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y Fl(RR)1004 4667 y Fu(\()p Fm(R)q Fu(\))h(+)f Fm(V)1282 4679 y Fl(r)r(R)1369 4667 y Fu(\()p Fm(r)n(;)c(R)q Fu(\))p Fm(;)194 b Fu(\(2\))-150 4816 y(the)21 b(in)n(teraction)f(of)h (the)g(electrons)f(with)i(eac)n(h)e(other)g Fm(V)1582 4828 y Fl(r)r(r)1652 4816 y Fu(\()p Fm(r)r Fu(\))i(the)-150 4911 y(in)n(teraction)32 b(b)r(et)n(w)n(een)g(the)h(n)n(uclei)g Fm(V)1037 4923 y Fl(RR)1142 4911 y Fu(\()p Fm(R)q Fu(\))g(and)g(the)g (in)n(terac-)-150 5007 y(tion)28 b(of)f(the)h(n)n(uclei)g(with)g(the)g (electrons)e Fm(V)1222 5019 y Fl(r)r(R)1310 5007 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\).)-67 5102 y(The)25 b(dynamics)f(of)g(the)h (system)f(is)h(giv)n(en)e(b)n(y)i(the)f(solution)g(of)-150 5198 y(the)k(time)g(dep)r(enden)n(t)g(Sc)n(hr\177)-42 b(odinger)26 b(equation)347 5389 y Fm(i)p Fu(\026)-42 b Fm(h)449 5333 y(@)p 434 5370 79 4 v 434 5446 a(@)5 b(t)522 5389 y Fu(\011\()p Fm(r)n(;)14 b(R)q(;)g(t)p Fu(\))23 b(=)g Fm(H)7 b Fu(\011\()p Fm(r)n(;)14 b(R)q(;)g(t)p Fu(\))p Fm(:)390 b Fu(\(3\))2042 -83 y(In)42 b(surface)g(hopping,)k (the)d(w)n(a)n(v)n(e)d(function)j(is)g(expanded)f(in)2042 12 y(an)i(orthonormal)e(basis)h(set)i(of)f(electronic)f(w)n(a)n(v)n(e)g (functions)2042 108 y Fm(\036)2091 120 y Fl(i)2119 108 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))2498 291 y(\011\()p Fm(r)n(;)g(R)q(;)g(t)p Fu(\))23 b(=)2940 212 y Fk(X)2988 389 y Fl(i)3074 291 y Fm( )3128 303 y Fl(i)3156 291 y Fu(\()p Fm(R)q(;)14 b(t)p Fu(\))p Fm(\036)3400 303 y Fl(i)3428 291 y Fu(\()p Fm(r)n(;)g(R)q Fu(\))350 b(\(4\))2042 530 y(where)35 b(the)h(n)n(uclear)e(p)r(osition)i Fm(R)g Fu(acts)f(as)g(a)g(parameter.)59 b(In-)2042 626 y(sertion)27 b(in)n(to)h(the)g(Sc)n(hr\177)-42 b(odinger)26 b(equation)i(\(3\),)g(m) n(ultiplication)2042 721 y(with)35 b Fm(\036)2287 691 y Fl(?)2287 743 y(j)2360 721 y Fu(from)e(the)i(left)g(and)f(in)n (tegration)f(o)n(v)n(er)g Fm(r)k Fu(leads)c(to)i(a)2042 817 y(set)27 b(of)h(coupled)f(equations)g(for)g(the)h Fm( )3266 829 y Fl(i)3294 817 y Fu('s:)2042 1193 y Fm(i)p Fu(\026)-42 b Fm(h)2144 1171 y Fu(_)2118 1193 y Fm( )2172 1205 y Fl(j)2207 1193 y Fu(\()p Fm(R)q(;)14 b(t)p Fu(\))23 b(=)g(\()p Fm(T)2594 1205 y Fl(R)2667 1193 y Fu(+)18 b Fm(V)2798 1205 y Fl(RR)2903 1193 y Fu(\))c Fm( )3003 1205 y Fl(j)3038 1193 y Fu(\()p Fm(R)q(;)g(t)p Fu(\))2141 1402 y(+)2220 1323 y Fk(X)2268 1500 y Fl(i)2354 1260 y Fk(")2402 1402 y Fm(V)2450 1414 y Fl(j)s(i)2527 1402 y Fj(\000)2628 1298 y Fn(3)p Fl(N)2614 1323 y Fk(X)2610 1499 y Fl(\013)p Fn(=1)2804 1346 y Fu(\026)-42 b Fm(h)2851 1308 y Fn(2)p 2761 1383 170 4 v 2761 1459 a Fu(2)p Fm(M)2884 1471 y Fl(\013)2941 1402 y Fm(D)3012 1368 y Fl(\013)3010 1422 y(j)s(i)3086 1402 y Fu(+)3187 1298 y Fn(3)p Fl(N)3173 1323 y Fk(X)3169 1499 y Fl(\013)p Fn(=1)3346 1346 y Fm(i)p Fu(\026)g Fm(h)p 3320 1383 128 4 v 3320 1459 a(M)3401 1471 y Fl(\013)3458 1402 y Fm(d)3501 1368 y Fl(\013)3501 1422 y(j)s(i)3574 1402 y Fu(\()p Fm(i)p Fu(\026)g Fm(h)o Fj(r)3751 1414 y Fl(R)3801 1422 y Fi(\013)3848 1402 y Fu(\))3880 1260 y Fk(#)3942 1402 y Fm( )3996 1414 y Fl(i)4024 1402 y Fu(\()p Fm(R)q(;)14 b(t)p Fu(\))p Fm(:)-106 b Fu(\(5\))2042 1641 y(where)30 b Fm(N)40 b Fu(is)30 b(the)i(n)n(um)n(b)r (er)e(of)h(n)n(uclei)g(and)f Fm(M)3514 1653 y Fl(\013)3592 1641 y Fu(the)h(mass)g(cor-)2042 1737 y(resp)r(onding)c(to)g(the)i Fm(\013)p Fu(-th)f(co)r(ordinate)f Fm(R)3369 1749 y Fl(\013)3416 1737 y Fu(.)38 b(Here)27 b(w)n(e)h(ha)n(v)n(e)e(in-)2042 1832 y(tro)r(duced)h(the)h(non-adiabatic)e(coupling)h(v)n(ectors)e Fm(D)3738 1802 y Fl(\013)3736 1854 y(j)s(i)3795 1832 y Fu(\()p Fm(R)q Fu(\))i(and)2042 1927 y Fm(d)2085 1897 y Fl(\013)2085 1949 y(j)s(i)2143 1927 y Fu(\()p Fm(R)q Fu(\))h(as)2461 2107 y Fm(D)2532 2073 y Fl(\013)2530 2128 y(j)s(i)2588 2107 y Fu(\()p Fm(R)q Fu(\))23 b(=)g Fj(h)p Fm(\036)2908 2119 y Fl(j)2944 2107 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))p Fj(jr)3236 2073 y Fn(2)3236 2128 y Fl(R)3286 2136 y Fi(\013)3332 2107 y Fj(j)p Fm(\036)3404 2119 y Fl(i)3433 2107 y Fu(\()p Fm(r)n(;)g(R)q Fu(\))p Fj(i)313 b Fu(\(6\))2042 2270 y(and)2462 2433 y Fm(d)2505 2398 y Fl(\013)2505 2453 y(j)s(i)2564 2433 y Fu(\()p Fm(R)q Fu(\))23 b(=)g Fj(h)p Fm(\036)2884 2445 y Fl(j)2919 2433 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))p Fj(jr)3211 2445 y Fl(R)3261 2453 y Fi(\013)3308 2433 y Fj(j)p Fm(\036)3380 2445 y Fl(i)3408 2433 y Fu(\()p Fm(r)n(;)g(R)q Fu(\))q Fj(i)p Fm(;)314 b Fu(\(7\))2042 2595 y(no)n(w)26 b(making)g(use)h(of)f (the)i(bra-k)n(et)d(notation)h(with)i(in)n(tegration)2042 2691 y(o)n(v)n(er)35 b(the)i(fast)g(co)r(ordinates)f Fm(r)j Fu(only)-7 b(.)65 b(The)37 b(matrix)f(elemen)n(ts)2042 2786 y Fm(V)2090 2798 y Fl(j)s(i)2149 2786 y Fu(\()p Fm(R)q Fu(\))28 b(of)f(the)h(electronic)f(Hamiltonian)g Fm(H)3462 2798 y Fl(r)3527 2786 y Fu(are)f(giv)n(en)h(b)n(y)2389 2949 y Fm(V)2437 2961 y Fl(j)s(i)2496 2949 y Fu(\()p Fm(R)q Fu(\))c(=)g Fj(h)p Fm(\036)2816 2961 y Fl(j)2852 2949 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))p Fj(j)p Fm(H)3144 2961 y Fl(r)3181 2949 y Fu(\()p Fm(r)n(;)g(R)q Fu(\))p Fj(j)p Fm(\036)3453 2961 y Fl(i)3481 2949 y Fu(\()p Fm(r)n(;)g(R)q 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Fl(RR)3636 5389 y Fu(\()p Fm(R)q Fu(\))p Fm(:)150 b Fu(\(13\))p eop %%Page: 3 3 3 2 bop 4043 -299 a Fu(3)-150 -83 y(The)25 b(equations)f(of)h(motion)f (go)n(v)n(erning)f(the)i(classical)e(particles)-150 12 y(are)k(then)h(giv)n(en)e(b)n(y)173 192 y Fm(M)254 204 y Fl(\013)319 171 y Fu(\177)301 192 y Fm(R)364 204 y Fl(\013)412 192 y Fu(\()p Fm(t)p Fu(\))d(=)g Fj(\000r)751 204 y Fl(R)801 212 y Fi(\013)861 192 y Fu([)p Fm(V)932 204 y Fl(j)s(j)998 192 y Fu(\()p Fm(R)q Fu(\))c(+)f Fm(V)1276 204 y Fl(RR)1381 192 y Fu(\()p Fm(R)q Fu(\)])c Fm(:)176 b Fu(\(14\))-150 372 y(The)22 b(motion)g(of)f(the)i(n)n(uclei)e(along)g (a)g(tra)5 b(jectory)21 b Fm(R)q Fu(\()p Fm(t)p Fu(\))h(is)f(at)h(an)n (y)-150 468 y(time)28 b Fm(t)g Fu(giv)n(en)e(b)n(y)h(the)h(forces)f(b)r (et)n(w)n(een)g(the)h(classical)e(particles)-150 563 y Fj(\000r)-16 575 y Fl(R)38 563 y Fm(V)86 575 y Fl(RR)219 563 y Fu(and)i(the)g(p)r(oten)n(tial)f Fm(V)922 575 y Fl(j)s(j)1016 563 y Fu(of)h(a)f(single)g(state)g Fm(\036)1663 575 y Fl(j)1699 563 y Fu(.)-67 660 y(There)47 b(are)g(some)g(lib)r (erties)h(in)g(ho)n(w)f(to)h(implemen)n(t)g(the)-150 755 y(hopping)c(b)r(et)n(w)n(een)g(the)h(p)r(oten)n(tial)g(energy)e (surfaces)g(result-)-150 851 y(ing)c(in)h(v)-5 b(arious)39 b(surface)f(hopping)i(v)n(ersions.)71 b(F)-7 b(or)39 b(example,)-150 946 y(in)30 b(their)f(original)f(pap)r(er)h(T)-7 b(ully)29 b(and)h(Preston)e(allo)n(w)n(ed)g(jumps)-150 1042 y(only)f(at)h(distinct)h(p)r(oin)n(ts)e(in)i(con\014guration)d (space,)h(namely)h(at)-150 1137 y(the)39 b(crossings)e(of)i(the)h (diabatic)e(p)r(oten)n(tials)h([28)o(].)71 b(This)39 b(has)-150 1232 y(the)34 b(adv)-5 b(an)n(tage)32 b(of)h(eliminating)g (the)h(need)f(to)g(adjust)h(the)g(ki-)-150 1328 y(netic)c(energy)e (during)g(a)h(hop,)h(but)g(determining)f(the)g(crossing)-150 1423 y(lines)g(of)g(higher)f(dimensional)h(energy)f(surfaces)f(can)i(b) r(e)h(a)e(dif-)-150 1519 y(\014cult)35 b(problem)f(to)g(solv)n(e.)56 b(In)34 b(con)n(trast)f(to)h(that,)j(the)d(few)n(est)-150 1614 y(switc)n(hes)c(algorithm,)g(in)n(tro)r(duced)g(b)n(y)g(T)-7 b(ully)31 b(in)g(Ref.)g([18)o(],)g(al-)-150 1710 y(lo)n(ws)e(switc)n (hes)h(at)g(an)n(y)f(p)r(osition)h(but)h(requires)e(some)h(form)f(of) -150 1805 y(energy)d(adjustmen)n(t)i(to)g(reco)n(v)n(er)d(energy)h (conserv)-5 b(ation.)-67 1902 y(In)23 b(this)f(w)n(ork)f(w)n(e)i(will)f (fo)r(cus)h(on)f(the)h(few)n(est)f(switc)n(hing)g(algo-)-150 1997 y(rithm.)49 b(The)32 b(electronic)e(w)n(a)n(v)n(e)g(function)j Fm(\036)p Fu(\()p Fm(r)n(;)14 b(t)p Fu(\))32 b(is)g(expanded)-150 2092 y(in)37 b(a)f(set)g(of)g(orthonormal)f(basis)g(functions)i Fm(\036)1388 2104 y Fl(i)1416 2092 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))37 b(with)g(a)-150 2188 y(parametric)e(dep)r(endence)j(on)e (the)h(n)n(uclear)f(co)r(ordinates)g(and)-150 2283 y(co)r(e\016cien)n (ts)27 b Fm(c)306 2295 y Fl(i)334 2283 y Fu(\()p Fm(t)p Fu(\))357 2484 y Fm(\036)p Fu(\()p Fm(r)n(;)14 b(t)p Fu(\))25 b(=)684 2405 y Fk(X)732 2582 y Fl(i)818 2484 y Fm(c)854 2496 y Fl(i)882 2484 y Fu(\()p Fm(t)p Fu(\))14 b Fm(\036)1039 2496 y Fl(i)1067 2484 y Fu(\()p Fm(r)n(;)g(R)q Fu(\()p Fm(t)p Fu(\)\))p Fm(:)361 b Fu(\(15\))-150 2741 y(In)50 b(order)e(to)h(determine)h(the)f(time)h(ev)n(olution)f(of)g (the)h(co-)-150 2836 y(e\016cien)n(ts)j Fm(c)251 2848 y Fl(j)286 2836 y Fu(,)59 b(one)52 b(considers)949 2804 y Fl(@)p 936 2818 65 4 v 936 2865 a(@)t(t)1011 2836 y Fm(A)1073 2806 y Fl(?)1073 2858 y(j)1111 2836 y Fm(A)1173 2848 y Fl(j)1261 2836 y Fu(and)h(iden)n(ti\014es)f Fm(c)1857 2848 y Fl(j)-150 2958 y Fu(with)27 b Fm(A)100 2970 y Fl(j)135 2958 y Fm(e)189 2905 y Fi(i)p 184 2914 33 4 v 184 2948 a Fh(\026)-31 b Fi(h)226 2928 y Fl(S)267 2936 y Fi(j)302 2958 y Fu(.)36 b(Then)27 b(the)f(time)h(ev)n(olution)f(of)g (the)h(co)r(e\016cien)n(ts)-150 3053 y(is)g(giv)n(en)g(b)n(y)h([24)o(,) f(25)o(])74 3312 y(_)-34 b Fm(c)99 3324 y Fl(i)126 3312 y Fu(\()p Fm(t)p Fu(\))51 b(=)e Fj(\000)464 3233 y Fk(X)508 3410 y Fl(j)597 3170 y Fk(\()664 3312 y Fm(c)700 3324 y Fl(j)735 3312 y Fu(\()p Fm(t)p Fu(\))843 3233 y Fk(X)882 3408 y Fl(\013)977 3220 y Fk(h)1044 3291 y Fu(_)1016 3312 y Fm(R)1079 3324 y Fl(\013)1127 3312 y Fu(\()p Fm(t)p Fu(\))p Fm(d)1264 3278 y Fl(\013)1264 3333 y(ij)1337 3312 y Fu(\()p Fm(R)q Fu(\()p Fm(t)p Fu(\)\))1560 3220 y Fk(i)1599 3170 y(\))385 3574 y Fj(\000)469 3518 y Fm(i)p 460 3555 48 4 v 460 3631 a Fu(\026)-42 b Fm(h)531 3495 y Fk(X)576 3672 y Fl(j)665 3574 y Fm(V)713 3586 y Fl(ij)786 3574 y Fu(\()p Fm(R)q Fu(\()p Fm(t)p Fu(\)\))14 b Fm(c)1058 3586 y Fl(j)1093 3574 y Fu(\()p Fm(t)p Fu(\))p Fm(:)535 b Fu(\(16\))-150 3842 y(W)-7 b(e)36 b(no)n(w)f(can)g(de\014ne)h(the)g (comp)r(onen)n(ts)f(of)h(the)g(densit)n(y)f(ma-)-150 3937 y(trix)29 b Fm(a)55 3949 y Fl(ij)139 3937 y Fj(\021)c Fm(c)265 3907 y Fl(?)265 3959 y(i)304 3937 y Fm(c)340 3949 y Fl(j)404 3937 y Fu(with)30 b(the)f(diagonal)f(elemen)n(ts)h(b)r (eing)g(the)h(o)r(c-)-150 4033 y(cupation)j(probabilities)f(of)h(the)g (corresp)r(onding)e(states.)52 b(The)-150 4128 y(time)28 b(deriv)-5 b(ativ)n(e)27 b(of)g(the)h Fm(a)701 4140 y Fl(ij)787 4128 y Fu(is)g(giv)n(en)f(b)n(y)-108 4345 y Fm(i)p Fu(\026)-42 b Fm(h)9 b Fu(_)-33 b Fm(a)12 4357 y Fl(k)q(j)107 4345 y Fu(=)194 4266 y Fk(X)244 4445 y Fl(l)314 4345 y Fj(f)p Fm(a)400 4357 y Fl(lj)456 4345 y Fu([)p Fm(V)527 4357 y Fl(k)q(l)594 4345 y Fj(\000)5 b Fm(i)p Fu(\026)-42 b Fm(h)760 4300 y Fu(_)751 4324 y Fm(~)740 4345 y(R)829 4323 y(~)804 4345 y(d)847 4357 y Fl(k)q(l)909 4345 y Fu(])p Fj(g)5 b(\000)g Fm(a)1093 4357 y Fl(k)q(l)1153 4345 y Fu([)p Fm(V)1224 4357 y Fl(lj)1286 4345 y Fj(\000)g Fm(i)p Fu(\026)-42 b Fm(h)1452 4300 y Fu(_)1443 4324 y Fm(~)1432 4345 y(R)1518 4323 y(~)1495 4345 y(d)1538 4357 y Fl(lj)1595 4345 y Fu(])p Fj(g)20 b Fm(:)42 b Fu(\(17\))-150 4603 y(F)-7 b(or)29 b(the)g(diagonal)f (elemen)n(ts)h Fm(a)864 4615 y Fl(ii)915 4603 y Fu(,)g(this)h(equation) e(can)h(b)r(e)h(sim-)-150 4699 y(pli\014ed)e(to)665 4895 y(_)-34 b Fm(a)698 4907 y Fl(ii)772 4895 y Fu(=)860 4816 y Fk(X)905 4993 y Fl(j)994 4895 y Fm(b)1030 4907 y Fl(ij)1745 4895 y Fu(\(18\))-150 5163 y(where)27 b(the)h Fm(b)269 5175 y Fl(ij)355 5163 y Fu(are)e(de\014ned)i(as)270 5389 y Fm(b)306 5401 y Fl(ij)388 5389 y Fu(=)488 5333 y(2)p 485 5370 V 485 5446 a(\026)-42 b Fm(h)543 5389 y Fj(=)14 b Fu([)p Fm(a)684 5401 y Fl(ij)742 5389 y Fm(V)790 5401 y Fl(ij)849 5389 y Fu(])19 b Fj(\000)f Fu(2)p Fj(<)1090 5297 y Fk(h)1128 5389 y Fm(a)1172 5401 y Fl(ij)1230 5389 y Fm(d)1273 5401 y Fl(ij)1359 5368 y Fu(_)1332 5389 y Fm(R)1396 5297 y Fk(i)1449 5389 y Fm(:)273 b Fu(\(19\))2042 -83 y(Note)31 b(that)h(the)g Fm(b)2613 -71 y Fl(ij)2702 -83 y Fu(are)f(an)n(ti-symmetric)f(under)h(index)h(in)n(ter-)2042 12 y(c)n(hange)22 b Fm(b)2348 24 y Fl(ij)2429 12 y Fu(=)h Fj(\000)p Fm(b)2618 24 y Fl(j)s(i)2698 12 y Fu(follo)n(wing)g(from)f (the)i(fact)f(that)h(the)f Fm(d)3891 24 y Fl(ij)3973 12 y Fu(are)2042 108 y(an)n(ti-symmetric)j(as)h(w)n(ell.)2125 203 y(A)n(t)i(an)n(y)f(giv)n(en)h(time)g(the)g(system)g(is)g (considered)f(to)h(b)r(e)g(in)h(a)2042 299 y(single)24 b(basis)h(state,)g(the)g(curren)n(tly)f(o)r(ccupied)h(state,)h(whic)n (h)f(w)n(e)2042 394 y(will)i(denote)g(with)g Fm(\036)2702 406 y Fl(occ)2799 394 y Fu(.)36 b(The)27 b(motion)g(of)g(the)g(slo)n(w) f(degrees)f(of)2042 490 y(freedom)i(is)g(go)n(v)n(erned)f(b)n(y)h(that) h(state)2160 678 y(\177)2142 699 y Fm(R)2205 711 y Fl(\013)2253 699 y Fu(\()p Fm(t)p Fu(\))23 b(=)2479 643 y Fj(\000)p Fu(1)p 2468 680 128 4 v 2468 756 a Fm(M)2549 768 y Fl(\013)2606 699 y Fj(r)2675 711 y Fl(R)2725 719 y Fi(\013)2785 699 y Fj(f)p Fm(V)2875 711 y Fl(RR)2994 699 y Fu(\()p Fm(R)q Fu(\()p Fm(t)p Fu(\)\))2252 922 y(+)2331 809 y Fk(Z)2427 922 y Fm(dr)j(\036)2582 888 y Fl(?)2582 943 y(occ)2693 922 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\()p Fm(t)p Fu(\)\))h Fm(H)3071 934 y Fl(r)3121 922 y Fu(\()p Fm(r)n(;)f(R)q Fu(\()p Fm(t)p Fu(\)\))h Fm(\036)3479 934 y Fl(occ)3589 922 y Fu(\()q Fm(r)n(;)f(R)q Fu(\()p Fm(t)p Fu(\)\))3884 805 y Fk(\033)3960 922 y Fm(:)-46 b Fu(\(20\))2042 1142 y(Note)28 b(that)g(this)g(set)f(of)h(equations)f(w)n(as)f(deriv)n(ed,)h (in)h(a)g(sligh)n(tly)2042 1238 y(di\013eren)n(t)g(manner,)f(b)n(y)h(T) -7 b(ully)28 b(in)g(Ref.)h([18)o(])f(with)h(the)f(assump-)2042 1333 y(tion)22 b(that)h(the)f(slo)n(w)g(particle)f(mo)n(v)n(emen)n(t)h (is)g(giv)n(en)f(b)n(y)h(a)g(tra)5 b(jec-)2042 1429 y(tory)33 b(and)h(that)g(the)g(electronic)g(w)n(a)n(v)n(e)e(function)j(is)f (expanded)2042 1524 y(according)26 b(to)h(eq.)h(\(15\).)2125 1620 y(One)38 b(problem)f(with)i(the)f(mixed)g(quan)n(tum)g(classical)f (sur-)2042 1715 y(face)i(hopping)g(is)h(the)g(dep)r(endence)g(on)g(the) g(c)n(hoice)e(of)i(basis)2042 1811 y(functions.)59 b(Usually)35 b(w)n(e)f(will)h(use)g(the)g(adiabatic)f(basis,)j(but)2042 1906 y(dep)r(ending)25 b(on)g(the)g(particular)f(system)g(it)i(is)e (not)h(alw)n(a)n(ys)e(clear)2042 2001 y(what)k(w)n(ould)g(b)r(e)h(an)g (optimal)f(basis)g([20)o(].)2125 2097 y(W)-7 b(e)45 b(will)f(no)n(w)g (discuss)g(ho)n(w)g(the)g(o)r(ccupied)h(state)f(is)g(de-)2042 2192 y(termined)39 b(in)f(the)h(few)n(est)g(switc)n(hes)f(algorithm.)68 b(The)39 b(equa-)2042 2288 y(tions)19 b(\(16\))f(and)h(\(14\))g(are)f (in)n(tegrated)g(sim)n(ultaneously)g(and,)i(b)r(e-)2042 2383 y(t)n(w)n(een)30 b(in)n(tegration)f(steps,)i(switc)n(hes)f(b)r(et) n(w)n(een)g(the)h(electronic)2042 2479 y(states)38 b(are)g(p)r (erformed,)j(in)e(order)e(to)h(main)n(tain)h(the)g(correct)2042 2574 y(statistical)f(o)r(ccupation)f(probabilities.)69 b(This)38 b(metho)r(d)h(min-)2042 2670 y(imises)e(the)g(n)n(um)n(b)r (er)g(of)g(switc)n(hes)g(p)r(erformed.)64 b(Ho)n(w)37 b(this)g(is)2042 2765 y(ac)n(hiev)n(ed)h(is)i(explained)f(nicely)h(in)g ([18)o(])g(and)g(w)n(e)f(will)h(follo)n(w)2042 2861 y(that)26 b(presen)n(tation.)35 b(If)27 b(w)n(e)f(consider)f(a)h(sw)n(arm)e(of)j Fm(N)35 b Fu(tra)5 b(jecto-)2042 2956 y(ries)24 b(with)i(o)r(ccupation) e(probabilities)g Fm(a)3315 2968 y Fl(ii)3391 2956 y Fu(at)h(time)g Fm(t)g Fu(and)g Fm(a)3934 2926 y Fg(0)3934 2978 y Fl(ii)4010 2956 y Fu(at)2042 3052 y(time)k Fm(t)2262 3022 y Fg(0)2309 3052 y Fu(=)24 b Fm(t)19 b Fu(+)g(\001)p Fm(t)29 b Fu(then)f Fm(N)2915 3064 y Fl(i)2967 3052 y Fu(=)c Fm(a)3100 3064 y Fl(ii)3151 3052 y Fm(N)37 b Fu(and)29 b Fm(N)3494 3022 y Fg(0)3485 3073 y Fl(i)3541 3052 y Fu(=)24 b Fm(a)3674 3022 y Fg(0)3674 3073 y Fl(ii)3725 3052 y Fm(N)37 b Fu(are)28 b(the)2042 3147 y(n)n(um)n(b)r(er)k(of)h (tra)5 b(jectories)30 b(in)j(state)f Fm(i)h Fu(at)f(times)h Fm(t)f Fu(and)h Fm(t)22 b Fu(+)f(\001)p Fm(t)p Fu(.)2042 3243 y(The)k(net)g(c)n(hange)f(of)h(the)g(n)n(um)n(b)r(er)g(of)g(tra)5 b(jectories)23 b(in)i(state)g Fm(i)f Fu(is)2042 3338 y(then)k(giv)n(en)f(b)n(y)2491 3509 y(\001)p Fm(N)2627 3521 y Fl(i)2678 3509 y Fu(=)22 b Fm(N)2841 3475 y Fg(0)2832 3529 y Fl(i)2883 3509 y Fj(\000)c Fm(N)3033 3521 y Fl(i)3083 3509 y Fu(=)23 b(\()p Fm(a)3247 3475 y Fg(0)3247 3529 y Fl(ii)3317 3509 y Fj(\000)18 b Fm(a)3444 3521 y Fl(ii)3495 3509 y Fu(\))c Fm(N)t(:)302 b Fu(\(21\))2042 3680 y(F)-7 b(or)19 b(short)g(time)i(steps)e(\001)p 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b(their)g(n)n(um)n(b)r(er)g(is)h(giv)n(en)e(b)n(y)h Fm(N)9 b Fu(\001)p Fm(tb)4003 4524 y Fl(j)s(i)4061 4512 y Fu(.)2042 4607 y(If)35 b(w)n(e)g(no)n(w)g(divide)g(that)h(b)n(y)f (the)g(n)n(um)n(b)r(er)g(of)h(tra)5 b(jectories)33 b(in)2042 4703 y(state)23 b Fm(i)g Fu(w)n(e)g(get)g(the)h(probabilit)n(y)f Fm(p)3147 4715 y Fl(ij)3228 4703 y Fu(for)g(a)g(tra)5 b(jectory)22 b(to)h(jump)2042 4798 y(from)k(state)g Fm(i)h Fu(in)n(to)f(state)g Fm(j)2639 5012 y(p)2681 5024 y Fl(ij)2762 5012 y Fu(=)2860 4955 y Fm(N)9 b Fu(\001)p Fm(tb)3071 4967 y Fl(j)s(i)p 2860 4992 270 4 v 2909 5068 a Fm(a)2953 5080 y Fl(ii)3004 5068 y Fm(N)3162 5012 y Fu(=)23 b(\001)p Fm(t)3359 4955 y(b)3395 4967 y Fl(j)s(i)p 3359 4992 96 4 v 3359 5068 a Fm(a)3403 5080 y Fl(ii)3464 5012 y Fm(:)450 b Fu(\(23\))2042 5222 y(Note)25 b(that)h(the)g(switc)n(hing)f (probabilities)g(are)g(prop)r(ortional)e(to)2042 5318 y(the)39 b(time)h(step,)j(th)n(us)c(the)h(total)f(switc)n(hing)f (probabilit)n(y)h(for)2042 5413 y(a)f(tra)5 b(jectory)37 b(when)i(passing)f(through)g(a)g(certain)g(region)f(is,)p eop %%Page: 4 4 4 3 bop 4043 -299 a Fu(4)-150 -83 y(apart)40 b(from)g(an)g (discretisation)f(error,)j(indep)r(enden)n(t)g(of)e(the)-150 12 y(time)35 b(steps)f(used.)57 b(Note)35 b(also)e(that)i(the)g (probabilities)e(giv)n(en)-150 108 y(b)n(y)i(eq.)g(\(23\))g(could)g(in) h(principle)f(get)g(arbitrarily)f(large,)i(but)-150 203 y(usually)24 b(the)h(time)g(step)g(required)f(for)g(accurate)f(in)n (tegration)g(of)-150 299 y(eq.)e(\(16\))g(is)h(small)f(and)g(switc)n (hes)g(will)h(b)r(e)g(improbable)e(\(see)i(also)-150 394 y(Ref.)28 b([19)o(]\).)-67 493 y(The)38 b(pro)r(cedure)f(for)g(the) h(sim)n(ulation)g(of)f(a)h(single)f(surface)-150 588 y(hopping)22 b(tra)5 b(jectory)20 b(with)i(the)h(few)n(est)f(switc)n (hing)f(metho)r(d)i(can)-150 684 y(b)r(e)28 b(roughly)e(divided)i(in)n (to)g(the)g(follo)n(wing)e(\014v)n(e)h(steps.)-150 855 y Ff(Step)32 b(1)h Fu(Set)53 b(initial)g(conditions)g(for)f(the)i(n)n 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Fs(is)g(the)f(v)n(elo)r(cit)n(y)2042 1311 y(b)r(efore)35 b(the)f(hop)g(and)2763 1292 y(_)2738 1311 y Fe(R)2796 1319 y Fd(T)2879 1311 y Fs(and)3063 1292 y(_)3037 1311 y Fe(R)3095 1319 y Fd(R)3180 1311 y Fs(are)h(the)f(t)n(w)n(o)h(new)g(v)n(elo)r(cities)2042 1398 y(corresp)r(onding)26 b(to)g(the)f(t)n(w)n(o)h(solutions)h(for)f Fe(\025)f Fs(of)h(24.)36 b(The)25 b(radius)h(of)2042 1485 y(the)d(circle)i(is)f(giv)n(en)g(b)n(y)f(the)g(v)n(elo)r(cit)n(y)h (after)g(the)g(jump,)f(where)h Fe(E)3971 1453 y Fd(k)q(in)3967 1498 y(new)2042 1572 y Fs(is)i(the)f(new)h(kinetic)g(energy)-6 b(.)2042 1859 y Fu(used.)47 b(The)32 b(solution)e(with)i(the)g(smaller) e(absolute)g(v)-5 b(alue)32 b(cor-)2042 1955 y(resp)r(onds)f(to)i(a)f (transmission)f(of)h(the)h(surface)f(normal)f(to)h Fm(d)4025 1967 y Fl(ij)2042 2050 y Fu(and)27 b(the)h(solution)g(with)g(the)g (larger)e(v)-5 b(alue)27 b(is)h(a)f(re\015ection.)37 b(As)2042 2146 y(is)d(suggested)f(in)i([31)o(,)h(34)o(])e(w)n(e)g(used) g(the)h Fm(\025)g Fu(with)f(the)h(smaller)2042 2241 y(absolute)f(v)-5 b(alue.)59 b(This)35 b(corresp)r(onds)e(to)i(the)g(suppression)f(of) 2042 2337 y(an)n(y)27 b(quan)n(tum)g(re\015ection)g(at)h(the)g(p)r (osition)f(of)h(the)g(switc)n(h.)2125 2435 y(Another)41 b(problem)g(is)g(that)h(quan)n(tum)f(mec)n(hanics)g(allo)n(ws)2042 2531 y(the)35 b(in)n(trusion)e(of)i(the)g(w)n(a)n(v)n(e)d(function)j (in)n(to)g(classical)e(forbid-)2042 2626 y(den)23 b(areas.)34 b(This)23 b(results)f(in)i(hops)e(where)h(there)g(is)g(no)g(solution) 2042 2722 y(for)39 b(eq.)g(\(24\).)71 b(In)40 b(the)f(case)g(of)g(suc)n (h)g(an)g(ev)n(en)n(t)g(one)g(has)f(to)2042 2817 y(either)33 b(sacri\014ce)f(consistency)-7 b(,)35 b(suggesting)d(a)h(hop,)i(or)d (energy)2042 2913 y(conserv)-5 b(ation.)70 b(In)40 b(the)f(few)n(est)g (switc)n(hes)g(algorithm)f(energy)2042 3008 y(forbidden)26 b(hops)g(are)f(rejected)i(th)n(us)f(partially)g(giving)f(up)i(con-)2042 3104 y(sistency)-7 b(.)34 b(There)19 b(is)h(an)f(extensiv)n(e)g 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b(=)2576 4129 y Fl(S)2538 4154 y Fk(X)2544 4331 y Fl(i)p Fn(=1)2671 4233 y Fm(c)2707 4245 y Fl(i)2735 4233 y Fu(\()p Fm(t)p Fu(\))p Fm(\036)2878 4245 y Fl(i)2920 4233 y Fu(\()q Fm(r)n(;)14 b(R)q Fu(\()p Fm(t)p Fu(\)\))19 b(+)f Fm(\036)3366 4245 y Fl(C)3436 4233 y Fu(\()p Fm(r)n(;)c(R)q Fu(\()p Fm(t)p Fu(\)\))g Fm(:)170 b Fu(\(25\))2042 4495 y(The)27 b(equation)g(of)h(motion)f(for) g Fm(\036)i Fu(is)e(giv)n(en)g(b)n(y)2478 4723 y Fm(i)p Fu(\026)-42 b Fm(h)2580 4667 y(@)p 2565 4704 79 4 v 2565 4780 a(@)5 b(t)2653 4723 y(\036)p Fu(\()p Fm(r)n(;)14 b(t)p Fu(\))24 b(=)f Fm(H)3049 4735 y Fl(r)3100 4723 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\()p Fm(t)p Fu(\)\))g Fm(\036)p Fu(\()p Fm(r)n(;)g(t)p Fu(\))p Fm(;)291 b Fu(\(26\))2042 4936 y(whic)n(h)32 b(is)g(a)g(time)h(dep)r(enden)n(t)g(Sc)n(hr\177)-42 b(odinger)30 b(equation)i(due)g(to)2042 5031 y(the)37 b(time)h(dep)r(endence)g(of)f Fm(R)q Fu(\()p Fm(t)p Fu(\).)67 b(The)37 b(electronic)g(Hamilto-)2042 5127 y(nian)k Fm(H)2309 5139 y Fl(r)2387 5127 y 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b(co)r(ordinates)g(of)h(b)r(oth)h(adsorbate)e(and)h(substrate)g(at)-150 4171 y(su\016cien)n(tly)e(long)f(propagation)e(times)j(to)g(correctly)e (describ)r(e)-150 4267 y(thermalization)d(and)g(dissipation)g (e\013ects.)-67 4363 y(One)54 b(of)f(the)h(problems)g(encoun)n(tered)f (when)g(mo)r(delling)-150 4458 y(DIET)34 b(pro)r(cesses)g(with)h (surface)e(hopping)h(algorithms)g(is)g(the)-150 4554 y(h)n(uge)22 b(n)n(um)n(b)r(er)h(of)f(electronic)g(adsorbate)f(and)i (substrate)f(states)-150 4649 y(in)n(v)n(olv)n(ed)f(whic)n(h)g(need)h (to)g(b)r(e)h(tak)n(en)e(in)n(to)h(accoun)n(t.)34 b(Due)22 b(to)g(the)-150 4745 y(large)k(n)n(um)n(b)r(er)h(of)g(and)h(similar)e (shap)r(es)h(of)h(the)f(corresp)r(onding)-150 4840 y(p)r(oten)n(tials,) i(it)g(is)f(clear)g(that)h(it)g(is)f(neither)h(feasible)f(to)h(explic-) -150 4936 y(itly)e(include)f(all)g(these)g(substrate)g(states)g(in)n (to)g(our)f(sim)n(ulation)-150 5031 y(nor)34 b(is)g(it)g(necessary)f (since)h(the)h(reaction)e(dynamics)h(is)g(dom-)-150 5127 y(inated)g(b)n(y)g(a)g(few)h(adsorbate)d(states,)k(whic)n(h)e(m)n(ust)g (b)r(e)h(tak)n(en)-150 5222 y(in)n(to)d(accoun)n(t.)49 b(The)32 b(main)f(e\013ect)i(of)f(the)g(substrate)f(states)h(is)-150 5318 y(coupling)j(di\013eren)n(t)g(adsorbate)f(states)h(either)g(to)g (eac)n(h)g(other)-150 5413 y(or)f(to)h(an)f(external)g(electromagnetic) f(\014eld.)59 b(W)-7 b(e)35 b(mo)r(del)g(this)2042 -83 y(e\013ect)26 b(collectiv)n(ely)e(b)n(y)h(com)n(bining)f(ideas)h(from)g (T)-7 b(ully's)25 b(few)n(est)2042 12 y(switc)n(hing)f(algorithm)g([18) o(])h(and)f(generalised)g(surface)g(hopping)2042 108 y(metho)r(d)k([37)o(])g(with)g(those)f(of)g(Brenig)g([38)o(])g(and)h (Saalfrank)e([39)o(])2042 203 y(who)36 b(in)n(tro)r(duced)g(optical)f (p)r(oten)n(tials)h(to)g(the)h(description)e(of)2042 299 y(DIET)27 b(pro)r(cesses.)2125 394 y(As)f(with)g(all)f(metho)r(ds)h (b)r(efore)g(the)g(Hamiltonian)g Fm(H)32 b Fu(is)26 b(split)2042 490 y(in)n(to)31 b(the)h(kinetic)f(energy)g Fm(T)2953 502 y Fl(R)3038 490 y Fu(of)g(the)h(n)n(uclear)e(co)r(ordinates)g Fm(R)2042 585 y Fu(and)e(an)g(electronic)g(part)g Fm(H)2945 597 y Fl(r)2981 585 y Fu(,)h(where)f(the)h(electronic)f(part)f(de-)2042 681 y(p)r(ends)22 b(explicitly)g(on)f(the)h(electronic)f(co)r (ordinates)g Fm(r)j Fu(and)e(para-)2042 776 y(metrically)27 b(on)g(the)h(p)r(osition)f(of)h(the)g(n)n(uclei)g Fm(R)2603 931 y(H)7 b Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))23 b(=)g Fm(T)3039 943 y Fl(R)3111 931 y Fu(+)18 b Fm(H)3263 943 y Fl(r)3300 931 y Fu(\()p Fm(r)n(;)c(R)q Fu(\))p Fm(:)414 b Fu(\(31\))2042 1085 y(Just)25 b(as)f(in)h(the)g(generalised)f (surface)g(hopping)g(algorithm,)h(the)2042 1181 y(electronic)20 b(w)n(a)n(v)n(e)g(function)i(\011)f(is)g(expanded)g(in)n(to)g(the)h (explicitly)2042 1276 y(treated)j(excited)h(adsorbate)e(states)i Fm(\036)3269 1288 y Fl(i)3323 1276 y Fu(and)g(a)f(collectiv)n(e)g (state)2042 1372 y Fm( )39 b Fu(con)n(taining)c(the)h(molecular)f (ground)g(state)h(together)f(with)2042 1467 y(the)28 b(con)n(tin)n(uum)f(of)h(substrate)f(excitations)2235 1622 y(\011\()p Fm(r)n(;)14 b(R)q(;)g(t)p Fu(\))23 b(=)2677 1543 y Fk(X)2725 1720 y Fl(i)2811 1622 y Fm(c)2847 1634 y Fl(i)2875 1622 y Fu(\()p Fm(t)p Fu(\))p Fm(\036)3018 1634 y Fl(i)3046 1622 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))19 b(+)f Fm(\036)3397 1634 y Fl(C)3453 1622 y Fu(\()p Fm(r)n(;)c(R)q(;)g (t)p Fu(\))p Fm(:)194 b Fu(\(32\))2042 1853 y(The)36 b(in\015uence)h(of)f(this)g(collectiv)n(e)g(state)f Fm(\036)3478 1865 y Fl(C)3571 1853 y Fu(on)h(the)h(rest)e(of)2042 1948 y(the)i(electronic)f(system)g(can)g(b)r(e)h(tak)n(en)f(in)n(to)g (accoun)n(t)g(b)n(y)g(an)2042 2044 y(e\013ectiv)n(e)f(non-Hermitian)f (Hamiltonian)h(\(see)g(c)n(hapter)f(16)g(in)2042 2139 y([40)o(]\))2257 2294 y Fm(H)2326 2306 y Fl(ef)7 b(f)2439 2294 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))23 b(=)g Fm(T)2799 2306 y Fl(r)2854 2294 y Fu(+)18 b Fm(V)2985 2306 y Fl(ef)7 b(f)3099 2294 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))k(+)g Fm(i)p Fu(\001\()p Fm(r)n(;)c(R)q Fu(\))p Fm(;)216 b Fu(\(33\))2042 2448 y(where)28 b Fm(T)2332 2460 y Fl(r)2398 2448 y Fu(is)h(the)g(kinetic)g(energy)f(op)r(erator)g(of)h(the)g (electrons.)2042 2544 y(The)g(e\013ectiv)n(e)h(p)r(oten)n(tial)f Fm(V)2937 2556 y Fl(ef)7 b(f)3080 2544 y Fu(and)30 b(the)g(optical)f(p) r(oten)n(tial)g(\001)2042 2639 y(are)22 b(real)g(functions)i(of)f Fm(r)k Fu(and)c Fm(R)q Fu(.)35 b(In)23 b(a)g(Newns-Andersson)f(pic-) 2042 2735 y(ture)e(\001)g(is)g(related)f(to)h(the)g(lifetime)h (broadening)d(of)i(a)g(resonance)2042 2830 y(state)27 b(whic)n(h)h(can)f(b)r(e)h(determined)g(via)f([41)o(])2491 3005 y(\001\()p Fm(E)5 b Fu(\))51 b(=)f Fm(\031)2948 2927 y Fk(X)2988 3105 y Fc(k)3082 3005 y Fj(j)p Fm(V)3153 3017 y Fc(k)3197 3005 y Fj(j)3220 2971 y Fn(2)3258 3005 y Fm(\016)s Fu(\()p Fm(E)23 b Fj(\000)18 b Fm(")3536 3017 y Fc(k)3580 3005 y Fu(\))p Fm(:)302 b Fu(\(34\))2042 3243 y(With)57 b(the)f(e\013ectiv)n(e)g(Hamiltonian)g(and)g(a)g (diabatic)g(\(i.e.)2042 3339 y Fj(r)2111 3351 y Fl(R)2165 3339 y Fm(\036)2214 3351 y 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Fl(e)3034 4014 y Fu(+)18 b Fm(V)3165 4026 y Fl(ef)7 b(f)3279 4014 y Fu(\()p Fm(r)n(;)14 b(R)q Fu(\))p 3493 4038 V 32 w Fm(\036)3560 4026 y Fl(i)3588 4014 y Fj(i)2492 4135 y Fu(\001)2561 4147 y Fl(j)s(i)2670 4135 y Fj(\021)49 b(h)p Fm(\036)2865 4147 y Fl(j)p 2915 4158 V 2932 4135 a Fu(\001\()p Fm(r)n(;)14 b(R)q Fu(\))p 3215 4158 V 32 w Fm(\036)3282 4147 y Fl(i)3309 4135 y Fj(i)42 b Fm(;)531 b Fu(\(36\))2042 4289 y(resp)r(ectiv)n(ely)-7 b(.)71 b(F)-7 b(or)38 b(the)i(diagonal)e(elemen)n(ts)h(of)g(the)g (densit)n(y)2042 4385 y(matrix)27 b Fm(a)2356 4397 y Fl(j)s(i)2442 4385 y Fu(this)h(leads)f(to)2504 4585 y(_)-33 b Fm(a)2538 4597 y Fl(j)s(j)2627 4585 y Fu(=)2714 4506 y Fk(X)2763 4683 y Fl(i)2848 4585 y Fm(b)2884 4597 y Fl(j)s(i)2961 4585 y Fu(+)3044 4506 y Fk(X)3092 4683 y Fl(i)3191 4529 y Fu(2)p 3187 4566 48 4 v 3187 4642 a(\026)-42 b Fm(h)3245 4585 y Fj(<)14 b Fu([)p Fm(a)3386 4597 y Fl(j)s(i)3444 4585 y Fu(\001)3513 4597 y Fl(j)s(i)3572 4585 y Fu(])g Fm(;)305 b Fu(\(37\))2042 4829 y(with)34 b Fm(b)2273 4841 y Fl(j)s(i)2365 4829 y Fj(\021)2476 4796 y Fn(2)p 2474 4810 38 4 v 2474 4858 a(\026)-34 b Fl(h)2522 4829 y Fj(=)14 b Fu([)p Fm(a)2663 4841 y Fl(j)s(i)2721 4829 y Fm(V)2769 4841 y Fl(j)s(i)2828 4829 y Fu(])34 b(and)g(the)h(densit)n(y)f(matrix)f(elemen)n(ts)2042 4925 y(de\014ned)27 b(as)e(b)r(efore.)37 b(Note)26 b(that)h(for)f(a)g (normalised)f(w)n(a)n(v)n(e)g(func-)2042 5020 y(tion)37 b(\011)f(the)i(o)r(ccupation)e(probabilit)n(y)g Fm(a)3385 5032 y Fl(cc)3485 5020 y Fu(for)h(the)g(\\rest")e(is)2042 5115 y(simply)27 b(giv)n(en)g(b)n(y)h Fm(a)2684 5127 y Fl(cc)2770 5115 y Fj(\021)23 b Fu(1)18 b Fj(\000)3001 5053 y Fk(P)3088 5140 y Fl(j)3137 5115 y Fm(a)3181 5127 y Fl(j)s(j)3247 5115 y Fu(,)27 b(leading)g(to)2415 5329 y(_)-34 b Fm(a)2448 5341 y Fl(cc)2535 5329 y Fu(=)22 b Fj(\000)2701 5250 y Fk(X)2745 5427 y Fl(j)2845 5329 y Fu(_)-34 b Fm(a)2878 5341 y Fl(j)s(j)2967 5329 y Fu(=)23 b Fj(\000)3133 5273 y Fu(2)p 3130 5310 48 4 v 3130 5386 a(\026)-42 b Fm(h)3201 5250 y Fk(X)3234 5427 y Fl(ij)3335 5329 y Fj(<)14 b Fu([)p Fm(a)3476 5341 y Fl(j)s(i)3534 5329 y Fu(\001)3603 5341 y Fl(j)s(i)3662 5329 y Fu(])g Fm(;)215 b Fu(\(38\))p eop %%Page: 6 6 6 5 bop 4043 -299 a Fu(6)-150 -83 y(since)51 -145 y Fk(P)139 -58 y Fl(j)s(i)211 -83 y Fm(b)247 -71 y Fl(j)s(i)329 -83 y Fu(=)22 b(0.)36 b(The)26 b(classically)e(treated)h(n)n(uclear)g (co)r(ordi-)-150 12 y(nates)31 b Fm(R)i Fu(ob)r(ey)e(formally)g(the)h (same)g(Newtonian)f(equation)g(of)-150 108 y(motion)c(as)g(for)g(the)h (generalised)e(surface)h(hopping)364 297 y(\177)346 318 y Fm(R)d Fu(=)530 262 y Fj(\000)p Fu(1)p 530 299 107 4 v 538 375 a Fm(M)646 318 y Fj(r)729 201 y Fk(\024)783 262 y Fj(h)p Fm(\036)864 274 y Fl(occ)p 975 283 4 84 v 992 262 a Fm(H)1061 274 y Fl(e)p 1111 283 V 1128 262 a Fm(\036)1177 274 y Fl(occ)1273 262 y Fj(i)p 783 299 523 4 v 851 375 a(h)p Fm(\036)932 387 y Fl(occ)p 1043 396 4 84 v 1060 375 a Fm(\036)1109 387 y Fl(occ)1206 375 y Fj(i)1316 201 y Fk(\025)1373 318 y Fm(;)349 b Fu(\(39\))-150 528 y(where)34 b Fm(\036)146 540 y Fl(occ)277 528 y Fu(is)g(the)g (curren)n(tly)g(o)r(ccupied)g(state.)56 b(In)35 b(order)e(to)-150 624 y(determine)28 b(b)r(oth)g(the)h(classical)d(and)i(the)g(quan)n (tum)g(dynamics)-150 719 y(self-consisten)n(tly)-7 b(,)30 b(the)g(equations)g(\(35\))f(and)h(\(39\))g(need)g(to)g(b)r(e)-150 815 y(in)n(tegrated)37 b(sim)n(ultaneously)g(while)i(the)f(curren)n (tly)f(o)r(ccupied)-150 910 y(state)e(is)g(determined)g(via)g(the)g (few)n(est)g(switc)n(hing)g(algorithm.)-150 1006 y(The)c(probabilities) f Fm(P)556 1018 y Fl(j)s(i)646 1006 y Fu(for)g(hops)h(b)r(et)n(w)n(een) f(the)i(di\013eren)n(t)f(p)r(o-)-150 1101 y(ten)n(tials)c Fj(h)p Fm( )229 1113 y Fl(occ)p 339 1122 V 356 1101 a Fm(H)425 1113 y Fl(e)p 475 1122 V 492 1101 a Fm( )546 1113 y Fl(occ)642 1101 y Fj(i)14 b Fm(=)g Fj(h)p Fm( )830 1113 y Fl(occ)p 940 1122 V 957 1101 a Fm( )1011 1113 y Fl(occ)1108 1101 y Fj(i)27 b Fu(for)f(the)i(classical)d(mo-)-150 1197 y(tion)j(are)e(for)h(jumps)i(b)r(et)n(w)n(een)e(explicitly)h (treated)f(states)493 1405 y Fm(p)535 1417 y Fl(j)s(i)616 1405 y Fu(=)714 1349 y(\001)p Fm(t)14 b Fu(2)g Fj(=)p Fu(\()p Fm(a)1019 1361 y Fl(ij)1077 1349 y Fm(V)1125 1361 y Fl(ij)1184 1349 y Fu(\))p 714 1386 503 4 v 886 1462 a(\026)-42 b Fm(ha)978 1474 y Fl(j)s(j)1226 1405 y Fm(;)496 b Fu(\(40\))-150 1617 y(and)19 b(the)g(probabilit)n(y)e Fm(p)594 1629 y Fl(j)s(C)700 1617 y Fu(to)h(go)g(from)g(state)h Fm(j)24 b Fu(to)18 b(the)h(collectiv)n(e)-150 1712 y(state)27 b Fm(\036)102 1724 y Fl(C)186 1712 y Fu(is)h(giv)n(en)f(b)n(y)439 1915 y Fm(p)481 1927 y Fl(j)s(C)590 1915 y Fu(=)688 1859 y Fj(\000)p Fu(2\001)p Fm(t)p 688 1896 206 4 v 712 1972 a Fu(\026)-42 b Fm(h)o(a)803 1984 y Fl(j)s(j)917 1836 y Fk(X)965 2013 y Fl(i)1051 1915 y Fm(a)1095 1927 y Fl(j)s(i)1153 1915 y Fu(\001)1222 1927 y Fl(j)s(i)1281 1915 y Fm(:)441 b Fu(\(41\))-150 2152 y(As)46 b(in)g(the)g(few)n(est)g(switc)n(hes)f (algorithm,)50 b(surface)45 b(hopping)-150 2248 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the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def % small fishscales 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 16 16 true [ 16 0 0 -16 0 16 ] {<008000800080014001400220 0c187007c001800080004001 40012002180c0770>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P17 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin %%Page: 1 1 10 setmiterlimit 0.06299 0.06299 sc % Polyline % % Begin Imported EPS File: exci_pot2.eps %%BeginDocument: exci_pot2.eps % n gs 1980 329 tr 18.551797 -18.603524 sc 270 rot -104 -132 tr sa n 104 132 m 558 132 l 558 605 l 104 605 l cp clip n countdictstack mark /showpage {} def % EPS file follows: %!PS-Adobe-2.0 EPSF-1.2 %%BoundingBox: 104 132 558 605 %%Title: exci_pot2.eps %%Creator: axel@AD777.Erding with xmgr v4.00 (Beta) %%CreationDate: Sun Jan 21 17:54:47 2001 %%EndComments 80 dict begin /languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix makepattern /Pat0 exch def <} image end } >> matrix makepattern /Pat1 exch def <} image end } >> matrix makepattern /Pat2 exch def <} image end } >> matrix makepattern /Pat3 exch def <} image end } >> matrix makepattern /Pat4 exch def <} image end } >> matrix makepattern /Pat5 exch def <} image end } >> matrix makepattern /Pat6 exch def <} image end } >> matrix makepattern /Pat7 exch def <} image end } >> matrix makepattern /Pat8 exch def <} image end } >> matrix makepattern /Pat9 exch def <} image end } >> matrix makepattern /Pat10 exch def <} image end } >> matrix makepattern /Pat11 exch def <} image end } >> matrix makepattern /Pat12 exch def <} image end } >> matrix makepattern /Pat13 exch def <} image end } >> matrix makepattern /Pat14 exch def <} image end } >> matrix makepattern /Pat15 exch def }{ /Pat0 { 0.000000 setgray } def /Pat1 { 0.062500 setgray } def /Pat2 { 0.125000 setgray } def /Pat3 { 0.187500 setgray } def /Pat4 { 0.250000 setgray } def /Pat5 { 0.312500 setgray } def /Pat6 { 0.375000 setgray } def /Pat7 { 0.437500 setgray } def /Pat8 { 0.500000 setgray } def /Pat9 { 0.562500 setgray } def /Pat10 { 0.625000 setgray } def /Pat11 { 0.687500 setgray } def /Pat12 { 0.750000 setgray } def /Pat13 { 0.812500 setgray } def /Pat14 { 0.875000 setgray } def /Pat15 { 0.937500 setgray } def /setpattern { } def } ifelse /m {moveto} bind def /l {lineto} bind def /s {stroke} bind def % Symbol fill /f { gsave fill grestore stroke } bind def % Opaque symbol /o { gsave 1.000000 1.000000 1.000000 setrgbcolor fill grestore stroke } bind def % Circle symbol /a { 3 -1 roll 0 360 arc } bind def /da { a s } bind def /fa { a f } bind def /oa { a o } bind def % Square symbol /sq { moveto dup dup rmoveto 2 mul dup neg 0 rlineto dup neg 0 exch rlineto 0 rlineto closepath } bind def /dsq { sq s } bind def /fsq { sq f } bind def /osq { sq o } bind def % Triangle symbols /t1 { moveto dup 0 exch rmoveto dup neg dup 2 mul rlineto 2 mul 0 rlineto closepath } bind def /dt1 { t1 s } bind def /ft1 { t1 f } bind def /ot1 { t1 o } bind def /t2 { moveto dup neg 0 rmoveto dup dup 2 mul exch neg rlineto 2 mul 0 exch rlineto closepath } bind def /dt2 { t2 s } bind def /ft2 { t2 f } bind def /ot2 { t2 o } bind def /t3 { moveto dup neg 0 exch rmoveto dup dup 2 mul rlineto neg 2 mul 0 rlineto closepath } bind def /dt3 { t3 s } bind def /ft3 { t3 f } bind def /ot3 { t3 o } bind def /t4 { moveto dup 0 rmoveto dup dup -2 mul exch rlineto -2 mul 0 exch rlineto closepath } bind def /dt4 { t4 s } bind def /ft4 { t4 f } bind def /ot4 { t4 o } bind def % Diamond symbol /di { moveto dup 0 exch rmoveto dup neg dup rlineto dup dup neg rlineto dup dup rlineto closepath } bind def /ddi { di s } bind def /fdi { di f } bind def /odi { di o } bind def % Plus symbol /pl { dup 0 rmoveto dup -2 mul 0 rlineto dup dup rmoveto -2 mul 0 exch rlineto } bind def /dpl { m pl s } bind def % x symbol /x { dup dup rmoveto dup -2 mul dup rlineto 2 mul dup 0 rmoveto dup neg exch rlineto } bind def /dx { m x s } bind def % Splat symbol /dsp { m dup pl dup 0 exch rmoveto 0.707 mul x s } bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.24 0.24 scale 1 setlinecap mark /ISOLatin1Encoding 8#000 1 8#054 {StandardEncoding exch get} for /minus 8#056 1 8#217 {StandardEncoding exch get} for /dotlessi 8#301 1 8#317 {StandardEncoding exch get} for /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis /ISOLatin1Encoding where not {256 array astore def} if cleartomark /makeISOEncoded { findfont /curfont exch def /newfont curfont maxlength dict def /ISOLatin1 (-ISOLatin1) def /curfontname curfont /FontName get dup length string cvs def /newfontname curfontname length ISOLatin1 length add string dup 0 curfontname putinterval dup curfontname length ISOLatin1 putinterval def curfont { exch dup /FID ne { dup /Encoding eq { exch pop ISOLatin1Encoding exch } if dup /FontName eq { exch pop newfontname exch } if exch newfont 3 1 roll put } { pop pop } ifelse } forall newfontname newfont definefont } def /Times-Roman makeISOEncoded pop /Times-Bold makeISOEncoded pop /Times-Italic makeISOEncoded pop /Times-BoldItalic makeISOEncoded pop /Helvetica makeISOEncoded pop /Helvetica-Bold makeISOEncoded pop /Helvetica-Oblique makeISOEncoded pop /Helvetica-BoldOblique makeISOEncoded 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b(including)40 b(the)g(remaining)-150 4840 y(molecular)21 b(degrees)g(of)i(freedom)f(in)g(the)h(sim)n (ulations,)g(has)f(only)-150 4936 y(a)27 b(small)f(in\015uence)i(on)e (the)i(desorption)e(dynamics,)g(as)h(T)-7 b(able)26 b(I)-150 5031 y(sho)n(ws.)40 b(The)29 b(same)g(is)g(also)e(true)i(for)g(the)g (transition)g(from)f(3D)-150 5127 y(to)39 b(7D.)h(Ho)n(w)n(ev)n(er,)g (including)f(recoil)g(pro)r(cesses)f(of)h(the)h(sub-)-150 5222 y(strate,)25 b(b)n(y)g(coupling)g(the)g(motion)g(of)h(the)f (adsorbate)f(molecule)-150 5318 y(to)e(a)h(surface)e(oscillator,)h(c)n (hanges)f(the)i(outcome)f(of)h(the)g(tra)5 b(jec-)-150 5413 y(tory)20 b(calculations)f(signi\014can)n(tly)-7 b(.)34 b(While)21 b(the)g(total)f(desorption)2118 1157 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 2267 @rwi @setspecial %%BeginDocument: NONiO_RotMom_7Dv0.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: NONiO_RotMom_7Dv0.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Mon Apr 12 19:55:23 2004 %%DocumentFonts: (atend) 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vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { 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stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont /Helvetica reencodeISO def /MFshow {{dup dup 0 get findfont exch 1 get scalefont setfont [ currentpoint ] exch dup 2 get 0 exch rmoveto dup dup 5 get exch 4 get {show} {stringwidth pop 0 rmoveto}ifelse dup 3 get {2 get neg 0 exch rmoveto pop} {pop aload pop moveto}ifelse} forall} bind def /MFwidth {0 exch {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont 5 get stringwidth pop add} {pop} ifelse} forall} bind def /MLshow { currentpoint stroke M 0 exch R MFshow } bind def /MRshow { currentpoint stroke M exch dup MFwidth neg 3 -1 roll R MFshow } def /MCshow { currentpoint stroke M exch dup MFwidth -2 div 3 -1 roll R MFshow } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 240 scalefont setfont 1.000 UL LTb 720 504 M currentpoint gsave translate 90 rotate 0 0 moveto [ [(Helvetica) 240.0 0.0 true true (Rot. 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b(dissipation)f(tak)n(es)f(place)h(while)h(the)g (molecule)f(is)g(in)h(its)-150 3637 y(ground)27 b(state)h(and)f(only)h (the)g(time)g(during)g(whic)n(h)f(desorption)-150 3732 y(still)c(o)r(ccurs)f(is)g(shortened)g(b)n(y)h(the)g(friction.)35 b(Without)23 b(friction,)-150 3828 y(there)29 b(is)g(a)g(long)f(trail)h (of)g(molecules)f(desorbing)g(at)h(v)n(ery)f(large)-150 3923 y(times,)23 b(whic)n(h)e(con)n(tribute)g(a)g(substan)n(tial)g (amoun)n(t)g(to)g(the)h(total)-150 4019 y(desorption)h(probabilit)n(y) -7 b(.)35 b(This)25 b(tail)f(is)g(cut)h(o\013)f(b)n(y)h(the)f(dissipa-) -150 4114 y(tiv)n(e)j(e\013ects)g(and)f(th)n(us)h(lo)n(w)n(ering)e(the) i(desorption)f(probabilit)n(y)-7 b(.)2125 -83 y(The)32 b(e\013ect)h(of)g(the)g(friction)f(on)h(the)g(momen)n(tum)f(distribu-) 2042 12 y(tion)i(and)g(the)g(vibrational,)g(oscillator)e(and)i (rotational)e(state)2042 108 y(o)r(ccupation)44 b(probabilities)g(of)h (the)g(desorbing)e(molecules)h(is)2042 203 y(negligible,)38 b(apart)d(from)h(an)g(o)n(v)n(erall)e(scaling)i(due)g(to)g(the)h(re-) 2042 299 y(duced)32 b(desorption)e(probabilit)n(y)-7 b(.)48 b(Most)31 b(tra)5 b(jectories)30 b(that)h(do)2042 394 y(desorb)38 b(lea)n(v)n(e)f(the)j(surface)e(at)g(times)i(usually)e (smaller)g(than)2042 490 y(the)29 b(corresp)r(onding)e(deca)n(y)h(time) i Fm(\034)3180 502 y Fl(D)3269 490 y Fu(of)f(the)g(oscillator.)39 b(Th)n(us)2042 585 y(the)25 b(amoun)n(t)f(of)h(energy)f(lost)g(is)h (rather)e(small)i(and)f(th)n(us)h(the)g(\014-)2042 681 y(nal)h(o)r(ccupation)g(probabilities)g(are)g(almost)g(una\013ected.)37 b(Note)2042 776 y(that)28 b(this)g(is)f(true)h(for)f(all)g(frictional)g (constan)n(ts)g(used.)2631 872 y Fp(IV.)88 b(CONCLUSIONS)2125 1120 y Fu(W)-7 b(e)29 b(ha)n(v)n(e)f(implemen)n(ted)i(a)f(mixed)g(quan) n(tum-classical)e(sur-)2042 1215 y(face)k(hopping)g(sc)n(heme)f(for)h (the)h(sim)n(ulation)e(of)i(electronically)2042 1311 y(non-adiabatic)27 b(pro)r(cesses)g(at)i(surfaces.)39 b(In)28 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