Azulene is a homoaromatic bicycle, isoelectronic with naphthalene. Azulenic patterns have been studied in Mathematical Chemistry in connection with ((5,7)3) ...
MATCH Communications in Mathematical and in Computer Chemistry
MATCH Commun. Math. Comput. Chem. 65 (2011) 173-182
ISSN 0340 - 6253
2PHJD3RO\QRPLDOLQ$]XOHQLF'HQGULPHUVx 09'LXGHDD $(9L]LWLXD)*KRODPL1H]KDGEDQG$5$VKUDILE D )DFXOW\RI&KHPLVWU\DQG&KHPLFDO(QJLQHHULQJ ³%DEHV%RO\DL´8QLYHUVLW\&OXM5RPDQLD
E
'HSDUWPHQWRI0DWKHPDWLFV8QLYHUVLW\RI.DVKDQ .DVKDQ,5,UDQ HPDLO DVKUDIL#NDVKDQXDFLU 5HFHLYHG-XO\
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
x
7KLVSDSHULVGHGLFDWHGWR(GZDUG&.LUE\7KH5HVRXUFH8VH,QVWLWXWH3LWORFKU\3HUWKVKLUH6FRWODQG8.IRUKLV EULJKWFRQWULEXWLRQWRWKHGHYHORSPHQWRI&KHPLFDO*UDSK7KHRU\
-174 ,QWURGXFWLRQ $VHWRIGLVMRLQWIDFHVRIDWULYDOHQWSRO\KHGUDOPDS0IRUPLQJDIDFWRU LHDUHJXODU YDOHQWVSDQQLQJVXEJUDSKEXLOWXSRQDOOYHUWLFHVRIDSRO\KHGUDOPDS0 LVFDOOHGDSHUIHFW &ODU3&VWUXFWXUH,WLVFRPSOHPHQWDU\WRD)ULHVVWUXFWXUHZKLFKLVD.HNXOpVWUXFWXUH KDYLQJWKHPD[LPXPSRVVLEOH Y QXPEHURIEHQ]HQRLG DOWHUQDWLQJVLQJOHGRXEOHHGJH IDFHV$.HNXOp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
E /H 2S &D 7 Y J DVIDFWRU
)LJXUH'LVMRLQWIDFHVHPEHGGHGLQWKHVSKHUH D DQG RSHQ WHWUDKHGURQ E
3DWWHUQV RWKHU WKDQ EHQ]HQH QDSKWKDOHQH FRURQHQH VXPDQHQH RU D]XOHQH LH D
SDLURISHQWDJRQDOKHSWDJRQDOFDUERQULQJV KDYHDOVREHHQFRQVLGHUHG 'LVMRLQWIDFHV FDQ EH HPEHGGHG RQ WKH VSKHUH LQ VXUIDFHV RI YDULRXV JHQHUD VHH )LJXUH E\ PDS RSHUDWLRQVDVLPSOHPHQWHGLQRXURULJLQDOVRIWZDUH&DJH9HUVDWLOH
-175'HQGULPHUV DUH K\SHUEUDQFKHG PDFURPROHFXOHV ZLWK D ULJRURXV DHVWKHWLFDOO\ DSSHDOLQJ DUFKLWHFWXUH 7KH\ FDQ EH V\QWKHVL]HG LQ D FRQWUROOHG PDQQHU HLWKHU E\ GLYHUJHQWPHWKRGVRIE\FRQYHUJHQWRQHV$VHOIDVVHPEO\SURFHVVZRXOGEHH[SHFWHGIRU GHQGULPHUV LIDQ\ REWDLQDEOHE\DJJUHJDWLRQRIYDSRUL]HGJUDSKLWH
7KHVH ULJRURXVO\ WDLORUHG VWUXFWXUHV UHDFK UDWKHU VRRQ D VSKHULFDO VKDSH WKXV
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
$]XOHQLFVWUXFWXUHV $]XOHQH &+ LV D KRPRDURPDWLF ELF\FOH LVRHOHFWURQLF ZLWK WKH QDSKWKDOHQH WR ZKLFKFDQUHDUUDQJHE\KHDWLQJEXWLWLVEOXHLQFRORUDQGLWVSK\VLFRFKHPLFDOSURSHUWLHV GLIIHUIURPWKRVHRIQDSKWKDOHQH$]XOHQLFSDWWHUVZHUHH[WHQVLYHO\VWXGLHGE\.LUE\ $ WRWDOUHVRQDQWD]XOHQRLGVWUXFWXUHZDVGHILQHGDVDFDUERQQHWZRUNZLWKDWOHDVWRQH.HNXOp VWUXFWXUHHTXLYDOHQWWRDVHWRID]XOHQHXQLWVDOOLQWHUFRQQHFWHGE\HVVHQWLDOO\VLQJOHERQGV ,WLVWKHUHDOL]DWLRQRIDJUDSK*WKDWHYHU\ZKHUHLVORFDOO\SODQDUDQGDOORZVDIDFWRURI GLVMRLQWF\FOHVGHILQHGRQHYHU\YHUWH[
-1767KHSDWWHUQJHQHUDWHGE\WKHVHTXHQFHRIPDSRSHUDWLRQV 2S /H 2S &D 0 LVD GLVMRLQW D]XOHQRLG REH\LQJ WKH .LUE\¶V GHILQLWLRQ )LJXUH LOOXVWUDWHV DQ D]XOHQLF $]X SDWWHUQDQGLWVFRPSOHPHQW DWULQDSKWK\OHQLF713SDWWHUQ HPEHGGHGLQWKHWHWUDKHGURQ
D 2S /H 2S &D 7 $]X Y H I J
E 2S /H 2S &D 7 713
)LJXUH$]XOHQLFSDWWHUQ D DQGLWVWULQDSKWK\OHQLF FRSDWHUQ E
$XQLWOLNHWKDWLQ)LJXUHDFDQMRLQLWVHOIWRJLYHHLWKHUDOLQHDUVWUXFWXUH )LJXUH
D RUDVXSUDGRGHFDKHGURQ )LJXUHE RUFDQDUUDQJHLQDGHQGULWLFWUHHRISURJUHVVLYH GHJUHHS )LJXUH E 6XSUDGRGHFDKHGURQE\$]XXQLWV Y J
D ,QILQLWH$]XOLQHDUVWUXFWXUH Y
)LJXUH7ZRSRVVLEOHZD\VLQHYROXWLRQRIWKHD]XOHQLFXQLWLQ)LJXUHD
-177 D 0RQRFHQWULF$]XGHQGULPHU Y P
E 'LFHQWULF$]XGHQGULPHU Y P
)LJXUH0RQRFHQWULF D DQGGLFHQWULF E D]XOHQLFGHQGULPHUVDWWKHILUVWJHQHUDWLRQVWDJH
2PHJDSRO\QRPLDO &RXQWLQJRIPRQRPHULFXQLWVDQGWKHLUFRQVLVWLQJVXEVWUXFWXUHVLVWKHVXEMHFWRI&KHPLFDO *UDSK 7KHRU\ 6HYHUDO FRXQWLQJ SRO\QRPLDOV KDYH EHHQ SURSRVHG IRU FKDUDFWHUL]LQJ WKH WRSRORJ\RIQDQRVWUXFWXUHVSDUWLFXODUO\RIGHQGULPHUV /HW * 9( EH D FRQQHFWHG ELSDUWLWH JUDSK ZLWK WKH YHUWH[ VHW 9 * DQG HGJH VHW ( * 7ZR HGJHV H [\ DQG I XY RI * DUH FDOOHG ³RSSRVLWH´ H RS I LI WKH\ DUH WRSRORJLFDOO\SDUDOOHO G [ Y
G [ X G \ Y G \ X
$ VHW RI RSSRVLWH HGJHV & H ^ I ( * I RS H` ZLWKLQ WKH VDPH IDFHULQJ HYHQWXDOO\ IRUPLQJ D VWULS RI DGMDFHQW IDFHVULQJV LV FDOOHG DQ RSSRVLWH HGJH VWULS RSV 8QGHU RSV FORVXUH ( * LV WKH XQLRQ RI GLVMRLQW RSV & & & N DQG &L &M
IRU
L z M L M N 7KHUHODWLRQRSVLVQRWQHFHVVDULO\WUDQVLWLYH /HWP *F EHWKHQXPEHURIRSVRIOHQJWKF 2PHJDSRO\QRPLDOLVGHILQHGDV
: * [
¦ F P * F [F
7KHILUVWGHULYDWLYH LQ[ HTXDOVWKHQXPEHURIHGJHVLQWKHJUDSK
:c *
¦ P * F F F
H
( *
$WRSRORJLFDOLQGH[FDOOHG&OXM,OPHQDX&, &, * ZDVGHILQHGRQ2PHJDSRO\QRPLDO
-178&, *
^>:c * @ >:c * :cc * @`
,WLVHDVLO\VHHQWKDWIRUDVLQJOHRSVRQHFDOFXODWHVWKHSRO\QRPLDO : * [ WKHLQGH[ &, *
F F F F
F F
u [ F DQG
,QWUHHJUDSKVWKH2PHJDSRO\QRPLDOVLPSO\FRXQWVWKHQRQRSSRVLWHHGJHVEHLQJ LQFOXGHGLQWKHWHUPRIH[SRQHQWF 0RUHDERXW2PHJDSRO\QRPLDOWKHUHDGHUFDQILQGLQ UHIV
2PHJDSRO\QRPLDOLQQDQRGHQGULPHUV 7KH2PHJDSRO\QRPLDOVWULFWO\IROORZVWKHSRO\JRQDOFRYHULQJRIVWUXFWXUHDQGFRQVLVWVRI IRXUWHUPV7KHWHUP DW WKHKLJKHVWH[SRQHQWF FRXQWVWKHRSSRVLWHHGJHVEHWZHHQWZR MRLQHG WHWUDSRGDO XQLWVPRQRPHUV P )LJXUH WKH QXPEHU RI VXFK VWULSV EHLQJ LW IROORZV WKDW D P 1H[W WHUP DW F FRXQWV QDSKWK\OHQLF XQLWV UHODWHG WR WKH WULQDSKWK\OHQHV )LJXUHE VL[IRUHDFKIDFHRIWKHWHWUDKHGURQDQGWKUHHVKDUHGEHWZHHQ WZR IDFHV uu ZKLOH DFFRXQWLQJ IRU WKH ERQGHG EUDQFKHV DQG FURVVMXQFWLRQ D P 7KH WHUP DW F DFFRXQWV IRU LVRODWHG RSSRVLWH HGJHV EHLQJ uu LQ RQH PRQRPHU WR ZKLFK VWULSV DUH DGGHG IRU HDFK MXQFWLRQ WR JLYH D P )LQDOO\ WKH QRQRSSRVLWH HGJHV DUH FRXQWHG uu SHU XQLW DQG D P )RUPXODV DUH OLVWHGLQ7DEOH
)LJXUH'LFHQWULFD]XOHQLF$]XGHQGULPHU DWJHQHUDWLRQU Y P VWULSF
7DEOH&RPSRVLWLRQUXOHVIRU2PHJDSRO\QRPLDOLQ$]XGHQGULPHUVGHVLJQHGE\ 2S /H 2S &D 7 RSHUDWLRQV
-179 )RUPXODV : ' D]X [
D ; D ; D ; D ;
&, 'D]X >:c ' @ >:c ' :cc ' @
0RQRFHQWULF'HQGULPHU : ' P [ P ; P ; P ; P ; D]X :c ' P :cc ' P D]X
D]X
&, ' P P D]X Y ' P D]X 5>@ ' P 5>@ ' P 5>@ '
D]X D]X D]X P 'LFHQWULF'HQGULPHU : ' P [ P ; P ; P ; P ; D]X
: ' D]X N [ N ; N ; N ; N ; 1XPEHURIPRQRPHUVPDWWKHVWDJHJHQHUDWLRQU U S U P ' S U S S L SU
¦
P 0' D]X
U
L
S
P '' D]X
U S U
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
-180&RQFOXVLRQV 'HQGULPHUVDUHKLJKO\RUGHUHGK\SHUEUDQFKHGVWUXFWXUHVDWWKHQDQRGLPHQVLRQV&RPSOH[ QDQRGHQGULPHUVFDQEHGHVLJQHGE\XVLQJVHTXHQFHVRIPDSRSHUDWLRQV&RPSRVLWLRQRI WKH JOREDO SRO\QRPLDO IXQFWLRQ RI PRQRPHULF FRQWULEXWLRQV ZDV GHULYHG IRU ERWK PRQR DQGGLFHQWULFGHQGULPHUVDFFRUGLQJWRWKHLUWRSRORJ\&RXQWLQJIRUPXODVIRUWKHQXPEHU RIPRQRPHUVLQGHQGULPHUVJURZQDWYDULRXVJHQHUDWLRQVZHUHDOVRJLYHQ
$FNQRZOHGJHPHQWV 7KHDUWLFOHLVVXSSRUWHGE\WKH5RPDQLDQ*UDQW31,,1R
5HIHUHQFHV >@ 3:)RZOHU73LVDQVNL-&KHP6RF)DUDGD\7UDQV >@ (&ODU3RO\F\FOLF+\GURFDUERQV$FDG3UHVV/RQGRQ >@ (&ODU7KH$URPDWLF6H[WHW:LOH\1HZ@ .)ULHV/LHELJV$QQ&KHP >@ , *XWPDQ 6 - &\YLQ ,QWURGXFWLRQ WR WKH 7KHRU\ RI %HQ]HQRLG +\GURFDUERQV 6SULQJHU%HUOLQ >@ -5'LDV-&KHP,QI&RPSXW6FL >@ ' - .OHLQ + =KX LQ $ 7 %DODEDQ (G )URP &KHPLFDO 7RSRORJ\ WR 7KUHH 'LPHQVLRQDO*HRPHWU\3OHQXP3UHVV1HZ@ (&.LUE\0$7&+&RPPXQ0DWK&RPSXW&KHP > @ - 9 .QRS : 5 0ĦOOHU . 6]\PDQVNL 6 1LNROLü 1 7ULQDMVWLü 0$7&+ &RPPXQ0DWK&RPSXW&KHP >@ 09'LXGHD3(-RKQ$*UDRYDF03ULPRUDF73LVDQVNL&URDW&KHP$FWD
>@ 09'LXGHD)RUPD 7RN\R
-181>@ 09'LXGHD0ùWHIX3(-RKQ$*UDRYDF&URDW&KHP$FWD >@ 0ùWHIX09'LXGHD3(-RKQ6WXGLD8QLY%DEHV%RO\DL >@ 09'LXGHD-&KHP,QI0RGHO >@ 0ùWHIX09'LXGHD³%DEHV%RO\DL´8QLY >@ * 5 1HZFRPH 9 . *XSWD * 5 %DNHU = 4@ '$7RPDOLD$OGULFKLPLFD$FWD >@ &-+DZNHU-0-)UHFKHW-$P&KHP6RF > @ &-+DZNHU./:RROH\-0-)UHFKHW-&KHP6RF3HUNLQ7UDQV >@ *51HZFRPH&10RRUHILHOG*5%DNHU0-6DXQGHUV6+*URVVPDQ $QJHZ&KHP,QW(G(QJO >@ * + (VFDPLOOD * 5 1HZFRPH $QJHZ &KHP ,QW (G (QJO >@ *51HZFRPH&10RRUHILHOG)9RJWOH'HQGULWLF0DFURPROHFXOHV &RQFHSWV 6\QWKHVHV3HUVSHFWLYHV9&+:HLQKHLP >@ 07DQJ&75HGHPDQ)&6]RND%LRFRQMXJDWH&KHP >@ %)3DQ';&XLҘ3;X7+XDQJ4/L5+H)*DR-%LRPHG3KDUPDFHXW (QJ >@ 09'LXGHD*.DWRQD$GYDQ'HQGULWLF0DFURPRO 09'LXGHD0$7&+&RPPXQ0DWK&RPSXW&KHP 09'LXGHD0$7&+&RPPXQ0DWK&RPSXW&KHP $63IDX3$3ODWWQHU+HOY&KLP$FWD '3LHVVH&KHP1HZV
-182'3LHVVH&RPSW5HQG 3$3ODWWQHU$63IDX+HOY&KLP$FWD ( & .LUE\ LQ 0 9 'LXGHD (G 1DQRVWUXFWXUHV 1RYHO $UFKLWHFWXUH 1RYD 1HZ
