TitleEnergy density concept: A stress te.

TitlEnrgy dnsity concpt: A strss t Author(s) Tchibn, Akitomo Cittion Journl of Molculr Structur: THE 3): 138-151 Issu Dt 010-03-15 URL http://hdl.hndl.nt/433/14569 RightCopyright 009 Elsvir B.V. Typ Journl Articl Txtvrsion uthor Kyoto Univrsity

Enrgy dnsity concpt: strss tnsor pproch Akitomo Tchibn Dprtmnt of Micro Enginring, Kyoto Univrsity, Kyoto 606-8501, JAPAN E-mil : kitomo@scl.kyoto-u.c.jp (rcivd: 1 August 009) Abstrct Concptul insights from th dnsity functionl thory hv bn normously powrful in th filds of physics, chmistry nd biology. A nturl outcom is th concpt of nrgy dnsity s hs bn dvlopd rcntly: drop rgion, spindl structur, intrction nrgy dnsity. Undr xtrnl sourc of lctromgntic filds, chrgd prticls cn b cclrtd by Lorntz forc. Dissiptiv forc cn mk th stt of th chrgd prticls sttionry. In quntum mchnics, th nrgy ignstt is nothr rul of th sttionry stt. Tnsion dnsity of quntum fild thory hs bn formultd in such wy tht it cn compnst th Lorntz forc dnsity t ny point of spc-tim. This formultion cn giv mchnicl dscription of locl quilibrium lding to th quntum mchnicl sttionry stt. Th tnsion dnsity is givn by th divrgnc of strss tnsor dnsity. Elctronic spin cn b cclrtd by torqu dnsity drivd from th strss tnsor dnsity. Th spin torqu dnsity cn b compnstd by forc dnsity, clld zt forc dnsity, which is th intrinsic mchnism dscribing th sttionry stt of th spinning motion of lctron. Ky Words nrgy dnsity; strss tnsor; spin torqu 1

1. Introduction Th id of th dnsity functionl thory (DFT) by Hohnbrg-Kohn [1] is tht with th xtrnl potntil vr givn, th lctronic ground stt nrgy is uniqu functionl of th lctron numbr dnsity n r : 3 E nr Fnr d rv r n r, (1.1) whr 3 drnr F nr N, (1.) is th univrsl functionl of nr intgrtd to giv th totl lctron numbr N. Th xtrnl potntil vr is th clmpd-nucli on-lctron lctrosttic potntil. Th id ws xtndd to grnd cnonicl nsmbl by Mrmin []. Concptul insights from th DFT hv bn normously powrful in th filds of physics, chmistry, nd biology [3]. A nturl outcom is th rliztion of nr nd nr, which mns tht th wv function nd th dnsity mtrix r givn s th functionls of nr. Bsd on this rliztion, th concpt of "nrgy dnsity" hs bn dvlopd rcntly using "strss tnsor" mchinris [4-1]. Th nrgy dnsity concpt hs bn ssntil in th fild thory with th Hmiltonin dnsity oprtor s shown in Fig. 1 nd th strss tnsors r usd ubiquitously for dscription of intrnl forcs of mttr. Thy hv bn originlly formultd by Puli [13] in quntum mchnicl contxt with th diffrntil forc lw showing tht it cn b drivd from th divrgnc rltions pplid to th nrgy-momntum tnsor undr gnrl situtions in th prsnc of lctromgntic filds, whil th bsic id dts bck to Schrödingr [14], nd rcntly dvlopd by Bdr [15] in his Atoms-in-Molculs (AIM) thory nd Epstin [16] in his hyprviril thory.

Fild thory of nrgy dnsity concpt Hmiltonin dnsity Hmiltonin oprtor H ( r oprtor ) 3 H d rh ( r ) xpcttion vlu < > intgrl 3 d r xpcttion vlu < > Enrgy dnsity Enrgy n ( ) ( ) H r H r 3 ( E drn ) H r H Fig. 1. Enrgy dnsity concpt in th fild thory. According to th Hohnbrg-Kohn thorm [1], th nonrltivistic ground stt wv function is uniqu functionl of th lctron dnsity, so is th "strss tnsor" in th fild thory, nd morovr, th nrgy dnsity, Lorntz forc dnsity, spin torqu dnsity, nd so on. Spcificlly for th rgumnttion of th rgionl chrg trnsfr procsss of chmicl rction coordints, nothr bsic strtgy is th us of "Apprtus" oprtors [17]; thn, with invoking th Mrmin ntropy principl for th Onsgr locl quilibrium hypothsis, th rgionl dnsity functionl thory hs bn dvlopd [18] nd succssfully pplid to th lctron flow of chmicl rctions [19,10-1]. Thus, w hv dvlopd th nrgy dnsity concpt usful in chmicl pplictions of th DFT. Th prformnc of th strss tnsor hs bn provd powrful thrin. Th bsic r of th fild thory is dp in bsic scinc vn though rgrttbly th univrsl functionl is not known yt. So tht our pproch is forml. Howvr, sinc th concpt is pprntly nturl nd xtrmly ppring, our pproch should nhnc our motivtion to sk for th novl brkthrough in th DFT. Th logic of th fild thory is fundmntl, nd is brifly rviwd hr s follows. Throughout in this ppr, w us th Gussin systm of units. Th spc-tim coordints r dnotd s x x x x x x x x ct x y z ct r, (1.3) 0 k 0 1 3,,,,,,,, whr c dnots th spd of light nd th Grk lttr runs fron 0 to 3 nd th Ltin from 1 to 3. 3

Th contrvrint vctor is trnsformd to th covrint vctor through g, (1.4) whr th Einstin summtion convntion for duplict indics is usd with th mtric tnsor 1 0 0 0 0 1 0 0 g g. (1.5) 0 0 1 0 0 0 0 1 k Th divrgnc of tnsor dnsity T is dfind in this ppr by th rul k k div T T. (1. 6) Th Lgrngin dnsity oprtor is composd of th fild oprtors of gug potntils nd mttr prticls crrying chrgs nd spins. Not only th lctronic spin but lso th nuclr spin r dscribd by stndrd modl of lmntry prticls. Th nuclr spin is intrcting with lctron s dmonstrtd by NMR chmicl shift or th Knight shift [0]. Positron mission tomogrphy hs bn widly known in chmistry community nd similr pplictions of positron-lctron nnihiltion in smiconductor industris [1]. Th mttr prticls r spin 1/ chirl Frmions in th stndrd modl []. Th mttr prticls r bound by gug bosons obying gug principl. Using th stndrd modl, th gug filds of QCD (quntum chromodynmics) is rducd from grnd unifid thory(gut)s SU (3) SU () U (1) SU (3) U (1), whr th Higgs fild brks th c w y c QED Winbrg-Slm lctrowk gug group SU () w U (1) y down to U (1) QED, but th color nd chrg symmtris rmin intct. Qurks r bound by gluons G whil lctron cquirs its chrg nd mss through th Higgs mchnism with th Yukw coupling, whn msslss photon A s wll s th mssiv bosons 0 Z nd W r mrging using th Glshow-Winbrg-Slm thory of spontnously brokn gug symmtry s shown in Appndix A. For lctrons, th ction through mdium hs bn formultd in QED (quntum lctrodynmics) using th gug-invrint Lgrngin dnsity oprtor L ( x ), 4

1 L F ( ) ( ) (, x F x L D ; x), (1.7) 16 whr F is th photon fild dnsity oprtor using Ablin trnsvrsl Coulomb gug potntils A, F A A v v v 0 E ( ) ( ) x x Ey x Ez E ( ) 0 ( ) x x Bz x By, k k ( ) 0 ( ) ( ) 0 A x, (1.8) Ey x Bz x Bx E ( ) ( ) z x By x Bx 0 nd whr L (, D ; x) is th Lgrngin dnsity oprtor of lctron, L (, D ; x) c i D m c. (1.9) with 0 bing th Dirc conjugt to. Th dnots th Dirc mtrics 0 0 1 0,, (1.10) 1 0 0 in th chirl rprsnttion with th Puli spin mtrics 0 1 0 i 1 0,, x 1 0 y i 0 z 0 1, (1.11) nd D is th covrint drivtiv for lctron, Z D ( ) x i A, Z 1 c, (1.1) with m nd Z bing th mss nd chrg numbr of lctron, rspctivly. QED hs th Poincr nd gug symmtris. Th gug symmtry of th fild thory is rlizd by th Bcci-Rout-Stor-Tyutin (BRST) symmtry of th Lgrngin: 4 1 d x L 0, (1.13) c whr dnots th BRST oprtor. It follows tht th physicl contnt of th gug thory is consistnt with th cohomology of th BRST oprtor. 5

QED llows th clmpd-nucli Hmiltonin, whr th tomic nucli r clmpd in spc nd r trtd s xtrnl sttic sourc of forc for lctrons. But in chmicl rction systms, th rrrngmnt of tomic configurtion is of primry intrst, nd hnc th dynmicl trtmnts of tomic nucli hv bn formultd by th Riggd QED thory [4], which is n Ablin gug thory, using th gug-invrint Lgrngin dnsity oprtor: 1 ( ) ( ) Lx F xf 16. (1.14) L (, D ; x) L (, D, D ; x) 0 k 0 k L (, D, D ; x) is th Lgrngin dnsity oprtor of th tomic nuclus: L (, D0, D ; x) i cd0 D, (1.15) m whr D is th covrint drivtiv of th tomic nuclus, Z D ( ) x i A, (1.16) c with m nd Z bing th mss nd chrg numbr of th th tomic nuclus, rspctivly. Th cnonicl quntiztion rul of th Schrödingr fild is nticommuttion rltionship for Frmions nd commuttion rltionship for Bosons. Th Riggd QED thory is gug invrint nd prsrvs trnsltionl nd rottionl symmtry but violts th Poincr symmtry. This is bcus th prsnc of th Schrödingr filds violts Lorntz invrinc of th Lgrngin dnsity. If w nglct th Schrödingr filds, thn w rcovr th convntionl QED with th Poincr symmtry s wll s th gug symmtry. In th prsnt ppr, w dmonstrt th nrgy dnsity, forc dnsity, nd spin torqu dnsity concpts in Ablin nd non-ablin gug thoris (s Appndix A). In clssicl mchnics, th Lorntz forc is th drivtiv of th kintic momntum with rspct to tim. In th quntum fild thoris, th drivtiv of th kintic momntum dnsity oprtor with rspct to tim givs th Lorntz forc dnsity oprtor plus th tnsion dnsity oprtor: s Eqs. (.3) nd (.38) in th Ablin gug fild thory nd Eq. (A.3) in th non-ablin gug fild thory. Th tnsion dnsity oprtor is givn by th divrgnc of th strss tnsor dnsity oprtor: s Eqs. (.34) nd (.39) in th Ablin gug fild thory nd Eq. (A.5) in th non-ablin gug fild thory. Th drivtiv of th Frmion spin ngulr momntum dnsity oprtor with rspct to tim givs th spin torqu dnsity oprtor nd forc dnsity oprtor, clld th zt forc dnsity oprtor: s Eq. (4.4) in th Ablin gug fild thory nd Eq. (A.31) in th non-ablin gug fild thory. Th torqu 6

dnsity oprtor is origintd from th strss tnsor dnsity oprtor: s Eq. (4.5) in th Ablin gug fild thory nd Eq. (A.3) in th non-ablin gug fild thory. Th zt forc dnsity oprtor is origintd from th grdint of th zroth componnt of th chirl currnt dnsity oprtor: s Eq. (4.7) in th Ablin gug fild thory nd Eq. (A.34) in th non-ablin gug fild thory. Phnomnologicl mdium ffcts r trtd in Sction 5.. Lorntz forc dnsity.1. Gnrl sttings In th Riggd QED thory, th chrg dnsity consrvtion lws r th continuity qution ( ) div ( ) 0 t x j x (.1) for lctron, nd ( ) div x j 0 t (.) for th tomic nuclus. As whol, div j 0 t, (.3) Whr stnds for lctron nd stnds for th tomic nuclus; ( x) is th chrg dnsity oprtor nd ( j x) is th chrg currnt dnsity oprtor. Th ( x) is dcomposd into, (.4) ZN, (.5) whr is th lctronic chrg dnsity oprtor nd ( x ) is th chrg dnsity oprtor of th tomic nuclus, nd whr N ( x ) nd N ( x ) is th position probbility dnsity oprtor of lctron nd th tomic nuclus, rspctivly: 0 N ( ) ( ) ( ) x x x, (.6) N. (.7) 7

Th ( j x) is dcomposd into j j j j, (.8) j Z v, (.9) whr j ( x ) is th lctronic chrg currnt dnsity oprtor nd j is th chrg currnt dnsity oprtor of th tomic nuclus, nd v dnots th vlocity dnsity oprtor: ( ) ( ) v ( ) x c x x, (.10) 1 m v i D i D. (.11) By Gordon dcomposition, w hv 1 0 v ( ) ( ) ( ) ( ) ( ) ( ) x i x D x x i D x x m. (.1) 0 rot i m m t Th v ( r ) my lso b writtn s th flux dnsity oprtor S ( r ) s follows: v ( r ) S ( r ). (.13).. Equtions of motion of filds Th qutions of motion of filds is obtind using stndrd vritionl principl. First, th Mxwll qutions of motion r found for th lctromgntic filds: 1 Ax ( ) Ex ( ) grd A0, div Ax ( ) 0, (.14) c t nd ( B x ) rot A ( x ) 1 Bx ( ) rot Ex ( ) 0 c t (.15), (.16) 8

div Bx ( ) 0, (.17) div E 4, (.18) 1 Ex ( ) 4 rot B j. (.19) c t c Scond, th Dirc spinor fild stisfis i D mc, (.0) 0 i D mc. (.1) Third, th Schrödingr fild stisfis i D Z A, (.) t 0 m 0 i x D x x Z A x x. (.3) t ( ) ( ) ( ) ( ) ( ) m.3. Equtions of motion of momntums Th momntum of th lctromgntic fild is th Poynting vctor dnsity oprtor Gx ( ) dfind s 1 Gx ( ) Ex ( ) Bx ( ) 4 c stisfis th qution of motion 1 1 Gx ( ) G Lx ( ) L div t (.4). (.5) In this xprssion, ( x) is th Mxwll strss tnsor dnsity oprtor nd Lx ( ) forc dnsity oprtor s follows: is th Lorntz 9

nd whr 1 ij i j j ( ) ( ) ( ) ( ) ( ) i x E x ij E x E x E x E 8 1 ( ) i ( ) j ( ) j ( ) i B x ij B x B x B x B 8 (.6) L L L, (.7) 1 L E j B, c (.8) 1 L E j B, c (.9) L ( x ) is th lctronic Lorntz forc dnsity oprtor nd L is th Lorntz forc dnsity oprtor of th tomic nuclus. It should b notd tht ( x) is symmtric: ij ji. (.30) Nxt, th lctronic kintic momntum dnsity oprtor ( r ) 1 ( ) ( ) ( ) ( ) ( ) ( ) x i x D x x i D x x, (.31) stisfis th qution of motion ( ) ( ) x L x. t (.3) Asid from th lctronic Lorntz forc dnsity oprtor L, th dnots th lctronic tnsion dnsity oprtor givn s th divrgnc of th lctronic strss tnsor dnsity oprtor s follows: whr ( ) div x, (.33) 10

nd k ic 0 ( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) x Dl x x Dk x x x Dk x Dl x 0 0 k l k l D D D D ] 1 ( j B) c k 0 kl ic ( ) [ ( ) ( ) ( ) ( ) ( ) x x Dk x x Dk x x ]. It should b notd tht is Hrmitn: (.34) (.35) ( ) x. (.36) Lstly, th kintic momntum dnsity oprtor mv ( x ) of tomic nuclus stisfis th qution of motion ( ) ( ) S mv x Lx. (.37) t Asid from th Lorntz forc dnsity oprtor L, th S dnots th tnsion dnsity oprtor givn s th divrgnc of th strss tnsor dnsity oprtor S ( x ) s follows: S ( ) div S x, (.38) whr Sk 4m [ ( xd ) ( xd ) D ( xd ) k k D D D D k k 1 ( j B) c k, (.39) [ ( xd ) ( xd ) D ( xd ) Skl k l k l 4m D D D D ] k l l k. (.40) 11

It should b notd tht th strss tnsor dnsity oprtor S ( x ) is Hrmitn nd symmtric: whr S ( ) S x, (.41) Skl Slk ( ) x. (.4) As whol, w obtin Lx ( ) t, (.43) Lx ( ) div ( ) x m v, (.44) ( ) ( ) S x x, (.45) ( ) ( ) S x x. (.46) Likwis, w obtin 1 ( ) ( ) ( ) div ( ) ( ) Gx G t x x x x, (.47) which is th momntum consrvtion lw of th Riggd QED systm. 3. Enrgy dnsity Th QED Hmiltonin dnsity oprtor H ( x ) is composd of th Hmiltonin dnsity QED oprtor of th lctromgntic fild H ( x ) nd th Dirc lctronic Hmiltonin dnsity EM oprtor H Dirc( x ) intrcting with th lctromgntic fild s follows []: H H H, (3. 1) QED EM Dirc H H A, (3. ) EM 0 H M A, (3. 3) Dirc 0 1

whr H is th lctromgntic fild nrgy dnsity oprtor nd M ( x ) is th lctronic mss dnsity oprtor: 1 H x 8 E x B x ( ) ( ) ( ), (3. 4) k M c i D m c, (3. 5) k Th lctronic mss dnsity oprtor M ( x ) my b writtn s th nrgy dnsity oprtor of lctron H ( x ) s follows: M H. (3. 6) Thus, th H ( x ) rducs from Eq. (3.1) to QED H H H. (3. 7) QED Th Riggd QED Hmiltonin dnsity oprtor dnotd s H ( x ) is drivd s Riggd QED follows: H H H, (3. 8) Riggd QED QED tom whr th nrgy dnsity oprtor H tom of tomic nucli intrcting through th lctromgntic fild nd th lctron fild is ddd to H QED( x ). Th H tom ( x ) is purly th kintic nrgy dnsity oprtor of tomic nucli: H T, (3. 9) tom 1 T ( ) ( ) ( ) ( ) ( ) ( ) x x D x x D x x m. (3.10) Th nrgy flow is found s follows: 1 1 H ( x ) c div G ( x ) G ( x ) E ( x ) j ( x ) j ( x ) E ( x ) t, (3.11) 13

1 ( ) H x c div ( x ) E ( x ) j ( x ) j ( x ) E ( x ) t, (3.1) 1 Htom div s E j j E t, (3.13) with lding to D k D 1 ( ) ( ) ( ) k D x x Dk x s, (3.14) i m D ( ) ( ) ( ) k x x D x ( ) ( ) ( ) xdk xd x 1 H x c G x G x c x s x t Riggd QED( ) div ( ) ( ) ( ) ( ), which is th nrgy consrvtion lw of th Riggd QED systm. Th viril thorm hs bn found to b [7]: 3 ERiggd QED d r Hriggd QED x dr mc x x T x 3 which in th nonrltivistic limit bcoms (3.15), (3.16) 3 Enon-rltivistic Riggd QED d r T x. (3.17) It should b notd tht th strss tnsor dnsity hs th dimnsion of th nrgy dnsity, nmly nrgy pr volum, sinc it hs dimnsion of forc pr r. Indd, th trc of th strss tnsor dnsity bcoms k k 1 c k 0 k [ i D ( ) ( ) ( ) ( ) k x x i Dk x x ], (3.18) M M m c 14

( ) ( ) ( ) ( ) ( ) ( ) S k x D x x D x x x k. (3.19) 4m D ( ) ( ) ( ) x x D x This is quivlnt to two tims th kintic nrgy dnsity; in th nonrltivistic limit, th intgrl of th trc of th strss tnsor dnsity givs two tims tht of th kintic nrgy dnsity s follows: ( ) ( ) 3 k S k dr k x k x dr dr T x nonrltivistic limit 3 S k 3 k. (3.0) Hnc, th strss tnsor dnsity givs locl pictur of two tims th kintic nrgy dnsity. Th ignvlu is th principl strss nd th ignvctor is th principl xis s summrizd s follows: S11 S1 S13 S ( ) S1 ( ) S ( ) S3 x x x ( x ) S31 S3 S33 0 0 S11 dig S 0 0 S 33 0 0. (3.1). (3.) S11 S S33 W shll nlyz th nrgy dnsity of chmicl bonds in th lctronic sttionry stt. Undr xtrnl lctromgntic filds, chrgd prticls cn b cclrtd by Lorntz forc. Dissiptiv forc cn bring bout th sttionry stt of th chrgd prticls. In quntum mchnics, nrgy ignstt is nothr rul lding to th sttionry stt. As hs bn shown in this ppr, tnsion of th quntum fild thory cn compnst th Lorntz forc, which is th bsic mchnism lding to th quntum mchnicl sttionry stt: s qutions (.3), nd (.37). Th ignvlu of th strss tnsor dnsity givs msur of th kintic nrgy dnsity nd th ignvctor, th principl xis, msur of th dirction of th bond. If th locl principl strss is positiv, it is clld th tnsil strss, whil if it is ngtiv, comprssiv. Sinc th mtric tnsor ij g hs ngtiv ignvlus, (-1, -1, -1), w should not tht th comprssiv strss givs positiv contribution to th kintic nrgy dnsity, whil th tnsil strss ngtiv. This indicts th nw pictur of th locl chmicl intrction nrgy dnsity. Th tomic lctron dnsity xhibits positiv kintic nrgy dnsity nd forms th lctronic drop rgion R D sprtd from th lctronic 15

tmosphr rgion R A by th intrfc S, lding to th comprssiv strss. This tndncy should of cours b intct in btwn ionic spcis intrctions. Th sitution should chng drmticlly for covlnt bond formtion, whr pir of lctrons should b bound tightly, thrby crts tnsil strss. Mny systms xhibit such gnric ftur, which is clld th spindl structur [5]. If in th lctronic drop rgion R D in btwn pir of toms thr crosss th Lgrng surfc of th lctronic tnsion, thn th distribution of th lctronic strsss nd th principl xis is of grt intrst to study th chrctr of th chmicl bond. Th locl gomtry is schmticlly shown in Figs. nd 3. For ppliction to chmicl rction coordints, w hv lso dvlopd concpt of th intrction nrgy dnsity [10-1]. Locl quilibrium in th sttionry stt Rpulsiv lctronic tnsion drivs th quntum mchnicl lctronic diffusion Lgrng surfc Rpulsiv lctronic tnsion compnsts th ttrctiv lctric fild of tomic nuclus S A B R D R A Fig.. Lgrng surfc of th lctronic tnsion in btwn pir of toms A nd B. Also shown r th lctronic drop rgion R D whr th lctronic kintic nrgy dnsity is positiv, th lctronic tmosphr rgion R A whr th lctronic kintic nrgy dnsity is ngtiv, nd th intrfc S which sprts thm. 16

Spindl structur of th covlnt bond Elctronic tnsil strss binds pir of th lctronic drop rgions R D s whr th comprssiv strss is prdominnt: covlnt bond visuliztion! Comprssiv strss Tnsil strss Comprssiv strss Fig. 3. Spindl structur schm for covlnt bond visuliztion in hydrogn molcul modl. Elctronic tnsil strss binds pir of th lctronic drop rgions R D s whr th comprssiv strss is prdominnt. 4. Torqu of spin ngulr momntum Th ngulr momntum dnsity oprtor ux ( ) of th lctromgntic fild dfind s ( ux ) rgx ( ), (4.1) stisfis th qution of motion 1 1 ( ) ( ) uxu x r Lx ( ) L +div t, (4.) 1 r L L div r whr w hv usd th symmtric chrctr of ( x). Th lctronic spin ngulr momntum dnsity oprtor 1 ( x ) with 17

( ) ( ) x x (4.3) stisfis th qution of motion 1 ( x ) t ( x ) ( x ), (4.4) t whr t dnots th spin-torqu dnsity oprtor dfind s [10] k n t ( ) x nk. (4.5) Th ( x ) dnots th zt forc dnsity oprtor dfind s k k 1 k ( ) ( ) x ck x ; no sum ovr k. (4.6) Th ltrntiv form is obtind s follows: k, (4.7) 0 5 5 Z k 5 j, (4.8) whr j ( x ) dnots th zroth componnt of th chirl currnt dnsity oprtor 0 5 j cz, (4.9) 5 5 0 1 3 5 i. (4.10) Th lctronic orbitl ngulr momntum dnsity oprtor dfind s r, (4.11) stisfis th qution of motion ( ) x rl div t. (4.1) r L div r t Sum of qutions (4.4) nd (4.1) lds to 18

1 r L div t t. (4.13) rl div r Th th nuclr orbitl ngulr momntum dnsity oprtor ( x ) dfind s rm v (4.14) stisfis th qution of motion: ( ) ( ) div S x rl x r, (4.15) t whr w hv usd th symmtric chrctr of S ( x ). For th ngulr momntum s whol, w hv 1 r L r div ( r) t t, (4.16) rl div r nd if th ux ( ) is furthr ddd to, w finlly obtin 1 1 ( ) ( ) ( ) ux u x x t rdiv t, (4.17) div r which is th ngulr momntum consrvtion lw of th Riggd QED systm. Elctronic spin cn thrfor b cclrtd by th torqu drivd by th strss tnsor, which is lso compnstd by th zt forc, which is nothr bsic mchnism lding to th sttionry stt of lctronic spin. It should b notd tht Puli Hmiltonin givs qution of motion of lctronic spin: s,.g., qution (11.155) of Jckson s txtbook on clssicl lctrodynmics [3]. Th BMT (Brgmnn-Michl-Tlgdi) qution nd Thoms prcssion r lso th txtbook mttrs. Our prsnt rsult incorports ll of thm in closd form plus th fild thorticl compnstion mchnism lding to th sttionry stt of lctronic spin. 19

5. Effctiv chrg dnsity of lctromigrtion 5.1. Elctromgntic nrgy dnsity in mgntodilctric mdi In th Riggd QED thory, th phnomnologicl intrction of systm A nd its nvironmnt bckground mdium M is trctbl using rgionl chrg nd currnt dnsitis. It is sily gnrlizd for systm A mbddd in th surrounding mdium M. For phnomnologicl forc concpts in mgntodilctric mdium such s chmicl rction systms in condnsd phs, w my usully rly on clssicl nlogy of prlll-plt cpcitor filld with dilctric s shown in Fig. 4. D( r), B( r) + + + + + + + + + + M P( r), M( r) E( r), H( r) A - - - - - - - - - - Fig. 4. Prlll-plt cpcitor filld with dilctric: phnomnologicl modl of chmicl rction systm A mbddd in n nvironmntl bckground mdium M. Th corrsponding gug potntils r th rgionl intgrls of th chrg nd trnsvrsl currnt dnsitis, dfind s follows [7]: 3 ( ct, s) A0 ( ct, r) d s A, (5.1) r s A 3 ( ct, s) A0 ( ct, r) d s M, (5.) r s nd M 0

1 3 jt ( cu, s ) AA ( ct, r ) d s c, (5.3) r s A 1 3 jt ( cu, s ) AM ( ct, r) d s c, (5.4) r s M whr th subscript A or M of th intgrl sign dnots th rgionl intgrls confind to th rgion r s A or M, rspctivly, nd whr u t. c Sinc th rgions A nd M ltogthr spn th whol spc, w hv A A A, (5.5) 0 0A 0M A A A A. (5.6) A M rdition Th lctric fild E is dcomposd into th lctric displcmnt Dx ( ) of th mdium M nd th polriztion Px ( ) of th systm A, dfind rspctivly s 1 Dx ( ) grd A0 A M M, c t (5.7) 1 1 Px ( ) grd A 0 A A A, 4 4 c t (5.8) so tht w hv 1 Ex ( ) grd A0 Ax ( ) c t. (5.9) 1 Dx ( ) 4 Px ( ) Ardition c t It should b notd tht th Px ( ) gnrlizs th usul dipol fild. Likwis, lt th mgntic fild H of th mdium M nd th mgntiztion M of th systm A b dfind rspctivly s H rot A, (5.10) M M 1 M ( x ) rot AA ( x ), (5.11) 4 1

thn w hv Bx ( ) rot Ax ( ) H 4 M, with. (5.1) H H rot A M rdition Th rgionl chrg dnsitis r thn rprsntd rspctivly s 1 A A0 A, (5.13) 4 1 M A0 M, (5.14) 4 with th Lplcin x y z, (5.15) nd hnc. (5.16) A M Likwis, th rgionl chrg currnt dnsitis r rprsntd s c 1 ja grd A0 A A A 4 c t, Px ( ) crot M t (5.17) c 1 jm grd A0 A M M, 4 c t (5.18) with th D'Almbrcin 1 c t, (5.19) nd hnc j j A jm. (5.0) Px ( ) crot M jm t Th rgionl dcomposition of th longitudinl nd trnsvrsl componnts of th currnt dnsitis r rprsntd s follows: j j j, (5.1) L T with j j j, (5.) L LA LM

j j j, (5.3) T TA TM whr c 1 j L grd A A 0, A 4 c t (5.4) c 1 j L grd A M M, 4 c t (5.5) c jt A A A, 4 (5.6) c jt A M M. 4 (5.7) Using Eqs. (5.4)-(5.7), w hv th ltrntiv forms of Eqs. (5.17) nd (5.18) rspctivly s j j j, (5.8) A LA TA j j j. (5.9) M LM TM Th linr rspons proprtis of th systm A undr th intrction with th nvironmnt mdium M my formlly b rprsntd with obvious nottion s follows: 1 Px ( ) Dx ( ) Ardition c t ( xex ) ( ), nd, (5.30) M H m 1 Dx ( ) Ardition 14 Ex ( ) c t 1 Ex ( ) 1 4, (5.31) ( xex ) ( ) B (14 m) H, ( xh ) ( x ) (5.3) nd 3

1 j xt D Ardition c t xt E. (5.33) ( xex ) ( ) int 5.. Effctiv chrg numbr of lctromigrtion Elctromigrtion is th phnomn of nuclr currnt inducd by lctric currnt in condnsd phs [4]. Th nucli ccpt diffusiv forc from th surrounding mdium ovr nd bov th Lorntz forc [5]. In our modl, th tnsion is th origin of th mdium ffcts [9]. Th linr rspons of th forc dfins th ffctiv chrg numbr tnsor dnsity oprtor Z * of th chrgd prticl s * 1 1 ( ) ( ) S Z x D x Ardition N j B L. (5.34) c t c Sinc th right hnd sid of this qution is 1 S ( ) ( ) ( ) ( ) ( ) ( ) S L x x E x x j x B x, c (5.35) w thn conclud * 1 ( ) ( ) S Z x D x Ardition N E c t 1 ( ) 4 ( ) S Dx Px Ardition ZN c t. (5.36) 1 ( ) S Dx Ardition ZN 4 PxZN ( ) c t Now w dfin nd w conclud ( ) ( ), (5.37) * Z x Z Z wind x Z Z Z, (5.38) wind sttic wind dynmic wind Z 4 Z, (5.39) sttic wind 1 S Z dynmic wind D Ardition N. (5.40) c t 4

dictts th cs of lctronic conduction. In It should b notd tht th formultion for this cs, th usul txtbook pproch dmonstrts th mdium ffct s th dissipsiv forc ginst Lorntz forc: s,.g., Eq. (1.16) of Ashcroft-Mrmin s txtbook on solid stt physics [6]. In our prsnt rsult, th dissiptiv forc mrgs from th tnsion dnsity s th fild thorticl forc dnsity compnsting th Lorntz forc dnsity. 6. Exmpl W shll dmonstrt th nrgy dnsity concpt using hydrogn. W hv lrdy publishd th H tom's R D, R A nd S [4-8] nd th spindl structur for covlnt H bond formtion [5] s schmticlly shown in Figs. nd 3. Th dnsity proprty of spin torqu my hr b ddd to. Sinc th lctric fild of proton is sphriclly symmtric, thr pprs no spin torqu dnsity for th ground nd xcitd stts nor th zt forc dnsity. Th sum of th two dnsitis should thn of cours zro, t ( r) ( r) 0, t ch point in spc s it should b for th sttionry stt, Eq.(4.4). Anothr xmpl with chmicl rction of hydrogn conomy my lso b ddd to. Rcntly, hydrogn-storg crbon mtrils for ful cll pplictions hv rcivd much ttntion nd th bsic rction mchnism hs not bn thoroughly known yt [7-4]. Sinc thr should b crtin nrgy brrir for H tom dsorption on grphn sht [9,30], our id hr is th chrg trnsfr from htro toms such s Al my nhnc th dformtion of grphn which my thn nhnc stbiliztion of H tom on th grphn surfc. W hv prformd b initio molculr-dynmics (MD) simultions using rstrictd Hrtr-Fock (RHF) lctronic wv function xpndd by 6-31+G* bsis st for Al nd Dunning-Huzing full doubl zt typ ons for H nd C for dynmic bhviors of H toms on modl grphn sht with Al tom [43]. W shll prsnt hr som of th prliminry rsults for dmonstrtion of th nrgy dnsity concpt. 5

0. 0. 0. -0.45-0.45 0.40-0.45 0. 1.364A 1.073A 11.0-0.45 0. -0.45 0.40 0.05 1.49A 10.0 119.0 0.05 0.40 1.430A 1.405A 10.0 0.05-0.45 0. -0.45 0.05 0.05-0.45 0. 0.40 0.05 0.40 0. -0.45-0.45 0. -0.45 0.40-0.45 0. 0. 0. () H H H C A B (b) Fig. 5. Modl grphn sht () th optimizd structur with Mullikn chrgs, nd (b) th ttcking sits A, B, nd C by H. 6

W hv shown in Fig. 5 () th optimizd modl grphn sht with Mullikn chrgs, nd (b) th thr ttcking sits A, B, nd C by H on it. H 0.5 1.098A 0. 0. -0.46 0.44 0. 1.517A 0. 1.531A -0.46-0.41 0. 0.35-0.3 0.10-0.8-0.3 1.517A 0. -0.46-0.0-0.3-0.46 0.10-0.41 Al 0.49 0. 0.40-0.46 0.05-0.0 0.35-0.46 0. 0. -0.46 0.44-0.46 0. 0. 0. B.E. = 0.90 V A-sit optimizd structur () 0. 0. 1.531A H 0.4 1.097A 0. -0.45 0. -0.49 0.45-0.47-0.05 0.30 0.6-0.37-0.5-0.17 0. 1.50A 1.50A 0. -0.67-0.5-0.45 0.07 0.6-0.37 0. 0.39-0.05 0.30 0. Al 0.5 0. -0.45-0.47 0.46-0.49-0.45 0. 0. 0. B.E. = 0.98 V B-sit optimizd structur (b) 7

0. 0.3 1.541A H 0.0 1.090A 0. 0. -0.45 0.40-0.46 0.00-0.74 0.33 0.11 0.08 0.19 1.090A -0.48 1.50A 0.5 0. -0.44-0.45 0.06-0.01-0.43-0.46 Al 0.41 0. 0.40-0.46 0.04 0.40-0.45 0.3 0. -0.46 0.40-0.45 0. 0. 0. B.E. =.357 V C-sit optimizd structur (c) Fig. 6. Thr of vrious locl minim corrsponding to () A, (b) B, nd (c) C ttcking sits by H in Fig. 5 with th binding nrgis (B.E.'s). Th Al tom situtd t th bck sid of th grphn sht stbilizs th dducts through chrg trnsfr. In Fig. 6 r shown th thr of vrious locl minim corrsponding to ths sits with th binding nrgis (B.E.'s), whr th Al tom situtd t th opposit sid of th grphn sht stbilizs th dducts through chrg trnsfr. 8

Fig. 7. Elctronic nrgy dnsitis on cross sction of snpshot in th MD simultion. () th positiv kintic nrgy dnsity in gry rgion nd th tnsion dnsity in rd rrow, nd (b) th 3 rd ignvlu of th strss tnsor dnsity in rd for positiv "tnsil" strss, whil in blu ngtiv "comprssiv" strss, nd th ignvctor in blck lin. Th xs of ordint nd bsciss r lbld in Bohr. Fig. 7 shows snpshot of lctron nrgy dnsity plots in th MD simultion drwn by Molculr Rgionl DFT progrm pckg dvlopd in our lbortory [44]. In this snpshot th bond lngth btwn th H nd th nrst C tom is 1.03 A. Th kintic nrgy dnsity in Fig. 7 () rprsnts th chrg trnsfr through th mrgd R D btwn H nd Al toms nd th grphn sht. Th tnsion dnsity rprsnts th forc which hlp th sp hybrid orbitl chng to sp 3 stbilizing th floppy surfc of th grphn sht. Th strss tnsor dnsity in Fig. 7 (b) shows th chrctristic chmicl bonding proprtis. Th strss tnsor dnsity btwn th H nd C toms rprsnts positiv "tnsil" strss nd th spindl structur is formd btwn thm. This dictts th covlnt bond formtion nd hnc thy r tightly bound with ch othr. On th othr hnd, th rgion btwn th Al nd C toms rprsnts ngtiv "comprssiv" strss. This corrsponds to th sitution tht th Al tom movs with no tightly covlnt trpping points on th grphn sht. Dtild discussions including th dynmics with th spin torqu for hydrogn storg procsss undr lctric strsss of th xtrnl lctromgntic filds in th surrounding mdium 9

should b of considrbl intrst, nd will b publishd lswhr. 7. Conclusion Undr xtrnl sourc of lctromgntic filds, chrgd prticls cn b cclrtd by Lorntz forc. Dissiptiv forc cn mk th stt of th chrgd prticls sttionry. In quntum mchnics, nrgy ignstt is nothr rul of th sttionry stt. In this ppr, tnsion dnsity of quntum fild thory is formultd in such wy tht it cn compnst th Lorntz forc dnsity t ny point of spc-tim. This formultion cn giv mchnicl dscription of locl quilibrium lding to th quntum mchnicl sttionry stt. Th tnsion dnsity is givn by th divrgnc of strss tnsor dnsity. Elctronic spin cn b cclrtd by torqu dnsity drivd from th strss tnsor dnsity. Th torqu dnsity cn b compnstd by forc dnsity, clld th zt forc dnsity, which is nothr bsic mchnism lding to th sttionry stt of th spinning motion of lctron. Th xtrnl ffct for chmicl rction systms is rlizd whr chmicl rction systm A mbddd in th nvironmntl mdium M is modld s prlll-plt cpcitor filld with dilctric. Th vibronic intrction tht gos byond th dibtic pproximtion hs bn incorportd s wll s th lctronic spin-dpndnt intrction. Acknowldgmnts This work hs bn supportd in prt by Gnt-in-Aid from th Ministry of Eduction, Cultur, Scinc nd Tchnology of Jpn, for which w xprss our grtitud. APPENDIX A A.1. Th nrgy dnsity in th non-ablin gug fild thory According to th stndrd modl, th intrction of th mttr prticls is mditd by th gug Bosons. Th known lctrowk, lctromgntic, nd strong intrctions hv ll turnd out to b govrnd by th principls of th gug invrinc undr locl gug trnsformtions. Th quntum fild thoris tht hv bn provd usful in dscribing th rl world r non-ablin gug thoris. Ths r mor gnrl thn th simpl Ablin gug invrinc of th QED. 30

In th stndrd modl, th Lgrngin is invrint undr gug trnsformtion of th Frmion fild of th mtril prticl s follows: i t, (A.1) whr is n rbitrry rl function nd whr t is th Hrmitn oprtor which gnrts th unitry trnsformtion. Th t stisfis th Li lgbr: t, t ict, (A.) C ; compltly ntisymmtric, (A.3) whr C dnots th structur constnt of th Li lgbr nd is compltly ntisymmtric. Th djoint rprsnttion A t dfind s t A ic (A.4) stisfis A A A t, t ict. (A.5) Sinc dos not trnsform lik, w must introduc gug potntil A nd us it to construct th covrint drivtiv s follows: D ia t (A.6) with th gug trnsformtion proprty D ( ) ( ) ( ) ( ) x x i x t D x, (A.7) if with A C A A D, (A.8) A ( ) ( ) A D x ia x t. (A.9) 31

Th gug potntil A lds to th gug fild dnsity oprtor F s follows: ( ), ( ) D x D x itf, (A.10) F A A C A A, (A.11) with th gug trnsformtion proprty A F i t F. (A.1) Th totlly ntisymmtric C gurnts th following rltionship A ( ), A A ( ) D x D x it F, (A.13) nd th Jcobi idntity A A ( ) ( ) ( ) ( ) A D x F x D x F x D F 0. (A.14) Th lst idntity is n lmnt of th Mxwll qutions. Th gug-invrint Lgrngin dnsity oprtor Lx ( ) with th non-ablin gug filds is gnrlly writtn s 1 L F ( ) ( ) (, x F x LM D ; x), (A.15) 16 whr L (, ; ) M D x is th Lgrngin dnsity oprtor of th mttr prticl. Th qutions of motion of filds is obtind using th vritionl principl. First, th rmining lmnts of th Mxwll qutions of motion r obtind for th non-ablin filds in th gug-invrint form using th Eulr-Lgrng qution s follows: A 4 D F J, (A.16) c whr J is th currnt of th mttr fild: L (, M D ; x) J c it D. (A.17) Th J stisfis th consrvtion lw 3

A ( ) D x J 0. (A.18) Scond, lt th mttr spinor fild Lgrngin dnsity oprtor b givn s follows: L (, ; ) ( ) ( ) ( ) M D x c x i D x mc x, (A.19) whr m dnots th mss of th mttr prticl. Th mttr fild stisfis th qution of motion i D mc. (A.0) A.. Th Lorntz forc dnsity nd th spin torqu dnsity in th non-ablin gug fild thory W shll driv th strss tnsor in non-ablin gug fild thory nd dmonstrt its ctiv rol in th qution of motion of th kintic momntum of th mttr prticl. Th kintic momntum dnsity oprtor of th mttr prticl dfind s 1 i ( xdx ) ( ) i Dx ( ), (A.1) 1 k k k i D i D, (A.) stisfis th qution of motion Lx ( ), (A.3) t whr Lx ( ) dnots th mtril Lorntz forc dnsity oprtor nd th dnots th mtril tnsion dnsity oprtor. Th Lx ( ) is givn s follows: k ( ) ( ) L x c F x t. (A.4) k Th is givn s th divrgnc of th mtril strss tnsor dnsity oprtor ( ) div x, (A.5) 33

k k l, (A.6) whr 0 k k ic [ D ( ) ( ) ( ) ( ) k x x D x x ] It should b notd tht is Hrmitn:. (A.7) ( ) x. (A.8) Similr rltionship holds in th Ablin gug fild thory: s Eqs. (.37) nd (.43). Th Frmion spin ngulr momntum dnsity oprtor 1 ( x ) with, (A.9) 1 k 1 ij ij 1 i j,, ijk J J 4 i, (A.30) stisfis th qution of motion 1 ( x ) t ( x ) ( x ), (A.31) t whr t dnots th spin-torqu dnsity oprtor dfind s k ij t. (A.3) ijk Th ( x) dnots th zt forc dnsity oprtor dfind s k k 1 k ( ) ( ) x ck x ; no sum ovr k. (A.33) Th ltrntiv form is obtind s follows: k, (A.34) k 5 0 5 j5, (A.35) q whr q dnots th chrg q Z of th mttr prticl nd j 0 5 ( x ) dnots th zroth componnt of th chirl currnt dnsity oprtor 34

j cq, (A.36) 5 5 0 1 3 5 i. (A.37) Similr rltionship holds in Ablin gug fild thory: s Eqs. (4. 3) - (4.10). A.3. Spontnously brokn symmtry Using th stndrd modl, th gug filds of QCD is rducd from th grnd unifid thory SU (3) SU () U (1) SU (3) U (1), whr th Higgs fild brks th (GUT)s c w y c QED Winbrg-Slm lctrowk gug group SU () w U (1) y down to U (1) QED, but th color nd chrg symmtris rmin intct. Th Hmiltonin dnsity oprtor in th fild thory is obtind by cnonicl quntiztion using Lgrngin dnsity oprtor: Lx ( ) L Gug filds LMtril prticl filds. (A.38) L L Higgs fild Yukw coupling Th mttr prticls r spin 1/ chirl Frmions in stndrd modl, ccompnying th thr gug symmtris. First, qurks hs intrnl color symmtry of th SU (3) c gug: f q q q rd blu grn f, (A.39) with th gug trnsformtion proprty c gc xp ( ) f ic x f c, (A.40) nd th covrint drivtiv g c ( ) D x i G ( x ). (A.41) c Flvour f r u, d, c, s, t, b, (A.4) RL, RL, RL, RL, RL, RL, 35

whr R, L dnots chirlity 1 1 1 5 ( ), 1 5 ( ) R x x L. (A.43) Scond, SU () w gug of wk isospin doublts (Cbibbo-rottd qurk d cl ): t u ( ) L k g u x L t 1 1 xp i ( ) : t x tk t3, t ( ) ( ) d x c d x cl cl, (A.44) t k g t 1 1 xp i ( ) : t x tk t3, t ( ) c x L L nd singlts ur, dcr, R : t 0. (A.45) Th qurks r mixd through th first nd scond gnrtions by th Cbibbo ngl d cos c c sin c d, (A.46) s sin c c cos c s or mor gnrlly through th first, scond, nd third gnrtions by th Cbibbo-Kobyshi-Mskw mtrix V V V d' ud us ub d s' Vcd Vcs Vcb s b' Vtd Vts V tb b c. (A.47) Third, U(1) y gug of wk hyprchrg: g y y y u R, L 4 1 u xp ( ) ( ) :, RL, iy x u x y RL, u y R ul c 3 3 y g y y d R, L 1 d xp ( ) ( ) :, cr, L iy x d x y cr, L d y cr dcl c 3 3. (A.48) y g y y xp i ( ) ( ) :, 1 y x x y c g y y y R, L xp ( ) ( ) :, 1 RL, iy x x y RL, y R L c for th 1 st gnrtion ud,,, nd similr rltionships for th nd gnrtion cs,,, nd 3 rd gnrtion tb,,,. Th chrg q Z of SU () w U (1) y r found to b 36

y Z t3. (A.49) 1 Zu, Z, 0, 1 RL, d Z Z crl, RL, 3 3 for th 1 st gnrtion ud,,, nd similr rltionships for th nd gnrtion cs,,, nd 3 rd gnrtion tb,,,. Th mttr prticls r bound by gug bosons obying gug principl rprsntd by th covrint drivtiv oprtor s gc ( ) ( ) g gt k y y D ( ) x i G x i tkw x i X. (A.50) c c c Qurks r bound by gluons G whil lctron cquirs its chrg nd mss through Higgs mchnism with Yukw coupling, whn msslss photon A s wll s mssiv bosons 0 Z nd W r mrging using Glshow-Winbrg-Slm thory of spontnously brokn gug symmtry s follows: whr g g t k y y ( ) gt i t ( ) sin kw x i X x i WZA c c c gt 1 0 i t3 sin W Z Z c cos, (A.51) W gt 1 ( ) i tw x tw c 3 cosw sinw sin cos A x X x W x Z x X x W x 0 3 W W 1 1 1 W x W x iw x g g cos W, sin g g g g t t it t y W t y t y Thn, th chrg of lctron mrgs s. (A.5) g sin ; chrg of lctron ( (-1)). (A.53) t W 37

Th Higgs fild H ( x ) itslf is n isospinor sclr fild with wk hyprchrg y=+1/: H, (A.54) H x ( ) 0 ( ) H x spontnously brokn if th potntil V H ( x ) hs th minimum t s with rl 0 H ( x ) v ( x), (A.55) nd ( x ). As rsult, th Yukw coupling Lgrngin dnsity L ( x ) turnd out to b Yukw coupling L G H H c. (A.56) Yukw coupling R L L R G ( ) ( ) v x c x It follows tht th mss of lctron mrgs s m G v; mss of lctron. (A.57) Morovr, th kintic trm of th Higgs fild Lgrngin dnsity L ( x ) with rl Higgs fild nd turnd out to b: LHiggs fild D H D H V H 1 W x W x v x 1 gt ( ) 4 c 0 Z x v x 1 gt 1 4 c cos W ( ) V H( x ). (A.58) This dmonstrts tht th mssiv bosons 0 Z nd W r mrging whil th msslss photon A rmind intct. 38

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