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Application of The Electron Nuclear Dynamics Theory to Hydrogen Abstraction and Exchange Reactions of H + HOD and D2 + NH3+


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L *(+0&$)",&+,&.%(N,(%+. T 5Y 5 Y 9 T 5 5 & A B AKB !&)*)*(3"--(.5",%&,2+0&$)",[.(=#+)&",."10")&",1"-)*(3",7#2+)(0"0(,)+ +. Y 5T T 5 -& AKB Y 5 T -9 T AKB =#+)&", K .*"!.)*+) 5 &.+3",.)+,)"10")&",+,%)*+))*()")+$+,2#$+0"0(,)#0&.3",.(-6(%*(0+2,&)#%("1)*()")+$+,2#$+-0"0(,)#0"1)*( 3"$$&.&",&.+$."-($+)(%)")*(&,&)&+$&05+3)5+-+0()(-+,%)*(&,&)&+$3"$$&.&", 6($"3&): < T T .&, 9 AKHB *(N,+$ 9 1"-)*(3"$$&.&",43+$$(%.3+))(-&,2+,2$(43+,:(3+$3#$+)(%:#.&,2 C5-(..&", II +,% K A!&)* T B+,%)*((=#+)&", fi"9 T fiY 9 Y AKJB !&)*)*(0&,#..&2,3"--(.5",%&,2)"+))-+3)&6(.3+))(-&,2+,%)*(5".&)&6(.&2,)" -(5#$.&6(.3+))(-&,2",.&%(-&,2)*+))*(&,)(2-+$&..00()-&3!-))*(%&.)+,3( "13$".(.)+55-"+3*A B!(3+,!-&)( 9 T 8 A B AKFB =#+)&", KF &.)*(3$+..&3+$(C5-(..&",1"-)*(.3+))(-&,2+,2$(2(,(-+)(%:+ 3(,)-+$5")(,)&+$&,)*(3(,)(-"10+..3""-%&,+)(..)(0

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H )"N,%)*+)A3"05$()&,2.=#+-(.B #T # > > 3".]AKKB !*(-(!(#.( $T% +,%%(N,()*(3"$$&.&",5+-+0()(> T%& i & i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HH v1fv'1fv1f Cos V cm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PAGE 72

GJ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.2 0.4 0.6 30 60 90 120 150 180 0Ekin/ eV. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.2 0.4 0.6 30 60 90 120 150 180 0Ekin/ eV. + : 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 30 60 90 120 150 180 0Ekin/ eV. 0.5 1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 30 60 90 120 150 180 0Ekin/ eV. 3 % &2#-(I<$#C6($"3&)3",)"#-0+51"-)*(+:.)-+3)&",5-"%#3)#.&,2+3"$$&.&", (,(-2&,)*("1G( (*+6(3$"3'!&.(<+B)")+$O#CW:B+5+-)&+$"1)*(2-&% !&)* "-&(,)+)&",$+:($43B.+0(!&)*"-&(,)+)&",W%B.+0(!&)*"-&(,)+)&", H*(-+%&+$%&0(,.&",&.)*(N,+$'&,()&3(,(-2&,)*(A(B+,%3"$"-. $+:($)*(*&.)"2-+03"#,)

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GF 0 1 2 3 4 5 6 0 5 10 15 20 Eint./ eV. 0 1 2 3 4 5 6 0 5 10 15 20 Eint./ eV. + : 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 Eint./ eV. 0 1 2 3 4 5 6 0 2 4 6 8 10 Eint./ eV. 3 % &2#-(K<&.)"2-+0.1"-&,)(-,+$(,(-2)-+,.1(-(%%#-&,2)*(3"$$&.&",1"-)*( +:.)-+3)&",5-"%#3)#.&,2+3"$$&.&",(,(-2&,)*("1G( (*+6(3$"3'!&.(< +B)")+$*&.)"2-+0W:B+5+-)&+$"1)*(2-&%!&)* "-&(,)+)&",$+:($43B.+0( !&)*"-&(,)+)&",W%B.+0(!&)*"-&(,)+)&",H

PAGE 74

GG 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 / Rad. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 / Rad. + : 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 / Rad. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 / Rad. 3 % &2#-(L<,2#$+-%&.)-&:#)&",*&.)"2-+0.1"-)*(+:.)-+3)&",5-"%#3)#.&,2+ 3"$$&.&",(,(-2&,)*("1G( (*+6(3$"3'!&.(<+B)")+$*&.)"2-+0W:B+ 5+-)&+$"1)*(2-&%!&)* "-&(,)+)&",$+:($43B.+0(!&)*"-&(,)+)&",W%B.+0( !&)*"-&(,)+)&",H

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GK 1 2 3 4 5 6 7 8 9 10 11 0.2 0.4 0.6 0.8 30 60 90 120 150 180 0Ekin/ eV. 1 2 3 4 5 6 7 8 9 10 11 0.2 0.4 0.6 0.8 30 60 90 120 150 180 0Ekin/ eV. + : 0.5 1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 0.8 30 60 90 120 150 180 0Ekin/ eV. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.2 0.4 0.6 0.8 30 60 90 120 150 180 0Ekin/ eV. 3 % &2#-(<$#C6($"3&)3",)"#-0+51"-)*(+:.)-+3)&",5-"%#3)#.&,2+3"$$&.&", (,(-2&,)*("1K( (*+6(3$"3'!&.(<+B)")+$O#CW:B+5+-)&+$"1)*(2-&% !&)* "-&(,)+)&",$+:($43B.+0(!&)*"-&(,)+)&",W%B.+0(!&)*"-&(,)+)&", H*(-+%&+$%&0(,.&",&.)*(N,+$'&,()&3(,(-2&,)*(A(B+,%3"$"-. $+:($)*(*&.)"2-+03"#,)

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GL 0 2 4 6 8 0 5 10 15 20 25 30 Eint./ eV. 0 2 4 6 8 0 5 10 15 20 25 Eint./ eV. + : 0 2 4 6 8 0 1 2 3 4 5 6 7 8 Eint./ eV. 0 2 4 6 8 0 5 10 15 20 Eint./ eV. 3 % &2#-(<&.)"2-+0.1"-&,)(-,+$(,(-2)-+,.1(-(%%#-&,2)*(3"$$&.&",1")*(+:.)-+3)&",5-"%#3)#.&,2+3"$$&.&",(,(-2&,)*("1K( (*+6( 3$"3'!&.(<+B)")+$*&.)"2-+0W:B+5+-)&+$"1)*(2-&%!&)* "-&(,)+)&",$+:($4 3B.+0(!&)*"-&(,)+)&",W%B.+0(!&)*"-&(,)+)&",H

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I 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 / Rad. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 / Rad. + : 0 0.5 1 1.5 2 2.5 3 0 5 10 15 / Rad. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 / Rad. 3 % &2#-(<,2#$+-%&.)-&:#)&",*&.)"2-+0.1"-)*(+:.)-+3)&",5-"%#3)#.&,2+ 3"$$&.&",(,(-2&,)*("1K( (*+6(3$"3'!&.(<+B)")+$*&.)"2-+0W:B+ 5+-)&+$"1)*(2-&%!&)* "-&(,)+)&",$+:($43B.+0(!&)*"-&(,)+)&",W%B.+0( H

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I 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.2 0.4 0.6 0.8 1 30 60 90 120 150 180 0Ekin/ eV. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 0.8 1 30 60 90 120 150 180 0Ekin/ eV. + : 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.2 0.4 0.6 0.8 1 30 60 90 120 150 180 0Ekin/ eV. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.2 0.4 0.6 0.8 1 30 60 90 120 150 180 0Ekin/ eV. 3 % &2#-(H<$#C6($"3&)3",)"#-0+51"-)*(+:.)-+3)&",5-"%#3)#.&,2+3"$$&.&", (,(-2&,)*("1( (*+6(3$"3'!&.(<+B)")+$O#CW:B+5+-)&+$"1 )*(2-&%!&)* "-&(,)+)&",$+:($43B.+0(!&)*"-&(,)+)&",W%B.+0(!&)* "-&(,)+)&",H*(-+%&+$%&0(,.&",&.)*(N,+$'&,()&3(,(-2&,)*(A(B +,%3"$"-.$+:($)*(*&.)"2-+03"#,)

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IH 0 2 4 6 8 10 12 0 5 10 15 20 25 30 35 Eint./ eV. 0 2 4 6 8 10 12 0 0.5 1 1.5 2 Eint./ eV. + : 0 2 4 6 8 10 12 0 1 2 3 4 5 6 Eint./ eV. 0 2 4 6 8 10 12 0 5 10 15 20 25 30 35 Eint./ eV. 3 % &2#-(J<&.)"2-+0.1"-&,)(-,+$(,(-2)-+,.1(-(%%#-&,2)*(3"$$&.&",1")*(+:.)-+3)&",5-"%#3)#.&,2+3"$$&.&",(,(-2&,)*("1( (*+6( 3$"3'!&.(<+B)")+$*&.)"2-+0W:B+5+-)&+$"1)*(2-&%!&)* "-&(,)+)&",$+:($4 3B.+0(!&)*"-&(,)+)&",W%B.+0(!&)*"-&(,)+)&",H

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IJ 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 / Rad. 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 / Rad. + : 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 / Rad. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 / Rad. 3 % &2#-(F<,2#$+-%&.)-&:#)&",*&.)"2-+0.1"-)*(+:.)-+3)&",5-"%#3)#.&,2+ 3"$$&.&",(,(-2&,)*("12-+0( (*+6(3$"3'!&.(<+B)")+$*&.)"2-+0W :B+5+-)&+$"1)*(2-&%!&)* "-&(,)+)&",$+:($43B.+0(!&)*"-&(,)+)&",W%B .+0(!&)*"-&(,)+)&",H

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IG Top View t = 820 a.u. t = 1500 a.u. (a) Mulliken PopulationTime(a.u.) 0 1 2 3 4 0 500 1500 2500 a N b N a D1 b D1 a D2 b D2(b) 0 5 10 15 20 25 0 500 1000 1500 20002500 D-D N-D1 N-D2 Time(a.u.)Interatomic Distance(c) &2#-(G<5&3+$)-+7(3)"-"1 !&)*+ &,&)&+$(3$&5.(%AHB"-&> (,)+)&",$(+%&,2)"+:.)-+3)&",&0(>(6"$#)&","1+)"0&33*+-2(.+,%&,)(-+)"0&3 %&.)+,3(.*(3"$$&.&",(,(-21"-)*()-+7(3)"-!+.(

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II Top View 0 1 2 3 4 0 500 1000 1500 2000 a N b N a D1 b D1 a D2 b D2 (a)(b) (c)Mulliken PopulationTime(a.u.) 0 2 4 6 8 10 12 14 0 400 800 1200 1600 D-D N-D1 N-D2 2000Time(a.u.)Interatomic Distance t = 1000 a.u. t = 1400 a.u. &2#-(I<5&3+$)-+7(3)"-"1 !&)*+ &,&)&+$(3$&5.(% AHB"-&(,)+)&",$(+%&,2)"%&-(3)(C3*+,2(&0(>(6"$#)&","1+)"0&33*+-2(.+,% &,)(-+)"0&3%&.)+,3(.*(3"$$&.&",(,(-21"-)*()-+7(3)"-!+.(

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IL 1 2 3 4 5 6 0.5 1 1.5 30 60 90 120 150 180 0Ekin/ eV. 0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 30 60 90 120 150 180 0Ekin/ eV. + : 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 1 1.5 30 60 90 120 150 180 0Ekin/ eV. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.5 1 1.5 30 60 90 120 150 180 0Ekin/ eV. 3 % &2#-(K<$#C6($"3&)3",)"#-0+51"-)*(+:.)-+3)&",5-"%#3)#.&,2+3"$$&.&", (,(-2&,)*("1G( (*+6(3$"3'!&.(<+B)")+$O#CW:B+5+-)&+$"1 )*(2-&%!&)* "-&(,)+)&",$+:($43B.+0(!&)*"-&(,)+)&",W%B.+0(!&)* "-&(,)+)&",H*(-+%&+$%&0(,.&",&.)*(N,+$'&,()&3(,(-2&,)*(A(B +,%3"$"-.$+:($)*(*&.)"2-+03"#,)

PAGE 88

K 0 5 10 15 0 5 10 15 20 25 Eint./ eV. 0 5 10 15 0 2 4 6 8 10 12 Eint./ eV. + : 0 5 10 15 0 0.5 1 1.5 2 Eint./ eV. 0 5 10 15 0 2 4 6 8 10 12 Eint./ eV. 3 % &2#-(L<&.)"2-+0.1"-&,)(-,+$(,(-2)-+,.1(-(%%#-&,2)*(3"$$&.&",1")*(+:.)-+3)&",5-"%#3)#.&,2+3"$$&.&",(,(-2&,)*("1G( (*+6( 3$"3'!&.(<+B)")+$*&.)"2-+0W:B+5+-)&+$"1)*(2-&%!&)* "-&(,)+)&",$+:($4 3B.+0(!&)*"-&(,)+)&",W%B.+0(!&)*"-&(,)+)&",H

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K 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 / Rad. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 / Rad. + : 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 / Rad. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 / Rad. 3 % &2#-(<,2#$+-%&.)-&:#)&",*&.)"2-+0.1"-)*(+:.)-+3)&",5-"%#3)#.&,2+ 3"$$&.&",(,(-2&,)*("12-+0G( (*+6(3$"3'!&.(<+B)")+$*&.)"2-+0W :B+5+-)&+$"1)*(2-&%!&)* "-&(,)+)&",$+:($43B.+0(!&)*"-&(,)+)&",W%B .+0(!&)*"-&(,)+)&",H

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0 0.02 0.04 0.06 0.8 1 12 14 0 200 400 600 800 1000 Energy (a.u.) x10-3 Time (fs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F 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 200 400 600 800 1000 1200 1400 1600 Energy (a.u.)x10-3 Time (fs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�+-(.&0&$+-)"")*(--(.#$).&,)*($&)(-+)#-(R LK S#.&,2$+-2(:+.&..().*(3"05#)(%5"$+-&@+:&$&)&.,")(C+3)$)*(%,+0&3",(.&,3(&) -(5-(.(,).)*(-(.5",.()"++#..&+,.*+5(%5#$.(!&)*+1-(=#(,3.5-(+%+$>

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G 1000.0500.00.0500.0 Detuning (a.u.) 0.000 0.010 0.020 0.030Absorbed Energy (a.u.) MD NPMD WP &2#-(JJ<:."-:(%(,(-24&, '#,&).4+.+1#,3)&","1%()#,&,2A&, +#B 1"-3+$3#$+)(%!&)*+,%5#$.(%N($%!&)*&,)(,.&) ;30&.#.(%

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I 600.0400.0200.00.0200.0400.0 Detuning (a.u.) 0.000 0.010 0.020 0.030 0.040Absorbed Energy (a.u.) MD NPMD WP &2#-(JF<:."-:(%(,(-24&, '#,&).4+.+1#,3)&","1%()#,&,2A&, +#B 1"-&3+$3#$+)(%!&)*+,%+#..&+,5#$.(%N($%!&)*&,)(,.&) HF ;30&.#.(%

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K -1.15 -1.1 -1.05 -1 -0.95 -0.9 -0.85 -0.8 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 Dipole (D.) Distance (a.u.) END NPMD &2#-(JG<"$(3#$+-%&5"$(6.:",%%&.)+,3()-+3(%%#-&,2)&0((6"$#)&",1"&+)%()#,&,2GK +#+,%N($%&,)(,.&)"1 ;301"-A$&,(B +,%A".3&$$+)"-5+-+0()-&33#-6(B*(3+$3#$+)(%(=#&$&:-�:",%%&.)+,3( &.HGG+#

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L 0.315 0.32 0.325 0.33 0.335 0.34 0.345 0.35 0.355 1.5 1.6 1.7 1.8 1.9 2 2.1 Dipole(D.) Distance(a.u.) END NPMD &2#-(JI<"$(3#$+-%&5"$(6.:",%%&.)+,3()-+3(%%#-&,2)&0((6"$#)&",1"+)%()#,&,2GK +,%N($%&,)(,.&)"1 ;301"-A$&,(B+,% A".3&$$+)"-5+-+0()-&33#-6(B*(3+$3#$+)(%(=#&$&:-�:",%%&.)+,3(&.II +#

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�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

PAGE 129

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

PAGE 130

0.005 0.003 0.0010.0010.0030.005 Field (a.u.) 0.319 0.329 0.339 0.349Dipole (a.u.) &2#-(JK<&5"$(+.+1#,3)&","1)*(N($%1"-1"-%&?(-(,)&,)(-,#3$(+-%&.> )+,3(.A%&?(-(,)$&,(.B*(N($%&,)(,.&)#.(%&. ;30+,%)*(%()#,&,2&. JK +#

PAGE 131

H 1.601.701.801.90 Internuclear Distance (a.u.) 3.0 3.5 4.0 4.5 5.0 5.5 6.0Polarizability (a.u.) &2#-(JL<"$+-&@+:&$&)A ( A_BB6.&,)(-,#3$(+-%&.)+,3(1"-#.&,2+N($%"1 ;30+,%%&?(-(,)%()#,&,2.

PAGE 132

J 2.02.53.03.54.04.55.0 Internuclear Distance (a.u.) 10.0 20.0 30.0 40.0Polarizability (a.u.) &2#-(J<"$+-&@+:&$&)A ( A_BB6.&,)(-,#3$(+-%&.)+,3(1"-&#.&,2+N($%"1 ;30+,%%&?(-(,)%()#,&,2.

PAGE 133

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JG Cluster +ICLUST: = 1* +Find_Cluster(Delta:Complex): ICLUST SD_State +ZOCA: Complex +R: Real +P: Real +Orth_Sim(ZOCA:Complex): ZOCA Complex +Orth_HF(ZOCA:Complex): ZOCA Complex +Modulus(ZOCA:Complex): Real Gen_Config +Gen_Next_Config() +Gen_Init_State() +Gen_Final_State() Charge State:Charge State Spin State:Spin State Config State:Config State Charge_State +Charge State: Charge State Spin_State +Spin State: Spin State Config_State +Config State: Config State Product +Overlap: Complex Basis +Rawbasis: Function +Rawdelta: Real +Transform: Complex 2 1..* &2#-(F<$+..%&+2-+01"-)*(5-"7(3)3"%(

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JK *(2(,(-+$&@+)&","1)*(5-(6&"#.+$2"-&)*01"-+2(,(-+$+$5*+>:()+3",N2> #-+)&",&.,")*+-%+,%*+.:((,&05$(0(,)(%,(",$,((%.)"3",.&%(-)*+) 1"-(+3*(C3&)+)&",3",N2#-+)&","1",('&,%A:()+;+$5*+B4+$$5"..&:$((C3&)+)&", 3",N2#-+)&",.1"-)*(")*(-'&,%A+$5*+;:()+B+-(2(,(-+)(% :Base Config create :Config 1:Valid First Config 1.1:Create Lower Rank Configs :Last Config 2:Valid Last Config 3.1: Create Lower Rank Configs 3*:[not last config] Next Config :Lower Rank Config e.g. spin configurations for charge configuration generation algotithm. First and last configurations are constructed based on heuristic assumptions. Generates next configuration in lexicographical order. Probability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F :Base Config create : Config 2:Valid Config : Config 1:Valid Config : Overlap :Create : Overlap 3.3.1a.1:Create 3.3.1a.2:Determinant 3.3.1a*:[not last config] Next Config 3.2:Determinant 3.3.1b:[last config] Next Config 3*:[not last config] Next Config 3.3:Next Config Numbers represent a single cycle over beta spin configuration space. :Probability 3.3.1a.3: Add Compute the products and add the probabilities. Next Config considers all excitations over all clusters. Last configuration is controlled by an excitation parameter. &2#-(FH<",N2#-+)&",2(,(-+)&",+$2"-&)*01"-(C3&)+)&",.

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Fevolve>p0.141003003 #Restart# ------------------------------------------------------------------------------@PROJDRV-I,Usinginputfileendyne.inp Fragment:12 Charges:0.00000002.0000000 Fragment:12 Excessalfa:00 Multiplicity:11 Probabilityforthespinstate:.51229D+00 ------------------------------------------------------------------------------Probabilityforthechargedstate:.51229D+00 Cumulativeprobability:.51229D+00 ------------------------------------------------------------------------------Fragment:12 Charges:1.00000001.0000000 Fragment:12 Excessalfa:1-1 Multiplicity:22 Probabilityforthespinstate:.19160D+00 ------------------------------------------------------------------------------Fragment:12 Excessalfa:-11 Multiplicity:22 Probabilityforthespinstate:.19160D+00 ------------------------------------------------------------------------------Probabilityforthechargedstate:.38321D+00 Cumulativeprobability:.89550D+00 ------------------------------------------------------------------------------Fragment:12 Charges:2.00000000.0000000 Fragment:12 Excessalfa:2-2 Multiplicity:33 Probabilityforthespinstate:.10305D-01 ------------------------------------------------------------------------------Fragment:12 Excessalfa:00 Multiplicity:11 Probabilityforthespinstate:.71662D-01 ------------------------------------------------------------------------------Fragment:12 Excessalfa:-22 Multiplicity:33 Probabilityforthespinstate:.10305D-01 ------------------------------------------------------------------------------Probabilityforthechargedstate:.92273D-01 Cumulativeprobability:.98777D+00 -------------------------------------------------------------------------------&2#-(FJ<-"7(3)"#)5#)1"-)*( ..)(0+)+3"$$&.&",(,(-2"1 (;+0##.&,2+,&05+3)5+-+0()(-"1+#

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FHFragment:12 Charges:3.0000000-1.0000000 Fragment:12 Excessalfa:3-3 Multiplicity:44 Probabilityforthespinstate:.19833D-04 ------------------------------------------------------------------------------Fragment:12 Excessalfa:1-1 Multiplicity:22 Probabilityforthespinstate:.38543D-02 ------------------------------------------------------------------------------Fragment:12 Excessalfa:-11 Multiplicity:22 Probabilityforthespinstate:.38543D-02 ------------------------------------------------------------------------------Fragment:12 Excessalfa:-33 Multiplicity:44 Probabilityforthespinstate:.19833D-04 ------------------------------------------------------------------------------Probabilityforthechargedsate:.77483D-02 Cumulativeprobability:.99552D+00 ------------------------------------------------------------------------------048381 evolve>&2#-(FJ<3",)&,#(%

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FJevolve>p0.17103030322 #Restart# ------------------------------------------------------------------------------@PROJDRV-I,Usinginputfileendyne.inp Fragment:12 Charges:0.00000002.0000000 Fragment:12 Excessalfa:00 Multiplicity:11 Probabilityforthespinstate:.10999D+00 ------------------------------------------------------------------------------Probabilityforthechargedstate:.10999D+00 Cumulativeprobability:.10999D+00 ------------------------------------------------------------------------------Fragment:12 Charges:1.00000001.0000000 Fragment:12 Excessalfa:1-1 Multiplicity:22 Probabilityforthespinstate:.19949D+00 ------------------------------------------------------------------------------Fragment:12 Excessalfa:-11 Multiplicity:22 Probabilityforthespinstate:.19949D+00 ------------------------------------------------------------------------------Probabilityforthechargedstate:.39898D+00 Cumulativeprobability:.50897D+00 ------------------------------------------------------------------------------Fragment:12 Charges:2.00000000.0000000 Fragment:12 Excessalfa:2-2 Multiplicity:33 Probabilityforthespinstate:.21284D-01 ------------------------------------------------------------------------------Fragment:12 Excessalfa:00 Multiplicity:11 Probabilityforthespinstate:.36180D+00 ------------------------------------------------------------------------------Fragment:12 Excessalfa:-22 Multiplicity:33 Probabilityforthespinstate:.21284D-01 ------------------------------------------------------------------------------Probabilityforthechargedstate:.40436D+00 Cumulativeprobability:.91334D+00 -------------------------------------------------------------------------------&2#-(FF<-"7(3)"#)5#)1"-)*( ..)(0+)+3"$$&.&",(,(-2"1 (;+0##.&,2+,&05+3)5+-+0()(-"1+#

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FFFragment:12 Charges:3.0000000-1.0000000 Fragment:12 Excessalfa:3-3 Multiplicity:44 Probabilityforthespinstate:.41331D-03 ------------------------------------------------------------------------------Fragment:12 Excessalfa:1-1 Multiplicity:22 Probabilityforthespinstate:.38601D-01 ------------------------------------------------------------------------------Fragment:12 Excessalfa:-11 Multiplicity:22 Probabilityforthespinstate:.38601D-01 ------------------------------------------------------------------------------Fragment:12 Excessalfa:-33 Multiplicity:44 Probabilityforthespinstate:.41331D-03 ------------------------------------------------------------------------------Probabilityforthechargedstate:.78029D-01 Cumulativeprobability:.99136D+00 ------------------------------------------------------------------------------03534&2#-(FF<3",)&,#(%

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Permanent Link: http://ufdc.ufl.edu/UFE0000311/00001

Material Information

Title: Application of The Electron Nuclear Dynamics Theory to Hydrogen Abstraction and Exchange Reactions of H + HOD and D2 + NH3+
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0000311:00001

Permanent Link: http://ufdc.ufl.edu/UFE0000311/00001

Material Information

Title: Application of The Electron Nuclear Dynamics Theory to Hydrogen Abstraction and Exchange Reactions of H + HOD and D2 + NH3+
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0000311:00001


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APPLICATION OF THE ELECTRON NUCLEAR DYNAMICS
THEORY TO HYDROGEN ABSTRACTION AND EXCHANGE
REACTIONS OF H + HOD AND D2 + NH+












By

MAURiCIO D. COUTINHO NETO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2001















ACKNOWLEDGMENTS


I would like to thank Drs. Y Ohm and E. Deumens for their guidance, patience

and relentless support over the many years I spend at QTP. I certainly enjoi' my

time and developed an even greater appreciation for good science. I would also

like to thank Prof. J. Broeckhove for helping with the calculations performed on

LiH and HF.

During the time I spent at QTP I had the privilege to share time and office space

with some wonderful and intelligent people. I would like to give special thanks to

Anatol Blass, Magnus Hedstr6m, Jorge Morales and Benny Mogensen for their

friendship and support. I would also like to thank past and present members

of the Ohrn/Deumens group for their camaraderie, friendship and help: Juan

Oreiro, Denis Jacquemin, Remigio Cabrera-Trujillo, Ben Killian, David Masiello

and Svetlana Malil ',--1.: li- I also would like to thank several people outside the

END group who made the environment at QTP such an enjo-'1.-,-l one, particularly

staff members Judy Parker, Coralu Clements, Arlene Rodriguez and Cindy Leprell.

I would like to thank past and present members of the Brazilian community in

Gainesville that helped to create the Brazilian Student Association (BRASA):

Fabiano Toni and Daniela Lopes, Uilson Lopes and Karina Gramacho, Dirceu and

Patricia Mattos, Ricardo Matos, Eliana Kampf, Carlos da Costa, Ilka Vasconcelos,

Francisco and Cleisa Cartaxo. I thoroughly enjoi, my extracurricular experiences

with these people.

Life would not be the same without its little pleasures: I would like to thank

Fabela, Vandinha and Juliet for their affection.










I can truly I- that I would not be in science if not for the support, guidance and

encouragement of my parents: Marfcio D. Coutinho Filho and Maria da Conceicao

D. P. de Lyra. I thank them sincerely. I would like to thank my dearest wife

Fabfola for her love, support, caring and encouragement.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS . . ii

ABSTRACT . . . vii

CHAPTERS

1 INTRODUCTION . . . 1

1.1 Background Theory ........................... 5
1.1.1 Second Quantization .......... ....... ..... 5
1.1.2 Diabatic, Adiabatic and END Representation . 6
1.2 Solutions to the Time-Dependent Schodinger Equation ...... 15
1.2.1 Time-Dependent Close Coupling Method .... 16
1.2.2 Classical Path Method ............ .. .. 16
1.2.3 Trajectory Surface Hopping Method. . 17
1.2.4 Car-Parrinello Method . ...... ... 18
1.2.5 Time-Dependent Hartree-Fock Based Methods ... 19
1.3 Electron Nuclear Dynamics (END) Method . 20
1.3.1 Narrow Wavepacket Limit .................. .. 24
1.3.2 Molecular Dynamics ............ ... .. 24
1.4 Cross Sections Calculations ................... .. 27
1.4.1 Differential Cross Section .............. .. 27
1.4.2 Classical Collisions ...... . .... 29
1.4.3 Laboratory-Center of Mass Coordinate Transformation 32
1.4.4 Semiclassical Differential Cross Section . .... 36
1.5 Measurements ............ . ...... 40
1.5.1 Total Cross Section Calculation . 41
1.5.2 Background on Guided Ion Beam Experiments ... 43

2 THE NH+ + D2 SYSTEM AT HYPERTHERMAL ENERGIES 46

2.1 Introduction ..... . ..... ............... 46
2.2 Experimental Results for Reaction NH+ + D2 ........... 47
2.3 Computational Details ......... ............... 53
2.3.1 Structures at Extrema Points of the NH+ Potential 53
2.3.2 Collision Grid and Initial Conditions . 57
2.4 Results ...... .......... .............. 60










2.4.1 Detailed Results for Collision at 6 eV .. ......
2.4.2 Detailed Results for Collision at 8 eV .. ......
2.4.3 Detailed Results for Collision at 12 eV ........
2.4.4 Detailed Results for Collision at 16 eV ........
2.5 Discussion .........................
2.6 Conclusions . . . .

3 H+HOD COLLISION AT 1.575 EV..... .

3.1 Introduction: . . . .
3.2 Computational Details .. .................
3.3 R results . . . .
3.3.1 Results for HOD -Vibrational Ground State ...
3.3.2 Results for HOD(0, 0, 1) -OH Local Mode Excited
3.4 Improving the Basis Description ...............
3.5 Conclusions . . . .

4 LIH AND HF DYNAMICS UNDER INTENSE FIELDS .

4.1 Introduction . . . .
4.2 Theory ...........................
4.2.1 END with External Fields .. ............
4.2.2 Initial Conditions .. ................
4.3 Results and Discussion .. .................
4.3.1 Static Properties .. .. .. .. ... .. .. .
4.3.2 Dynamics Results ................ ...
4.3.3 Effect of the Basis Set .................
4.4 Conclusions . . . .

5 FINAL STATE ANALYSIS .. ...........


5.1 Introduction .........
5.2 Background Theory .. ............
5.2.1 Unitary Transformations Using Basis
5.2.2 Thouless Determinant .. ......
5.2.3 Basis Set Transformations ......
5.2.4 Final State Analysis .. .......
5.3 Simple Application .. ...........
5.4 Implementation .. .............
5.4.1 D esign . . .
5.4.2 Algorithm s .. ...........
5.4.3 Numerical Examples .. .......
5.5 Conclusions .. .. .. ... .. .. .. .


Operators


. 89


. 105










6 CONCLUSION . . . 159


REFERENCES . . . 162

BIOGRAPHICAL SKETCH . . 168
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

APPLICATION OF THE ELECTRON NUCLEAR DYNAMICS
THEORY TO HYDROGEN ABSTRACTION AND EXCHANGE
REACTIONS OF H + HOD AND D2 + NH+

By

Maurfcio D. Coutinho Neto

December 2001


C ,i iln iil: Y. Ohrn
Major Department: C(i!. i-I -ry

The field of quantum molecular dynamics have flourished in the last 20 years.

Methods that propose the solution of the time dependent Scrh6dinger equation

for a molecular reactive process abound in the literature. However the 1 i i iry

of these methods focus on solving the nuclear Schr6dinger equation subject to a

known electronic potential. The electron nuclear dynamics (END) method pro-

poses a framework of a hierarchy of approximations to the Schr6dinger equation

based on the time dependent variational Principle (TDVP). A general approach is

sought to solve the electronic and nuclear problem simultaneously without making

use of the Born-Oppenheimer approximation.

The purpose of this work is to apply the minimal END to areas where its unique

qualities can give new insight into the relevant dynamics of a chemical or physical

process. Minimal END is a method for direct non-adiabatic dynamics. It describes

the electrons with a family of complex determinantal wave-functions in terms of

non-orthogonal spin orbitals and treats the nuclei as classical particles.










In the first two studies, we apply the END method to hydrogen abstraction

and exchange reactions at hyper-thermal collision energies. We investigate the

D2+NH+ reaction at collision energies ranging from 6 to 16 eV and the H+HOD

reaction at a collision energy of 1.575 eV. Collision energies refer to center of mass

energies. Emphasis is put on the details of the abstraction and exchange reaction

mechanisms for ground state reactants. In a final application we use minimal END

to study the interaction of a strong laser field with the diatomic molecules HF and

LiH. Effects of the polarization of the electronic potential on the dynamics are

investigated.

Emphasis is also placed on the development of a general method for interpreting

the final time dependent wave-function of the product fragments. The purpose is

to analyze the final state wave-function in terms of charge transfer channels as well

as individual contributions pertaining to excited states.















CHAPTER 1
INTRODUCTION

The reaction rate constant is usually understood to be one of the most fun-

damental quantities in chemistry. However, in order to calculate rate constants

from a fist principles theory we need to understand reactive processes at the sin-

gle molecule level. Rate constants are obtained from first principles formalism

as weighted averages over all different processes working at the molecular level.

To understand the chemical reactions is to understand how molecules evolve and

transform in time. Over the last twenty years chemists and physicists have devel-

oped experiments that investigate reactions at the single molecule level under well

controlled conditions. Experimentalists have been able to select initial states of

reactants and to perform molecular collisions at specific energies. The level of con-

trol over the experiments has reached the point that theoretical results from first

principles calculations can the directly be compared to, and aid in the interpreta-

tion of experiments. However, a different perspective is needed in order to better

understand current cross beam, guided ion beam and photochemical experiments

to name a few.

This work lies within the realm of time-dependent phenomena in chemistry

and physics. The time evolution of a system is governed by the time-dependent

Schr6dinger equation. In order to describe the time evolution of a molecular sys-

tem, approximations to the time-dependent Schr6dinger equation have to be de-

veloped as analytical solutions are known only to a restricted set of model systems.

The field of quantum molecular dynamics has seen unparalleled development over

the last two decades. Methods for the solution of the time-dependent Schr6dinger










equation for a molecular reactive process abound in the literature. However, the

i i ii i y of these methods make use of the central dogma in both time-dependent

and time-independent calculation of molecular properties and reaction rates: the

Born-Oppenheimer (BO) approximation. The BO approximation in invoked to

separate the Schr6dinger equation for the whole system into two separate ones to

be solved independently: one for the electrons and another one for the nuclei.

We are interested in the application of time-dependent methods that solve ap-

proximately the Schr6dinger equation for the study of reactive collisions processes.

Traditionally time-independent approaches have been used to make qualitative

and sometimes quantitative predictions about chemical reactions and mechanisms.

Simple models have been used to predict rate constants, usually the most desired

chemical quantity to be calculated, from time-independent results. C('i! 11 -I want

to know if a reaction can happen and how fast it happens. The first question can

be addressed by time-independent methods. The answer to the second question,

however, needs to include a time-dependent description in an explicit or implicit

form. Simple models, commonly based on transition state theory, can produce

very good results when reaction conditions under consideration are "mild". In the

time-independent picture mechanistic interpretation is based on extrema features

of the electronic potential energy surface (PES).

Because of the success of the use of time-independent methods for the cal-

culation of reaction rates and the difficulty associated with time-dependent de-

scriptions, explicit time-dependent calculations or reaction processes have not be-

come common practice. However, there is a lot to gain from time-dependent ap-

proaches as a much clearer picture of reactions mechanism can be drawn from

time-dependent simulations. For example, in reactions involving charge transfer,










following how charge changes with nuclear configuration can offer insight into the

reaction mechanism and reveal which are the most important nuclear modes to the

reaction. In addition explicit time-dependent descriptions can mimic experimental

conditions more closely than time-independent descriptions. With time-dependent

methods one can for example

* Consider initial energy distributions that match experimental temperatures.

* Consider pure initial states as in experiments that measure state to state cross

sections.

* Consider non-equilibrium initial conditions as in the case of photochemical re-

actions.

Time-dependent methods can be classified in two groups based on the way that

they approximate the molecular time-dependent Schr6dinger equation. The first

group consists of methods that solve the time-dependent Schr6dinger equation for

the nuclear degrees of freedom subject to a known PES while the second group

consists of direct dynamics methods. Direct dynamics in this context means that

the electronic and nuclear degrees of freedom are propagated simultaneously. This

is different from finding the solution for the electronic problem along a nuclear

trajectory. The electron nuclear dynamics (END) method explored in this work is

a direct dynamics method.

The electron nuclear dynamics (END) method provides a framework to build

a hierarchy of approximations to the Schr6dinger equation based on the time-

dependent variational principle (TDVP). A general approach is sought to solve

the electronic and nuclear problem simultaneously without invoking the Born-

Oppenheimer approximation. The dynamical equations of motion are obtained

by using the TDVP with wavefunction parametrized according to coherent states










(CS). Direct dynamics methods and in particular the END method have several

advantages when compared to other methods, the most obvious ones being

* No need to use cumbersome coordinate systems, cartesian coordinates are used

throughout.

* No need to construct potential surfaces.

* Excited electronic states are considered in an approximate and consistent man-

ner.

Because of its qualities the END method can be applied to a wide range of chemical

and physical processes.

In the first two studies, we apply the END method to hydrogen abstraction

and exchange reactions at hyper-thermal collision energies. Hydrogen abstrac-

tion reactions are relatively simple and yet fundamental in detailed experimental

investigations. From a theory perspective these reactions provide a wealth of in-

formation for testing methods and facilitating the interpretation of results, when

comparisons to experiment are appropriate. We investigate the D2 + NH+ reac-

tion at collision energies ranging from 6 to 16 eV and the H + HOD reaction at

a collision energy of 1.575 eV. Collision energies refer to center of mass energies.

Emphasis is put on the details of the abstraction and exchange reaction mecha-

nisms for ground state reactants. In a final application we use minimal END to

study the interaction of a strong laser field with diatomic molecules, namely HF

and LiH. Effects of the polarization of the electronic potential on the dynamics

are investigated.








5

1.1 Background Theory

1.1.1 Second Quantization

In this section we introduce electron field operators expressed in a basis of spin

orbitals. Electron field operators obey the anti-commutation relations

[F(r), Ft(r')] = (r- r')

[F(r), F(r')]+ 0 (1.1)

[Ft(r), Ft(r')] =+ 0

and in our case are expanded in a basis of spin orbitals { } and basis creation

operators {bt} [1]
K
Ft(r) (r)b bt. (1.2)
i=i 1
Our convention uses W as a row array and bt as a column array. The basis

creation operators obey the anti-commutation relation

[b,,bl]+ = 6,s (1.3)

[bt, b = 0 (1.4)

[br, b]+ 0 (1.5)

A single determinantal state of N electrons can be constructed as an ordered

product of the basis creation operators
N
IPo) =f b vac). (1.6)
h=1
where Ivac) is the no-particle state, with the following mapping between Slater

determinants and products of creation operators being valid
N
S...N bht Ivac) (.7)
h=l1










i.e., there is a one to one correspondence between a slater determinant and a

ordered product of basis creation operators. With this mapping and the commu-

tation relations given in Equations 1.3-1.5 expectation values and matrix elements

can be calculated for fermionic states (see for example Linderberg and Ohrn [1])

using second quantization algebra.



1.1.2 Diabatic, Adiabatic and END Representation

The distinction between diabetic and adiabatic parametrization of a product

molecular wavefunction is at the core of the approximations to the time-dependent

and time-independent Schr6dinger equations. The original construction of this

particular product form is due to Born and Oppenheimer [2] with generalizations

by Born and Huang [3]. We will limit at first our discussion to the time-independent

Sch6dinger equation.

1.1.2.1 Adiabatic representation

Start with the total molecular Hamiltonian

H = H(x,X) + T (1.8)

where H,(x, X) is the electronic Hamiltonian including electronic kinetic energy,

internuclear repulsion and electron-electron repulsion. In Equation 1.8 T, is the

nuclear kinetic energy operator, x corresponds to the electronic degrees of freedom

while X corresponds to the nuclear degrees of freedom. The time-independent

Shr6dinger equation for the whole system is given by

(H- E)E(x,X)= 0 (1.9)

Consider the following expansion in a product form (the Born-Huang expansion)
N
E(x, X; Xo) = (x, Xo)X (X) (1.10)
i










where ,' (x, Xo) are electronic wavefunctions depending on a parameter Xo and

Xi(X) are nuclear wavefunctions. The set given by {,' (x, Xo)} is composed by the
solutions of the N-electron electronic problem (Equation 1.11) and forms a basis

for Expansion 1.10

(H (x, Xo) e (Xo)> (x, Xo) 0, i = 1,..., N. (1.11)

Substituting the Born-Huang (BH) expansion into the molecular Schr6dinger equa-

tion we obtain
N
(T, + H,(x, Xo)- E) (x, Xo)X(X) = 0 (1.12)

Using the fact that the electronic basis set forms an orthonormal set, we can elim-

inate the explicit dependence on the electronic degrees of freedom by multiplying

from the left by Wb(x, Xo) and integrate over all electronic variables to obtain
N
(Qj (x, Xo) Txi X)',' (x, Xo)) + (e(Xo) E)xj(X) =0 j 1..N (1.13)

Equation 1.13 is a coupled set of second order differential equations and has

as solutions the nuclear stationary states of the molecular system. The adiabatic

choice for the electronic basis consists of making Xo0 X in Equation 1.13 so

that the electronic wavefunction depends parametrically on the nuclear degrees of

freedom.

The first term in Equation 1.13 (without loss of generality I will use V2 as

the nuclear kinetic energy operator) produces the following terms

(bj(x, X) TXi(X)'' (x,X))= (1.14)

2 6j V2X(X) + 2(j(x, X) |VI' (x, X)) Vx (X)
2m +(j(x, X) |V (x, x)) x(X)










The non-adiabatic coupling elements are defined from Equation 1.14 as

r7i= (j (x, X) V (x,X)) (1.15)

and

7 = (X4(x,X) V2 ,' (x,X)) (1.16)

where Equation 1.15 refers to couplings of the first kind and Equation 1.16 refers

to couplings of the second kind.

The non-adiabatic coupling element of the first kind are believed to be the most

important [4]. Their effects on the dynamics can be non-negligible when adiabatic

electronic states change rapidly with changes in the nuclear configuration (X) as

in an avoided crossing situation. As this term is also multiplied by Vxi(X), the

total coupling also depends on how fast the nuclear wave function changes with

X. The Schrodinger equation for the nuclei in the electronic adiabatic basis can

now be written as

1 1 _I .(7(x + T(2)
m X(X)+ (e(X) E)Xj(X) -(27 v + )x(X) 0 (1.17)
i
If one considers only diagonal terms (i = j) in Equation 1.14, Equation 1.13

becomes effectively a set of uncoupled equations for the nuclear degrees of freedom.

At this level of approximation one can solve for the nuclear wavefunction for a single

(adiabatic) potential energy surface.

The Born-Oppenheimer equation for the nuclei (the approximation to Equation

1.17) is obtained if we neglect all coupling term from Equation 1.14, resulting in

the uncoupled set of equations

t V2(X + (ej(X) E) (X) = 0, j 1,...,N. (1.18)
2m










The molecular Schrodinger equation in an adiabatic basis can we written in a

more succinct form by considering the relations between the non-adiabatic terms

given in Equations 1.15 and 1.16 [5].

Consider the derivative of an overlap matrix element

V (j (x, X) (x, X)) = 0 (1.19)

(Vj (x,X)'' (x,X)) = -(j (x,X) |I V (x,X))

or

() (= ). (1.20)
ij i

For real functions the non-adiabatic coupling elements of the first kind form an

antisymmetric matrix with zeros on the diagonal. For complex functions the T 1)

matrix is anti-hermitian.

We can deduce the relationship between the non-adiabatic coupling elements

of the first and second kind by considering

V71) =V((x,X) V',' (x, X)) (1.21)

then

(Vj (x,X) V 1 j (x,X)) + (j (x,X) V2 ,' (x,X)) =(1.22)

(Vj (x,X) |V1' (x,X)) + r

or that

S= -W(V7(x, X)V',' (Xx, X)) + V1 (1.23)

using the resolution of the identity and the fact that is antisymmetric we get
using the resolution of the identity and the fact thatT) is antisymmetric we get
'ij satsmercw e


(2) ( 1)2 (1)
'i 7, \'n '


(1.24)










The last relation can be used in the nuclear Schrdinger equation in an adiabatic

basis to obtain

1
2 V2(X) + (e(X) E)(X) (1.25)

-1 () i(X) Z(2 V + VT )XidX) = 0
i i
which in matrix form reads

SV2(X)+(e(X)-E- (l)2)y(X)- (2T()7V+VT())y(X) 0. (1.26)
2m 2m 2m

Equation 1.26 can be reorganized to

(V + T(1))2X(X) + (e(X) E)X(X) = 0 (1.27)
2m

In the matrix notation equations, X is a vector of dimension N, while e, E and

T1) are square matrices of dimension N2.

The non-adiabatic coupling elements have a direct effect on the momentum of

the nuclei. For this reason they are called dynamical coupling elements [5]. To

solve the nuclear Schridinger equation non adiabatically (the full coupled set of

differential equations), one needs all nonzero dynamical coupling elements between

the relevant electronic states.

1.1.2.2 Diabatic representation

The diabetic construction is obtained if we consider Xo to be a constant in

Equation 1.12. As in the previous case, the electronic functions {bj(x, Xo)} form

an orthonormal set. Originally the diabetic electronic set was constructed by

solving the time-independent electronic Sch6dinger equation (Equation 1.11) at

an equilibrium geometry Xo. Approximate solutions to the nuclear Schr6dinger

equation at positions X were sought using perturbation theory [3]. Later the

concept of a diabetic representation was extended to include any electronic basis









set constructed so as to minimize the coupling elements given by Equations 1.15
and 1.16.
The diabetic equivalent to Equation 1.17 can be obtained by a similar proce-
dure. Multiplying Equation 1.13 by rb*(x, Xo) from the left and integrating over
all electronic space and spin we obtain

(- V2- E) X(X) + (j (x, Xo) H,(x,X)' (x,Xo)) i(X) 0 (1.28)

Using the identity He(x,X) = He(x, Xo) + V(x,X) V(x, Xo) and

( j (x, Xo) H,(x, X)',' (x, Xo)) (1.29)

6ijoe + ( (x,Xo) V(x, X) V (x, Xo)' (x, Xo))

we can reorganize the Sch6dinger equation in the diabetic representation to be

V2 E) x(X) + vj(X,Xo)Xi(X)= 0 (1.30)

where we use the definition vi = eijei + ( j(x, Xo)lV(x, X) V(x, Xo)' (x, Xo)).
In matrix form, Equation 1.30 has the form

( 7V2 (X) + ((X,Xo) E)x(X) 0 (1.31)

were the matrices f (the vij matrix) and E are of dimension N2 and x is a vector
of dimension N.
The non-adiabatic coupling terms in this case are potential coupling elements.
To solve the nuclear Schr6dinger equation non adiabatically, we need all nonzero
potential coupling elements between electronic states.

1.1.2.3 Adiabatic-diabatic transformation
The dynamical coupling elements of the adiabatic representation have the unde-
sired property of growing markedly near curve crossings. Their analytical behavior
with energy can be obtained by considering the expression












V (,' (x,X)I H (x, X) Ij(x, X)) = (1.32)

(V (<' (x,X)) H (x, X) I (x,X)) + ,' (x,X)I H (x, X)V (x,X))

+ (,' (x, X) H,(x, X) Ij(x, X))

which with the help of 1.11 and 1.20 can be reduced to

(i) ( (x, X) H (x, X) (x, X))
S(e(X) ej(X)) ( )

It is very difficult to construct a good adiabatic basis set at a fixed position Xo

that can be used for a global (all X) solution of the Schridinger equation. As a

result Equation 1.30 in its present form is not very useful.

An alternative solution is to define an adiabatic-to-diabatic representation trans-

formation at all nuclear configurations X. There are formal and empirical v--i-

to introduce such a transformation, see Baer [5] and Sidis [6] for a discussion on

the subject. The resulting Scrbdinger equation, although closely related to the

diabetic representation given by Equation 1.30, is in this case more convenient to

solve for arbitrary nuclear configurations. This new construction enables one to

define diabetic representations that also have nuclear configuration dependence.

The disadvantage is that, as in the adiabatic case, dynamical coupling elements

are also present in the Scrbdinger equation. However, unlike the diabetic case

dynamical coupling elements can be made to be slowly varying with respect to

changes in nuclear configuration. Constraints can be imposed on the transforma-

tion so that the resulting Schidinger equation has small or vanishing dynamical

coupling elements.










A time-independent Schridinger equation in terms of a modified adiabatic basis

has potential and dynamical coupling elements. In matrix notation it can be

written as



2 (V2 + (1))x(X) + (W(X, Xo) E)(X) = 0 (1.34)
2m

where

W AtvA (1.35)

with A being the diabetic to adiabatic transformation matrix.

1.1.2.4 Time-dependent parametrization

The extension of the BH expansion to a time-dependent form is usually done

in a way to leverage on the knowledge gained in solving the time-independent

Schr6dinger equation. The spatial dependence in the electronic and nuclear vari-

ables is kept and all the time dependence is introduced in an additional parameter

set {c}.
N
E(x, X, t; Xo) = c(t (x,Xo)(X) (1.36)

Because the same functional dependence is kept for the spatial dependence

part, all the coupling term definitions for adiabatic and diabetic representations

carry over to the time-dependent Schr6dinger description.

1.1.2.5 END diabetic representation

The electron nuclear dynamics method employs a diabetic parametrization of

the molecular wave function. The END diabetic product has as parameters the

average nuclear positions (R) and moment (P) in addition to electronic wavefunc-

tion parameters z. As in the BH expansion the END expansion can be written in

a general form as sums of products of electronic and nuclear parts, i.e.












N
E(x, X, t; z, R, P) = c(t' (x; z, R, P)X (X; R, P). (1.37)

Such a functional dependence implies that, as in Equation 1.34, dynamical equa-

tions derived from such parametrization will have both dynamical and potential

couplings.

In the simplest case of a single product (N = 1) approximation to the wave-

function, the END product can be written as

E(x, X, t; z, R, P) = (x; z, R, P)x(X; R, P) (1.38)

where R and P are average position and moment of the nuclei and z are the

electronic wavefunction parameters. Both Equations 1.37 and 1.38 depend on the

time-dependent parameters z, R and P. Equation 1.38, however, depends on time

only through the wavefunction parameters and defines the minimal parametriza-

tion for the END method. As we shall see the solution for time evolution of the

wavefunction parameters amounts to the time-dependent Hartree-Fock (TDHF)

method for electrons and nuclei.



1.2 Solutions to the Time-Dependent Schidinger Equation

In this section we review time-dependent methods that approximate the solu-

tion of the time-dependent Schr6dinger equation

t z(x, X, t) H(x, X, t)E(x, X, ) (1.39)

We place emphasis on methods that attempt to include non-adiabatic effects and

do not use predefined potential energy surfaces (PES).










1.2.1 Time-Dependent Close Coupling Method

The close coupling method attempts to solve the Schrodinger equation directly

by employing an expansion of the wave function usually given by [7, 8]




n 2m

exp ( R(t')) + 2) dt (1.40)

Expansion in terms of adiabatic electronic eigenstates is commonly used. As in

Equation 1.36, all time dependence is contained in the expansion coefficients. The

close coupling equations resulting from Expansion 1.40 are usually solved along

prescribed classical nuclear coulombic or rectilinear paths.

Equation 1.40 presents an expansion of the electronic wavefunction in terms of

an electronic basis {(} with time-dependent coefficients c. Un(R) is the electronic

potential energy, m is the electron mass and f(F, R(t)) is a parametric function

of the nuclear positions. Because of the limitations concerning the solution of the

nuclear part and the need for global PES with coupling elements, the close coupling

method has been used mostly to describe charge transfer reactions of small systems

at relatively high energies.


1.2.2 Classical Path Method

The variant of the classical path described here uses nuclear coordinates as

parameters and electronic variables as the internal quantum variables. With this

parametrization, the classical path method [9, 4] is very similar in spirit to the

close coupling approach as explained above. It uses a similar parametrization (see

Equation 1.41) of the electronic wavefunction, however, it does not contain explicit

terms to account for Galilean invariance of the electronic wavefunction. As in the










close coupling approach, the dependence of the solution on the nuclear degrees

of freedom is parametric. The principal difference between the two approaches is

that in the classical path method the solution of the classical nuclear equations of

motion is attempted simultaneously with the propagation of the electronic coupled

sets of equations. In the classical path method the forces acting on the classical

nuclei are computed as an ad-hoc average between the electronic states involved

in the dynamics.

The classical path method uses the parametrization

r, R) Y c(t)Q(r, R) exp J' U(R(t'))dt (1.41)

in terms of the electronic basis {1}, the time-dependent coefficients c and the

electronic potential energy Un(R).


1.2.3 Trajectory Surface Hopping Method

The trajectory surface hopping method (TSH) [10] (for a review on the subject

see C'!i ipin i:i [11]) attempts to solve the equations for the classical path method

with the parametrization described above by using a probabilistic scheme. Instead

of solving the coupled set of equations at all nuclear configurations, a hopping

probability for the electronic configuration is computed at each instant in time. The

resulting equation for the classical nuclei depends only on the hopping probability

between the states involved (usually two). All explicit electronic dynamics (and

phase information) is removed on the TSH method.


1.2.4 Car-Parrinello Method

The Car-Parrinello (CP) [12] method has been extensively used for structure

as well as dynamics simulations. The CP method tries to find minima for the

following energy functional












E[{,' (, )},{R}, {a}]

d3 2 + U(n(r), { a)}, a}' (f,) (1.42)
2m

where U(n(r), {R}, {a}) is the potential including the electronic energy, internu-

clear repulsion and the potential resulting from any external constraint, n(r) is the

electronic density defined according to Kohn-Sham theory as n(r) = ,' *(r)jr),

RI are the ions positions and {,' (r', )} is an electronic adiabatic single particle
basis set. The minimization of the functional above is equivalent to finding the

solution for the Kohn-Sham [13] problem.
In order to obtain equations of motion a Lagrangian is also introduced

L(K R, n(r)) -

d/d3j (I + M 2 + P E[{' (), {}, {a}](1.43)
i I v
where p and p, are arbitrary parameters and .3 are the ions masses.

The CP method attempts to minimize the above energy functional by employ-

ing a simulating annealing [14] procedure on wave function parameters (coefficients

of the KS orbitals), the ions positions and the constraint parameters) a. The La-

grangian function given above is used to define equations of motion for the varia-

tional parameters. From the equations of motion, a simulated annealing procedure

is implemented. It is important to note that the equations of motion derived from

the CP procedure are only meaningful when i is equal to zero. This guarantees

that propagation of the nuclei happens on the Born-Oppenheimer surface given by

the solution of the KS equation for the electrons. The CP method as explained
here is not suitable for describing any explicit electronic (non-Born-Oppenheimer)

dynamics.










1.2.5 Time-Dependent Hartree-Fock Based Methods

In the context of (time-independent) electronic structure calculations the time-

dependent hartree-fock (TDHF) method calculates the linear response of a system

described by a single Slater determinant. The equations for the TDHF method

can also be obtained by considering the second order (harmonic) expansion of the

single determinant expression for the energy in terms of the electronic wavefunction

parameters [1]. The TDHF or RPA approximation amounts to finding the normal

modes for the electronic degrees of freedom [15].

For a time-dependent description, the TDHF method appears as an approxima-

tion to the time-dependent Scr6dinger equation. It is a consequence of considering

a single term in the BH expression (cf. Equation 1.38) for the wave function with

nuclei considered as classical particles and the electronic wavefunction as a single

determinant. The TDHF method has a long history as it was already proposed

in the 30s by Dirac[16]. Early applications for scattering [17, 18] did not consider

the nuclear motion to be dynamic. These original applications did not consider

explicit electron nuclear coupling as prescribed coulombic or straight line trajecto-

ries were used. Later derivations obtained the TDHF equations for the evolution

of the wavefunction parameters by constructing a Lagrangian and applying the

time-dependent variational principle (TDVP) [19, 20, 21, 22] or by using an den-

sity matrix propagation scheme [23] with classical nuclei. If no approximations are

made this procedure ensures that instantaneous couplings between electronic and

nuclear degrees of freedom are considered during the dynamics. The later appli-

cations of the TDHF method are the only ones suitable for describing chemical

reactions.

The END method in its simplest implementation is a TDHF method for elec-

trons and nuclei. It consists of using a wave function parametrized according to

equation 1.38 and employing the time-dependent variational principle [24] in order










to obtain equations of motion for the wave function variational parameters c, z, R

and P. In its minimum implementation, only z, R and P are used as parameters.

The resulting equations are a generalization of the Time-Dependent Hartree-Fock

equations for moving nuclei.


1.3 Electron Nuclear Dynamics (END) Method

The complete END equations include all couplings between the electronic an

nuclear degrees of freedom. These couplings have the general form that depends

on the parametrization of the wavefunction (c.f. Equations 1.37 and 1.38 ). The

addition of interaction of matter with an external field to such a formalism is

straightforward if we consider the description of fields in the semiclassical limit

(high intensity).

The END theory has been the focus of many publications [25, 26, 27] and

will be briefly reviewed here. Emphasis is placed on the topics judged to be most

important for the understanding of the analysis presented in the following chapters.

At the core of the END theory is the time-dependent variational principle,

which is based on the definition of the quantum action [28]

A L((*,)dt, (1.44)

where the quantum mechanical Lagrangian (with symmetric time derivatives and

it acting on the left side) is defined as

i8 i8
L((*, )=((I-- Hi1)/((1() (1.45)
2 Ot 2 Ot

where H is the quantum mechanical Hamiltonian of the system and ( is a vector

of time-dependent complex parameters. For the states considered for the END

ansatz time dependence is obtained through the variational parameters. With this

constraint the quantum Lagrangian has the form (using the chain rule)










i OIdnS InS .
L 2 (: a alsP)- E (1.46)

where we used E = ((lHf()/((|() and S((*, () = (| <). The equations of motion

(EOM) are obtained by making the action stationary and imposing fixed endpoints

boundary conditions. The equations of motion in terms of general variational

parameters is given by

aE
i Co = O (1.47)
13
aE
-i ,C (( (1.48)
i3
Note the marked similarity between Equation 1.47 and the classical Hamwith S

defined as S((*,() = ((I ()ilton's equation. In Equation 1.47, the sums run over

the parameter set {(}. The Hermitian metric matrix C elements are given by

a 2 In S
C gaip (1.49)

The equations of motion in matrix form reads

C 0 ( aE
C a0 (1.50)
0 -C* (* L

The END ansatz is a molecular coherent state constructed from

() = -,R)IR, P) = z)1), (1.51)

where the nuclear part is a product of Gaussians of the type

1) b )R2 + k(Xk Rk)] (1.52)
k
and the electronic part is a single determinantal (unnormalized) coherent state
K N
Iz) exp[ 1 b*]lO ). (1.53)
j=N+1 i=1










Note that the electronic state |z) depends only on z. This fact ensures that the

structure of the equations of motion derived from such ansatz have the same form

as Equation 1.50.

The Lagrangian for the molecular system in the narrow wavepacket approxi-

mation for the nuclei is given by

i alnS alnS i alnS alnS
L = U1[Pj + R )]Rjl+ ( a )Pj
I v 2 9RjI R9nl 2 9lji OP'jl
i I lnS. 8lnS
2 azh Zph z*ph zX) (1.54)
p,h
with S (z, R', P'lz, R, P) and

E = P,/2 + (z, R', P'Hei z, R, P)/(z, R', P'|z, R, P). (1.55)
ji
where the electronic energy Hamiltonian is given by

H ZAZB+ V+ (1.56)
SrAB 2m, L i rij
A,B iAB 2j ri,
The equations of motion are obtained by minimizing the action as before. The

resulting EOM are a set of coupled differential equations given by

iC 0 iCR iCp z 9E/9z*

0 -iC* -iC, -iC*, OE/Oz
(1.57)
iCt -iC CRR -I + CR R OE/R

iCt -iCT I+ Cp Cpp )P E/9P

where the coupling terms in the narrow wave packet limit (zero width) are

a21nS
(Cxy)ij;kl = -21mn IR'R (1.58)
OXikOYji



(CxJ,)ph = az Xik R\'=R, (1.59)
azphOXik










and

a2inS
Cph;qg a z*,z R=R R (1.60)

The coupling elements given by Equations 1.58-1.60 correspond to terms related

tot the Galilean invariance, non-adiabatic coupling, and pure electronic coupling

between electronic states [29, 25] respectively.


1.3.1 Narrow Wavepacket Limit

As mentioned earlier the dynamical equations above were derived taking the

width of nuclear wavepackets to be zero. That approximation is comparable to the

frozen Gaussian approximation.

The effect of adding a width to the dynamical equations can be understood by

its individual effects on the terms of the Lagrangian. The expectation value of the

p2 operator of a Gaussian wavefunction is (p2) = (_2/ax2) p2 + 1/2b2, so the

frozen width term in that case is only a constant to be added to the kinetic energy.

There is no contribution to the dynamics of the system due to this term.

The electron nuclear attraction integrals will change their value modestly due

to the finite width of the nuclear charge distribution. Those effects however are

minor because the width of the electronic wavefunction is much greater than the

width of the nuclear wavefunction.

The overlap term can have nonzero off diagonal components due to the nuclear

wavefunction. Although this can happen, only high energy scattering processes

would make those terms of a sufficient size to be important. It is well-known that

the innermost orbital of second row elements hardly affect processes of chemical

interest.










Minimal END dynamics as presented in Section 1.3 therefore is completely

equivalent to dynamics with frozen Gaussian [30, 31] for the collision energies in

question.



1.3.2 Molecular Dynamics

Simplifications of the complete END equations are possible under certain cir-

cumstances and can greatly reduce the computational requirements. Such approx-

imations are useful in comparing the END method with other methods. The END

equations provide a general approach to molecular dynamics, so that standard

equations for dynamics on the potential surface are easily obtained from the END

equations.

In particular, if we require the stationary condition (z* z = 0) for the elec-

tronic wavefunction in equation (7) molecular dynamics (M!1)) equations are ob-

tained. The nuclear potential is given by the z parameters that satisfy

c 9 E(z, z*, R, P, )


or in the presence of an external electromagnetic field

aEo(z,z*,R,P) + pf,= iCRR. (1.62)


Equation 1.62 includes two terms that are truly dynamical. One is the dipole

term that depends on the time-dependent field, and the other is the boost term

that is related to a Galilean transformation of the electronic reference frame to the

moving nuclear frame. This boost term has an equivalent effect on the dynamics

as the electronic translation factors (ETF) usually employ, l1 to basis functions

that follow the nuclear motion. These two terms are the difference between the

electronic structure calculations performed in this work and the electronic potential

surface usually used for standard MD calculations.










The dynamical equations in this approximation have a form consistent with

the fact that z 0 and are

CRR -I R 9E/9R (1.63)
I 0 P 9E/9P

Note that one of the coupling terms is still present in the dynamical equations.

The CRR term (in this case) is just a friction like term related to the dr fo.vin. force

of the electrons on the center. The dynamics takes place on an adiabatic potential

surface. These are the equations referred in this work as molecular dynamics (\ 1)).

We can make two distinct approximations. The first is to neglect the depen-

dence on the center velocity of the electronic energy (neglect the boost term in

eqn. 1.61). This reduces the Equations for the electronic part to Hartree-Fock

(HF) equations. For consistency CRR is also set to zero in Equation 1.63.

Another approximation consists of neglecting the dipole contribution to the

electronic energy when constructing the electronic wavefunction. This is what is

usually done when a pre-calculated potential surface is used to study the dynamics

of the nuclear wavefunction under the influence of an electromagnetic field.

The exclusion of the dipole term from the calculation of the electronic energy

and density introduces some changes to the calculation of the force term (% ) in

Equation 1.63. Because an electronic state calculated by neglecting the effects

of the external field is not stationary, extra terms would be introduced in the

energy derivative expression. The non-polarized molecular dynamics (NPMD) is

a molecular dynamics method that neglects the effects of the external field in the

electronic configuration both in the energy and energy derivatives. As the name

indicates, the electronic wavefunction is not allowed to polarize in the presence of

the field.










As we shall see, the MDNP method is useful for comparison purposes because

with it we are able to introduce a hierarchy of methods classified with respect to the

behavior of the electrons under the action of a field. According to their dynamical

behavior we can have full dynamical electrons (END), polarizable electrons (\!l)),

or static electrons (\!I)NP).


1.4 Cross Sections Calculations

As an experimental observable, total cross section (TCS) is an important quan-

tity to measure and compute. As the theoretical definition follows from the con-

cept of flux, a direct experimental measurable quantity, straight comparisons can

be made between theoretical and experimental results. The TCS can be defined

in terms of an integral over the differential cross section (DCS). The DCS is an

important quantity for theory as it relates to the deflection function from both

quantum and classical formalisms. In this section we introduce the subject of

classical binary collision for a central potential. Later in the section a connection

between classical and quantum solutions for the scattering problem is made. A

small generalization to reactive scattering processes is also mentioned.



1.4.1 Differential Cross Section

The formal differential cross section (DCS) for a collision problem is defined as



() number of particles scattered into solid angle dQ per unit time .6
doL(E) = (1.64)
incident beam intensity

The binary collision problem has as constants of motion the collision energy and

the total angular momentum. For a structureless or nonrotating target the angu-

lar momentum is specified by the initial impact parameter b. Enforcing particle

conservation for a scattering by a central potential we obtain












27rlbdbl = 2rda(E Isin OdOe (1.65)
dO

with I being the incident beam intensity, 0 the scattering angle, E the collision

energy and b the impact parameter. The definition of the DCS for scattering by a
central potential is therefore given by

da(E) b db b
dO sin 0 dO sinOd d (10

The total cross section is defined as an integral over the DCS
[ da(E)
a(E) = 27 sin OdO. (1.67)
J dO
From classical trajectories obtained from any of the semiclassical methods de-

scribed previously in this document we can calculate the deflection angle O as a

function of the impact parameter b. The function resulting from this calculation is

called a deflection function (0(b)) and 1pl "i- a central role in scattering theory. As

experiments cannot make a distinction between positive and negative deflection

angles, the scattering angle used in Equations 1.65 and 1.66 is defined w.r.t the

deflection angle computed for trajectories such that

0 101 with 0 < 0 < T. (1.68)

The foremost feature of the classical DCS given by Equation 1.66 is that it has

a singularity for 0 equal 0, 7 and at the extrema of the deflection function 0(b).

At these values of the deflection angle quantum effects are important for a correct

description of the DCS. The cases where 0 = 0 or 0 = 7 are called glory scattering
while the cases where the deflection function has a maximum or minimum is called

rainbow scattering. Semiclassical corrections to the DCS for small angles were

derived by Shciff [32] with applications to scattering using the END method by










Cabrera-Trujillo et al. [33]. Rainbow scattering corrections were first developed

by Berry [34] and applied in the END context by Morales [35] for the H+ + H2

charge transfer reaction and by Hedstrom [36] for the H+ + H20 charge transfer

reaction.

Fortunately, for the reactive cases treated in this document we do not encounter

such difficulties. The deflection functions calculated are all repulsive in character

and do not posses a singularity on the cross section for all impact parameters

investigated. The standard classical expression was used to compute cross sections

in this document.

For probabilistic events coupled to classical nuclei, such as electron transfer

reaction, the more general expression for the classical DCS is to be used, i.e.

dk (E) bsPk(bs)
dO sin 0 1b=b

where we introduce a sum over trajectories with the same scattering angle. In

Equation 1.69, Pk(b) is the probability of an event labeled by k. In the END

context k usually labels a reactive or charge transfer channel. A brief review of

semiclassical theory and the relations between classical and semiclassical results is

presented in section 1.4.4


1.4.2 Classical Collisions

The classical description of a binary collision process introduces several fun-

damental concepts used throughout this document. In this section we outline the

collision of two structureless particles interacting via a potential that depends only

on the internuclear distance (V(R)).

The kinetic energy of two particles is given by

Tm -2 m2 7 (-2 ml + m2) -2 2
Ek = m2 2 = m V + PV (1.70)
2 2 2 2










where on the r.h.s we expressed the kinetic energy in terms of the relative velocity,

the center of mass velocity and the reduced mass where

U = J1-- 2 (1.71)

V = mi + mT2v (1.72)

P = MI (1.73)
(mi + m2)
This new set of coordinate is usually referred to as center of mass coordinates. The

kinetic energy associated with the collision process is given by

Ekin = 2. (1.74)

For a spherically symmetric potential it is useful to define a new set of coordi-

nates by centering the target at the origin and decomposing the velocity vector in

its radial and angular components

S= R R + Roi,. (1.75)

where 'R and to are the radial and angular unit vectors. The relative kinetic energy

in this new set of coordinates is given by

1 1 J2
Ek = (R2 R2) 2 + 2 (1.76)
2 2 2pR2

where we used the definition of the total angular momentum for the system

J = Rx p = R2O 1. (1.77)

The conjugate moment of R and 0 is (L = T U)
AL
pR a t PR, (1.78)
OR
aL
pe pR2 = JI, (1.79)
S

respectively, where J is the total angular momentum.










The Hamiltonian is defined as

H = RR + po0 L ( p + (R) (1.80)

with the corresponding Hamilton's equations of motion for the conjugate moment

as

aH p2 U
p -- (1.81)
R aR R3 R' (.8)
aH
O 0. (1.82)

Equation 1.82 shows that pe is a constant of motion and that the total angular

momentum is conserved. The magnitude of the total angular momentum of the

collision is also related to the initial impact parameter and the initial collision

velocity by

IJI= rn' ,,1= pRvl sin0t=o (1.83)

The final 101 for the collision, called scattering angle, can be calculated by using

Expression 1.77 and 1.80 (with H = E) and the equation

Rf HO (Rf 0
SdR (1.84)

with the minus sign corresponding to attractive scattering and the positive sign to

repulsive scattering. Considering that the integral is symmetric w.r.t the distance

of closest approach (Ro) we can write

0 2bb dR U (1.85)
o R2(1- b2 U(R) )
Equation 1.85 is the classical expression for the scattering angle generated by a

central potential in the center of mass coordinate system.










1.4.3 Laboratory-Center of Mass Coordinate Transformation

We demonstrate in the last section that the intrinsic features of a scattering

problem can be more clearly described in the center of mass coordinate frame.

Because of this most theory and several experimental results are reported in the

center of mass frame. We present in this section a brief review of laboratory-

center of mass (C' 1) transformation for the scattering angle and differential cross

sections. For reactive processes however there is no unique way of defining a global

set of internal coordinates for describing reactants and products. Some coordinate

representations used to describe reactive processes in polyatomics, e.g. the Jacobi

coordinate set, do not present a continuous parametrization for a reactive process.

The problem of using internal or center of mass coordinates is worsened for multi-

channel reactions where the coordinates for reactants and each one of the products

may not have a continuous parametrization. That is one of the reasons that a

cartesian set of coordinates for describing a reactive process is used with the END

theory.

The Laboratory-C\ I coordinate transformation can be better understood by

considering a vector diagram (a Newton diagram) of the velocities of the particles

involved on the collision process. In Figures 1.2 and 1.1 initial quantities are

labeled by i while final quantities are labeled by f. Primes quantities refer to

relative coordinates while umprimed quantities refer to laboratory coordinates.

Consider the triangle depicted in Figure 1.1. By using simple trigonometric

rules we can relate laboratory quantities to C'\ quantities. Transformations from

C'\ I to laboratory scattering angle can be found by simple algebraic manipulations.

Consider the equation

vlf sinp = vif sin (1.86)

Vlf cos = vlf cos + Vc, (1.87)











31




























Vlf / ..--"V' \
-I 4













SV2
4
4
4
































...-.... ...
I 4

















II


4 g
4 .
i r .
n n u
4
4
4
4
4
4 O
4P
4c i
4O
44i


Figur 1.:Nwo iga o olso rceswt h agtiiilya et

Onl fia uniisaepotd










to find that (completing squares)

S= +p + 2p cosO (1.88)

where we use Vm = v vi and define the collision parameter p = P, in terms

of the initial collision velocity vii and the final projectile velocity in the center of

mass frame. With Equation 1.88 we can define the laboratory scattering angle in

terms of center of mass quantities

cos + p
cos+- O+pc (1.89)
1 + p2 + 2pcos
The inverse transformation (laboratory to C \! scattering angle ) can be found

by considering the relations

vf sinO = vif sin (1.90)

v/f cos = vlf cos Vcm. (1.91)

With similar manipulations we obtain the C\ I scattering angle in terms of labora-

tory quantities, i.e.

cos s p'
cos pO- =- (1.92)
\/1 + p2 2p' cos p
where we define a new collision parameter p'

p' = lV (1.93)

Equation 1.92 defines the C '\ scattering angle in terms of laboratory quantities

only.

A transformation for the differential cross section between laboratory and C'\

frames can be found by imposing flux conservation of particles


2wrlj(0) sin OdO = 2wlc(p) sin opdp


(1.94)

















Vlf Cos(p
Figure 1.2: Velocity vectors from Newton diagram (Fig. 1.1) used to construct the
C'\!-I i oratory transformation.
which can be rearranged to

d(cos )95)
a(9) a(S) d (1.95)
d(cos o)
Equation 1.95 can be used together with Equation 1.94 to find a transformation be-

tween the differential cross section represented in terms of center of mass quantities

to one in terms of laboratory quantities [37].


1.4.4 Semiclassical Differential Cross Section

A formal relationship between the quantum and classical differential cross sec-

tion (DCS) can be obtained by means of the semiclassical theory. We start by

reviewing a few results for the quantum scattering problem under the influence of

a spherical potential (V(r)). Due to the symmetry of the problem L2, L, are good

quantum numbers. Therefore we seek solutions that are also solutions of L2, L,

and H. Our search starts by investigating the solution for the free particle problem

in terms of spherical waves. The quantum and semiclassical treatment followed in

this section deals only with elastic scattering processes.

The time-independent Schr6dinger equation for scattering off a central potential

is given by

2mT
[V2 2V(r) + k2]0(l = 0 (1.96)
V










where the momentum f= h1 defines the collision energy
2
E = (1.97)
2m

An interesting expression for the DCS that enables us to connect semiclassical and

quantum theories is defined with respect to the scattering amplitude and is given

by

da
da A(O)|2, (1.98)

which is related to the amplitude of the wavefunction at large distances (r)(Q is

the solid angle element). The ansatz of the scattering solution is usually taken to

be



d(r) = eik + A k (1.99)
r

This ansatz can be understood as being a superposition of an unperturbed part

related to the incoming wave and a outgoing spherical scattered wave. It is easy to

see that such wavefunction has a flux trough an element of area dS = r2 sin(O)d

given by [38] Al) 2, resulting in a DCS given by Equation 1.98.

The partial wave expansion is a standard method to solve the quantum scat-

tering problem. In this method the wavefunction I(rf is expanded in terms of

Legendre polynomials

S(r- (, ,(r)P(cos0) (1.100)
l=0
where cQ are expansion coefficients, '. (r) are the solutions to the radial only prob-

lem at a fixed angular momentum I (partial waves) and P (cos 0) are the Legendre

polynomials. The functions '. (r) have to satisfy the radial equation for all 1, i.e.

d2 2 2m 1(1 r+ 1)
S2 + k- 2 V(r) 2 ]()= 0 (1.101)










The solution for the partial waves can be found by considering the solution of the

radial Schr6dinger equation in the limit r oo [38]. For potentials that decay

faster then the .,-- ,iii d tic solution for the radial Schrodinger equation converges

to the free particle solution given by


8(r] Alt 0C [e-i(kr-l7/2) i(kr-l-F/2) YCOS ), (1102)
1 =0
AT A, B, LC-kr1w2) 1 2)] P,(cos 0), (1.102)

with the difference between the solutions without or with the potential given by

the different boundary conditions at small r (actually at r = 0). It is possible

to show that in the latter case the influence of the potential changes only the

relative phase of the outgoing wave when compared to the free particle solution.

The general solution for the partial waves in the ..-i ,!l' 'l ic region is given by


I-o
A( A, t B, [L-i(kr- w/2) i(kr-/w2) t P] (cos 0), (1.103)
l/0
where all the influence of the potential is included in the phase shift term ei26. The

latter equation has a more familiar form given by

1 7
A (r A sin(kr + 61). (1.104)
r 2

The overall normalization constant for the solution is not of interest in this case.

Instead, by expanding Equation 1.99 in terms of partial waves, using
1 00
eikz (21 + 1)P,(cos0) (1.105)
1=0
and comparing with Equation 1.103 we can obtain an expression for the scattering

amplitude [39]

A(0) (21 + 1() e2iS 1)P~(cos 0), (1.106)
1=0
where we used the expansion given in Equation 1.103 and the orthogonality relation

for the Legendre polynomials.










The relationship between the quantum result and semiclassical theory can be

brought out by considering the expression for the phase shift in the JWKB (Jeffreys-

Wentzel-Kramers-Brioullin) theory [38]. Consider the JWKB approximation to the

central potential problem (Eqn. 1.101) in the classical region

S(r=-- l sin kl(r)dr + (1.107)
(k, (r)) 2 al 4
with al the classical turning point and were we use
ki(r) k2 _V(r) ( 1.108)
2m r2

Direct comparison between the JWKB solution and Equation 1.102 -1-'-1 -1-;

that the JWKB phase shift is given by

F 1 P00
1 (1 + 2) + (k (r) kr)dr (1.109)

Equation 1.109 has a more explicit form as

= ki(r)dr- ki(, (r, ) 0)dr. (1.110)
J J(l+)/k
By considering the derivative of the phase shift w.r.t L = (1 + 1) we obtain

j6 L dr j A L 2dr. (1.111)
OL I/~ [k2 L /r221/2 J, [k2- V(r)/2m L2 /r2]1/2

Then substituting the classical variables given by

L = (1+ )h mvb= khb, (1.112)
2
k2h2
E 2= (1.113)
2m

we obtain

06 7 1t
Sb dr. (1.114)
9L 2 a r [1 V(r)/E b2/r2]

Equation 1.114 is exactly | of the classical deflection function [37]. Noting that

the variation in L is equivalent to a variation in 0 through the collision momentum










and the deflection function we arrive at the relationship

aa1 1
a -2e(b). (1.115)
00 2

The relationship given by Equation 1.115 is of great importance for correcting

results obtained with the END theory when the need arises. By knowing the clas-

sical deflection function and its relation to the semiclassical solution for the elastic

scattering problem, semiclassical DCS can be computed via Equation 1.106 from

purely classical deflection functions. It is important to note that the deflection

function (and resulting phase shift) as used in the END theory do include inelas-

tic processes and presents a generalization over the JWKB results as presented

in this section. Most importantly, deflection functions calculated by the END

method can introduce effects of excited electronic states that are fundamental for

the description of charge transfer scattering processes.


1.5 Measurements

Equation 1.116 gives the velocity dependent rate constant in terms of the re-

action cross section for a process R

k(v) = v7R(v). (1.116)

The attenuation of an incident beam in the x direction due to process R for a

A + B collision is given by

dl
d- k(v)UnAB = I(x)nBUR(v), (1.117)
dx

where I(x) is the incident flux in the x direction and nA/B is the number density

for particle A/B. The reactive cross sections therefore determine the microscopic

rate for the reaction. Temperature dependent reaction rates can be constructed

by convoluting a Maxwell-Boltzmann velocity distribution with aR(v).










The opacity function is a theoretical quantity that can also help with mecha-

nism interpretation. It is defined in terms of the cross section derivative w.r.t the

impact parameter as

1 daR
P(b) db (1.118)
27b db

and it gives the fraction of reactive collisions for a given impact parameter b.

The reaction cross section can be given as a weighted average over the opacity

function:

JR = 27bP(b)db. (1.119)
0


1.5.1 Total Cross Section Calculation

The integral cross section can also be calculated by integrating over the initial

conditions provided that we know the opacity function for the process of interest.

The integral cross section is defined by

Uk(ELab) 27 bPk(b; ELab)db. (1.120)

To calculate the opacity function Pk(b, ELab) we have to average over the many

initial orientations possible for a non spherical target, i.e.,

1 /27r /27r r7r
Pk(b, ELab) 82 pk(a, 3, 7, b) sin /3dad/3d7, (1.121)

where the function pk(a, 3, 7, b) is the reaction or charge transfer probability de-

fined for channel k for a single trajectory calculation. The trajectory initial con-

dition is specified by the target orientation labeled by angles a, 3 and 7 and by

an impact parameter b. If the projectile is not spherically symmetric one needs

to introduce it's orientation average in the probability function. This approach is










sufficient if we are interested in total cross sections only. Product angular distri-

butions (DCS) are not explicitly considered in this analysis.

In order to evaluate the average integral numerically, we need to consider pk (a, 3, 7, b)

a smooth function of its parameters and construct a polynomial interpolation for

the probability function over the angular variables. From that point we do a nu-

merical integration to compute the integral. Consider the trapezoidal rule

f f(x)dx hi f(xi), (1.122)
JO i=0
where we used f(O) = f(27). This integration implies zero order expansion for the

integrand, therefore any available value in the interval can be used to compute the

integral.

To compute the integral in 3 some care has to be taken as the weight function

goes to zero for 3 = 0 and 3 = One solution is to evaluate the integral in 3

analytically. Using a first order expansion for the integrand function we have


J 2-F 2-F
f (/3) sin3d/3 f(/3) sin/3d/3, (1.123)

where

f(/3) =fi + hi(3 i) (fi+i + fi), (1.124)

and hi = (P3i+ Pi). Substituting in Equation 1.123 we obtain

Sf(/) sin /d/ -fi(hipi cos 1 + cosP + cos Pi+ hAi+l co(1.125)

+3i+1 + hi sin 3 +ji+ -hifi sin 3) + fi+ (hi sin 3i+i

-hifi+l cos 0 i+ + hiAi cos 0ji+ hi sin pi)

This procedure is used throughout this work to compute total cross sections for

the several reactions studied.










1.5.2 Background on Guided Ion Beam Experiments

The absolute cross sections for the abstraction and exchange channels NDH+

and NDH+ in the reactive collision NH --+ D2 are measured using a guided ion

beam (GIB) apparatus [40, 41, 42]. These experiments are best suited for measur-

ing ion-molecule reactions at hyper-thermal energies such as the one considered in

this work. The main distinction between these experiments from the traditional

crossed beam experiments is that the former can measure total cross sections but

due to experimental design cannot measure product angular distributions. The

final result of such experiments are time-of-flight (TOF) profiles that show ion

count (intensity) versus time. The TOF profiles are integrated producing the total

intensity. Branching ratios are obtained as the fraction of summed intensity in

a particular product channel. Relative cross section are the ratio of a product

intensity over the reactant intensity. In addition, this methodology allows for the

measurement of product ion laboratory velocity parallel to the incoming beam.

Typical absolute errors for total cross sections measured in this scheme are in the

i,'. range while relative errors are smaller [41].

The special features that the GIB setup shares with the crossed beam experi-

ments are [43] as follows,

* Single collisions provided that the pressure is low enough.

* Reactants are velocity (energy) selected.

* Supersonic nozzle expansion guarantees that reactant molecules have low tem-

perature.

* Reactants can be state selected by use of electric, magnetic or laser fields (by

coupling lasers to the setup).

* Reactants can be oriented and polarized to study steric effects.

And unlike most crossed beam experiments, absolute cross sections are measured.











Schematics of the GIB apparatus are pictured in the Figure 1.5.2 from Dresseler

[41]. The experimental setup is divided into

* An ion source generator.

* An octopole ion guide responsible for selecting the ion's collision energy and

guiding the beam to the reaction chamber. A quadrupole mass filter can be

used to mass select the reactant ions before they enter into the octopole ion

guide.

* A field free collision chamber filled with the second reactant.

* A second octopole ion guide responsible for collecting the nascent product.

* The detection part with a quadrupole mass filter.

GAS INLET
EXTRACTION ELECTRON
COLLISION LENS MULTIPLIER
INJECTION CHAMBER MULI
LENS \


OCTOPOLE 1 OCTOPOLE 2
MANOMETER QUADRUPOLE
DEFLECTOR OUTLET
ELECTRODE
Figure 1.3: Pictorial representation of the interaction region of the guided ion
beam experiment.

Some improvements of the GIB experiment by Poutsma [44] over the Quadrupole

Tandem Mass Spectrometer (QTMS) setup used by Morrison [45] are

* The octopole guide would better control the divergence of the slow particles.

* The second octopole better collects both forward and side scattered products

than a quadrupole guide of T:\ S setups.

The net result is that GIB experiments with octopole guides have better sensitivity

than Quadrupole guided T\ IS experiments.

The GIB apparatus can be combined with a resonance-enhanced multiphoton

ionization (REMPI) operating on the ion source to select the initial vibrational

state of the reactant ion. By first electronically exciting the system to a state of








42

similar nuclear configuration as the ion (usually a Rydberg state) and later ionizing

the excited molecule, REMPI is capable of producing vibrationaly excited ions at

selected vibrational states. In the case of ammonia the C' can be used as it has a

planar nuclear configuration like the ion. Note that this technique is used also to

generate ground state NH+ molecules when the Av 0 transition is targeted.














CHAPTER 2
THE NH+ + D2 SYSTEM AT HYPERTHERMAL ENERGIES

2.1 Introduction

Our research involves the theoretical study of reactive scattering processes.

More specifically we have been studying the reaction of NH+ with D2 at hyper-

thermal energies. Poutsma [44] and coworkers recently investigated this reaction in

a molecular beam study providing experimental results in the form of relative cross

sections for the two most important channels, the abstraction channel (NH3D+ +

D) and the exchange channel (NH2D+ + D + H).

The NH+ with D2/H2 reaction presents serious challenges for both experiment

and theory. It is amenable enough for experimental investigation but exhibits non-

Arrhenius temperature dependence of the reaction rate at low temperatures and

two competing channels at higher collision energies that have strong dependence

on NH+ internal excitation. Moreover the NH+ + H2 reaction is believed to

have some importance in the interstellar synthesis of ammonia. In this work we

investigate the NH+ + D2 reaction at hyper thermal collision energies ranging from

6 to 16 eV in the center of mass (COM).

This chapter is divided in four sections and one appendix. Section 2.2 pro-

vides some background information about guided ion beam experiments and a

brief review of experimental results for the reaction of interest whose initial con-

ditions closely relate to the ones used in our calculations. Emphasis is placed on

experiments performed by R. N. Zare's group. Section 2.3 presents a few theo-

retical results for electronic structure calculations for the NH+ molecule and also

presents the initial conditions grid and basis functions used for our cross section










calculations. Sections 2.4 presents results for total cross section, differential cross

section and flux-velocity contour plots at collision energies of 6, 8, 12 and 16 eV in

the center of mass frame. Sections 2.5 and 2.6 present the discussion conclusions

respectively.



2.2 Experimental Results for Reaction NH+ + D2

Two investigations performed at R. N. Zare's group explore the NH+ + D2

reaction at collision energies ranging from 0.5 to 10 eV in the COM. In the earlier

study Morisson at al. [45] utilize a T\ i setup with quadrupole ion guides and

considers initially excited NH+ ions in the umbrella bending mode with v2 = 0-

9. The highest excited state corresponds to an internal energy of 1.1 eV. In the

more recent study Poutsma at al. [44] use a GIB setup and considers two almost

isoenergetic vibrational states as well as the ground vibrational state of the ion.

The two excited states considered are the vlv2 1025 and vlv2 1122 where vi

corresponds to the symmetric stretching mode and v2 corresponds to the umbrella

bending mode. The excited vibrational states 1025 1122 and have 0.60 and 0.63 eV

of internal energy respectively.

The main interest of both studies is to gain insight into the mechanism of the

reaction by considering several initial conditions with different energy components

(vibrational,kinetic). By studying the energy flow from reactants to products one

hopes to gain insight into the mechanisms of the generation of the exchange and

abstraction products and their interrelationship.

At the experimental conditions considered the two main channels present for

the reaction are:

NH+ + D2 NH3D+ + D (I)


and











NH3 + D2 -+ NH2D+ + HD (II)

or the equivalent for the present experiment

NH+ + D2 NH2D+ + D + H. (III)

Even though GIB experiments can provide results for total cross sections, only

relative cross sections were reported. Due to its design the sensitivity of the exper-

iment depends on the kinetic energy of the ion source, i.e. sensitivity is collision

energy and nascent product kinetic energy dependent. The result is that prefer-

ential detection of a particular product can occur if the channels differ markedly

in final kinetic energy. The mentioned experimental weakness w.r.t sensitivity is

not as pronounced for the newer GIB setup as compared to the older quadrupole

guide Ti\ [S experiment but still seem to be an issue for the reaction under study.

The kinematics of the system guarantees that the products are forward scat-

tered in the LAB frame, however for isotropically scattered products the collection

efficiency for the GIB setup is only 10' Therefore depending on the mechanism

in operation and the resulting angular distribution, the collection efficiency can be

different for the different reaction channels.

Another consideration is that the experiment measures the long time limit for

the abstraction and exchange cross sections. With a traveling time of 40 to 50

pis from the reaction chamber to the detector, excited state nascent products can

further react or dissociate and substantially affect cross section values. Energy

spread for the COM collision energies seem to be well under control and reported

to be of 0.2 eV in the COM or 1 eV in the laboratory frame.

The main qualitative experimental insight from Zare's work can be summarized

as: channel (I) was found to be very sensitive to increase in collision energy while

channel (II)/(III) was found to be very sensitive to increase in vibration excitation

at collision energies above 4 eV in the COM.










Others have reached similar conclusions while studying the same reaction sys-

tem under different initial conditions. Following Eisele et al. [46], Morrison sug-

gests that the formation of the exchange product happens through a two step

process with (I) followed by (IV). More recently another alternate mechanism

was -Ii---.- -1. I by Poutsma (V). Winniczek at al. [47] explored reactions (I) and

(II)/(III) using a cross ion beam setup. The experiment used a fixed collision en-

ergy of 0.5 eV in the COM and vibrationaly excited excited ammonia cations with

energies ranging from 3.3 to 4.9 eV. Winniczek found that excess vibrational energy

appears in the product final kinetic energy and that the abstraction reaction pro-

ceeds trough a direct rebound mechanism (not spectator stripping) while exchange

proceeds trough the formation of a long lived complex. Winniczek -ii-.-., -I that

for the low collision energy considered in the experiment formation of the exchange

product is not due to a two step process having the abstraction product as an in-

termediate, but rather due to the formation of a weakly bound complex. RRKM

estimates -,.; -1 a lifetime consistent with a well depth of 0.2-0.7 eV relative to

the reactants. In his work Winniczek -r-.i-i. -I; that that this energy is compatible

with a NH+ D2 complex and much less then the binding energy for the complex

NH3D+ D. However more recent theoretical calculations [48] i-.I; -1 -that com-

plex NH3D+ D has a biding energy of 0.85 eV while complex NH+ D2 has a

biding energy of only 0.06 eV when compared to reactants. Ischtwan at al. [49]

also -i .i: -i a possible reaction coordinate cutting thought extrema points of the

NH3D2 potential surface, without the formation of a NH3D+ D complex, but

with an internal NH+ rotation w.r.t the hydrogen that would lead to the exchange

channel (II). Therefore, as opposed to what Winniczek claims, the dynamics of

exchange and abstraction processes may not be so dramatically different.










Kemper and Bowers [50] considered COM collision energies in the 0.07-1 eV

range and vibrational excitations in the 1-5 eV range in a tandem ICR (ion cy-

clotron resonance) experiment. Although operating in a rather different collision

regime his conclusions are similar to Zare's: kinetic energy enhances (I) while vi-

brational energy enhances (II)/(III). Bowers sI -'-' -1 that the exchange reaction

proceeds trough a combination of direct and complex formation mechanisms while

the abstraction reaction proceeds trough a direct mechanism at higher tempera-

tures (above 200-300 K). Low temperature experiments [51] -i t-- -1 a low reaction

barrier of 0.09 eV at the entrance channel for the abstraction reaction.

In Section 2.2 we introduce the spectator str':'i':rI model model used by Zare's

group to explain some of the experimental results. Though simple, the model

can help connect results obtained in the experiments with results obtained by

theoretical calculations. We make use of some of its results in the next paragraph.

An important threshold energy for the present study is the dissociation energy

for the process

NH3D+ NH2D+ + H, (IV)

and an equivalent process for the loss of a D atom. The accepted threshold

value for such process is 5.5 eV Using the spectator stripping model we can predict

at what collision energy we could have the formation of NH2D+ as a two step

process of reaction (I) followed by reaction (IV). Assuming reactants initially

in their ground state and using AHrx, = 1.04 eV the predicted value for the

COM collision energy necessary for the production of NH2D+ from dissociation

of NH3D+ is 8.0 eV in the COM. In other words, for collision energies above 8.0

eV we should see an increase in NH2D+ formation due to the opening of a new

reaction channel.

NH3 + D2 NH2D + H --- NHD+ + D + H. (V)










Results presented by Zare's group [45, 44] seem to corroborate the two step

mechanism and spectator stripping model with two facts:

* There is an insensitivity of the cross section to a particular vibrational mode

being excited.

* There is a steep increase in channel (II) (III) products at collision energies above

8 eV in the COM concomitant with a depletion of channel (I).

Two of the best accepted limiting mechanisms for a direct reaction (faster then the

rotation of the formed reaction complex) are the spectator ./,.l.'..:j'. mechanism

and the rebound mechanism. In the rebound mechanism the reaction occurs at

small values of the impact parameter b and products are back scattered in the

COM frame. Cross sections are usually small due to the small range of reactive

impact parameters. In this section we will describe the second limiting mechanism,

the spectator stripping, model in detail.

The spectator stripping model is known to give reasonable estimates for internal

energy deposition in product fragments for collision energies above 1 eV The model

assumptions are rather crude but it can help to shed some light on the reaction

mechanism of the abstraction channel and its relationship to the exchange channel

for the title reaction.

In the spectator stripping model the projectile NH, picks a D atom while

the other D acts as an spectator. The main assumption of the model is that the

spectator D and the reactant fragment NH+ do not change momentum during

the collision. With that assumption of momentum conservation one can derive the

internal energy being deposited in the abstraction product NH3D+. In this model

the product NH3D+ velocity is given by (v is the original relative velocity)














mNH3D+


mNH +MD2 mDmNH
M M
NH3+ (MD2 MD) V


(2.1)

(2.2)


where M is the total mass. Energy conservation under these conditions takes the

form (AHr,, being the energy liberated during the reaction)

MNHTnD2V 2 NHD+mD'2
+ Eot ND --- + AHr, + Et. (2.3)
2M 2M


where we can use the fact that the new relative velocity is given by v


it NH m (~D2NH3D+ nmDmNH+ nt
Eint 3 2 AHrm + Et.
2M AmNH3D+

Equation 2.4 can be rewritten in a more useful form as


(mD2mNH3D+ TnDTnNH+
Eint Ecm
E coll. NH
mD2 mNH3D+


AHrxn + E nt


mNH+
= u-M


(2.4)


(2.5)


or numerically


Eint Ecolm 0.5526 + AHrxn + E t.


(2.6)


The values predicted by the spectator stripping model for collision energies

considered in our simulations are given in Table 2.1.



2.3 Computational Details

2.3.1 Structures at Extrema Points of the NH+ Potential

Early calculations involving characterization of NH+ ground state structure

were performed by Bugaets and Zhogolev [52] and by Kassab et al. [53]. Ischtwan

[48] did a more comprehensive study of the NH+ electronic potential surface with

calculation of several minima and transition states. Large changes in energy were










Effects of electronic correlation on extrema structures and energies are sizable

but do not change the principal features of the potential surface. The largest effect

among the structures considered by Ischtwan [48] was on the transition state (TS)

for the abstraction process. Using MP4 as the correlated method this structure

corresponds to an activation barrier of 0.28 eV, while in Hartree Fock (HF) the

barrier is 0.67 eV.

Discussion of reaction mechanism at hyper-thermal collision energies is not as

straight forward as at lower, thermal, collision energies. At thermal collisions one

expect the low energy path, sometimes referred as the reaction path, to dominate

and a single mechanism can be -ii-:.- -I. 1 for a reaction. Transition state theory

can be used to rationalize changes in transition state energy and geometry with

changes in the reaction rate. For hyper-thermal energies there is no unique path,

as reactive trajectories explore a much grater region of the potential. Extrema

structures however (transition states (TS) and minima) can still help to interpret

trajectories according to limiting cases.

Additional complexity is added at hyper thermal collision energies due to the

possibility of electronic excitations during or after the reactive collision process.

Electronic excitation can change not only the final energy partition but the out-

come of the reaction process itself. We don't expect electronic excitation related to

the charge transfer channel to pl iv an important role during the initial approach for

the NH++D2 reaction. Electronic excited states are usually important to describe

charge transfer processes and can influence reactive channels as well, however, in

our case the charge transfer channel (NH+ + D+) is very high in energy. Electronic

excitation is allowed in our simulation and often happens shortly after a reactive

encounter when the product molecule vibrates violently. As a consequence some









non negligible amount of energy can flow to electronic degrees of freedom. A more
detailed discussion about this topic is presented in the discussion section.







a b














c d


Figure 2.2: Minima structures for reactants products and complexes.

In Figure 2.2 we show the structure of some minima's for the Hartree Fock
potential surface. Structures a and b are for reactant and product while c and
d are minima structures along the abstraction reaction path. Figure 2.3 shows
transition states (TS) found for the Hartree Fock potential surface. The relative
energy differences between structure a in Figure 2.2 and structures in Figures 2.2
and 2.3 are given in Table 2.2 and 2.3. Edss is the the D2 dissociation energy.










COD


8
0D


0



i


Figure 2.3: Transition state structures along abstraction (f,g) and exchange (e, i
and h) reaction paths.


(W'










Among the TS structures, f and g lead to the abstraction product while e, i and

h lead to the exchange product.
Table 2.2: Energies differences relative to reactants (structure a) and structures
given in Figure 2.2 in eV. B1 is a N:6-311G*, H:6-311 basis; B2 is a DZP basis
with polarization only on N.


Basis EG/eV Eb/eV Ec/eV Ed/eV
B1 0 0.779 -0.696 0.065
B2 0 -0.768 -0.786 -0.039


Table 2.3: Energies differences relative to reactants (structure a in Figure 2.2) and
structures given in Figure 2.3 in eV. B1 is a N:6-311G*, H:6-311 basis; B2 is a DZP
basis with polarization only on N.


Basis E,/eV Ef/eV E,/eV Eh/eV Ei/eV Edis/eV
B1 2.881 -0.065 0.629 -0.680 2.613 3.553
B2 -0.039 0.954 -0.776 2.527 3.557




2.3.2 Collision Grid and Initial Conditions

Often during the dynamics of reactive trajectories we observe structures sim-

ilar to the ones pictures in Figures 2.2 and 2.3. Reactive trajectories at collision

energies used in this simulation however can explore a much more diverse range

of structures then the extrema ones pictured in Figures 2.2 and 2.3. Our initial

condition grid has to be general enough to explore the whole possibilities of re-

active trajectories at all collision energies as it is very hard to anticipate how the

dynamics will progress. For the experiment in question there is no orientation

preference on either reactants consequently an average over initial orientations is

necessary for comparison between theoretical and experimental results.

The grid of initial conditions used to compute the cross sections has 10 unique

orientations for the target NH+ and 3 unique orientations for the projectile ( one










along the C3 symmetry axis of NH+, and two other orientations perpendicular

to it). The total number of unique orientations considered is therefore 30. We

considered impact parameters ranging from 0 to a maximum around 3 a.u. (typi-

cally we only run 6 impact parameters per orientation) with steps of 0.4 a.u. The

total number of trajectories needed to generate a cross section is around 200. As

mentioned in the previous section we used a 6-311G* basis for the nitrogen and a

6-311G basis for the hydrogen and deuterium. Figure 2.5 shows the initial NH+

orientations while Figure 2.4 shows the initial D2 orientations used in our simula-

tions. The NH+ orientations are generated by rotations parametrized according

to Euler angles a, /3 and 7. An example of a collision configuration is pictured in

Figure 2.6.

Orientation on the D2 grid can be classified according to similarities with struc-

tures from Figures 2.2 and 2.3. Trajectories starting from orientation 1 (axial) can

reach TS g in Figure 2.3, with the external D atom leaving in the backward scat-

tering direction (in the laboratory frame). One can -v without much questioning

that structures similar to g are important for reactive processes having a rebound

mechanism. Trajectories starting from orientation 3 (eclipsed) form a molecular

complex with structure similar to g but with the D2 rotated 90 degrees and allow

for the external D atom to leave in the forward direction. Orientation 3 also allows

for initial configurations that have D2 and NH bonds being initially eclipsed and

for configurations similar to e, f and i. Trajectories starting from orientation 2

(-1 I.:: red) can explore regions of the configuration space similar to e, f and i as

well as some unusual configurations, e.g. with D2 and NH bonds being initially

-1 ,--.. red at 90 degrees.
























Top Top Top
1 2 3





-If





Side Side Side
1 2 3


Figure 2.4: Initial D2 orientations used in the cross section calculations.


















1-F

0.0.0 0.60.0 0.120.0






60.0.0 60.60.0 60.120.0






0.60.60 0.120.60 60.60.60



,m .

60.120.60 60.0.60 0.0.60


Figure 2.5: Initial NH+ orientations used in cross section calculation. The first
10 di-I ,1 i1 ,1 orientations are unique. Orientations are labeled by Euler angles a,
3 and 7.

























Figure 2.6: Example of collision configuration picturing the impact parameter b
and the laboratory scattering angle 0.
2.4 Results

For the problem at hand we are interested in the reactive cross section for

channels (I) and (III). As it turns out, channel (II) was not important for the en-

ergies considered in the simulation. As mentioned before, cross beam experiments

can provide flux dependent angular distributions. Experimental differential cross

sections can be measured and provide more detailed information about reaction

mechanism. GIB experiments can also provide some angular distribution infor-

mation by measuring the velocity vector along the propagation direction of the

octopole guide. If final velocities are also measured, cross beam raw experimental

data can be represented in terms of intensity contour maps or flux-:' /.. ..:,/ dis-

tributions [43]. Flux-velocity distributions can be used as a tool by both theory

and experiment as a probe for the mechanism in action as idealized mechanisms

provide distinctive profiles in flux-velocity plots. Collision complexes formation

produces flux-velocity plots that are symmetric w.r.t to forward/backward scat-

tering direction. A rebound mechanism produces a flux- ... .:,; plot having

pronounced intensity over angles 180 while a reaction following the spectator

stripping model produces mostly forward scattering with defined final momentum.










The particular procedure we use for calculating the total cross section is dis-

cussed in C'! lpter 1 Section 1.5.1. Results for cross sections at four different

energies are given in table 2.4.

From this point on experimental results referred in the text correspond to

results from Poutsma et al. [45] and Morrison et al. [44] unless otherwise stated.

Because the experimental flux was reported as a percentage of the total flux and

the flux is proportional to the cross section, Table 2.4 shows experimental results

normalized to the total theoretical cross section.

The primary fact one appreciates is that the same experimental trend is ob-

served on the theoretical results: abstraction reaches a maximum and then decay

with collision energy while exchange ahv--l- increase with collision energy. How-

ever, our results overestimate abstraction cross sections and underestimate ex-

change cross sections. The abstraction cross section maximum is reached at a

higher collision energy then in the experiment, 8 eV vs. the experimental value of

6 eV.
Table 2.4: Total cross sections for abstraction and exchange. Experimental values
are calculated as a percentage of the theoretical total reactive cross section.
2coll 2 aep 2 p 2
EcoMlev Uexch/la absta ch O /aa abst/ 0
6 0 5.38 0.62 4.86
8 0.14 6.46 2.42 4.15
12 1.57 4.34 na na
16 2.25 3.92 na na



Table 2.5 contains results for the total cross section divided according to the

different contributions arising from the D2 grid (Figure 2.4). There are numerous

important results one can read from the partial cross section results in Table 2.5.

So all the trajectories with orientation 1 (axial) averaged over all orientations of










NHS are labeled by superscript 1, and similarly for orientations 2 (-1 I.---, red) and

3 (eclipsed). The following is a list of important features obtained from Table 2.5:

* At 6 eV of collision energy there is no exchange and abstraction comes from

contributions for D2 orientations 1 and 3 only. Note that although orientation

1 leads to trajectories that can reach TS g in Figure 2.3, orientation 3 is the

one that contributes most for the total cross section. This is true for all energies

considered.

* Contributions from orientation 1 show an intriguing behavior for the abstraction

reaction being basically constant between 6 and 8 eV, dropping at 12 eV then

increasing again at 16 eV.

* Results for orientation 3 do not change much between 6 and 12 eV but drop

dramatically at 16 eV. Maximum contribution to abstraction cross section from

this orientation and maximum integral reactive cross section are reached at 8 eV.

This happens because different parts of the NH+ grid are reactive at different

collision energies.

* Direct exchange threshold is close to 8 eV of collision energy. Exchange integral

cross section picks up at 12 eV with the contributions coming from orientation

3 being the most important.
Table 2.5: Breakdown of the total cross sections for abstraction and exchange in
terms of D2 initial collision orientation. Labels refer to D2 orientations as pictured
in Figure 2.4.
1 3 1 2 3
EgoeM/eV xch/ao cexch /ao cch /ao abst. /a abst. lo abst. ao
6 0 0 0 1.68 0 3.71
8 0.094 0.0 0.049 1.710 0.716 4.010
12 0.516 0.288 0.765 0.132 0.716 3.497
16 0.449 0.721 1.075 1.724 0.232 1.967










2.4.1 Detailed Results for Collision at 6 eV

For this collision energy the spectator stripping model (SSM) predicts the ki-

netic energy loss into internal excitation to be 3.31 eV (see Table 2.1) for the

abstraction reaction (reaction (I)). When compared to the computed values from

the END simulations, shown in Figure 2.8-a and Table 2.6, the SSM value is a

good estimate for the average. The computed distribution is nevertheless broad

with values ranging from 0.2 to 5 eV. The broadening is expected as trajectories

that do not strictly follow the SSM have different momentum for outgoing D atom

and as a result different internal excitations.

Average values for the kinetic energy transfer contribution to the internal ex-

citation as well as kinetic contribution of SSM estimates are given in Table 2.6.

The SSM predictions for the kinetic energy contribution to the internal excitation

agrees well with the average value computed from ENDyne trajectories. As ex-

pected at higher energies the SSM estimate compares well with the value computed

from trajectories with D2 initially in the eclipsed orientation 3. Most important,

the agreement between the SSM and END computed values validates the analysis

of Morrison et al. [54] based on estimates for the internal excitation of NH3D+.

In particular this is so because he considered the SSM estimate to be the aver-

age internal excitation. SSM based predictions helped explain the lack of mode

specificity of the title reaction.

Again if we analyze our results in terms of contributions coming from different

parts of the D2 grid (Figure 2.4) a much more interesting picture emerges. Figure

2.8-a shows two peaks, one at higher energy coming from D2 orientation 1 pictured

in Figure 2.8-b and another one at lower energy coming from D2 orientation 3 pic-

tured in Figure 2.8-d. There is consequently a very strong dependence in product

internal excitation with the particular reactive collision configuration. ('I. ,,I Illy










Table 2.6: Average internal excitation from kinetic energy transfer, given in eV,
from simulation results. Labels 1, 2 and 3 refer to D2 orientations as pictured in
Figure 2.4. Also included are the results predicted by the SSM model in eV for
kinetic energy transfer.
~o~,lnrle l, ] rlk2 3 rin
EcOM/eV ave ve Eave Eave ESSM
6 2.67 4.03 4.23 1.77 3.31
8 3.92 5.56 4.32 2.89 4.42
12 5.77 8.78 7.62 4.82 6.63
16 7.44 7.16 9.63 7.52 8.84


this conveys that orientations 1 and 3 have different reaction mechanisms. Orien-

tation 1 exhibit a rebound type mechanism with a large amount of kinetic energy

loss to internal excitation while orientation 3 display a mechanism more according

to the SSM. Deviations from the SSM occur because the outgoing D and the newly

formed NH3D+ interact. As the D + NH3D+ potential is mostly repulsive the

SSM predicted value is an upper estimate for the energy loss to internal excitation

for the reaction. This generalization of the SSM mechanism is called the spectator

stripping recoil mechanism (SSR).
Table 2.7: Average scattering angle (Ocom.), given in Radians, from simulation
results. Labels 1, 2 and 3 refer to D2 orientations as pictured in Figure 2.4

E5M/eV Oce 0/ve 0ce 03g
UCOM/C Oave ae ave a ave
6 1.94 2.58 1.76 1.53
8 1.94 2.55 2.27 1.50
12 1.43 1.87 1.84 1.25
16 1.15 1.38 1.5 0.82


A chief attribute from Figure 2.8 is that all reactive trajectories from orientation

1 have internal energy above the dissociation threshold for reaction (IV). Hence,

theory supports that formation of the abstraction product, reaction (III), at this

collision energy can be due to the opening of a new channel given by (V). The

predicted threshold value by the SSM model in Section 2.2 is 8 eV COM. Actual

simulation results place this value somewhat below 6 eV as trajectories that can










undergo dissociation have a rebound type mechanism resulting in a grater then

SSM kinetic energy loss to internal excitation. The experimental threshold for the

exchange reaction seems to be around 4.5 eV of collision energy COM.

As shown in Table 2.4, there is no direct formation of the exchange product

NH2+ at this collision energy and orientation 2 only leads to one reactive trajec-

tory.

Figure 2.7 shows the flux ,. ... :, '/distribution as a contour map of trajectories

count. The polar variables are the final NHDD+ kinetic energy and the scattering

angle in the COM frame. Here again, the distinct differences between trajectories

starting with orientation 1 and 3 can be seen. Figure 2.7-b shows slow backward

scattered products while Figure 2.7-d shows products scattered at a higher velocity

and with scattering angle peaked around 900. A histogram of the C'\ scattering

angle is pictured in Figure 2.9 and also di-~ 'i' a two peak profile. Orientation

1 produces a larger, closer to 1800 degrees scattering angle, while orientation 3

produces a peak around 900 degrees.



2.4.2 Detailed Results for Collision at 8 eV

Increasing the collision energy to 8 eV COM results in a few changes on internal

energy excitation histograms and flux velocity plots. For D2 orientation 1, Figures

2.10-b, 2.11-b as well as Tables 2.6 and 2.7 indicate that an increase in collision

energy resulted in an increase almost solely of internal excitation. D2 orientation 3

shows an increase in both internal excitation and exit product velocity. The signif-

icant result at 8 eV is the appearance of contributions coming from D2 orientation

2.

As one might expect, a much larger fraction of trajectories contains internal

excitation above the threshold limit. At 8 eV of collision energy a substantial




























06


190


150


05 1 15


0 180


90
06
120 60




1 50/ \ 30






180


Ekn/ eV.


2 25 3 35 4 45 05 1 15


Ekin/eV


2 25 3 35 4 45




b


06
120 60




150 \30






180

Ekin/eV.


06
120 60




150 / 30






0 180 0

Ekin/eV


01 02 03 04 05 06 07 08 09 05 1


Figure 2.7: Flux velocity contour map for the abstraction product using a collision

energy in the COM of 6 eV. We have clockwise: a) total flux; b) a partial of the grid

with D2 orientation label 1, c) same with orientation 2; d) same with orientation

3. The radial dimension is the final kinetic energy in the COM (eV) and colors

label the histogram count.


15 2 25 3 35


05 1


































1 2 3
Eint /eV.

a


















1 2 3
Eint/eV.

C


4 5 600


4 5 600


1 2 3
Eint /eV.


1 2 3
Eint /eV.


4 5 6


4 5 6


Figure 2.8: Histograms for internal energy transferred during the collision for the
abstraction product using a collision energy in the COM of 6 eV. We have clockwise:
a) total histogram; b) a partial of the grid with D2 orientation label 1, c) same
with orientation 2; d) same with orientation 3.




































0.5 1 1.5
0/ Rad.

a


0.5 1 1.5
0 / Rad.


I2 25 0
2 2.5 3 0


2 2.5 3 0


0.5 1 1.5
S/ Rad.


0.5 1 1.5
S/ Rad.


2 2.5 3


2 2.5 3


Figure 2.9: Angular distribution histograms for the abstraction product using a
collision energy in the COM of 6 eV. We have clockwise: a) total histogram; b) a
partial of the grid with D2 orientation label 1, c) same with orientation 2; d) same
with orientation 3.


0.8










fraction of trajectories from orientation set 2 and 3 have internal excitation above

threshold for the new channel (V). Orientations 1 and 3 are clearly responsible

for the two peak profile in Figure 2.11-a and 2.15-a. As in the 6 eV collision,

trajectories with initial D2 orientation 1 produces more internal excitation due to

the rebound mechanism then trajectories with initial D2 orientation 3. Trajectories

with initial D2 orientation 2 produces mostly backward scattered products with

internal excitation in between orientations 1 and 3.

Mostly important at 8 eV is that the cross section for a direct exchange reaction,

not due to a two step process terminated by reaction (V), is non zero as shown

by Table 2.4. A direct exchange reaction mechanism is usually characterized by a

billard ball type collision in which a D atom ejects an H atom. A two step exchange

mechanism is unlikely to be observed in our dynamics as the propagation time in

our simulations is short, with propagation times ranging from 80 to 200 ps, and

not long enough to allow for the dissociation of metastable NH3D+ molecules.

Scattering angle histograms continue to have a multi peak profile, with tra-

jectories with D2 initially at orientation 1 and 2 producing mostly back-scattered

products while 3 produces a histogram peaked at less then 900 degrees. As in the

6 eV case, D2 orientation 1 produce reactive trajectories trough a rebound mech-

anism while D2 orientation 3 produce reactive trajectories more according to the

SSM. The predicted value by the SSM for kinetic energy loss to internal excitation

overestimates, as in the 6 eV case, the computed number for most trajectories

having initial D2 given by orientation 3.



2.4.3 Detailed Results for Collision at 12 eV

At 12 eV of collision energy the picture is somewhat different then at 6 eV and

8 eV. The direct exchange cross section is large and D2 orientation 1 produces


























90
08
120 1 60


S-.1 "


E n/eV.


1 2 3 4 5 6 7 8 9 10 11




a



90
08













180

E kn/eV.


0 180

E kn/eV.


1 2 3 4 5 6 7 8 9 10 11


08
120 60




150 E 30






0 180 -

E kineV.


15 2 25 3 35 05 1 15 2 25 3 35 4 45


Figure 2.10: Flux velocity contour map for the abstraction product using a collision

energy in the COM of 8 eV. We have clockwise: a) total flux; b) a partial of the grid

with D2 orientation label 1, c) same with orientation 2; d) same with orientation

3. The radial dimension is the final kinetic energy in the COM (eV) and colors

label the histogram count.


120


150


05 1









69









30 25

25 20

202

15
15
10
10

5 5 E


0 2 4 6 80 2 4 6 8
Eint/ eV. Eint/ eV.

a b



8 20

7-

6 15

5

4 10

3-

2- 5-

1- --1

0 2 4 6 8 0 2 4 6 8
Ent / eV. Eint/ eV.

c d



Figure 2.11: Histograms for internal energy transferred during the collision for
the abstraction product using a collision energy in the COM of 8 eV. We have
clockwise: a) total histogram; b) a partial of the grid with D2 orientation label 1,
c) same with orientation 2; d) same with orientation 3.



































0.5 1 1.5
I/Rad.


0.5 1 1.5
I/Rad.

C


2 2.5 3 0


2 2.5 3 0


0.5 1 1.5
I/Rad.


0.5 1 1.5
I/Rad.


2 2.5 3


2 2.5 3


Figure 2.12: Angular distribution histograms for the abstraction product using a
collision energy in the COM of 8 eV. We have clockwise: a) total histogram; b) a
partial of the grid with D2 orientation label 1, c) same with orientation 2; d) same
3.










very little abstraction product. Given the increase in internal excitation when

going from 6 eV to 8 eV, trajectories with D2 initial orientation given by 1 at

12 eV of collision energy seem to be unable even to form metastable NH3D+ in

large numbers. A few reactive trajectories form a highly excited NH3D+ product

(Figure 2.14-b).

As in the previous lower energy cases, the SSM prediction for kinetic energy

loss to internal excitation for trajectories with D2 initially at orientations 2 and

3 overestimates for most trajectories the computed values produced by the dy-

namics (see Figures 2.14-c and 2.14-d and Table 2.6). However, the theoretical

computed distribution of internal excitation at 12 eV has several trajectories with

internal excitation around and above the SSM prediction of 6.6 eV, specially from

trajectories with D2 initially at orientation 2 as pictured in Figure 2.14-c.

Figures 2.15-a trough 2.15-d show that trajectories with D2 initial orientation

1 show mostly backward scattering, consistent with a rebound mechanism, while

orientation 3 shows a forward scattering peaked around 600 degrees. Trajectories

with D2 at initial orientation 2 shift to smaller values when compared to trajectories

at 8 eV of collision energy and peak around 900 degrees. All angular information

in addition to exit velocities is conveyed in the flux velocity plots (Figure 2.13).

There it is clear to notice that an increase in 4eV of collision energy resulted in

very little increase in the final exit velocity for the outgoing NH3D+. Most excess

energy went into internal excitation and D exit velocity. Because trajectories with

D2 initially at orientation 1 don't form the abstraction product in great numbers,

the total flux is mostly forward scattered in the COM as shown in Figure 2.13-a.

In Figures 2.16 and 2.17 we present typical reactive trajectories for abstraction

and exchange respectively at 12 eV of collision energy. The abstraction trajectory

shown is very similar to other reactive trajectories at lower energies, while direct






























120


150


,a
r. i~l -


i..


E kn/eV.


05 1 15 2 25 3 35 4 45 5 55




a



90

120 60

08



150 / 30



02





E eV -
Ekin/eV.


30


90
1
120 60

08


060
150 30







0 180 0 I o

E k/eV.


01 02 03 04 05 06 07 08 09





b



90


120


Jaw,


30







- 0


0 180


E kn/eV.


25 3 35 4 45 05 1 15 2 25


35 4 45 5 55


Figure 2.13: Flux velocity contour map for the abstraction product using a collision

energy in the COM of 12 eV. We have clockwise: a) total flux; b) a partial of

the grid with D2 orientation label 1, c) same with orientation 2; d) same with

orientation 3. The radial dimension is the final kinetic energy in the COM (eV)

and colors label the histogram count.


180


180


05 1 15




































2 4 6
Eint/ eV.

a


2 4 6
Eint/eV.

c


8 10 12 0
8 10 12 0


OL
8 10 12 0


2 4 6
Eint/eV.


2 4 6
Ent/ eV.


8 10 12


8 10 12


Figure 2.14: Histograms for internal energy transferred during the collision for
the abstraction product using a collision energy in the COM of 12 eV. We have
clockwise: a) total histogram; b) a partial of the grid with D2 orientation label 1,
c) same with orientation 2; d) same with orientation 3.



































0.5 1 1.5
S/ Rad.

a


















0.5 1 1.5
S/Rad.


2 2.5 3 0


2 25 3 0


0.5 1 1.5
S/ Rad.


0.5 1 1.5
S/Rad.


2 2.5 3


2 2.5 3


Figure 2.15: Angular distribution histograms for the abstraction product using a
collision energy in the COM of gram 12 eV. We have clockwise: a) total histogram;
b) a partial of the grid with D2 orientation label 1, c) same with orientation 2; d)
same with orientation 3.










abstraction reactions are only common at 12 eV and 16 eV. Mulliken population

plot for the abstraction trajectory (Figure 2.16) shows that after the reactive en-

counter and breakage of the D2 bond, the system is so excited that during the first

oscillation of the newly formed N-D bond a species with a biradical character is

formed. However, shortly after 1500 a.u. of time the energy is redistributed over

other vibrational and electronic modes leading to the formation of a metastable

NH3D+ species. For the exchange channel the Mulliken population plot shows an

adiabatic process corresponding to direct exchange reaction.


2.4.4 Detailed Results for Collision at 16 eV

At the highest collision energy investigated in this work, 16 eV, there is enough

energy that the dynamics can explore a new set of reactive NH+ orientations

having initial axial D2 configuration (1). As opposed to what happens at 6 eV

and 8 eV, the axial orientation at 16 eV does not lead to a rebound mechanism

(see Figures 2.20 table 2.7 for the average scattering angle). Most important, at

this energy the ,.-- r'ed (3) orientation that at lower energies contributes most

to the abstraction cross section is substantially reduced. Several trajectories form

NH3D+ for a very short time but the excess of internal excitation does not allow

for product formation. This shows that saturation of the abstraction channel for

the direct process is occurring at the 16 eV collision energy.

At 16 eV of collision energy the SSM prediction also overestimate the computed

values for kinetic energy loss. However, when compared to the prior estimates at

lower collision energies, the computed value by the SSM is not far off. Average

values for the kinetic energy loss and scattering angle shown in Tables 2.6 and

2.7 indicate that at this energy the SSM is a good approximation for the reaction










mechanism for all orientations. Figures 2.18 and 2.20 show that albeit having a

broad angular distribution, all the flux is forward scattered for all orientations.


2.5 Discussion

Structures similar to c and g from Figure 2.3 occur on abstraction trajectories

at 6 and 8 eV COM collision energy that have a rebound type mechanism. They

are generated when starting with D2 initially at orientation 1 (axial) as pictured

in Figure 2.4 and form abstraction products with a large amount of internal exci-

tation. Structure i from Figure 2.3 occurs in direct exchange trajectories at 8 and

12 and 16 eV of collision energy COM. Most of exchange trajectories have a billard

ball type mechanism, somewhat more constrained but similar to structure i, where

the incoming D2 molecule hits a H atom head on followed by the breakage of the

D2 bond, the ejection of a H atom and capture of a D atom by the nitrogen. The

configurations that contributes most to the abstraction cross section at all energies

are modifications of structure g where the D2 molecule is parallel or at a small

angle to the original NHS plane. These configurations are generated by starting

with D2 orientation 3 (eclipsed) as pictured in Figure 2.4. On these configurations

the external D atom escapes in the forward lab direction and the mechanism is

analogous to the SSM. Although the mechanism is SSM alike, SSM usually over

estimates the amount of internal excitation on the abstraction product molecule

NH3D+. We do not observe structure e in exchange trajectories as it's config-

uration implies a rebound type mechanism that probably would occur at lower

collision energies and high internal excitation.

The temporal scale used to generate experimental results considered in this

work are much larger then the reaction times and much larger then the simulation

times considered on the dynamics simulation. Time of flight in octopole guides for


























90














180









a



90
1 5
120 60




150 30





















1809
















Ekin/eV.


1 5
120 60




150 1 .30






0 180 0

E kn/eV.


05 1 15 2 25 3 35


1 5
120 1 60




150 /, '- 30






0 180 0

Ekin/ eV.


01 02 03 04 05 06 07 08 09 05 1 15 2 25


Figure 2.18: Flux velocity contour map for the abstraction product using a collision

energy in the COM of 16 eV. We have clockwise: a) total flux; b) a partial of

the grid with D2 orientation label 1, c) same with orientation 2; d) same with

orientation 3. The radial dimension is the final kinetic energy in the COM (eV)

and colors label the histogram count.


35 4 45 5 55









80









25 12


20 10

8

6-
10
4-

5-- 2-


0 5 10 15 0 5 10 15
Eint eV. E ntleV.

a b



2- -12-

10
1.5
8

1- -- 6-

4
0.5
2-

0 0
0 5 10 15 0 5 10 15
EntleV. EntleV.

C d



Figure 2.19: Histograms for internal energy transferred during the collision for
the abstraction product using a collision energy in the COM of 16 eV. We have
clockwise: a) total histogram; b) a partial of the grid with D2 orientation label 1,
c) same with orientation 2; d) same with orientation 3.



































0.5 1 1.5
S/Rad.

a


















0.5 1 1.5
S/ Rad.


2 2.5 3 0


2 2.5 3 0


0.5 1 1.5
S/Rad.


0.5 1 1.5
S/ Rad.


2 2.5 3


2 2.5 3


Figure 2.20: Angular distribution histograms for the abstraction product using a
collision energy in the COM of gram 16 eV. We have clockwise: a) total histogram;
b) a partial of the grid with D2 orientation label 1, c) same with orientation 2; d)
same with orientation 3.










the experiment considered here is around 50 ps from the reaction chamber to the

detector while simulation times range from 80 to 200 fs. Owing to the formation

of metastable abstraction products and the long time for product detection, the

raw numbers for reaction cross sections computed from our simulations might not

be the most accurate ones to to compare with experimental results. A solution

could be to just extend the simulation to longer times to estimate the fraction of

metastable NH3D+ products that undergo reaction (V). Albeit a good idea, this

would be an expensive computational procedure as we don't know for how long we

would have to propagate the simulation to obtain a good estimate for the fraction

of metastable products that dissociate into NH2D+. Trial simulations on highly

exited products reveal that 200 fs was not enough to reach a definitive conclusion.

In order to estimate the possible measured results from the END calculations

we compute three different relative cross section estimates by assuming that 50'.

7 .' or 101 1' of the metastable NH3D+ products actually dissociate by losing an

H or a D atom with equal probability before reaching the detector.

The results of this exercise are shown in Table 2.8 for the assumption that 7".'

of the abstraction products actually dissociate. We also show in Figure 2.21 the

experimental results [55] for collision energies between 6 and 10 eV compared with

our estimates from the END simulations. The theoretical "error -,i represent

the 50'- and 101'' breakups and the line is drawn for the 7 .' assumption. It

is interesting to note that at the lower collision energies the experimental results

are closer to the 50' dissociation rate, while they are closer to the 1CWl', rate

for the higher collision energies. Our estimate at 10 eV of collision energy are

interpolated results, since no END calculation were run for that energy. The results

shown in Table 2.8 have a remarkable agreement with experimental data from

[55]. Values for relative cross section for the exchange and abstraction channels










as well as trends are reproduced. Not shown in Table 2.8 is the fact that, when

corrected for dissociation, the abstraction cross section maxima for the energy

interval considered occurs at 6 eV and not at 8 eV as shown in Table 2.4. This

results agrees with experiments performed by Morrison et al. [54]. In order to

compute the internal energy, we use the values for kinetic energy loss to internal

excitation given in Figures 2.8, 2.11 and 2.14 and assume, as in the SSM, that

all AHr,, energy goes for internal excitation of NH3D+. Using the dissociation

threshold given in Table 2.3 the estimate for the reactive cross section considering

dissociation are shown in Table 2.8.
Table 2.8: Total cross sections for abstraction and exchange considering dissocia-
tion of metastable NH3D+ products. Theoretical values at 10 eV are interpolated.

EOM/eV Uexch/alo abst./l Qexch/O h abst / a
6 14 86 11 88
8 31 69 37 63
10 49 51 63 37
12 69 31 na na
16 79 21 na na





2.6 Conclusions

Minimal END theory with a single determinantal wavefunction for the electrons

and classical nuclei is capable of capturing the essential chemistry of the reactions

(I) through (III). Our results strongly -, -.I: -1 that the exchange product formation

is mostly due to a two step process given by channel (III). Contributions of a

direct process become appreciable at 10 and 12 eV of collision energy COM. By

making the assumption of dissociation of a substantial fraction of the metastable

abstraction products (NH3D+) the calculated relative cross sections agree well

with the experimental results [55]. This assumption is supported by the behavior











100
100 Exc. exp
S-*--Abs. exp



60-

40 -

20 -

0
5 7 9 11 13
Energy in eV (COM)


Figure 2.21: Experimental and theoretical relative cross sections. Theoretical val-
ues and error bars are obtained from dissociation probability estimates.
of sample trajectories and by the saturation of the abstraction channel seen at

16 eV of collision energy. Our predictions of the collision energy threshold for

the exchange reaction being lower then 6 eV and the maximum of the adjusted

abstraction cross section occurring at 6 eV for the corrected results also agrees

with the measured results.

We believe that for the energies used in the present study the inclusion of all

degrees of freedom is essential for the correct description of the chemical process.

Fast internal energy redistribution occurring on nascent NH3D+ products can en-

hance the life span of metastable molecules by redistributing vibrational excitation

among all modes. After the initial energy redistribution, dissociation can only hap-

pen in the relatively rare event of energy getting refocused in a particular stretch

mode.

We conclude that for the energies considered the principal process for the ab-

straction product formation (NH3D+) does not proceed via a rebound mechanism.

However, such a mechanism is very important for the exchange process at 6 and 8










eV via the reaction (III). It is also clear that the SSM estimates for internal exci-

tation are inadequate for this reaction since two very distinct mechanisms are in

operation. Interestingly the predicted SSM value turns out to be a good estimate

for the averaged internal excitation, but overestimates the calculated values for

END trajectories that have a mechanism more akin to that of the SSM assump-

tions, for instance reactive trajectories starting from orientations 2 and 3 in Figure

2.4.

Although we did not explore the effects of internal excitation on the reactivity

we find, due to the two step mechanism present with the formation of a long-lived

intermediate, that the title reaction has very little mode specificity at a hyper-

thermal collision regime in agreement with experiment.














CHAPTER 3
H+HOD COLLISION AT 1.575 EV.

3.1 Introduction:

In work done by Bronikowski and coworkers [56, 57, 58] the cross section ratio

(CS ratio), or branching ratio, for reactions (I) and (II) was measured for HOD

prepared in its ground molecular state.

H + HOD H2+ OD (I)


H + HOD HD+OH (II)

The experiment also measures the reaction where HOD was prepared with one

quanta of vibrational excitation in the local OD and OH stretching modes. Exper-

imental results show that enhancement due to the initial vibration was noticeable

and that selective chemistry could be achieved for this 4 atom system. This was

one of the first studies in which one could successfully control the selectivity of

a reaction by exciting one of the reactants. The reaction of H + HOD was also

experimentally studied by Sinha [59] and by Metz [60] where HOD was prepared

initially in higher vibrational states.

H + HOD D(H) + H20(HOD) (III)


H + H20 H2 + OH (IV)

The values measured by Bronikowski and coworkers indicated an enhancement

greater then 20 for TOD and greater then 5 for cOH when comparing to the cross

sections for HOD in its ground molecular state. The reaction enhancement is

calculated as the ratio between the cross section for OH and OD formation starting

with HOD in its ground state and the cross section starting with HOD in its










excited vibrational state. A branching ration of 1.38 0.14 is associated with the

ratio of the cross sections "OD for HOD in its ground state shows that an isotopic
TOH
effect exists even when no excitation is present in the nuclear motion. No absolute

cross section was obtained in Bronikowski's work. However, comparison with the

work of Kesseler and Kleinermanns [61], which measured the absolute cross section

of the reaction (IV) at the same collision energy made possible the estimation of

the total cross section. The estimates were obtained for reactions (I) (II) by

computing the ratio between the Laser induced Fluorescence (LIF) signal of the

species with known total reactive cross section, aoH from reaction (IV), and the

LIF signal of species with unknown total cross section.

Most, if not all, of the theoretical work done on this reaction use a common

potential surface, sometimes with modifications, prepared by Elgersma and Schatz

[62] over the ab-initio data points of Walch and Dunning [63] (the WDSE poten-

tial). There has been calculations using the quasiclassical trajectories method by

Schatz and coworkers [62, 64], full quantum (with and without constrained nuclear

degrees of freedom) by Zhang and Light [65] and by Wang and Bowman [66]. The

and close-coupling equations for the nuclear motion were solved by Clary [67]. The

reader may at this point ask what is different between the prior approaches and

ours. First, our method does not use a precalculated potential energy surface. This

has the advantage that one does not need to know the whole potential surface in

advance. Second, we allow for excited electronic states to participate in the dy-

namics. Although the first excited potential surface of HOD is far apart in energy

from the ground state one. This energy splitting is not known for all possible

geometries of the collision complex H + HOD. Also, our experience have shown

that one can perceive small cumulative effects on the differential cross section [68]

due to the influence of excited states on the dynamics. It is still uncertain if those










effects are important for the calculation of the total cross section of the considered

reaction. It is nevertheless clear that if the dynamics stay close to the ground state

potential the surface then minimal END is not of the same quality as dynamics on

a surface obtained from high quality ab-initio methods.

We also find it valuable to verify if conclusions reached in previous work would

hold for our study considering that we have a different description of the collision

process with the END theory. We realize that our description of the electronic

degrees of freedom is approximate, but the surface used in most calculations, the

WDSE surface, is also inexact. For example, for a collision on the WDSE surface

only one of the bonds is allowed to break due to the .,-v lii i li: nature of the

potential w.r.t. the activation barrier for abstraction. While experimental results

indicate that the nonreactive bond acts as a spectator bond during the reaction

and theoretical results do agree with that observation, the difference in activation

barrier between the two bonds must not be crucial for the simulations. As we

have a symmetric potential, there is no need to introduce ad-oc constraints. The

reactive bond is whatever one the dynamics choses.

Some experimental considerations need also be keep in mind when comparing

the experimental results with results from theoretical calculations. In order to

compute the products absolute concentration the experiment assumes that the

LIF signal used to detect the products is near its saturation limit of 50 This

means that the published enhancements are in fact a lower bound to the true

enhancement. On the other hand the source of hydrogens for the experiment

produces two atoms, one with 1.575 eV and another one with 0.6 eV of translational

energy. The last one does not have enough energy to react with unexcited HOD

but does have enough energy to react with the excited HOD molecule even in

its lowest vibrational states. As the hydrogens are produced from the photolysis










at a ratio of 1:1.8, the one with 1.575 eV being the most common, the effect on

the enhancement calculation might be non-negligible and the enhancement due

to vibrational excitation can from that point of view be overestimated. Those

considerations reveal the difficult nature of the experimental study undertaken by

Bronikowski and coworkers.



3.2 Computational Details

The basis function used throughout the calculation was a STO-3G (1s2s2p)

for the Oxygen and 4-31G (ls2s) for the hydrogens and deuterium. Although

small, this basis produces the correct behavior necessary for the description of the

reaction, i.e., the barrier for the H + HOD reaction is for some known reactive

orientations close to the published value of 0.9 eV, i.e., smaller then the hydrogen

translational energy of 1.575 eV. In an exploratory study we consider the initial

orientation of the H momentum vector to lie along the OD bond axis and with

the HOD molecule in the plane that contains the OH bond and the projectile

momentum vector, we refer, from now on, to this as the original collision plane.

Calculations showed that using this orientation and the O:STO-3G H:4-31G basis

the system undergoes a hydrogen abstraction reaction. This result is in agreement

with others that point to this particular orientation as important in explaining the

overall reactivity of the H + HOD reaction. An attempt to improve the basis used

in the first part of this work is developed in Section 3.4.

The calculation of the reaction cross section for the vibrationally excited HOD

requires new considerations not present in the calculation of the ground state

cross section. Due to the importance of the timing between the bond vibration

and the impact time, we need to introduce another average distinct from the ones

considered before. Specifically, we need to average over all possible conditions









generated when different initial phases are considered for the vibrational motion.
The integration of the phase angle is equivalent to integrating over different initial
internuclear separations [69, 70] with the limits of the integral given by the period
of vibration. In practice this is done by considering different impact distances
in a range given by the travelled distance of the projectile during a vibrational
period, i.e., we calculate the cross section as defined above considering different
initial impact distances and the same vibrational phase and average the result to
produce the final cross section. In our calculations we consider four distinct initial
distances for each cross section calculation of a vibrationally excited reactant. The
integral for the total cross section presented in chapter 1 becomes
0o
7k(ELab) 27 j bPk(b; ELab)db (3.1)

with an opacity function Pk(b, ELab) given by:
1 f2i f2i fV rT
Pk(b, ELab) 8 2 pk(a
where we have chosen to represent the phase integral over a vibrational period r.
A visual representation of the mentioned procedure is shown in Figure 3.1

X


b




Range of 180.128.0
initial distances
Y
Figure 3.1: Pictorial representation of the phase average procedure used to calcu-
late cross sections for initially excited HOD reactant.










A consideration absent in the 1i ii liy of the work done concerning this re-

action is the study of the role of excited states on the dynamics of the hydrogen

abstraction. It has been shown that the first dissociative state of H20 (A2 1B1) can

be accessed with 266nm (4.66 eV) lasers and even 2;4;:i"1 (4.30 eV) lasers provided

the system is far enough from equilibrium (see Vander [71] and also Plusquellic

[72] and references therein). For the ground vibrational state of H20 the absorp-

tion band starts around 180nm (6.88 eV). Approximate calculations using CIS

and our small basis placed the first excitation energy at 7.79 eV. However, the

quantity of interest here, at least for interpretation purposes, is really the lowest

excitation energy for the collision complex. Spectroscopic results for collision com-

plexes are hard to find and the present system is no exception. We estimated the

first excitation energy, using CIS (configuration interaction with single excitations

[73]), for the collision complex formed during the reaction starting with the ini-
tial orientation labeled by Euler angles 0.128.0 and impact parameter 0.2 to be

only 1.17 eV For the mentioned CIS calculations we used the same basis function

used throughout this work. Calculation using a larger basis for all atoms, namely

PVDZ, produced a first excitation energy of 2.61 eV for the same geometry of the

collision complex. In a less dramatic example, the collision complex formed during

the reaction that started with the initial orientation labeled by the Euler angles by

0.90.0 and impact parameter 1.0 had the first excitation energy at 2.62 eV for CIS

and our small basis and 4.50 eV when using CIS and a PVDZ basis. The study of

the nature of the excitation also revealed that there is only one configuration that

has major importance for the description of that particular excited state. This is

important, as it makes our method quite appropriate for describing the dynamics

that involves the ground and this particular first excited state.










3.3 Results

We are using throughout this work the notation [c, s3,7], where a, /3,7 are

Euler angles, to label the different initial target orientations of the collision. In

this notation [0, 0, 0] is the original orientation pictured in Figure 3.2. Vibrational

excitations of the HOD molecule are identified by tree numbers, the first one refers

to the excitation of the mode with character of OD stretching, the second one refers

to excitations of the bending mode, and the third one refers to excitation with the

character of the OH stretching mode.

The O:STO-3G H:4-31G basis has a total of 22 functions and corresponds

to 60 electronic parameters in the Thouless determinant. The grid used in the

cross section calculations consists of 120 different orientations generated by Euler

angle rotations. We used the y-counter-clockwise convention with the z axis lying

between the H and D atoms in the molecule plane and the y axis perpendicular to

z and to the HOD plane. For impact parameters we used an interval of 0.2 a.u.

ranging from 0.0 to 4.0 a.u. along the x axis. The impact parameter was measured

with respect to the center of mass of the target.

As mentioned earlier our method does not uses potential energy surfaces. Al-

though we consider this an advantage of the method there is no simple picture

that can be extracted from our mathematical description of the dynamical state.

To help to visualize the results obtained for the cross section we focus on key ori-

entations leading to reactive trajectories and contribute substantially to the total

cross section calculation. We believe that the direct interpretation of the results

produced by the dynamics, bypassing the step of interpreting potential energy sur-

face features, may be more meaningful especially when excited states participate

in the dynamics.