Symmetrizer: Algorithmic determination of point groups in nearly symmetric molecules

Hope College Hope College

Hope College Digital Commons Hope College Digital Commons

Faculty Publications

7-15-2012

Symmetrizer: Algorithmic determination of point groups in nearly Symmetrizer: Algorithmic determination of point groups in nearly

symmetric molecules symmetric molecules

R Jeffrey Largent

William F. Polik

Hope College

, [email protected]

J R. Schmidt

Follow this and additional works at: https://digitalcommons.hope.edu/faculty_publications

Part of the Chemistry Commons

Recommended Citation Recommended Citation

Largent, R. Jeffrey, William F. Polik and J. R. Schmidt. "Symmetrizer: Algorithmic Determination of Point

Groups in Nearly Symmetric Molecules." Journal of Computational Chemistry 33, no. 19.00 (2012):

1637-1642.

This Article is brought to you for free and open access by Hope College Digital Commons. It has been accepted for

inclusion in Faculty Publications by an authorized administrator of Hope College Digital Commons. For more

information, please contact [email protected].

#$%%&'()*&(+",-./0123415"67270418921/8"/:";/182"<0/=>?"18"@790-A"#A4472015"

%/-75=-7?"

!"#$%&&'%(#)*'+%,-.#/0110*2#3"#45106"

67>9024782"/:"B3741?20A"

C/>7"B/--7.7D"EF"&9?2"!G

"#20772D"C/--98HD"%)"IJIGGKJLLL"

$"#!"#78920:-"

'37/072159-"B3741?20A")8?212=27"98H"67>9024782"/:"B3741?20A"

M81N70?12A"/:"O1?5/8?18D"%9H1?/8D"O)"FEPLQ"

;<=-'*8->"#A44720A"1?"98"7R20747-A"=?7:=-"98H">/S70:=-"2//-"18"5/4>=2921/89-"53741?20AD"T/23"

:/0">07H15218."237">0/>70217?"/:"4/-75=-7?"98H":/0"?14>-1:A18."59-5=-921/8?U",-23/=.3"4723/H?"

:/0"H727041818."237">/182".0/=>?"/:">70:752-AK?A4472015"4/-75=-7?"907"S7--"V8/S8D":18H18."237"

5-/?7?2">/182".0/=>":/0"9"!"#$%&K?A4472015"4/-75=-7"1?":90"-7??"?2=H17HD"23/=.3"12">07?782?"498A"

=?7:=-"9>>-15921/8?U"W/0"231?"079?/8D"S7"1820/H=57"#A447201X70D"98"9-./01234"H7?1.87H"2/"

H72704187"9"4/-75=-7Y?"?A44720A"7-74782?"98H"5-/?7?2K4925318.">/182".0/=>?"T9?7H"/8"9"

=?70K9HZ=?29T-7"2/-709857D"98H"2378"2/"?A447201X7"2392"4/-75=-7"2/"9".1N78">/182".0/=>"

.7/4720AU")8"5/8209?2"2/"5/8N7821/89-"4723/H?D"#A447201X70"29V7?"9"T/22/4K=>"9>>0/953"2/"

?A44720A"H727521/8"TA"-/59218."9--">/??1T-7"?A44720A"7-74782?"98H"=21-1X18."231?"?72"2/"H7H=57"

237"4/?2">0/T9T-7">/182".0/=>?U"O7"7R>-918"231?"9>>0/953"18"H7291-D"98H"9??7??"237":-7R1T1-12AD"

0/T=?287??D"98H"7::151785A"/:"237"9-./01234"S123"07?>752"2/"N901/=?"18>=2">90947270?"/8"?7N709-"

27?2"4/-75=-7?U"O7"9-?/"H74/8?20927"98"9>>-15921/8"/:"#A447201X70"TA"18270:9518."12"S123"237"

O7T%["S7TKT9?7H"18270:957"2/"5/4>=2921/89-"53741?20A">95V9.7?"9?"9"?3/S59?7"/:"12?"79?7"/:"

1827.0921/8U"

?%(@5':=>"?A44720A\"?A44720A"7-74782\">/182".0/=>\".0/=>"237/0A\"9-./01234

A"###B,-'5:C8-05,"

" B/8?1H70921/8"/:"?A44720A"1?"7R20747-A"=?7:=-"18"5/4>=2921/89-"53741?20AD"T/23":/0"

>07H15218."4/-75=-90">0/>70217?"98H":/0".0792-A"?14>-1:A18."59-5=-921/8?U"'3=?"12"1?"31.3-A"

9HN9829.7/=?"2/"T7"9T-7"2/"9=2/4927"1H7821:15921/8"/:"237"?A44720A"7-74782?"98H"237">/182"

.0/=>?"/:"4/-75=-7?U"C/S7N70D":18H18."237"5-/?7?2">/182".0/=>"/:"!"#$%&"?A4472015"4/-75=-7?"

1820/H=57?"9HH121/89-"5/4>-7R12AU"'31?"1?"18":952"9":90"4/07"14>/02982"98H"182707?218.">0/T-74D"

9?"4/-75=-7?"/:278"H/"8/2"39N7">70:752-A"?A4472015"5/8:/04921/8?D"S372370"H=7"2/"237049-"

:-=52=921/8D"]938K'7--70"H1?2/021/8D"7R27089-">702=0T921/8D"/0"4707-A"8=470159-"14>0751?1/8U")2"1?"

23707:/07"=?7:=-":/0"98"9-./01234"2/":18H"237">/182".0/=>"2392"9"4/-75=-7"4/?2"5-/?7-A"492537?D"

S12318"9"=?70K?>751:17H"2/-709857U"

" %723/H?":/0"H727041818."237">/182".0/=>"/:"9">70:752-A"?A4472015"4/-75=-7"907"S7--K

V8/S8U"'37"5-9??15"9>>0/953">0/>/?7H"TA"WU",U"B/22/8"18"!JQE"1?"9"?2091.32:/0S90H"37=01?215":/0"

H727041818."237">/182".0/=>"/:"9"4/-75=-7"T9?7H"/8"237">07?7857"/0"9T?7857"/:"V7A"

5390952701?215?D"T=2"12"1?"9>>-159T-7"/8-A"2/"1H79-D">70:752-AK?A4472015"?20=52=07?U

",8/2370"

9>>0/953"20792?"?A44720A"9?"9"5/8218=/=?">0/>702A"092370"2398"9"H1?50727"/87"H79-18."?/-7-A"

S123"1H79-".7/472017?U

"#21--"/2370"9-./01234?"39N7"T778"H7N1?7H":/0"=?7"18"?/:2S907"9>>-15921/8?"

2392"185/0>/0927"237"1H79"/:"9"2/-709857":/0"700/0"18"?A44720AD

EDI

"T=2"237?7"/8-A">0/N1H7"237"

=?70"S123"237"31.37?2K/0H70"?=..7?27H">/182".0/=>U"'31?"1?"=8H7?109T-7"18"?12=921/8?"1?"S3153"

-/S70K/0H70"?A4472017?"907">07:7007HD"?=53"9?"955/=8218.":/0"237"]938K'7--70"7::752U""

" W/0"237?7"079?/8?D"S7"9H/>2"9"T/22/4K=>"9>>0/953"2/"?A44720A"H727521/8U"[=0"

9>>0/953"T7.18?"TA"-/59218."9--":79?1T-7"?A44720A"7-74782?">07?782"18"9"4/-75=-7D":0/4"S3153"

/87"598"H7H=57">-9=?1T-7">/182".0/=>?U"'31?"9>>0/953"49V7?"12"201N19-"2/"^=9821:A"237"H7N1921/8"

:0/4"9">90215=-90">/182".0/=>D"98H"12"9-?/">0/N1H7?"18:/04921/8"9T/=2"3/S"S7--"7953"18H1N1H=9-"

?A44720A"7-74782"_:12?`"9"4/-75=-7U",HH121/89--AD"39N18."-/5927H"9--"?A44720A"7-74782?"/:"9"

>/182".0/=>"=?18."/=0"4723/HD"1H7821:15921/8"/:"237"?A44720AK=81^=7"92/4?"98H"4/-75=-90"

?A447201X921/8"T75/47?"^=127"?2091.32:/0S90HU"

D"###EF%'F0%@#5&#;GG'5*89"

" O7"T7.18"S123"9"T017:"/N70N17S"/:"/=0"?A44720A"9-./01234U"'37"9-./01234"?2902?"TA"

?7905318.":/0"7953"2A>7"/:"?A44720A"7-74782"=?18."98"7R39=?21N7"?79053D"=21-1X18."01./0/=?"

237/074?"2/"8900/S"237"?79053">0/57??U","H7291-7H"H7?501>21/8"/:"237"?79053">0/57??"1?".1N78"18"

9"?=T?7^=782"?7521/8U";/278219-"?A44720A"7-74782?"907"27?27H"98H"7N9-=927H"9?":/--/S?+"'37"'23"

?A44720A"/>70921/8"9??/51927H"S123"237"7-74782"1?"9>>-17H"2/"7N70A"92/4"(

)

"18"237"4/-75=-7"2/"

.7870927"92/4"(

)

U"W/0"7953"(

)

, 237"5-/?7?2"92/4"/:"237"?947"2A>7D"*

)

D"1?":/=8HD"98H"2378"237"

700/0"9??/51927H"S123"2392"?A44720A"7-74782"1?"59-5=-927H"9?"

error =

max

! B

i=1

S3707"$

)

"1?"237">70>78H15=-90"H1?29857"T72S778"92/4"(

)

98H"237"?A44720A"7-74782 98H"n"1?"237"

/0H70"/:"237"7-74782U"'37"8=47092/0"1?"237"H1?29857"2392"98A".7870927H"92/4"-17?":0/4"/87"/:"

12?"?A44720A"7^=1N9-782"5/=8270>902?D"98H"237"H78/41892/0"?70N7?"2/"V77>"237"700/0"

18H7>78H782"/:"9"4/-75=-7Y?"?1X7U"):"237"H1?29857"T72S778"(

)

+98H"237"?A44720A"7-74782"1?"?49--"

aS12318"/87"bcD"2378"237"H78/41892/0"1?"?72"2/"=812A"2/">07N782"237"9??1.84782"/:"98"

700/87/=?-A"31.3"700/0"2/"237"?A44720A"7-74782U"'31?"700/0"1?"2378"5/4>907H"2/"9"=?70KH7:187H"

2307?3/-HD"98H"9"?A44720A"7-74782"1?"/8-A"9557>27H"1:"12"-17?"S12318"237"2307?3/-HU""

" '37"87R2"?27>"1?"2/"H72704187"9--"N19T-7">/182".0/=>?".1N78"237"?72"/:"9557>27H"?A44720A"

7-74782?U"'31?"?72"1?"5/4>907H"2/"237"?72"/:"7-74782?"2392"5/4>/?7"7953"V8/S8">/182".0/=>D"

98H"9">/182".0/=>"1?"9557>27H"1:"12?"7-74782?"907"9"?=T?72"/:"237"9557>27H"?A44720A"7-74782?U"

'37"-1?2"/:"9557>27H">/182".0/=>?"1?"2378"098V7H"T9?7H"/8"2S/"50127019+"'37"?72"H1::707857"

T72S778"237"2S/"?72?"98H"237"4798"700/0"/:"7953">/182".0/=>Y?"5/8?212=782"?A44720A"

7-74782?U";/182".0/=>?"907"098V7H"31.370"237"?49--70"237"H1::707857"T72S778"237"?72"/:"

9557>27H"?A44720A"7-74782?"98H"237"?72"/:"7R>7527H"?A44720A"7-74782?D"S123"217H">/182"

.0/=>?"098V7H"31.370"237"-/S70"23710"4798"7-74782"700/0U"'37"78H"07?=-2"1?"9"-1?2"/:">/??1T-7"

>/182".0/=>?"/0H707H":0/4"4/?2K?>751:15"2/"-79?2K?>751:15U"'37"=?70"598"53//?7"94/8."N901/=?"

>-9=?1T-7">/182".0/=>?"1:"237"?A44720A"/:"237"4/-75=-7"1?"=85-790D"/0"7N78"2/"?A447201X7"237"

4/-75=-7"2/"9"-7??K?>751:15">/182".0/=>U""

" C9N18."1H7821:17H"9"290.72">/182".0/=>D"237"4/-75=-7"598"T7"?A447201X7HU"d7:/07"

?A447201X18.D"3/S7N70D"237"1H7821:17H"?A44720A"7-74782?"4=?2"T7"1H79-1X7H"2/"074/N7"237"

700/0"07?=-218.":0/4"237"14>70:752".7/4720A"/:"237"/01.189-"4/-75=-7U"'37"?72"/:"92/4?"907"

07H=57H"2/"9"?49--"?=T?72"/:"?A44720AK=81^=7"92/4?D"98H"237?7"92/4?"907"9-1.87H"S123"07?>752"

2/"237"?A44720A"7-74782?U"'37"/>70921/8?"/:"237"1H79-1X7H"?A44720A"7-74782?"907"9>>-17H"2/"

237?7"92/4?"2/".7870927"237"?A447201X7H"4/-75=-7U""

,".09>3159-"/N70N17S"/:"237"?A447201X921/8">0/57H=07"598"T7"?778"T7-/S"18"W1.=07"!U"

H"###I%=8'0G-05,#5&#;1+5'0-92=#

H";"##7(22%-'(#J1%2%,-#I%-%8-05,#;1+5'0-92#

# O7"S1--"T017:-A"/=2-187"237"H7291-?"/:"237"?A44720A"7-74782"H727521/8"9-./01234U"O7"=?7"

237">309?7"_27?218.`"/0"_5375V18.`"?A44720A"7-74782?"2/"07:70"2/"59-5=-9218."237"700/0"

9??/51927H"S123"98"7-74782"98H"5/4>9018."12"2/"237"=?70"2307?3/-H"9?"H7?501T7H"18"?7521/8"GU"

C707"S7"=21-1X7"9"8=4T70"/:"237/074?"2/"?14>-1:A"237"?79053":/0"?A44720A"7-74782?D"129-151X7H"

18"237"?7521/8?"T7-/SU"

H";"A"###B:%,-0&08*-05,#5&#4'0,80G*1#;K%=#

# W10?2D"237"578270"/:"49??"1?"59-5=-927H"98H"9--"?A44720A"7-74782?"-/5921/8?"907"H7:187H"

07-921N7"2/"12U"@7R2D"237">01851>9-"4/4782?"98H">01851>9-"9R7?"/:"1870219"907"59-5=-927H"TA"

5/8?20=5218."98H"H19./89-1X18."237"1870219-"278?/0U"W189--AD"237">01851>9-"4/4782?"/:"1870219"907"

5/4>907H"2/"H72704187"23710"H7.787095AD"?>751:159--A"2/"H72704187"1:"237"4/-75=-7"1?"9"

?>370159-D"?A4472015D"/0"9?A4472015"2/>U

" '37">01851>9-"9R7?"598"T7"74>-/A7H"2/"H09492159--A"?>77H"=>"237"?=T?7^=782"?79053":/0"

237"4/-75=-90"?A44720A"7-74782?D"=21-1X18."237":952"2392"#+,&--".$&+#/',+0$+$"1%"2.'0!+3%#!"+

!0$-#%+-4,.+5"+#!+"'6"!7"2.0$+01+.8"+'!"$.'#%+."!,0$U"'31?"1?"7N1H782"T759=?7"9"?A44720A"

/>70921/8"4=?2"-79N7"9"4/-75=-7"a98H"23707:/07"237"4/4782"/:"1870219"278?/0c"=85398.7HU"'3=?"

237"278?/0"4=?2"T7"18N901982"S123"07?>752"2/"9--"?A44720A"/>70921/8?U"'31?"237/074"14>-17?"2392"

9"?A44720A"9R1?"/0">-987"4=?2"5/82918"92"-79?2"/87"/:"237">01851>9-"9R7?"1:"237"4/4782?"/:"

1870219"907"8/8KH7.7870927U"):"237"4/4782?"/:"1870219"907"H/=T-AKH7.7870927D"2378"98A"9R1?"/0"

>-987"4=?2"712370"5/82918"237"8/8KH7.7870927"9R1?"/0"T7"/023/./89-"2/"12U"):"98"7-74782"H/7?"

8/2"/T7A"237?7"50127019"S12318"9"?49--"2/-709857D"12"1?"1447H1927-A"07Z7527H"S123/=2":=02370"

27?218.U"'37?7"1H79?"907"=?7H"2/"/>2141X7"237"?=T?7^=782"?A44720A"7-74782"?790537?U"

H";"D"###L%=-#&5'#B,F%'=05,#M%,-%'"

" ,?"18N70?1/8"578270?"907">/182?"98H""7"$&+,&--".$&+"%"-"!.+-4,.+20!.#'!+.8"+2"!."$+01+

-#,,D"98"18N70?1/8"578270"4=?2"T7"92"237"578270"/:"49??D"1:"12"7R1?2?U"

H";"H""""L%=-#&5'#4'5G%'#!5-*-05,*1#;K%="

" ;0/>70"0/2921/8"9R7?"907"181219--A"5375V7H"9-/8."237">01851>9-"9R7?U"'37"9-./01234"2378"

?790537?":/0"9R7?"5/8291818."T/23"237"578270"/:"49??"98H"98"92/4D":/--/S7H"TA"9R7?"5/8291818."

237"578270"/:"49??"98H"41H>/182?"T72S778"-1V7K92/4?U"):"237">/182".0/=>"/:"237"4/-75=-7"1?"

5=T15D"H72704187H"TA"12?"4/4782?"/:"1870219"T718."201>-A"H7.7870927D"2378"98"9HH121/89-"?27>"

4=?2"T7"74>-/A7H"2/"-//V":/0">0/>70"0/2921/89-"9R7?">9??18."230/=.3"237"578270?"/:">/-A./89-"

:957?U"'31?"1?"955/4>-1?37H"TA"?7905318."#%0!6+104!9+:

;

+#/",+10$+<+#!9+=+6$043,D"98H"#%0!6+#%%+

20-5'!#.'0!,+01+2$0,,>3$0942.,+01+:

;

+#/",+10$+?+6$043,U"W189--AD"1:"237"4/-75=-7"1?"8/2"5=T15D"2378"

237"31.37?2K/0H70">0/>70"0/2921/89-"9R1?"1?"?7-7527H"98H"H78/27H"9?"237">01490A"9R1?U"

H";"N"###L%=-#&5'#B2G'5G%'#!5-*-05,*1#;K%="

" (%%+'-3$03"$+$0.#.'0!#%+#/",+-4,.+5"+20'!2'9"!.+@'.8+"/',.'!6+3$03"$+$0.#.'0!#%+#/","98H"

8#7"+#+9"6$""+"'.8"$+"A4#%+.0+0$+.@'2"+.8"+9"6$""+01+.8"+20'!2'9"!.+3$03"$+$0.#.'0!#%+#/',U"

H";"O"###L%=-#&5'#!%&1%8-05,#41*,%="

" '37":10?2"?27>":/0"-/59218."07:-7521/8">-987?"1?"2/"?79053":/0"23/?7"2392"907"/023/./89-"2/"

>07N1/=?-A"-/5927H">0/>70"0/2921/89-"9R7?U"W/0"["98H")">/182".0/=>?D"231?"?79053"1?"-14127H"2/"B

9R7?D"9?"#%%+$"1%"2.'0!,+#$"+0$.8060!#%+.0+:

;

+#/",+'!+.8","+6$043,U"W/0"9--"/2370".0/=>?D"9HH121/89-"

5375V?"907"07^=107H+";-987?"S123"8/049-?"5/8291818."237"578270"/:"49??"98H"98"92/4"907"

5375V7HD":/--/S7H"TA">-987?"S123"8/049-?"5/8291818."237"578270"/:"49??"98H"9"41H>/182"

T72S778"-1V7K92/4?U"

" #77"W1.=07"G":/0"9":-/S53902"/N70N17S"/:"237"?A44720A"7-74782"H727521/8"9-./01234U"

H"P"###450,-#Q'5CG#;==0+,2%,-#;1+5'0-92#

" '37">/182".0/=>"9??1.84782"9-./01234"492537?"237"9557>27H"?A44720A"7-74782?"2/"237"

7-74782?"5/829187H"18"7953"V8/S8">/182".0/=>U"C/S7N70D"?1857"S7"S982"=?70?"2/"T7"9T-7"2/"

?A447201X7"4/-75=-7?"2/"-/S70K/0H70">/182".0/=>?"18"9HH121/8"2/"237"31.37?2K/0H70">/182".0/=>"

:/=8HD"231?">0/57??"4=?2"T7"H/87"18"9"?>751:15":9?31/8U"'31?"18N/-N7?"78?=018."2392"0/2921/8?"

90/=8H"237">01490A"9R1?"18"9R19-".0/=>?"907"237"?947"98.-7":0/4"7953"/2370D"98H"2392"07:-7521/8?"

18"237">-987"/:"237"4/-75=-7"907">01/0121X7H"/N70"H137H09-"07:-7521/8?U"'37"9557>27H"?A44720A"

7-74782?"907"5/4>907H"2/"23/?7"/:">-9=?1T-7">/182".0/=>?D"98H"98A">/182".0/=>"2392"H1H"8/2"

39N7"9--"/:"12?"?A44720A"7-74782?"492537H"1?"1447H1927-A"H1?590H7HU"(7491818.">/182".0/=>?"

907"098V7H"T9?7H"/8"3/S"498A"9557>27H"7-74782?"5/=-H"8/2"T7"492537H"2/"2374D"98H"

?75/8H901-A"/8"237"4798"700/0"/:"237"7-74782?"2392"S707"492537H"2/"2374U""

H"M##7(22%-'0R*-05,#;1+5'0-92"

[857"9">/182".0/=>"39?"T778"H7?1.8927H":/0"9"4/-75=-7D"?A447201X921/8"2/"2392">/182"

.0/=>"598"/55=0U"'37"?A44720A"7-74782?":/=8H"2/"49253"2392">/182".0/=>"4=?2"T7":10?2"T7"

1H79-1X7H"S123"07?>752"2/"7953"/2370U"'31?"78291-?D":/0"7R94>-7D"0/29218."237"7R1?218."7-74782?"2/"

78?=07"2392"9--">701>3709-"9R7?"907"/023/./89-"2/"237">01490A"9R1?"98H"2392"237"07:-7521/8">-987?"

907"9>>0/>01927-A"/017827H"S123"07?>752"2/"7953"/2370U"'37"?A447201X921/8"0/=2187"9-?/"

H72704187?"237"-79?2"?72"/:"92/4?"2392".7870927?"237"/01.189-"92/4?"18"237"4/-75=-7"S378"9--"

?A44720A"/>70921/8?"907"9>>-17H"2/"2374D"S3153"907"07:7007H"2/"9?"237"?A44720AK=81^=7"

92/4?U"'31?"?72"/:"92/4?"4=?2"T7"/017827H"?/"2392"237A"-17"7R952-A"/8"237"1H79-1X7H"?A44720A"

7-74782?U",?"?=53D"92/4?"S12318"LU!"b"907"4/N7H"/82/"2374D"S31-7"23/?7":=02370"9S9A"907"

9??=47H"2/"8/2"-17"/8"?A44720A"7-74782?"98H"907"-7:2"=85398.7H"?1857"237">/?121/8?"/:"

?A44720AK=81^=7"92/4?"2392"H/"8/2"5/1851H7"S123"?A44720A"7-74782?"907"90T12090A"S123"

07?>752"2/"237">/182".0/=>U"W189--AD"237":=--"?72"/:"?A44720A"/>70921/8?"1?"9>>-17H"2/"7N70A"

?A44720AK=81^=7"92/4D".78709218."237":=--"?72"/:"92/4?"18"23710"?A4472015"5/8:1.=0921/8U"

N"###7-'C8-C'%#5&#7(22%-'0R%'#

# [=0"14>-74782921/8D"#A447201X70D"S9?"S012278"18"]9N9"98H"49V7?"=?7"/:"237"/TZ752K

/017827H">0/.094418.">909H1.4"2/">902121/8"237"5/H7"182/"N901/=?"5-9??7?U"'37?7"5-9??7?"598"T7"

79?1-A"1827.0927H"182/"]9N9"9>>-72?"/0"9>>-15921/8?"9?"9"?14>-7"-1T090A":/0">70:/0418."?A44720AK

07-927H"29?V?U""

'37"-1T090A"1?"578209--A"/017827H"90/=8H"237"B&--".$&"5-9??D"S3153"H/7?"237"T=-V"/:"237"

S/0V"/:"?A44720A"H727521/8"98H">/182".0/=>"4925318.U"'37"C0'!.D$043"5-9??"5/82918?"237"

?A44720A"7-74782?D"?A44720AK=81^=7"92/4?D"98H">/182".0/=>"8947"T9?7H"/8"9">/278219-">/182"

.0/=>":/0"9"4/-75=-7D"98H"29V7?"5907"/:"?A447201X18."237"4/-75=-7Y?"92/4?"2/"12?">/182".0/=>"

?A44720AU"'37"E%"-"!."?=>705-9??"H7:187?"98"18270:957":/0"237"=!7"$,'0!D"F"1%"2.'0!D"

C$03"$F0.#.'0!D"98H"=-3$03"$F0.#.'0!"5-9??7?D"S3153"5/82918"237">/?121/8"98H"H107521/8"

18:/04921/8"/:"7953"2A>7"/:"7-74782D"9?"S7--"9?"12?"9??/51927H"?A44720A"/>70921/8U"'37"(.0-D"

G0%"24%"D"C0'!.HID"98H"G#.$'/JI"5-9??7?"?70N7"9?"H929"?20=52=07?"98H"H/"49237492159-"

>0/57??18.D"98H"237"C$01'%"$"5-9??">0/N1H7?"?/47"N70A"T9?15"0=82147K>0/:1-18.":=8521/89-12AU""

#A447201X70"9-?/"185-=H7?"9"G#'!"5-9??"2392"9--/S?"12"2/"0=8"9?"9"?298HK9-/87"5/4498HK

-187"9>>-15921/8U"W/0"7R94>-7D"

]9N9""KZ90""?A447201X70UZ90""CG[URAX"

!L"

>0/H=57?"/=2>=2"?3/S8"18"f1?218."!U"#A447201X70"5/82918?"T9?15":=8521/89-12A":/0"079H18."98H"

S01218."g$*":1-7?D"9?"S7--"9?"9"5/4>07378?1N7"27?2"?=127"/:"7R94>-7"4/-75=-7?"98H"23710"18278H7H"

?A4472017?"92"N901/=?"2/-709857?U"#A447201X70"?/=057"5/H7"98H"T1890A"907"T718."H1?201T=27H"9?"

2//-?":/0"T/23"H7N7-/>70?"98H"07?7905370?"=8H70"237"<@M"f7??70"-1578?7D"S3153"9--/S?"=?70?"2/"

4/H1:A"237"5/H7":/0"23710"/S8">70?/89--AK291-/07H"07^=1074782?"1:"237A"?/"H7?107U

O"###B,-%+'*-05,#@0-9#E-9%'#4'5+'*2=#

" O31-7"#A447201X70"598"T7"0=8":0/4"237"5/4498H"-187D"12"1?"9-?/"7R>/?7?"9"

>0/.09449215"18270:957":/0"79?A"1827.0921/8"S123"/2370"]9N9KT9?7H">0/.094?U""

'37"5/07"/:"237"#A447201X70",;)"5/8?1?2?"/:"237"!"#$$%&'()*+&","98H"-'(./00%&'()*+&",$"

4723/H?"18"B&--".$&"98H"237"$122#)+'3#4&0#5"0#"98H"!#)6122#)+170#2#()$"4723/H?"18"

C0'!.D$043U"'37"-'(./00%&'()*+&",$+4723/H"18121927?"237"?A44720A"98H">/182".0/=>KH727521/8"

4723/H"H7?501T7H"790-170"18"237">9>70"2/"T=1-H"=>"9"-1?2"/:"9--">/278219-">/182".0/=>?"98H"23710"

?A44720A"7-74782?D"S31-7"237"!"#$$%&'()*+&","4723/H"14>-74782?"9":9?2D"H751?1/8K2077">/182"

.0/=>"H7270418921/8"9-./01234":/0"T9?15"/8K237K:-A">/182".0/=>"18:/04921/8U"",--"/:"237"H7?107H"

:=8521/89-12A":/0"?A44720AK07-927H"29?V?"598"T7">70:/047H"=?18."237?7":/=0"4723/H?U"C/S7N70D"

/2370"4723/H?"7R1?2":/0"/T291818."9HH121/89-"H7291-7H"?A44720A"H929"1:"877H7HU""

" ,?"9"59?7"?2=HAD"231?"-1T090A"S9?"1827.0927H"S123"9"S7TKT9?7H".09>3159-"18270:957":/0"

5/4>=2921/89-"53741?20A">0/.094?"59--7H"O7T%[U

""O7T%["=?7?"#A447201X70"2/">0/N1H7"079-K

2147"?A44720A"18:/04921/8"9T/=2"9"4/-75=-7"T718."4/H7-7HD"9?"S7--"9?"237"/>21/8"2/"

?A447201X7"9"4/-75=-7">01/0"2/"?=T41??1/8"2/"9"5/4>=2921/89-"78.187"?=53"9?"<9=??198D

<947??D

"/0"%[;,BU

"")8"9HH121/8D"O7T%["=?7?"237"&-74782"H929"?20=52=07?":0/4"

!!"

#A447201X70"2/"N1?=9-1X7"?A44720A"7-74782?"/8"98A"4/H7-7H"4/-75=-7U"'31?":792=07">0/N1H7?"9"

N70A"182=121N7">152=07"/:"4/-75=-90"?A44720A"98H"1?"=?7:=-":/0"7H=5921/89-">=0>/?7?U"

#50778?3/2?"/:"O7T%[Y?"?A44720A"59>9T1-1217?"598"T7"N17S7H"18"W1.=07?"E"98H"IU"

S"###!%=C1-=#*,:#I0=8C==05,#

" '31?"?A44720A"H727521/8"9-./01234"39?">0/N7H"2/"T7"9":-7R1T-7D"0/T=?2D"98H"07-921N7-A"

:9?2"4723/H"/:"H727041818."4/-75=-90"?A44720A">0/>70217?U"#/47"?94>-7"?2921?215?"/:"237"

9-./01234":/0"?7N709-"-90.7"4/-75=-7?"S123"-/SD"1827047H1927D"98H"31.3"?A44720A"9?".1N78"18"

'9T-7"!U""

,?"237"H929"18H15927?D"237">70:/049857"/:"237"9-./01234"1?"9"5/4>-7R":952/0"/:"?7N709-"

>90947270?D"185-=H18."8=4T70"/:"92/4?D"_=81^=787??`"/:"92/4?D"2A>7"/:">/182".0/=>D"98H"237"

N901/=?"2/-709857"2307?3/-H?"74>-/A7H"230/=.3"237">0/.094U"[TN1/=?-AD"237".079270"237"

8=4T70"/:"92/4?"18"9"4/-75=-7D"237"4/07">/?121/8?"877H"2/"T7"5375V7H":/0">/??1T-7"?A44720A"

7-74782?D"?/"5/4>=2921/89-"2147"185079?7?"S123"237"?1X7"/:"237"18>=2"4/-75=-7U",HH121/89--AD"237"

4/07"/:"/87"2A>7"/:"7-74782"23707"1?"S12318"237"4/-75=-7D"237"390H70"12"1?":/0"237"9-./01234"2/"

0=-7"/=2">/??1T-7"?A44720A"7-74782?"/8"237"T9?1?"/:"92/415">0/>70217?U"#/"9"N70A"3/4/.787/=?"

4/-75=-7"-1V7"d=5V418?270:=--70787"29V7?"5/8?1H709T-A"4/07"2147"2/"989-AX7"2398"9"4/07"

37270/.787/=?"4/-75=-7"-1V7"37R9>A01H187"10/8U",HH121/89--AD"S31-7"?/47"?A44720A"7-74782?"/:"

9R19-".0/=>?"598"T7"0=-7H"/=2"T9?7H"/8"1870219-"5/8?209182?D"5=T15".0/=>?"39N7"201>-A"H7.7870927"

4/4782?"/:"1870219"98H"?/"9--"/:"23710"?A44720A"7-74782?"907"1870219--A"N9-1HU"#141-90-AD"S31-7"

'9R/-"39?"9-4/?2"2S157"9?"498A"92/4?"9?"B

D"12"39?"8/"?A44720AD"98H"?/"/8-A"7-74782?"2392"

5/1851H7"S123"237">01851>9-"9R7?"/:"1870219"4=?2"T7"?790537HU"W189--AD"237"700/0"2/-709857?":/0"

!G"

N901/=?"59-5=-921/8?"9-?/"9::752"237"?>77H"/:"237"9-./01234">0/:/=8H-AU"W/0"7R94>-7D"-/S7018."

237"2/-709857"=?7H":/0"98.-7"5/4>901?/8?"S378"5375V18."1:"9"?A44720A"7-74782"1?"5/1851H782"

S123"237">01851>9-"9R7?"/:"1870219"598"H09492159--A"?>77H=>"237"59-5=-921/8D"T=2"92"237"7R>78?7"

/:">/??1T-A"41??18."9"?A44720A"7-74782"1:"237"4/-75=-7"1?"?7N707-A"H1?2/027HU"

T"###M5,81C=05,=#

" #A447201X70">0/N1H7?"9"?2091.32:/0S90HD":-7R1T-7D"98H"0/T=?2"9-./01234":/0"H727041818."9"

4/-75=-7Y?">/182".0/=>"98H"?A447201X18."12?".7/4720AU"O31-7"?141-90"18"914"2/">9?2"9>>0/9537?D"

231?"9>>0/953"1?"=81^=7"18"2392"12"=21-1X7?"8/2318."T=2"?>9219-".7/4720A"98H"185/0>/0927?"9"31.3"

H7.077"/:"5/820/-"/N70"237"H727521/8"98H"?A447201X921/8">0/57??U"#A447201X70"1?"9T-7"2/"398H-7"

8790-AK?A4472015"4/-75=-7?"9?"S7--"9?"1H79-"/87?D"598"?=..7?2"9--">-9=?1T-A"4925318.">/182"

.0/=>?"S12318"9"=?70K?72"2/-709857D"98H"T/23"-/5927?"952=9-"?A44720A"7-74782?"/8"9"4/-75=-7"

98H"/=2>=2?"18:/04921/8"9T/=2"2374"18"98"/0.981X7H":/0492U",HH121/89--AD"12"1?"14>-747827H"18"

]9N9"98H"?/"598"T7"79?1-A"185/0>/0927H"182/"S7T"9>>-72?"9?"S7--"9?"?298H9-/87"?/:2S907"

>95V9.7?U"

U"###;86,5@1%:+2%,-=#

" B/4>=218."07?/=057?"=?7H"18"231?"07?79053"S9?"?=>>/027H"TA"@921/89-"#517857"

W/=8H921/8"9S90H?"BC&KLFGLPLI"98H"BC&K!LEJJGFU""(]f"98H"OW;".0927:=--A"95V8/S-7H.7"

?=4470"?21>78H?":0/4"@921/89-"#517857"W/=8H921/8"9S90H"BC&KLQGIQLGU"

!E"

!%&%'%,8%="

!U" B/22/8D"WU",U":8"-'2#%+(33%'2#.'0!,+01+D$043+?8"0$&U"O1-7A+"@7S"$/0VD"!JQED"!Je!D"!JJLU"

GU" *9T0/H?VAD"CU\";7-7.D"#U\",N810D"6U"B/8218=/=?"#A44720A"%79?=07?U"K04$!#%+01+.8"+(-"$'2#!+

:8"-'2#%+B02'".&U"!JJED"AAOD"eGPeKeGeJU"

EU" )N98/ND"]U\"#53hh04988D"<U"#14>-7",-./01234?":/0"6727041818."237"%/-75=-90"#A44720AU"

K04$!#%+01+:8"-'2#%+=!10$-#.'0!+#!9+:0-34."$+B2'"!2",U"!JJJD"HVD"PGeKPEPU"

IU" ;1-921D"'U\"W/081D",U\"#$%%[f+",";0/.094"2/":18H"237"%9R14=4"#A44720A"<0/=>"18"98",2/4"

B-=?270D".1N78"9";07:1R7H"'/-709857U"K04$!#%+01+(33%'"9+:$&,.#%%06$#38&U"!JJeD"HAD"FLEKFLIU"

FU" f7N187D")U"@U"G0%"24%#$+B3"2.$0,203&U"O1-7A+"@7S"$/0VD"!JPFU"

QU" #A447201X70"1?"9N91-9T-7"9?"?=>>-747829-"4927019-"/0"92"

SSSUS7T4/U872iH/S8-/9Hi?A447201X70UX1>U"

PU" ;/-1VD"OU"WU\"#5341H2D"]U"(U"L"5G<+M"$,'0!+N;OPOPPJ"D"O7T%["ffB+"C/--98H"%)D"GL!GU"

eU" %U"]U"W01?53D"<U"OU"'0=5V?D"CU"dU"#53-7.7-D"<U"&U"#5=?7019D"%U",U"(/TTD"]U"(U"B377?7498D"<U"

#59-4981D"jU"d90/87D"dU"%788=551D"<U",U";7270??/8D"CU"@9V92?=Z1D"%U"B901592/D"gU"f1D"CU";U"

C09253198D",U"WU")X49A-/ND"]U"d-/18/D"<U"*378.D"]U"fU"#/8878T70.D"%U"C9H9D"%U"&3909D"kU"

'/A/29D"(U"W=V=H9D"]U"C9?7.9S9D"%U")?31H9D"'U"@9V9Z149D"$U"C/8H9D"[U"k129/D"CU"@9V91D"'U"

j07N78D"]U",U"%/82./470AD"]0UD"]U"&U";709-29D"WU"[.-190/D"%U"d790>90VD"]U"]U"C7AHD"&U"d0/2370?D"kU"

@U"k=H18D"jU"@U"#290/N70/ND"(U"k/T9A9?31D"]U"@/0498HD"kU"(9.39N953901D",U"(78H7--D"]U"BU"

d=0982D"#U"#U")A78.90D"]U"'/49?1D"%U"B/??1D"@U"(7.9D"]U"%U"%1--94D"%U"k-787D"]U"&U"k8/RD"]U"dU"

B0/??D"jU"d9VV78D"BU",H94/D"]U"]90941--/D"(U"</4>702?D"(U"&U"#20924988D"[U"$9XA7ND",U"]U"

,=?218D"(U"B9441D"BU";/47--1D"]U"OU"[53270?V1D"(U"fU"%90218D"kU"%/0/V=49D"jU"<U"*9V0X7S?V1D"

!I"

<U",U"j/23D";U"#9-N9H/0D"]U"]U"698878T70.D"#U"69>>0153D",U"6U"69817-?D"[U"W90V9?D"]U"dU"

W/07?498D"]U"jU"[021XD"]U"B1/?-/S?V1D"6U"]U"W/RU"D#4,,'#!+PQD"(7N1?1/8",ULGD"<9=??198D")85U+"

O9--18.:/0H"B'D"GLLJU"

JU" %U"OU"#5341H2D"kU"kU"d9-H01H.7D"]U",U"d/92XD"#U"'U"&-T702D"%U"#U"</0H/8D"]U"CU"]78?78D"#U"k/?7V1D"

@U"%92?=89.9D"kU",U"@.=A78D"#U"]U"#=D"'U"fU"O18H=?D"%U"6=>=1?D"]U",U"%/82./470AU"<947??"

N70?1/8"!GD"]98"GLLJ"a(EcU"]/=089-"/:"B/4>=2921/89-"B3741?20AU"!JJED"!ID"!EIPK!EQEU"

!LU" #27S902D"]U"]U";U"%[;,B"GLLJD"N70?1/8"!!ULFEfU"+]/=089-"/:"B/ 4 >= 27 0K,1H7H"%/-75=-7"67?1.8D+!JJLR+

IR+!O "

---------------------------------------------------------

Symmetry/testFiles/test/H2O.xyz Tolerance: 0.1

---------------------------------------------------------

Principal Axes are non-degenerate

+ ELEMENTS FOUND:

C2 with a rating of 0.098488

sigma with a rating of 0.000000

sigma with a rating of 0.098488

+ POINT GROUP FOUND:

{ E, 2sigma, C2 }

***************************

1. C2v: 0.06565896714675919

***************************

2. Cs: 0.0

3. C2: 0.09848845072013875

4. C1: 0.0

Search took 0.018000 seconds

"#$%#&'!()!!"##$%&'($&)*+##,-.)/'-$)'-%$&0,*$)+1%21%)0+&)-$,&/")3"##$%&'*)4,%$&5)

)

!"#$%&

'()"*+"*,-,&

',."."/0$*1%&

2,#"31/(4(*,&

5/$*&

67.810"%%&

9:,)(."%&;$/)7%"&

()

'#!

-").

01-"

,3!

<&$=&>?$)-&

''4!

,,!

2$!

2/!

@$(*?&A/$73&

#6!

5*B,/-($*&C-D&

/:///!

/://,!

/:///!

@/$3,/&E$?"?($*&C-D&

/:/&,!

/:/#&!

/:';2!

/:4&;!!

5)3/$3,/&E$?"?($*&C-D&

/:///!

/://#!

/:/',!

/:/#/!

E,=%,.?($*&C-D&

/:/'<!

/://;!

/://,!

/://$!

F1)),?/(G"?($*&C-D&

/:///!

/:/'2!

/:/</!

/:/<$!

!$?"%&!(),&C-D&

/:/$'!

/:/$&!

/:,</!

/:&/,!

!"0%,&HI&"=>?@ABC=+!=D!EF>>1GABH1A!@IJ=ABG8>!AK+GB>1C:!

"#!

!"#$%&'()*+",-.!

!"#$%&'/0'$%&'%(&)!*+!,-..&/'(01/(*2!3'*4&,,5!

!"#$%&'10!67*)481'/!*+!3*(2/!,-..&/'-!&7&.&2/!9&/&4/(*2!17:*'(/8.5!

!"#$%&'20';4'&&2,8*/!*+!1..*2(1!)(/8!'&+7&4/(*2!3712&,5!

!"#$%&'30!;4'&&2,8*/!*+!8&<13-'(9(2&('*2!)(/8!=

!1<&,!129!'&+7&4/(*2!3712&5!

"#!

!"#$%&'%%!"#$%&'()'**#+,'(#-#*#$+)((

((((((((((((((.$"()/00#)+(123$+(0,2/1)%

!"#$%('%%4#+#,*3$#()'**#+,'5/$36/#(.+2*)(

(((((((((((((((.$"(.-30$(73+8(,#)1#9+(+2()'**#+,'((

(((((((((((((((#-#*#$+)%

!"#$%)'%%:";/)+()'**#+,'(#-#*#$+)(

(((((((((((((((2&()#-#9+#"(123$+(0,2/1%

!"#$%*'%%<#$#,.+#()'**#+,39(*2-#9/-#(='%

%%%%%%%%%%%%%%%.11-'3$0(.--()'**#+,'(21#,.%2$)(

(((((((((((((((+2()'**#+,'5/$36/#(.+2*)(

"#$%&'!(!

"$!

"#$%&'!)!

%&!

"#$%&'!*!

%"!

"#$%&'!+!