z;j:Z::;R:F:::::R:::::::::>:C:IS:yyyA:;:::Ja@Na`>;B:::::::::::::::::::::::::::::::: ::::::::::::::1:" }{TEXT -1 273 "for each fundamental frequency and fo r each force constant. \nThe easiest way to do this would be to run th e simulation, in a fixed normal mode,\nas a function of \"k\", a force constant in the potential energy expression, plot the results,\nand t ake the derivative graphically." }{MPLTEXT 1 0 0 "" }}}}{MARK "0 22 2 " 273 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }