167 T componnt oftforc on atom B can b drvd as: F B =, E =,K (, ) (.2) wr w av usd 2 = ( ) =2 (.3) T scond drvatv: 2 E = K (, ) = K (1, ) + 3 (.4).2.2

Transcription

1 166 ppnd Valnc Forc Flds.1 Introducton Valnc forc lds ar usd to dscrb ntra-molcular ntractons n trms of 2-body, 3-body, and 4-body (and gr) ntractons. W mplmntd many popular functonal forms n our program..2 Bond Bond ntractons ar ncludd for any two atoms wc ar \connctd." No spcal squnc s assumd. Two typs of bond paramtrs ar currntly mplmntd; ty ar Harmonc (Typ=1) and Mors (Typ=2). Fgur.1: Bond ntracton.2.1 Harmonc Bond T smplst nrgy prsson s t armonc ntracton E = 1 2 K (, ) 2 (.1) wr t qulbrum bond dstanc s n ngstroms () and t forc constant K s n (kcal/mol)/ 2.

2 167 T componnt oftforc on atom B can b drvd as: F B =, E =,K (, ) (.2) wr w av usd 2 = ( ) =2 (.3) T scond drvatv: 2 E = K (, ) = K (1, ) + 3 (.4).2.2 Mors Bond T smplst nrgy prsson capabl of dscrbng bond dsrupton s t Mors ntracton E = D (, 1) 2 (.5) wt =,(,) (.6) Hr s t qulbrum bond dstanc (), D s t bond nrgy (kcal/mol), and s a paramtr rlatd to K (t curvatur or forc constant at ). T zro nrgy s dnd so tat E =0at. T paramtr s rlatd to K and D as

3 168 n Equaton.7: = r K 2D (.7) T componnt oftforc on atom B: F B =, E =,2D (, 1) = 2D (, 1) (.8) Hr w av usd: =,,(,) =, (.9) T scond drvatvs: 2 E =,2D (, 1).3 ngl =,2D, (2, 1) = 2D (2, 1) 2 + (, 1), (, 1) 2 + (, 1) 3, (, 1) (.10) ngl ntractons ar ncludd for anytwo atoms tat ar bondd to a common atom. urrntly four typs of angl functonals ar mplmntd; ty ar Harmonc osn (Typ=1), Harmonc osn wt Strtcng (Typ=11), Harmonc (Typ=21), and Harmonc wt Strtcng (Typ=31). s n Fg..2, w av

4 169 Fgur.2: ngl ntracton cos = (.11) Bfor w gt nto computaton of forcs and scond drvatvs, w'll drv two usful ntts. cos =, cos (.12) ( ) 2 and 2 cos =, cos ( ) 2 =, ( ) cos ( ) 4, cos ( ) 2,, cos ( ) 2 ( ) 2 (.13).3.1 Harmonc osn ngl Gnral as T cosn armonc potntal as t followng form: E = 1 2 (cos, cos ) 2 (.14)

5 170 wr s t qulbrum angl and s a paramtr rlatd to t forc constant. Drntat Equaton.14 wt rspct to cos, and w obtan E 0 = E 00 = E cos = (cos, cos ) 2 E 2 cos = (.15) (.16) Tus t nrgy drvatvs ar E 2 E 2 =,E 0 sn (.17) =,E 0 cos, E 00 sn cos = E 00 sn 2, E 0 cos (.18) Tn t forc constant K bcoms ladng to K = 2 E 2 = sn 2 (.19) = K sn 2 (.20) T zro of nrgy s dnd so tat E( ) = 0, nc t barrr toward lnarzaton s E barrr = E(180 o )= 1 2 (1 + cos ) 2 (.21) In gnral, t paramtrs n forc ld dnton ar n trms of (dgr) and K ((kcal/mol)/rad 2 ).

6 171 T componnt oftforc on atom s: F =, E =,(cos, cos ) cos =,(cos, cos ), cos ( ) 2 (.22) T scond drvatv on atom s: 2 E = (cos, cos ) cos = cos cos + (cos, cos ) 2 cos =, cos ( ) 2, cos ( ) 2,(cos, cos ) ( ) + cos ( 3 ( ), 2 2 ( ) ) 4,(cos, cos ), cos (.23) ( ) 2 ( ) 2 Smlarly, t forc on atom F =,(cos, cos ), cos ( ) 2 (.24) and t scond drvatv on atom 2 E =, cos ( ) 2, cos ( ) 2,(cos, cos ) ( ) 3 + cos ( ( ) 2, 2 ( ) 4 ),(cos, cos ), cos ( ) 2 ( ) 2 (.25)

7 172 T forc on atom B F B =,F, F (.26) and t scond drvatv on atom B 2 E B B =, 2 E, 2 E (.27) Lnar as s! ; sn! 0, and n Equaton.20 gos to nnty. onsquntly, for t lnar gomtry t Equaton.14 s rplacd by E = (1 + cos ) (.28) For angls clos to lnar, ',, cos ', (.29) ladng to E = (.30) Tus s now t forc constant of t lnar molcul. T componnt oftforc on atom s: F =, E = =, cos, cos ( ) 2 (.31)

8 173 T scond drvatv on atom s: 2 E =, cos =, 1 ( ) 2, ( ) 2, cos ( ) 2 ( ) 2 = Smlarly, t forc on atom s: +2 cos ( ), cos 4 ( ) 2, + ( ) cos ( ) 4, cos ( ) 2 F = T scond drvatv on atom s: 2 E, cos ( ) 2 =, cos ( ) 3 ( ), cos 4 ( ) 2 (.32) (.33) (.34) gan for atom B, F B =,F, F ; (.35) and 2 E B B =, 2 E, 2 E : (.36).3.2 Harmonc osn ngl oupld wt Bond Strtc Gnral as For good dscrpton of vbratonal frquncs, t s ncssary to us an angl trm tat dpnds on bond dstanc. T smplst suc potntal avng t propr sym-

9 174 mtry, E(, ) =E(+), s E(; ; )= 1 2 (cos, cos ) 2 + D(, )(cos, cos ) +E(, )(cos, cos )+F(, )(, ) (.37) T componnt oftforc on atom for ts nrgy prsson s: F =, E =, (cos, cos ) cos, D(, ) cos, E(, ) cos =,, D(cos, cos ), F (, ) (cos, cos )+D(, )+E(, ) cos, D(cos, cos ), F (, ) =, (cos, cos )+D(,)+E(,) (,cos ( ) ) 2, D(cos, cos ), F (, ) (.38) T scond drvatv at atom : 2 E = 2 cos + cos (cos, cos )+D(,)+E(,) cos + D + +D cos + D(cos, cos + F (, ), F ( ) 3 (, ) (.39) wr 2 cos = and cos = av bn drvd n prvous sctons.

10 Lnar as 175 For lnar molculs ( = ) w us t followng E = (1 + cos ) + D(, )+E(,) +F(, )(, ) (.40) T forc on atom s: F =, E =, cos + D(, )+E(,) + (cos +1)D +F(, ) =,,cos + T scond drvatv on atom : 2 E ( ) 2 D(cos +1)+F(, ) + D(, )+E(, ) = 2 cos + D(, )+E(,) +D cos +, ( ) 3 D(cos +1)+F(, ) (.41) (.42) T computatons of forcs and scond drvatvs on atom ar smlar to tat of atom ; forcs and scond drvatvs on atom B follow tat of prvous sctons..3.3 Smpl Harmonc ngl Gnral as E() = 1 2 K (, ) 2 (.43)

11 176 For t smpl armonc potntal Equaton.43, (dgr) s t qulbrum angl wl K ((kcal/mol)/rad 2 ) s t forc constant. Snc = cos cos =, 1 cos sn (.44) T forc on atom s: F = K(, )sn,1 cos (.45) T scond drvatv on atom 2 E =, K (, )sn,1 cos =, K(, )sn,1 2 cos + K sn,2, (, ) cos cos cos sn 3 (.46) wr w av usd sn = sn cos cos =, cos cos sn (.47) Lnar as ssumng =,, w av E = 1 2 K()2 (.48) Snc cos(, ) =, O( 3 ) (.49)

12 177 t forc on atom to scond-ordr s: F =, 1 2 K 2 cos cos =, 1 2 K 2 cos ( ) =, K cos (.50) T scond drvatv on atom s: 2 E = K 2 cos (.51).3.4 Harmonc ngl oupld wt Bond Strtc Gnral as T gnral prsson wc coupls a armonc angl wt bond strtc s: E = 1 2 K(, ) 2 + D(, )(, ) + E(, )(, )+F(, )(, ) (.52) T forc on atom s: F =, E cos =, K(, ) cos, cos cos, D(, )+E(, ) D(, )+F(, )

13 178 =K(, )sn,1 cos + D(, )+E(,), D(, )+F(, ) sn,1 cos (.53) T scond drvatv at atom s: 2 E Lnar as =, K(, )+D(,)+E(,) sn,1 2 cos cos K(, )+D(,)+E(,) cos cos,,, Ksn,1 cos + D(, )+F(,), Dsn,1 cos Tr s no dnton for lnar cas. + D sn,1 cos (, ( ) 3 ) sn 3 (.54).4 Torson Fgur.3: Torson ntracton torson ntracton s dnd wt rspct to four atoms I, J, K, L as n Fg..3.

14 179 T bonds ar I{J, J{K, and K{L. Dn: Lt's dn F = I, J G = J, K H = L, K ; (.55) and =F G B =H G: (.56) Tn cos = B B (.57) T torson nrgy s a functon of tr t ddral angl, or cos. For = I; J; K; or L, w'll drv t ntts usd n computng of forcs and scond drvatvs. cos = 1 = 1 = ( ) B B + ( B B ), 2, ( B B + 1 B B ) B 3 B + 1 B B = 1 (B ), ( ) B B 3 B + 1 B ( ), (B ) B B B 3, B B B 2, (B B ) B B 3 (.58) and 2 cos 1 =, ( ) 1 3 B B, (B B ) B + B 3

15 , ( ) 1 B 3 B + 1, ( ) B 3 B B + 1 B Furtr pandng, w av:, (B B ) B B 3, (B B ) B : (.59) B 3 ( 1 )=, ( 1 B )=, 1 B B B + 1 B B 3 B ( ) = (B B ) B B = B 3 B B + B 1 ) B 2 3, 3 5( ) B B + B 2 B B 3 B 3 1 ) B 3 B, 3 B B B5(B ) (.60) (.61) (.62) (.63) wr w av usd B = 1 ( ) = 2 = B B B (.64) (.65) Now lt's comput t ndvdual trms = (F G) = ( F G ) F G = G + F (.66)

16 181 wr s t Lv-vta vctor. Smlarly B H G = G + H (.67) For = I, wc s atom I, w av Smlarly I = I = G =, G =,( G) (.68) B I B I B B I For = J, wc s atom J, wav J B J B J B B J For = K, wc s atom K, wav =, (B G) (.69) =0 (.70) =0 (.71) =( G), (F ) =(B G), (F B) (.72) (.73) =, (H ) (.74) =, (H B) (.75) B K K B K =, ( F) (.76) =, (B F) (.77) =( G) +(H) (.78)

17 B B K 182 =(B G) +(HB) (.79) nd for = L, wc s atom L, w av L B L B L B B L = 0 (.80) = 0 (.81) =,( G) (.82) =,(B G) (.83) Trms usd n scond drvatvs 2 = 2 ( F G ) F G = + F G (.84) 2 I I 2 JJ 2 KK 2 LL = 0 (.85) = (,, ) =,, =0 (.86) = 0 (.87) = 0 (.88) Smlarly 2 B H G = + H G (.89) 2 B I I 2 B JJ = 0 (.90) = 0 (.91)

18 183 2 B KK 2 B LL = ( + ) = + =0 (.92) = 0 (.93) For = I I I = (F I G ) (F I G ) = G G = G G I B I B I B I = (, )G G = (G G), G G (.94) = 0 (.95) = 0 (.96) For = J J J = (, G + F ) (, G + F ) = G G, G F, F G + F F J B J B J B J = (G G), G G + (F F), F F + (G F), G F + (F G), F G (.97) = (, G + F ) H =, G H + F H =,G H + (G H)+ (F H), F H (.98) = H H = (H H), H H (.99)

19 184 For = K K K B K K B K B K = F F = F F = (F F), F F (.100) = (,F ) (, G, H ) = F G + F H = F G, (F G)+ (F H), H F (.101) = (, G, H ) (, G, H ) = G G + H G + G H + H H = (G G), G G, (H G)+G H, (G H)+H G + (H H)+H H (.102) For = L L L B L L B L B L = 0 (.103) = 0 (.104) = G G = G G = (G G), G G (.105).4.1 Pur Torson T gnral nrgy s dnd by: E() = nx =0 cos() (.106)

20 185 wr s t angl btwn plan IJK and JKL. T sum can nclud up to n = 12. Typcally, all of t ar zro, cpt 0 computd by t rcursv formula and 3. Knowng cos, cos(n) can b cos(n) = cos[(n, 1)] cos, cos[(n, 2)] (.107) In ordr to comput t forcs and scond drvatvs, w nd cos(n) cos = cos(n) = n sn(n) sn cos (.108) and 2 cos(n) 2 cos = n sn(n) sn cos sn(n) cos, n cos(n) sn = n sn 3 (.109) wr t sns can b computd from t cosns. Wn,! 0, cos(n) cos = n lm n,!0 = n2 (.110) and 2 cos(n) 2 cos = n lm,!0 o n(1, ), n(1, 1 2 n2 2 ) 3 = 1 2 n2 (n 2, 1) (.111) Havng t abov ntts and formulas n t prvous scton, w can comput t forc of atom, wt = I; J; K; or L as: F =, cos(n) =, cos(n) cos cos (.112)

21 186 and t scond drvatv as 2 cos(n) = cos(n) cos 2 cos + 2 cos(n) 2 cos cos cos (.113).4.2 ross ouplng For can molculs, an accurat dscrpton of tr rotatonal vbraton frquncs oftn rqurs t ntroducton of cross couplng trms. Tr typs of cross couplngs ar mplmntd, bond cross couplng, cosn angl cross coupng, and angl cross couplng. Bond ross ouplng In addton to t pur torson trms, t couplng nrgy of t two outr bonds (IJ and KL) s ntroducd as E = K bb ( IJ, IJ )( LK, LK ) (.114) wt addtonal forcs on t four atoms as and addtonal scond drvatvs as 2 E I I 2 E J J F I =,K bb ( LK, LK F J = K bb ( LK, LK F K =,K bb ( IJ, IJ F L = K bb ( IJ, IJ = K bb ( LK, LK ) = K bb ( LK, LK ) ) IJ IJ ) IJ IJ ) LK LK ) LK LK IJ, IJ IJ, IJ IJ ( IJ ) 3 IJ ( IJ ) 3 (.115) (.116) (.117) (.118) (.119) (.120)

22 2 E KK 2 E LL 187 = K bb ( IJ, IJ ) = K bb ( IJ, IJ ) LK, LK LK, LK LK ( LK ) 3 LK ( LK ) 3 (.121) (.122) osn ngl ross ouplng Smlar to bond cross couplng, ts s also an addtonal trm. T nrgy wc coupls cosns of t two angls ( IJK or JKL ) wt t torson angl s gvn as E = K aa (cos IJK, cos IJK )(cos JKL, cos JKL ) cos (.123) For = I; J; K, and L, t addtonal forc s gvn by: F =, E cos IJK =,K aa cos,k aa (cos IJK, cos IJK (cos JKL, cos JKL )(cos JKL, cos JKL ) cos and t addtonal trms n t scond drvatv ar: 2 E = K aa cos cos IJK (cos JKL, cos JKL ) + (cos IJK, cos IJK cos JKL ) cos IJK cos JKL cos IJK cos JKL +K aa cos + +K aa cos 2 cos IJK (cos JKL, cos JKL ) +K aa cos (cos IJK, cos IJK ) 2 cos JKL cos IJK +K aa cos +K aa (cos IJK, cos IJK (cos JKL, cos JKL (.124) ) + (cos IJK, cos IJK cos JKL ) )(cos JKL, cos JKL ) 2 cos ) + (cos IJK, cos IJK cos JKL ) (.125)

23 ngl ross ouplng 188 Smlar to cosn angl cross couplng, r t two angls ( IJK and JKL ) ar coupld wt t torson angl to gv an nrgy: E = K aa ( IJK, IJK )( JKL, JKL ) cos (.126) T addtonal forcs ar gvn by F =, E JKL, JKL = K aa sn IJK IJK, IJK +K aa sn JKL,K aa ( IJK, IJK cos IJK cos cos JKL cos )( JKL, JKL ) cos (.127) and addtonal trms for scond drvatvs ar gvn by 2 E JKL, JKL =,K aa sn IJK JKL, JKL,K aa sn IJK cos IJK 2 cos IJK + IJK, IJK sn JKL + IJK, IJK sn JKL cos JKL cos 2 cos JKL cos 1 1 cos IJK cos JKL cos IJK cos JKL +K aa cos + sn IJK sn JKL +K aa cos (JKL, JKL ) cos IJK cos IJK cos IJK sn 3 IJK +K aa cos (IJK, IJK ) cos JKL cos JKL cos JKL sn 3 JKL +K aa ( IJK, IJK )( JKL, JKL ) 2 cos (.128) wr can b I;J;K, or L and smlar drvaton of cos =, cos IJK =, cos JKL =, 2 cos IJK =, 2 cos JKL =, and 2 cos = can b found n prvous sctons.

24 .5 Invrson 189 onsdr a cas n wc tr (and only tr) atoms, J, K, and L ar bondd to a cntral atom I. T umbrlla moton n wc t angl of IL wt rspct to t plan IJK cangs from plus to mnus somtms rqurs a spcal ntracton trm; ts s t nvrson trm. Two typs of nvrson trms ar mplmntd. Ty ar mbr mpropr torson and spctroscopc nvrson..5.1 mbr Impropr Torson Fgur.4: mbr mpropr torson For atom J, K, and L bondd to atom I, wav tr ddral angls. T nrgy s t avrag of t tr \torson trms." For angl kjl, dnd as t angl btwn t LIJ and KIJ plans, t nrgy s takn as E LIJ,KIJ = 1 2 cos(n kjl) (.129) So E nv = 1 6 cos(n jkl ) + cos(n kjl ) + cos(n klj ) (.130) wr N = 2 for planar and N = 3 for ttradral. Wt N = 2, t potntal as a mnmum for planar ( = 180 o ) and t mama at =90;270 o. Tr s also a mnmum at =0 o. Wt N = 3, t potntal as a mamum at 180 o and mnmua

25 190 at 120 o and 240 o, followd by mama at 60 o and 300 o. Enrgy, forcs and scond drvatvs can b calculatd as torson ntracton..5.2 Spctroscopc Invrson Fgur.5: Spctroscopc nvrson T spctroscopc nvrson nrgy s dnd by E nv = 1 6 (cos j, cos ) 2 + (cos k, cos ) 2 + (cos `), cos ) 2 (.131) wt angl l as t angl btwn ln IL and plan IJK, k as t angl btwn ln IK and plan IJL, and j as t angl btwn ln IK and plan IKL. W'll only consdr trm cos l ; t otr two trms can b calculatd accordngly. Dnng = IJ IK,wav: sn l = cos l = IL IL q (.132) 1, sn 2 `: (.133) Lt: E` = 1 6 (cos `, cos ) 2 (.134)

26 191 For = I; J; K, or L, w av F = E l = 1 3 (cos l, cos ) cos l sn l sn l =, 1 3 (cos l, cos ) sn l cos l sn l (.135) and 2 E l = 1 3 (sn l ) sn 2 l sn l, 1 cos l 3 (cos l, cos ) sn l 2 sn l cos l, 1 3 (cos l, cos )cos,3 l sn l sn l (.136) Furtr sn l = 1 IL(IL + IL IL ), (IL IL )( IL ( IL ) 3 ) 1, (IL IL 3)( ) (.137) and 2 sn l = 1 + IL IL + + IL IL 1 IL IL 1 IL IL 1, ( IL IL ) IL, ( IL IL ) IL ( IL ) 3, ( IL IL ) IL, ( ) 3 1 ( IL ) 3 1 ( IL ) 3, ( ) 3, ( ) 3 (.138)

27 192 Furtr pand ts to gv: ( 1 IL IL ) =, 1 IL IL IL + 1 ( IL ) 3 IL 2 IL (.139) ( 1 ) =, (.140) ( IL IL ) IL = ( IL ) 3 IL IL IL ( IL ) + IL 2 IL IL 3 ( IL ) 3 + IL IL 1 IL ), 3 IL IL ) ( IL ) 3 ( IL ) 5(IL (.141) ( ) 3 = 1 + ) , 3 5( ) (.142) plac t ndvdual trms n t abov quatons by t followng for atom I: IL I =, (.143) I =, IK, IJ (.144) 2 I = + =0 (.145)

28 193 and for atom J: IL J = 0 (.146) J = IK (.147) 2 J J = 0 (.148) and for atom K: IL K = 0 (.149) K = IJ (.150) 2 K K = 0 (.151) Fnally, for atom L IL L 2 IL LL L = (.152) = 0 (.153) = 0 (.154) Usng ts, w can complt t trms of forcs and scond drvatvs orgnatd from E l. Trms of E j and E k can b computd accordngly..6 Strss ontrbutons Snc t valnc forc lds only dpnd on atomc postons, nrgy consrvaton appls. Tus f s t unt cll volum and t strss, w can wrt t valnc

29 194 strss contrbuton as = NX =1 F (.155) wr N s t numbr of atoms n t unt cll, F s t forc of atom, wl s t coordnats of atom.