Interaction Potentials for Ch4 on Benzene - Copy
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Abstract
A+$ #-+% +/ #,%#-- # $ "-+, *!#-%* ,',!$'6 !/"!,+#-- +' $#' #$!#/ T*! /!
,-!#' !-/ $ !*+,-#$ #""-+,#%+'/ +/ %*/ /%$'.- !',$#.! N#%$#- .#/6 *+,* "$+$+-,'/+/%/ &!%*#'!6 +/ # #-#-! #-%!$'#%+! !- +%* $ ##'%#.!/ #/ ,&"#$! +%* .#/-+'!6
'#&!-: *.! $- $!/!$!/6 - ,/%6 ,-!#' $'+'. ,*#$#,%!$+/%+,/ (- !&+//+' -!!-/ #' -
.$!!'*/! +&"#,%) #' .$!#%!/% !'!$. "!$ '+% // #-- %*! %*!$ *$,#$'/ H!!$6 /+',!
/"!$,$+%+,#- +' STP ,'+%+'/6 &!%*#'! #-#/ *#/ # !$ - !'/+% S%$+'. %*! .$!#%!/%
#&'%/ &!%*#'! +' # .+!' -+&+%! -&! +/ %*/ # $!#- ,*#--!'.! $ +%/ #""-+,#%+' % %* .#/
%$#'/"$%#%+' #' .#/ "!$! !*+,-!/ A&'. %*! #$+/ /%$#.! &!%*/ /!6 #/$"%+' /+'.
#""$"$+#%! %!$+#-/ +/ # %!,*'-.+,#-- +&"$%#'% &!%* A/$! '#%$#- .#/ (ANG) *#/ %*!
"%!'%+#- % !,&! %*! &/% ,/%;!!,%+! &!%* ,&"#$! % %*!$ &!%*/ /,* #/ -+!+!
'#%$#- .#/ (LNG) #' ,&"$!//! '#%$#- .#/ (CNG)
I' %*! $!,!'% !#$/ %*!$! *#/ !!' # /%$'. +'%!$!/% +' !!-"+'. %!$+#-/ $ /%$#.!
'#%$#- .#/ (&!%*#'!) A *+-! #.6 %*! A%-#'%# G#/ L+.*% A/$!'% R!/!#$,* G$" /..!/%! %*!%#$.!% 150 -&!/ .#/ (&!#/$! +' STP ,'+%+'/) "!$ -&! /%$#.! !//!- (
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%!&"!$#%$! H!!$6 +% +/ +&"$%#'% % ,'/+!$ %*#% *$.!' +/ ! ,#- .#/++,#%+'
$ #%!$ !-!,%$-/+/6 #' %*!/! "$,!//!/ .!'!$#%! #+$ "--%#'%/ +$!,%- $ +'+$!,%-
Natura# +as. N#%$#- .#/ (NG)6 *+,* "$+$+- ,'/+/%/ &!%*#'!6 +/ # #-#-!
#-%!$'#%+! !- +%* $ ##'%#.!/ #/ ,&"#$! +%* .#/-+'!6 '#&!-: *.! $-
$!/!$!/6 - ,/%6 ,-!#' $'+'. ,*#$#,%!$+/%+,/ (- !&+//+' -!!-/ #' - .$!!'*/!
+&"#,%) #' .$!#%!/% !'!$. "!$ '+% // #-- %*! %*!$ *$,#$'/ A '#%$#- .#/
!*+,-! (NG
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Fi+ure 4: T*! C-& "%!'%+#- T*! %%& ,$! $!"$!/!'%/ %*! +'%!$#,%+' !%!!' %
"#$%+,-!/ ""/+%! ,*#$.! %*! %" ,$! $!"$!/!'%/ %*! +'%!$#,%+' "#$%+,-!/ !#- ,*#$.!
1)> Rotation !atrices
I' -+'!#$ #-.!$#6 # $%#%+' %$+ +/ # %$+%*#% +/ /! % "!$$& #$%#%+'+' E,-+!#' /"#,!
$ !#&"-! %*! %$+
$%#%!/ "+'%/ +' %*!xy;C#$%!/+#' "-#'!,'%!$,-,+/! %*$.* #' #'.-! #% %*! $+.+'
%*!C#$%!/+#' ,$+'#%! //%!& T "!$$& %*! $%#%+' /+'. # $%#%+' %$+!6 %*! "/+%+'
!#,* "+'% &/% ! $!"$!/!'%! # ,-&' !,%$6 ,'%#+'+'. %*! ,$+'#%!/ %*! "+'% A
$%#%! !,%$ +/ %#+'! /+'. %*! %$+ &-%+"-+,#%+'! S+',! %$+ &-%+"-+,#%+' *#/ '
!!,% ' %*! ?!$ !,%$ (+!6 ' %*! ,$+'#%!/ %*! $+.+')6 $%#%+' %$+,!/ ,#' '- ! /! %
!/,$+! $%#%+'/ #% %*! $+.+' %*! ,$+'#%! //%!&
T*! --+'. %*$!! #/+, $%#%+' %$+,!/ $%#%! !,%$/ #% %*!x6y6 $"#+/6 +' %*$!!+&!'/+'/:
12
http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttp://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttp://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttp://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttp://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Matrix_multiplication -
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T*!/! $%#%+' %$+,!/ *#/ !!' /! +' %*+/ /% $ ,*#'.+'. %*! $!-#%+! $+!'%#%+' 5;/+%!
&!%*#'! &-!,-! !$ !'?!'! ,!'%$!6 $%#%+'. %*! &!%*#'! &-!,-! #-'. 6 #' ?;#+/
/%#$%+'. $& %*! +'+%+#- $+!'%#%+'
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) !ETHOD AND RESULT
)1 Introduction
I' %*+/ ,*#"%!$ +'%!$#,%+' "%!'%+#- ,$! #$! "-%%! $+%+'. /-- "$.$#&/ (/+'. C "$.$#&+'. ) $
%* '! /+%! #' +! /+%! &!- &!%*#'! #$+'. +/%#',! !%!!' %*! ,!'%$! &!%*#'! #' %*!
,!'%$! !'?!'! $+'. A-/ %*! +! /+%! &!- *#/ !!' .+!' #$+/ $+!'%#%+'/ !!"+'. %*! C;#%&
&!%*#'! + #! %*! !'?!'! $+'. ,!'%$! % +' %*! $+!'%#%+' +%* &+'+&& "//+-! !-- W! '!! %
+' %*! &+'+&& "//+-! !-- +' ,#/! &!%*#'! !'?!'! //%!& +' $!$ % ,*!, +%/ &/% /%#-!
$+!'%#%+'
) !ode#s o( !et"ane
Fi+ure >.M!%*#'! &!-/ /! +' %*! "$!/!'% ,#-,-#%+'/ T*! ' #'.-!/ #' ' -!'.%*/
/*' $ %*! +!;/+%! &!%*#'! &!- #$! /+&+-#$ $ #-- %*! '/ #' #'.-!/
T*!$! #$! % +!- /! ,-#//+,#- &!-/ $ &!%*#'!6 '#&!- %*! '!;/+%! #' K--' +!;/+%!
&!-/ T*! "#$#&!%!$/ $ %*! '!;/+%! &!- !$! !!-"! /+'. %*! #"$;-++ !+-+$+& #%#
*!$!#/ %*/! $ %*! K--' +!;/+%! &!- #/ #/! ' $!! !'!$. ,#-,-#%+'/ % "$!+,% /-#%+' #&!%*#'! &-!,-! +' - #%!$ B%* %*! &!-/ /,,!//-- $!"$,! %*! -+;"*#/! !*#+$
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&!%*#'! T*! ,*#$.!/ #' %*! LJ "#$#&!%!$/ $ %*! % &!-/ #$! /*' +' %*! T#-! 3 I' %*! $+.+
K--' +!;/+%! &!- +'%!$&-!,-#$ /+%!;/+%! +'%!$#,%+'/ #$! %$!#%! /+'. %*! 12; L!''#$;J'!/
+'%!$#,%+' "%!'%+#- #' C-&+, +'%!$#,%+'/ #$! /! $ +'%!$#,%+'/ !%!!' ,*#$.!/ B%* %*! +!;/+%!
#' '!;/+%! &!-/ #$! !"+,%! +' %*! +.$! 5
Tab#e 4 : L!''#$;J'!/ "#$#&!%!$/ #' C-&+, ,*#$.!/ $ '!;/+%! #' +!;/+%!
&!%*#'! &!-/
One8site
*ode#
Fi,e8site *ode#
O (A) 373 C
H
34
25
(K) 148 C
H
55055
7901
(!) 00 CH ;0015
)$ !ode# o( 'en?ene
Fi+ure @.B!'?!'!&!-/ /! +' %*! "$!/!'% ,#-,-#%+'/ T*! ' #'.-!/ #' ' -!'.%*/
/*' $ %*! +!;/+%! &!%*#'! &!- #$! /+&+-#$ $ #-- %*! '/ #' #'.-!/
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23
*!$!
O++/ %*! L!''#$;J'!/ +#&!%!$ $ %*! +%* /"!,+!/6
++/ %*! !-- !"%* $ %*! +%* /"!,+!/
1) Interaction o( one8site *et"ane 6it" ben?ene.
T*! "%!'%+#- ,$! $ '!;/+%! &!%*#'! #' !'?!'! +/ %#+'! "-#,+'. %*! &!%*#'! &-!,-!
#% +!$!'% "/+%+'/ #-'. %*! #+/ "#//+'. %*$.* %*! ,!'%$! %*! !'?!'! $+'. I' %*+/ //%!&
'- L!''#$;J'!/ +'%!$#,%+' +/ %*!$! #/ %- !#$-+!$ %*! ,-&+, "#$% +/ ?!$ $ &!%*#'!;
!'?!'! //%!& P$.$#& 1 (+' #""!'+ B) +/ $+%%!' ( +' C -#'.#.!) % ,#-,-#%! %*! "%!'%+#-
!'!$. #% #$' 100 +!$!'% "/+%+'/ #-'. %*! #+/ "#//+'. %*$.* %*! ,!'%$! %*! !'?!'!
$+'. /+'. %*! !#%+' *!$! %*! L!''#$;J'!/ "#$#&!%!$/ $ '!;/+%! &!%*#'! +/ .+!' +' %#-! 4
#' $ !'?!'! +' %#-! 5 T*! ,$! %#+'! %*$.* %*+/ "$.$#& +/ /*' +' +.$! 7 T*!
&+'+&& "%!'%+#- !'!$. !-- %#+'! +' %*+/ ,#/! +/ #$' ;4 J=&- #% # /!"!$#%+' 398
#'./%$& *+,* %,*!/ +%* %*!
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,-&+, "#$% +/ ?!$ $ &!%*#'!;!'?!'! //%!&6 % %*+/ //%!& .+!/ +!$!'% "%!'%+#- !'!$.
!-- $ +!$!'% $+!'%#%+' +%* $!/"!,% % !'?!'! $+'.6 '% -+! %*#% +' ,#/! '!;/+%! &!%*#'!
*+,* .+!/ '- # /+'.-! &+'+&& "%!'%+#- !-- S +' $!$ % +' %*! &+'+&& "%!'%+#- !--
$ +!;/+%! &!%*#'!;!'?!'! //%!&6 %*! &!%*#'! &-!,-! +/ "-#,! !$ %*! ,!'%$! !'?!'!
$+'. #' .+!' 1400 $#'& $%#%+'/ #% +%/ ,!'%$! // #' %*!' $ !!$ $%#%+' ( /%#$%+'.
$& %*! +'+%+#- $+!'%#%+' ) %*! "%!'%+#- !'!$. #% #$' 100 +!$!'% "/+%+'/ #-'. %*! #+/
"#//+'. %*$.* %*! ,!'%$! %*! !'?!'! $+'. +/ ,#-,-#%! /+'. %*! !#%+' *!$! %*! L!''#$;J'!/ "#$#&!%!$/ $ +!;/+%! &!%*#'! +/ .+!' +' %#-! 4 #' $ !'?!'! +' %#-! 5
M#' /-- "$.$#&/ #$! $+%%!' % "-% %*! "%!'%+#- !'!$. ,$!6 % +' %*! &+'+&&
!'!$. !-- #&'. #-- $+!'%#%+' "//+-! #' ,$$!/"'+'. /!"#$#%+' I /%#$%! ,#-,-#%+'. %*!
+'%!$#,%+' "%!'%+#- $& %*! +'+%+#- $+!'%#%+' &!%*#'! !'?!'! //%!& .+!' % &! /*' +'
+.$! 8 T*! ,$+'#%!/ +'+%+#- +!;/+%! &!%*#'! #' !'?!'! +/ .+!' +' ,$+'#%!;2 #'
,$+'#%!;1 (+' #""!'+ A) $!/"!,%+!-
Fi+ure B. I'+%+#- $+!'%#%+' &!%*#'! !'?!'! //%!&
I "$,!! %*$.* --+'. /%!"/ % +' %*! &+'+&& "%!'%+#- !-- $ +!;/+%! &!%*#'!;
!'?!'! //%!&
1 C#-,-#%+' %*! "%!'%+#- !'!$. #% #$' 100 +!$!'% "/+%+'/ #-'. %*! #+/ "#//+'.
%*$.* %*! ,!'%$! %*! !'?!'! $+'. $ %*! +'+%+#- $+!'%#%+' &!%*#'! &-!,-! ( CH ')!$ !'?!'! #/ /*' +' +.$! (/+'. "$.$#& 2)6/+'. %*! !#%+' *!$! %*! L!''#$;J'!/
"#$#&!%!$/ $ +!;/+%! &!%*#'! +/ .+!' +' %#-! 4 #' $ !'?!'! +' %#-! 5 T*! ,$! %#+'!
+/ /*' +' +.$! 9 T*! &+'+&& "%!'%+#- !'!$. !-- %#+'! +' %*+/ ,#/! +/ #$' ;4385
J=&- #% # /!"#$#%+' 415 #'./%$&/
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,$+'#%!;3 #' ,$+'#%!;1(#""!'+ A) L+! $ +'+%+#- $+!'%#%+' &!%*#'!6 $ %*+/
$+!'%#%+' #-/ %*! "%!'%+#- !'!$. +/ ,#-,-#%! #% #$' 100 +!$!'% "/+%+'/ #-'. %*! #+/
"#//+'. %*$.* %*! ,!'%$! %*! !'?!'! $+'. /+'. %*! !#%+' T*! ,$! %#+'! +/ /*' +'
+.$! 11 T*! &+'+&& "%!'%+#- !'!$. !-- %#+'! +' %*+/ ,#/! ,&!/ % % ! #$' ;043
J=&- #% # /!"#$#%+' 374 #'./%$&/ T*!/! #-!/ %,*!/ +%*
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Fi+ure 1$. C&"#$+/' "%!'%+#- ,$!/ $ 5;/+%! &!%*#'! !$ !'?!'! +%* +%/ CH;" #'
CH;' $+!'%#%+'
I' #-- ,#/!/ ! /!$! %*#% %*! /+'.-! /+%! &!-/ $ &!%*#'! "$!+,% # -!$ +'+'.
!'!$. *!' ,&"#$! +%* %*! +!;/+%! &!- (+%* CH ") #/ /*' +' +.$!14
Fi+ure 14. C&"#$+/' "%!'%+#- ,$!/ $ 5;/+%! &!%*#'! +%* +%/ CH;" #'
CH;' $+!'%#%+' #' 1;/+%! &!%*#'! !$ !'?!'!
T*! &/% /%#-! $+!'%#%+' &!%*#'!;!'?!'! //%!& +/ /*' +' +.$! 106 +%*
&+'+&& !'!$. !-- ;03 J=&- #' ,$$!/"'+'. +'%!$#%&+, +/%#',! 374
#'./%$&/
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V
%%interaction of C( of met'ane wit' t'at of ben"ene
$0+XF"(18917537"(398133903;&+'62)605)
0+XFZ4Z#Z("(=$0+X612);"(=$0+X6))
==interaction of )* of met'ane wit' C of ben"ene
$4+XF"(18917537"(2895814297;&+'62)605)
4+XFZ4Z,Z("(=$4+X612);"(=$4+X6))
==interaction of )+ of met'ane wit' C of ben"ene
F"(434318007;&+'62)
$11+XF"(594457539605)
$12+XF"(3721211605)
$13+XF"(497953977605)
$14+XF"(093487815605)
$15+XF"(224229038605)
$1+XF"(0231441154605)11+XF4Z,Z("(=$11+X612);"(=$11+X6))
12+XF4Z,Z("(=$12+X612);"(=$12+X6))
13+XF4Z,Z("(=$13+X612);"(=$13+X6))
14+XF4Z,Z("(=$14+X612);"(=$14+X6))
15+XF4Z,Z("(=$15+X612);"(=$15+X6))
1+XF4Z,Z("(=$1+X612);"(=$1+X6))
1+XF3Z(11+X12+X13+X14+X15+X1+X)
==total &otential enegy
+XF"Z(0+X1+X4+X)
$+XF398133903;&+'
,%Y['YYD+/%#',! (+' #&./%$') : Y$+X
,%Y[%YYE'!$. (+' J=&-):Y+X
F&+';+',
&+'F
\
&+'!-!F0X ==finding t'e minimum energy well and corres&onding distance
&+'+/F$0X
$(+'% F0F99)
V
+(X&+'!-!)
V
&+'!-!FX
&+'+/F$X
\
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-#% $11100X6$12100X6$13100X6$14100X6$15100X6$1100X611100X612100X613100X614100X6
15100X61100X6$21100X6$22100X6$23100X6$24100X6$25100X6$2100X621100X622100X623100X6
24100X625100X62100X6$31100X6$32100X6$33100X6$34100X6$35100X6$3100X631100X632100X6
33100X634100X635100X63100X6$41100X6$42100X6$43100X6$44100X6$45100X6$4100X641100X6
42100X643100X644100X645100X64100X
-#% *13X1X6*23X1X6*33X1X6*43X1X63X3X6'!13X1X6'!23X1X6'!33X1X6'!43X1X
-#% &+'!-!&!'%1400X6&+'+/%#',!1400X6&+'!-!6&+'+/6&&+'!-!&!'%6&&+'+/%#',!6$+!'%#%+'1400X6
$+!'%#%+'
,%YE'%!$ %*! &+' #' +/%#',! ,$+'#%! !'?!'! $& &!%*#'! ( + ,#$' &!%*#'! +/ #%
,$+'#%! ?F398 #' +/ #! !'?!'!): Y
,+'&+'+
+',F(#'.-!;&+'#'.-!)=1440
+',F(&+'+;)=100
,'F(138Z022)=1000 ==conversion of unit of energy from # to k$%mol
==rotation of met'ane along x-axis about its c-atom
$(+'% F0F1400)
V
&+'F&+'+
$(+'% +F0+F99+)
V
%%interaction of C( of met'ane wit' t'at of ben"ene
$0+XF"(18917537"(398133903;&+'62)605)
0+XFZ4Z#Z("(=$0+X612);"(=$0+X6))
==,$+'#%!/ H4 +%* $+.+' #% ,!'%$! C;#%& &!%*#'!
*4F0
*4F0
?*4F;1085524733
*1F;099995285
*1F;02502515
?*1F031841577
*2F0718721927*2F;0732812593
?*2F031841577
*3F0275273359
*3F0988837743
?*3F031841577
*40X0XF*4 *41X0XF*4 *42X0XF?*4
*10X0XF*1 *11X0XF*1 *12X0XF?*1
*20X0XF*2 *21X0XF*2 *22X0XF?*2*30X0XF*3 *31X0XF*3 *32X0XF?*3
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,%Y[%YY!'!$. (+' J=&-):Y+X
F&+';+',
&+'F
\
%%storing minimum energy for a &articular rotaion in an array
&+'!-!F0X
&+'+/F$0X
$(+'% F0F99)
V
+(X&+'!-!)
V
&+'!-!FX
&+'+/F$X
\
\
&+'!-!&!'%XF&+'!-!
&+'+/%#',!XF&+'+/
$+!'%#%+'XF&+'#'.-!
#'.-!F&+'#'.-!+',
&+'#'.-!F#'.-!
\
%%calculating minimum energy after all &ossible rotaion along x-axis
&&+'!-!&!'%F&+'!-!&!'%0X
&&+'+/%#',!F&+'+/%#',!0X
$+!'%#%+'F$+!'%#%+'0X
$(+'% +F0+F1400+)
V
+(&+'!-!&!'%+X&&+'!-!&!'%)
V
&&+'!-!&!'%F&+'!-!&!'%+X&&+'+/%#',!F&+'+/%#',!+X
$+!'%#%+'F$+!'%#%+'+X
\
\
%%finding t'e coordinates of .-site met'ane wit' minimum orientation &osition
10X0XF1 11X0XF0 12X0XF0 10X1XF0 11X1XF,/($+!'%#%+') 12X1XF;/+'($+!'%#%+')
10X2XF0 11X2XF/+'($+!'%#%+') 12X2XF,/($+!'%#%+')
%%coordinate of )+,),) and )*
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$+!'%#%+'F$+!'%#%+'+X
\
\
==finding t'e coordinates of .-site met'ane wit' minimum orientation &osition
10X0XF,/(&+'#'.-!) 11X0XF0 2X0XF/+'(&+'#'.-!) 10X1XF0 11X1XF1 12X1XF0
10X2XF;/+'(&+'#'.-!) 11X2XF0 12X2XF,/(&+'#'.-!)
==coordinate of )+,),) and )*
$(+'% F03)
V
$(+'% F01)
V
'!1XXF0
'!2XXF0
'!3XXF0
'!4XXF0$(+'% -F0-3-)
V
'!1XXF'!1XX(1X-XZ*1-XX)
'!2XXF'!2XX(1X-XZ*2-XX)
'!3XXF'!3XX(1X-XZ*3-XX)
'!4XXF'!4XX(1X-XZ*4-XX)
\
\
\
'!*1F'!10X0X
'!*1F'!10X1X
'!?*1F'!10X2X3981339030
'!*2F'!20X0X
'!*2F'!20X1X
'!?*2F'!20X2X3981339030
'!*3F'!30X0X
'!*3F'!30X1X
'!?*3F'!30X2X3981339030
'!*4F0
'!*4F0
'!?*4F'!40X2X3981339030
,%Y['YY['YY['YYM+'+&& !'!$. !-- #%!$ #-- "//+-! $%#%+' #-'. ;#+/ F
Y&&+'!-!&!'%Y J=&-Y
,%Y['YY#% # +/%#',! F Y&&+'+/%#',!Y #&./%$&Y
,%Y['YY$ #' $+!'%#%+' F Y$+!'%#%+'Z(180=314159254)Y !.$!!Y
,%Y['YY['YY#' %*! ,$+'#%!/ 5;/+%! &!%*#'! +%* &+'+&& $+!'%#%+' "/+%+' +/ : YY['Y
,%Y['YY,0YY[%Y0Y[%YY[%Y0Y[%YY[%Y3981339
,%Y['YYH1YY[%Y'!*1Y[%Y'!*1Y[%YY[%Y'!?*1
,%Y['YYH2YY[%Y'!*2Y[%Y'!*2Y[%Y'!?*2
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,%Y['YYH3YY[%Y'!*3Y[%Y'!*3Y[%Y'!?*3
,%Y['YYH4YY[%Y'!*4Y[%YY[%Y'!*4Y[%YY[%Y'!?*4
==storing t'e values of coordinates of met'ane wit' minimum energy
/%$!#& "
""!'(Y%%Y)
"Y['YY,0YY[%Y0Y[%YY[%Y0Y[%YY[%Y3981339
"Y['YYH1YY[%Y'!*1Y[%Y'!*1Y[%YY[%Y'!?*1
"Y['YYH2YY[%Y'!*2Y[%Y'!*2Y[%Y'!?*2
"Y['YYH3YY[%Y'!*3Y[%Y'!*3Y[%Y'!?*3
"Y['YYH4YY[%Y'!*4Y[%YY[%Y'!*4Y[%YY[%Y'!?*4
",-/!()
$!%$'(0)
\
%ro+ra* > . CALCULATING THE !INI!U! ENERG ELL IN INTERACTION OF >8SITE
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!ETHANE 31B= de+ree i)e0 CH u% ITH 'EN5ENE
+',-!+/%$!#&
+',-!+'+"
+',-!,/%+
+',-!,%*
+',-!/%$!#&
+',-!/%-+*
+',-!%+&!*
+',-!-+&+%/
/+'. '#&!/"#,! /%
/+'. /%::'&!$+,-+&+%/
+'% +'()
V
-#% #F590851756F35756,F2237896F326&+'6B,'6&+'#'.-!F3141592546+',
-#% *16*16?*16'!*16'!*16'!?*16*26*26?*26'!*26'!*26'!?*26*36*36?*36'!*36
'!*36'!?*36*46*46?*46'!*46'!*46'!?*4
-#% $100X6$0100X60100X61100X62100X63100X64100X6100X
-#% $11100X6$12100X6$13100X6$14100X6$15100X6$1100X611100X612100X613100X614100X6
15100X61100X6$21100X6$22100X6$23100X6$24100X6$25100X6$2100X621100X622100X623100X6
24100X625100X62100X6$31100X6$32100X6$33100X6$34100X6$35100X6$3100X631100X632100X6
33100X634100X635100X63100X6$41100X6$42100X6$43100X6$44100X6$45100X6$4100X641100X6
42100X643100X644100X645100X64100X
-#% *13X1X6*23X1X6*33X1X6*43X1X63X3X6'!13X1X6'!23X1X6'!33X1X6'!43X1X
-#% &+'!-!6&+'+/
,%Y!'%!$ %*! &+' #' +/%#',! !'?!'! $& &!%*#'! ( + ,#$' &!%*#'! +/ #% ,$+'#%!
?F398 #' +/ #! !'?!'!: Y
,+'&+'
+',F(&+';)=100
,'F(138Z022)=1000 ==conversion of unit of energy from # to k$%mol
%% +/( deg !01A1203 04 M51)A35 A037 8-A829
$(+'% +F0+F99+)
V
==interaction of c( wit' ben"ene
$0+XF"(18917537"(398133903;&+'62)605)
0+XFZ4Z#Z("(=$0+X612);"(=$0+X6))
==coordinates of )* wit' origin at centre of C-atom of met'ane
*1F;099995285
*1F;02502515
?*1F031841577
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4+XF41+X42+X43+X44+X45+X4+X
%%interaction of )+ wit' ben"ene
'!*1F'!10X0X
'!*1F'!10X1X
'!?*1F'!10X2X3981339030
$11+XF"("('!*1;13751793362)"('!*1002533910262)"('!?*1;&+'62)605)
$12+XF"("('!*1;05441062)"('!*112030771062)"('!?*1;&+'62)605)
$13+XF"("('!*1;070953277362)"('!*1;1178280762)"('!?*1;&+'62)605)
$14+XF"("('!*1070953277362)"('!*11178280762)"('!?*1;&+'62)605)
$15+XF"("('!*105441062)"('!*1;12030771062)"('!?*1;&+'62)605)
$1+XF"("('!*113751793362)"('!*1;002533910262)"('!?*1;&+'62)605)
11+XF4Z,Z("(=$11+X612);"(=$11+X6))
12+XF4Z,Z("(=$12+X612);"(=$12+X6))
13+XF4Z,Z("(=$13+X612);"(=$13+X6))
14+XF4Z,Z("(=$14+X612);"(=$14+X6))
15+XF4Z,Z("(=$15+X612);"(=$15+X6))
1+XF4Z,Z("(=$1+X612);"(=$1+X6))
1+XF11+X12+X13+X14+X15+X1+X
==interaction of ) wit' ben"ene
'!*2F'!20X0X
'!*2F'!20X1X
'!?*2F'!20X2X3981339030
$21+XF"("('!*2;13751793362)"('!*2002533910262)"('!?*2;&+'62)605)
$22+XF"("('!*2;05441062)"('!*212030771062)"('!?*2;&+'62)605)
$23+XF"("('!*2;070953277362)"('!*2;1178280762)"('!?*2;&+'62)605)
$24+XF"("('!*2070953277362)"('!*21178280762)"('!?*2;&+'62)605)$25+XF"("('!*205441062)"('!*2;12030771062)"('!?*2;&+'62)605)
$2+XF"("('!*213751793362)"('!*2;002533910262)"('!?*2;&+'62)605)
21+XF4Z,Z("(=$21+X612);"(=$21+X6))
22+XF4Z,Z("(=$22+X612);"(=$22+X6))
23+XF4Z,Z("(=$23+X612);"(=$23+X6))
24+XF4Z,Z("(=$24+X612);"(=$24+X6))
25+XF4Z,Z("(=$25+X612);"(=$25+X6))
2+XF4Z,Z("(=$2+X612);"(=$2+X6))
2+XF21+X22+X23+X24+X25+X2+X
=%interaction of ) wit' ben"ene
'!*3F'!30X0X
'!*3F'!30X1X
'!?*3F'!30X2X3981339030
$31+XF"("('!*3;13751793362)"('!*3002533910262)"('!?*3;&+'62)605)
$32+XF"("('!*3;05441062)"('!*312030771062)"('!?*3;&+'62)605)
$33+XF"("('!*3;070953277362)"('!*3;1178280762)"('!?*3;&+'62)605)
$34+XF"("('!*3070953277362)"('!*31178280762)"('!?*3;&+'62)605)
$35+XF"("('!*305441062)"('!*3;12030771062)"('!?*3;&+'62)605)
$3+XF"("('!*313751793362)"('!*3;002533910262)"('!?*3;&+'62)605)
31+XF4Z,Z("(=$31+X612);"(=$31+X6))
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&+'F&+'+
$(+'% +F0+F99+)
V
%%interaction of c( wit' ben"ene
$0+XF"(18917537"(398133903;&+'62)605)
0+XFZ4Z#Z("(=$0+X612);"(=$0+X6))
%%coordinates of )* wit' origin at centre of C-atom of met'ane
*4F0
*4F0
?*4F1085524733
*1F;099995285
*1F02502515
?*1F;031841577
*2F0718721927
*2F0732812593
?*2F;031841577
*3F0275273359
*3F;0988837743
?*3F;031841577
*10X0XF*1 *11X0XF*1 *12X0XF?*1
*20X0XF*2 *21X0XF*2 *22X0XF?*2
*30X0XF*3 *31X0XF*3 *32X0XF?*3
*40X0XF*4 *41X0XF*4 *42X0XF?*4
0X0XF,/(&+'#'.-!) 1X0XF;/+'(&+'#'.-!) 2X0XF0 0X1XF/+'(&+'#'.-!)
1X1XF,/(&+'#'.-!) 2X1XF0 0X2XF0 1X2XF0 2X2XF1
$(+'% F03)
V
$(+'% F01)
V
'!1XXF0
'!2XXF0
'!3XXF0
'!4XXF0
$(+'% -F0-3-)V
'!1XXF'!1XX(X-XZ*1-XX)
'!2XXF'!2XX(X-XZ*2-XX)
'!3XXF'!3XX(X-XZ*3-XX)
'!4XXF'!4XX(X-XZ*4-XX)
\
\
\
%%interaction of )* wit' ben"ene
'!*4F'!40X0X
'!*4F'!40X1X
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'!?*4F'!40X2X3981339030
$41+XF"("('!*4;13751793362)"('!*4002533910262)"('!?*4;&+'62)605)
$42+XF"("('!*4;05441062)"('!*412030771062)"('!?*4;&+'62)605)
$43+XF"("('!*4;070953277362)"('!*4;1178280762)"('!?*4;&+'62)605)
$44+XF"("('!*4070953277362)"('!*41178280762)"('!?*4;&+'62)605)
$45+XF"("('!*405441062)"('!*4;12030771062)"('!?*4;&+'62)605)
$4+XF"("('!*413751793362)"('!*4;002533910262)"('!?*4;&+'62)605)
41+XF4Z,Z("(=$41+X612);"(=$41+X6))
42+XF4Z,Z("(=$42+X612);"(=$42+X6))
43+XF4Z,Z("(=$43+X612);"(=$43+X6))
44+XF4Z,Z("(=$44+X612);"(=$44+X6))
45+XF4Z,Z("(=$45+X612);"(=$45+X6))
4+XF4Z,Z("(=$4+X612);"(=$4+X6))
4+XF41+X42+X43+X44+X45+X4+X
%%interaction of )+ wit' ben"ene
'!*1F'!10X0X'!*1F'!10X1X
'!?*1F'!10X2X3981339030
$11+XF"("('!*1;13751793362)"('!*1002533910262)"('!?*1;&+'62)605)
$12+XF"("('!*1;05441062)"('!*112030771062)"('!?*1;&+'62)605)
$13+XF"("('!*1;070953277362)"('!*1;1178280762)"('!?*1;&+'62)605)
$14+XF"("('!*1070953277362)"('!*11178280762)"('!?*1;&+'62)605)
$15+XF"("('!*105441062)"('!*1;12030771062)"('!?*1;&+'62)605)
$1+XF"("('!*113751793362)"('!*1;002533910262)"('!?*1;&+'62)605)
11+XF4Z,Z("(=$11+X612);"(=$11+X6))
12+XF4Z,Z("(=$12+X612);"(=$12+X6))
13+XF4Z,Z("(=$13+X612);"(=$13+X6))14+XF4Z,Z("(=$14+X612);"(=$14+X6))
15+XF4Z,Z("(=$15+X612);"(=$15+X6))
1+XF4Z,Z("(=$1+X612);"(=$1+X6))
1+XF11+X12+X13+X14+X15+X1+X
%%interaction of ) wit' ben"ene
'!*2F'!20X0X
'!*2F'!20X1X
'!?*2F'!20X2X3981339030
$21+XF"("('!*2;13751793362)"('!*2002533910262)"('!?*2;&+'62)605)
$22+XF"("('!*2;05441062)"('!*212030771062)"('!?*2;&+'62)605)
$23+XF"("('!*2;070953277362)"('!*2;1178280762)"('!?*2;&+'62)605)
$24+XF"("('!*2070953277362)"('!*21178280762)"('!?*2;&+'62)605)
$25+XF"("('!*205441062)"('!*2;12030771062)"('!?*2;&+'62)605)
$2+XF"("('!*213751793362)"('!*2;002533910262)"('!?*2;&+'62)605)
21+XF4Z,Z("(=$21+X612);"(=$21+X6))
22+XF4Z,Z("(=$22+X612);"(=$22+X6))
23+XF4Z,Z("(=$23+X612);"(=$23+X6))
24+XF4Z,Z("(=$24+X612);"(=$24+X6))
25+XF4Z,Z("(=$25+X612);"(=$25+X6))
2+XF4Z,Z("(=$2+X612);"(=$2+X6))
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"$+!'%#%+'%XZ(180=314159254)Y[%Y&+'!-!&!'%%XY['Y
\
",-/!()
#'.-!F&+'#'.-!+',
&+'#'.-!F#'.-!
\
&&+'!-!&!'%F&+'!-!&!'%0X
&&+'+/%#',!F&+'+/%#',!0X
$+!'%#%+'F$+!'%#%+'0X
!F&+'!-!&!'%0X
F&+'+/%#',!0X
F$+!'%#%+'0X
%%minimum energy after all &ossible rotaion along "-axis
$(+'% +F0+F72+)
V+(&+'!-!&!'%+X&&+'!-!&!'%)
V
&&+'!-!&!'%F&+'!-!&!'%+X
&&+'+/%#',!F&+'+/%#',!+X
$+!'%#%+'F$+!'%#%+'+X
\
\
,%Y['YY['YY['YYM+'+&& !'!$. !-- #%!$ #-- "//+-! $%#%+' #-'. ?;#+/ F
Y&&+'!-!&!'%Y J=&-Y
,%Y['YY#% # +/%#',! F Y&&+'+/%#',!Y #&./%$&Y
,%Y['YY$ #' $+!'%#%+' F Y$+!'%#%+'Z(180=314159254)Y !.$!!Y
%%maximum energy well among all i&ossible rotaion along "-axis
$(+'% F0F72)
V
+(&+'!-!&!'%X!)
V
!F&+'!-!&!'%X
F&+'+/%#',!X
F$+!'%#%+'X
\
\
,%Y['YY['YY['YYM#+&& !'!$. !-- #%!$ #-- "//+-! $%#%+' #-'. ?;#+/ F Y!Y
J=&-Y
,%Y['YY#% # +/%#',! F YY #&./%$&Y
,%Y['YY$ #' $+!'%#%+' F YZ(180=314159254)Y !.$!!Y
,%Y['YY['YYT*! + +' #' &+' !'!$. !-- #%!$ $%#%+' #-'. ?;#+/ F Y!;
&&+'!-!&!'%Y J=&-Y
=%finding t'e coordinates of .-site met'ane wit' minimum orientation &osition
10X0XF,/($+!'%#%+') 11X0XF;/+'($+!'%#%+') 12X0XF0 10X1XF/+'($+!'%#%+')
11X1XF,/($+!'%#%+') 12X1XF0 10X2XF0 11X2XF0 12X2XF1
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$!%$'(0)
\
Re(erences
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