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1 Physicl Chmistry Chmicl Physics Volum 2 Numbr August 200 Pgs ISSN COVER ARTICLE Pchuki nd Koms Rovibrtionl lvls of HD COMMUNICATION Emsly t l. Ab initio simultion of proton spin diffusion

2 PAPER Physicl Chmistry Chmicl Physics Rovibrtionl lvls of HDw Krzysztof Pchucki* nd Jck Koms* b Rcivd 3th April 200, Accptd th Jun 200 DOI: 0.039/c0cp00209g Th dissocition nrgis of ll rottion vibrtionl stts of th molculr HD in th ground lctronic stt r clcultd to high ccurcy by including nondibtic, rltivistic 2, nd quntum lctrodynmic 3 ffcts, with pproximt trtmnt of smll highr ordr 4, nd finit nuclr siz corrctions. Th obtind rsult for th ground molculr stt of (0) cm is in smll disgrmnt with th ltst most prcis xprimntl vlu. I. Introduction Sinc th bginning of quntum mchnics molculr hydrogn nd its isotopomrs hv bn ground for tsting nd dvloping xprimntl tchniqus nd thorticl modls. In dtrmintion of th dissocition nrgy (D 0 ), xprimntl nd thorticl msurmnts hv diminishd thir individul uncrtintis to blow 0 3 cm nd r in good grmnt. In prticulr, th ltst thorticl D 0 = (0) cm of H 2, obtind by Piszcztowski t l., grs vry wll with (37) cm drivd xprimntlly by Liu t l. 2 Anlogous rsults obtind lst yr for D 2 r (9) cm from thory nd (60) cm from xprimnt. 3 Th tiny diffrnc of cm fits wll within both rror stimts. To chiv this 0 3 cm lvl of ccurcy, th thory must hv tkn into ccount, with sufficint prcision, not only th lctron corrltion but lso th finit nuclr mss, rltivistic, nd quntum lctrodynmics (QED) ffcts. Prticulrly chllnging is th ccurt inclusion of nondibtic ffcts. On possibl pproch is to obtin nondibtic wv function (dpnding xplicitly on nuclr coordints) by minimizing th nonrltivistic nrgy. For H 2 such clcultions, using xplicitly corrltd Jms Coolidg functions, wr ttmptd by Ko"os nd Wolniwicz in 963 4,5 nd 5 yrs ltr by Bishop nd Chung. 6 Th sm uthors prformd purly nondibtic clcultions for HD. Ko"os nd Wolniwicz obtind D 0 = cm, 7 whrs Bishop nd Chung rportd D 0 = cm. 8 Clcultions in similr spirit, but using xtnsivly optimizd xplicitly corrltd Gussin functions, wr prformd by Stnk t l. 9 Thir nondibtic wv function ws furthr mployd to comput prturbtivly th rltivistic corrction to th nondibtic nrgy. An pprnt drwbck of ths mthods is thir dcrsing ccurcy obsrvd for th highr xcitd stts, prticulrly thos lying clos to dissocition thrshold. For such stts th prturbtiv trtmnt of rltivistic ffcts my b indqut. As n xmpl, th v = 4, J = 4 stt of Institut of Thorticl Physics, Univrsity of Wrsw, Hoż 69, Wrsw, Polnd. E-mil: krp@fuw.du.pl b Fculty of Chmistry, A. Mickiwicz Univrsity, Grunwldzk 6, Poznn, Polnd. E-mil: koms@mn.poznn.pl w Elctronic supplmntry informtion (ESI) vilbl: Extnsiv tbls of ll 400 bound rovibrtionl stts of HD. S DOI: 0.039/c0cp00209g H 2 bcoms rsonnc ftr th inclusion of rltivistic ffcts on th lvl of th potntil nrgy curv (PEC). Morovr, crtin proprtis lik th ortho pr mixing or th scttring lngth, r inccssibl within th dirct nondibtic pproch. In contrst, th nondibtic prturbtion thory (NAPT) pproch mployd hr, rlis on solving th rdil, vribl-mss Schro dingr qution with th PEC for th nucli constructd from th dibtic potntil ugmntd by R-dpndnt nondibtic, rltivistic nd QED corrctions. Th thory of th nondibtic potntils hs bn dvlopd in rf. 0 nd, whrs th rltivistic nd QED corrctions to th PEC r vlutd on th bsis of th nonrltivistic quntum lctrodynmics (NRQED). 2 4 Ths corrctions r unmbiguously idntifid by n xpnsion of bound tomic or molculr stt nrgy in powrs of th fin structur constnt : E = E (0) + 2 E (2) + 3 E (3) + 4 E (4) +, () whr E (3) nd highr ordr trms my dditionlly dpnd on ln. Th first trm of th xpnsion rprsnts th nonrltivistic nrgy, 2 E (2) is th lding rltivistic contribution, trms proportionl to 3 nd 4 dscrib th QED ffcts of th lding nd highr ordr, rspctivly. In this ppr w rport on ppliction of this pproch to ll rovibrtionl lvls of th ground lctronic stt of HD molcul. Uncrtinty of our rsults coms minly from th nglct of th finit nuclr mss corrctions of th ordr 2 m/m to th rltivistic contribution to th PEC, nd from th pproximt trtmnt of th 4 corrction. Th nglct of highr ordr nondibtic trms proportionl to (m/m) 3 lso incrss th ovrll uncrtinty. II. Nonrltivistic Hmiltonin W considr two-lctron ditomic molcul in th rfrnc frm ttchd to th gomtricl cntr of th two nucli. Th totl wv function f is solution of th sttionry Schro dingr qution with th Hmiltonin Hf = Ef, (2) H = H l + H n, (3) 988 Phys. Chm. Chm. Phys., 200, 2, This journl is c th Ownr Socitis 200

3 split into th lctronic nd nuclr prts. In th lctronic Hmiltonin H l ¼ X whr V is th Coulomb intrction r 2 2m þ V; ð4þ whr ~H n 0 ¼ H0 n þ l2 r l r R m 2 r r RðVÞ; r r R ¼ H 0 n þ l2 m r 2 R þ l2 2 ri r j r i R rj R ðvþ ð4þ V ¼ r A r B r 2A r 2B þ r 2 þ R ; th nucli hv fixd positions ~R A (proton) nd ~R B (dutron), nd ~R = ~R A ~R B. Th nuclr Hmiltonin is H n ¼ r2 R r2 l r R r l 2m n 2m n M B M A ¼Hn 0 þ H00 n ; whr r l ¼ P 2 r, m n = (/M A +/M B ) is th nuclr rducd mss, nd Hn 0, H00 n r vn nd odd prts with rspct to th invrsion. In ordr to simplify th clcultion of nondibtic corrctions w introduc unitry trnsformtion of th form ð5þ ð6þ H = U + HU (7) U ¼ l r r R with ~r = P ~r nd th nuclr mss symmtry prmtr Th trnsformd Hmiltonin is ð8þ l ¼ m : ð9þ 2 M B M A ~H ¼H þ l½h; r r RŠþ l2 2 ½½H; r r RŠ; r r RŠþOðl 3 Þ; ð0þ whr th highr ordr trms in th lctron nuclr mss rtio O[(m /M A,B ) 3 ] r nglctd, so tht ~H ¼H l þ H 0 n þ l½v; r r RŠ þ 2 l2 m ½r l r R; r r RŠþ l2 2 ½½H l; r r RŠ; r r RŠ; ðþ nd th odd O[(m /M A,B ) 2 ] trms r nglctd s wll. Th intrnl commuttor in th lst trm of qn () is ½H l ; r r RŠ¼ r r RðVÞ 2 m r l r R; ð2þ so tht th trnsformd Hmiltonin cn b dcomposd s ~H ¼ H l þ ~H 0 n þ ~H 00 n : ð3þ ~H 00 n ¼ lr r RðVÞ: ð5þ Both th nuclr Hmiltonins involv th drivtiv of th Coulomb oprtor V, which is r RðVÞ ¼ 2 r A r 3 A þ r B r 3 B r 2A r 3 þ r 2B 2A r 3 2B n R 2 ð6þ with ~n = ~R/R, whil th scond drivtiv of V is furthr trnsformd in qn (47) (49). III. Adibtic pproximtion In th dibtic pproximtion th totl wv function of th molcul f (~r, ~R) =f l (~r)w(~r) (7) is rprsntd s product of th lctronic wv function f l nd th nuclr wv function w. Th lctronic wv function obys th clmpd nucli lctronic Schro dingr qution [H l E l (R)] f l i = 0, (8) whil th wv function w is solution to th nuclr Schro dingr qution with th ffctiv potntil gnrtd by lctrons r2 R þ E ðrþþe l ðrþ E jwi ¼0; 2m n ð9þ whr E (R) is th so-clld dibtic (or digonl) corrction dfind s E ðrþ ¼hf l jh 0 n jf li l ¼ 2m n ðhr Rf l jr Rf l i l hf l jr 2 l jf l i l Þ: ð20þ Sprtion of th ngulr vribls in qn (9) lds to th wll-known rdil nuclr R R 2m ¼ E w J ðrþ: JðJ þ Þ þ 2m n R 2 þ E l ðrþþe ðrþ w J ðrþ ð2þ Solving this qution givs n dibtic nrgy lvl E nd n dibtic rdil nuclr wv function w J. This journl is c th Ownr Socitis 200 Phys.Chm.Chm.Phys., 200, 2,

4 IV. Nondibtic nuclr Schrödingr qution Following th NAPT formlism introducd rcntly, 0, w cn obtin nrgy lvls E including lding nondibtic corrctions by solving th following nondibtic vrsion of th rdil Schro dingr qution " R JðJ þ Þ R 2 2m k 2m? ðrþr 2 þ YðRÞ w J ðrþ ¼Ew J ðrþ; ð22þ whr Y(R) is nondibtic potntil nrgy function. In th nonrltivistic limit YðRÞ ¼E l ðrþþe ðrþþde n ðrþþde 0 n ðrþ; ð23þ with th nondibtic corrction constructd from th homonuclr prt de n (R), dfind in our prvious work on H 2, 0, nd th htronuclr prt proportionl to l 2 de 0 n ¼l2 f l r 2 R m þ 2 ri r j r i R rj R ðvþ f l l # þ f l r r RðVÞ ðe l H l Þ 0 r r RðVÞ f l ; l ð24þ which is obtind from qn (4) nd (5). Aprt from th nondibtic potntil Y(R), th diffrnc btwn qn (22) nd (2) lis in th ffctiv msss usd. In th dibtic qn (2) th rducd nuclr mss m n ppring in both trnsltionl nd rottionl kintic trms is constnt, whil in th nondibtic qn (22) it is givn by two diffrnt functions of th intrnuclr distnc. Ths two ffctiv rducd mss functions 2m k ðrþ þ W k ðrþ l2 2m n m 2m? ðrþ þ W? ðrþ l2 2m n m ð25þ ð26þ r dfind with th hlp of dditionl rdil functions W k ðrþ ¼ m 2 n r Rf l n ðe l H l Þ 0 n r Rf l ð27þ nd W? ðrþ ¼ m 2 n ðd ij n i n j Þ 2 r i R f le l H l rj R f l l : ð28þ l In totl, thr rdil functions r ndd to construct th nondibtic rdil Schro dingr qn (22) for ditomic molculs: two functions, dfind by qn (27) nd (28), to dscrib th vribl ffctiv rducd msss of qn (25) nd (26), nd th nondibtic potntil Y. This potntil, in turn, is xprssd by nothr four functions: BO nrgy E l, dibtic E, nondibtic homonuclr de 0 n nd htronuclr de 0 n corrctions (s qn (23)). V. Sprtd toms limit At lrg intrnuclr distncs th ffctiv rducd mss functions (25) nd (26) r xpctd to pproch vlu corrsponding to th rducd mss of sprt H nd D toms ¼ þ : ð29þ m A m p þ m m d þ m Bcus W J (R) nd W > (R) tnd to m (4m 2 n), whn R - N, w hv 2m k ðþ ¼ 2m? ðþ ¼ m 2m n 4m 2 l2 n m ð30þ ¼ m þ m ; ð3þ 2 m p m p m d m d which r xctly th lding trms of th xpnsion of th tomic rducd mss (29) in th lctron nuclr mss rtio " ¼ m þ m 2 ð32þ 2m A 2 m p m p m p þ m d m þ m 2 # : ð33þ m d m d In th sprtd toms limit, th nonrltivistic nrgy of th systm (th dissocition thrshold) E(N) is simply sum of th nrgis of hydrogn nd dutrium toms xprssd by thir rducd msss EðÞ ¼ m H 2 m D 2 : ð34þ Th xpnsion of E(N) in th lctron to nuclus mss rtio is of th form EðÞ ¼ þ m þ m 2 m p m d 2 m 2 m 2 p þ m2 m 2 d þ: ð35þ Subsqunt trms of this xpnsion coincid with th R - N limits of corrsponding componnts of th nondibtic potntil Y(R) of qn (23), E l (N) =, (36) E ðþ ¼ m 2m n ; ð37þ de n ðþ ¼ m 2 ; ð38þ 2m n de 0 n ðþ ¼ l2 : ð39þ In prticulr, th sum of qn (38) nd (39) is qul to th third trm in th xpnsion (35). VI. Rltivistic nd rditiv corrctions Th rltivistic corrction to th dibtic potntil for singlt stt is givn by th xpcttion vlu with th 990 Phys. Chm. Chm. Phys., 200, 2, This journl is c th Ownr Socitis 200

5 nonrltivistic wv function of th Brit Puli Hmiltonin 5 2 ^H BP ¼ X p 4 8 þ p X X Z A dðr AÞþp X dðr bþ 2 ob X p 2 ob A p b þ p r b r b r 3 b r b p b : ð40þ Th xpcttion vlu E (2) (R) =hf l Hˆ BP f l i l s function of R, ws computd for H 2 to high ccurcy by Wolniwicz 6 in 993 nd hs rcntly bn rclcultd in rf.. In th prsnt clcultions, s in ll th prvious ons, w hv omittd th smll rltivistic rcoil corrctions, nmly thos proportionl to 2 m /M. Anothr 2 ffct, which cn b sily incorportd into th rltivistic potntil, rsults from th sptil distribution of th nuclr chrg. Th nrgy shift cusd by this ffct is givn by th formul * + E fs ðrþ ¼ 2p 2 X Z A r 2 ch 3 ðaþ f X l dðr AÞ f l ; ð4þ l 2 C A whr l C ¼ 386: fm is th Compton wvlngth ovr 2p nd r ch (A) is th root mn squr chrg rdius of th nucli A, with vlus of r ch (p) = (69) fm nd r ch (d) = 2.402(28) fm. 7,8 For th dissocition nrgy of th ground rovibrtionl lvl this ffct is quit smll nd mounts to cm with tndncy to diminish to zro for highr lvls. Th lding ordr QED corrction is givn by 9 E ð3þ ðrþ ¼ X ( þ 4 3 ln hf l jdðr bþjf l i l ob 7 6p f l P ) f l þ 3 X A r 3 b X ln ln k 0 4Z A 3 hf ljdðr AÞjf l i l : l l ð42þ Th numricl vlution of E (3) hs bn dscribd in dtil in rf.. W only mntion hr tht this vlution includs such trms s th Bth logrithm ln k 0 nd th xpcttion vlu of th Arki Suchr distribution P(/r 3 ). 20 As prviously, th highr ordr QED contribution 4 hs bn stimtd by th corrsponding on-loop lctron slf-nrgy corrction E ð4þ ðrþ p X X ln 4 hf 96 l jdðr AÞjf l i l : A ð43þ Th lrg-r bhviour of th bov rltivistic nd QED potntils hs bn dtrmind using symptotic constnts rportd in rf. nd 2. Th rltivistic nd QED corrctions cn b computd dirctly, s xpcttion vlus with th dibtic wv function. It is mor convnint nd mor ccurt, howvr, to includ thm into th nondibtic Schro dingr qn (22) by dding prtinnt rdil functions into th Y(R) potntil (23). In such n pproch, th ignvlu of th Schro dingr qution rprsnts totl nrgy including ll th mntiond finit nuclr mss, rltivistic nd QED ffcts. VII. Computtionl dtils Th rdil nondibtic qn (22), prt from th clmpd nucli nrgy E l nd th dibtic corrction E, involvs W, W >, nd th potntils de n nd de 0 n in qn (24). Th numricl vlus for ll but th lst rdil functions wr obtind for H 2 nd simpl rscling by th first or scond powr of th rducd mss rtio convrts thm to th prtinnt HD functions. For this rson, w shll omit dtild dscription of how ths functions wr obtind, rfrring th rdr to our prvious work on H 2. 0, Blow w giv only bsic informtion on ths functions nd thn concntrt on th nw trms which rsult from th nuclr mss symmtry in HD. Th lctronic nrgy, E l, usd in this work is xctly th sm s th on rportd in rf.. Its nlytic form is bsd on th nrgy points clcultd by Sims nd Hgstrom 22 using Hyllrs wv function nd by Cnck 23 using n xplicitly corrltd Gussin (ECG) wv function. Th rltiv ccurcy of ths clcultions is of th ordr of 0 2, which corrsponds to bout 0 0 of th rltiv ccurcy of th Born Oppnhimr potntil. Th ground stt dissocition nrgy obtind by numriclly solving th dibtic Schro dingr qn (2) in th Born Oppnhimr pproximtion with this nlytic potntil is cm (s lso Tbl ). Also th rltivistic nd QED corrctions to th potntil obtind for H 2 in rf. pply dirctly to HD bcus thy do not dpnd on th nuclr mss. Th dibtic corrction E hs bn vlutd nlyticlly by mns of nw mthod dscribd in rf. 0 nd. Th rdil function E prviously obtind for H 2 hs bn rscld to HD by th rtio of th rducd msss of nucli m H 2 n =mhd n E HD ¼ m p þ m d E H 2 2m d ð44þ nd ld to th dibtic dissocition nrgy of th ground stt qul to cm. Similrly, th nondibtic potntils de n, W, nd W > wr obtind for H 2 in rf. nd hr r rscld to HD by th squr of th rducd mss rtio mpþm d 2m d 2. Numricl Tbl Componnts of D 0 (in cm ) for th v =0,J = 0 stt of HD. Uncrtintis of 2 nd 3 com from th nglct of nuclr rcoil corrctions nd tht of 4 from th pproximt formul Componnt D 0 BO () Adibtic corrction () Nondibtic corrction (2) 0 subtotl (2) 2 corrction (4) 2 finit nuclr siz corrction 0.000(0) subtotl (5) 3 corrction 0.964(2) 4 corrction 0.006(8) Totl (0) This journl is c th Ownr Socitis 200 Phys.Chm.Chm.Phys., 200, 2,

6 vlus of th nuclr msss m p = m nd m d = m usd in this study r bsd on th CODATA 2006 compiltion of fundmntl physicl constnts 7 nd wr tkn from th NIST Wb Pg. 8 Th nuclr rducd mss of HD is m n = m nd th nuclr mss symmtry prmtr l = m. Th only nwly vlutd function of R is th htronuclr nondibtic corrction de 0 n, qn (24), rsulting from thos trms of th Hmiltonin H, which contin l [s qn (4) nd (5)]. de 0 n compriss thr prts. Th first prt is nlogous to th nuclr kintic nrgy trm in th dibtic corrction (20) nd rquirs vlution of th drivtiv of th lctronic wv function ovr th nuclr vribl ~R. This diffrntition cn b ccomplishd with th hlp of th following formul 24 Rf l ¼ n ðe l H l Þ f l i R n L nf l : ð45þ In th bov qution, th first trm givs th prlll componnt nd rquirs n dditionl bsis st of S + g symmtry to vlut th rducd rsolvnt. Th prpndiculr componnt is obtind by vlution of th xpcttion vlu of n oprtor rsulting from th lst trm, which involvs th nuclr ngulr momntum oprtor L n ¼ ir r R. Hr w md us of th following idntity vlid for th S stts: ~L n f l = ~L l f l, whr ~L l is th lctronic ngulr momntum oprtor L l ¼ i P r r. In this nw formultion, it is possibl to void th involvmnt of P symmtry functions th prpndiculr componnt is obtind dirctly from th lctronic ground stt wv function s R 2 hf ljl 2 l jf li l : ð46þ Th scond prt of de 0 n contins oprtors which r difficult in numricl vlution, so w trnsform it to mor convnint form using th following idntity r i Rr j R(V) =(r i Rr j R r i lr j l)(v) +r i lr j l(v). (47) Th first trm on th right hnd sid of qn (47) is ðr i R rj R ri l rj l ÞðVÞ R j d ij R 2 ¼3Ri R 5 4p 3 dij d 3 ðrþ; ð48þ (th d 3 (R) prt cn b nglctd), whil th scond trm is vlutd using intgrtion by prts hf l r i r j r i lr j l(v) f l i l = R d~rvr i lr j l(r i r j f 2 l). (49) Th third prt of th htronuclr nondibtic corrction de 0 n ; qn (24), is gin scond ordr quntity, which rquirs vlution of th rsolvnt in th bsis st of S + u symmtry. All ths xpcttion vlus s wll s th scond ordr quntitis wr vlutd in th bsis of xponntilly corrltd Gussins (ECG) functions 25 c k ðr ; r 2Þ¼ðþ ^P 2 Þð ^iþx " # xp X2 A k;ij ðr i s k;iþðr j s k;jþ ; i;j¼ ð50þ whr th mtrics A k nd vctors ~s k contin nonlinr prmtrs, 5 pr bsis function, to b vritionlly optimizd with rspct to ithr th lctronic nrgy or prtinnt Hyllrs functionl. Th ntisymmtry projctor ( + Pˆ2 ) nsurs singlt symmtry, th sptil projctor ð ^iþ nsurs th grd (+) or ungrd ( ) symmtry, nd th X k prfctor nforcs S stts whn qul to, or P stts whn qul to y i (th prpndiculr Crtsin componnt of th lctron coordint). For th scond ordr mtrix lmnts w gnrtd 600-trm ECG bsis st of S + g or S + u symmtris. Th nonlinr prmtrs of this bsis wr optimizd by minimizing th functionl corrsponding to this mtrix lmnt. Finlly, th totl potntil Y in th Schro dingr qn (22) rds YðRÞ ¼E l ðrþþe ðrþþde n ðrþþde 0 n ðrþþeð2þ ðrþ þ E fs ðrþþe ð3þ ðrþþe ð4þ ðrþ: ð5þ All its componnts wr shiftd by subtrcting corrsponding tomic vlus (s sction V nd rf. ) so tht thy symptoticlly tnd to zro. VIII. Rsults nd discussion Tbl shows th dissocition nrgy of th ground rovibrtionl lvl dcomposd into ll th known significnt contributions. Prticulr corrctions hv bn computd s diffrnc btwn th ignvlus obtind dding succssivly corrsponding contributions to th potntil Y, qn (5). For instnc, th 2 rltivistic corrction hs bn vlutd from two ignvlus: on obtind with Y ¼ E l þ E þ de n þ de 0 n þ Eð2Þ nd th othr with Y ¼ E l þ E þ de n þ de 0 n. Rltivistic nd QED corrctions cn lso b obtind without th nondibtic potntil de n þ de 0 n. Th diffrnc for th ground stt is quit smll 0 6 cm, howvr for xcitd stts th diffrnc cn b lrgr. Thr r svrl possibl sourcs of th uncrtinty in th finl dissocition nrgy. Th thr dominting r (i) th missing rltivistic nd QED rcoil trms of O(m /M), (ii) th nglct of th nondibtic trms of O[(m /m n ) 3 ], nd (iii) th pproximt trtmnt of th 4 contribution. Although th formuls for th omittd rltivistic rcoil trms r xplicitly known, 24 no numricl clcultions hv bn prformd so fr. Th rror cusd by th nglct of this trm cn b stimtd s m /m n tims th 2 corrction (s rf. ) nd, nlogously, tims th 3 corrction to ccount for th missing QED rcoil trm. For D 0 of th ground rovibronic lvl ths two contributions r cm nd cm, rspctivly. In similr fshion, th contribution to th rror budgt from th missing highr ordr nondibtic trms cn b pproximtd s proportionl to m /m n tims th scond ordr nondibtic corrction, which mounts to cm t th ground lvl. Th lst mningful prt of th uncrtinty rsults from th incomplt trtmnt of th highr ordr QED ffcts. As prviously, (rf. ) w consrvtivly stimt tht th trms omittd in E (4), qn (43), contribut c. 50% of th on-loop trm, which yilds cm of th uncrtinty. 992 Phys. Chm. Chm. Phys., 200, 2, This journl is c th Ownr Socitis 200

7 Th qudrtic sum of ths four rror componnts lds to th ovrll uncrtinty on th ground stt D 0 of lss thn cm. For th rottionlly nd vibrtionlly xcitd lvls, th uncrtinty chngs in ccord with th siz of th corrctions. Its stimtion for individul lvls is listd in th ESI.w In totl, thr r 400 bound lvls with th vibrtionl quntum numbr v rnging from 0 to 7. Th numbr of th rottionl lvls dcrss with growing v from 37 for v =0to only 2 in th highst v = 7 stt. Th full st of th totl dissocition nrgis is prsntd in Tbl 5. Morovr, dtild spcifiction, similr to tht in Tbl, hs bn prprd for ch bound rovibrtionl lvl nd is vilbl in th ESI.w For ch combintion of th vibrtionl nd rottionl quntum numbrs thr r 8 ntris corrsponding to: six componnts of th dissocition nrgy, th totl D 0, nd th stimtd uncrtinty of th totl D 0. Th six componnts of th totl D 0 r, rspctivly: th Born Oppnhimr, dibtic, nondibtic, 2 rltivistic (including finit nuclr siz), 3 QED, nd 4 QED. Tbl 2 ssmbls svrl xprimntl nd thorticl nondibtic vlus of D 0 obtind ovr th yrs for th ground rovibrtionl lvl. Mor dtils on th progrss in dtrmining th dissocition nrgy of HD cn b found in brif rviw by Stoichff. 26 Th first vritionl nondibtic clcultion for HD hs bn prformd by Bishop nd Chung. 8 Thy usd 858 bsis functions, ch bing product of n lctronic Jms Coolidg function nd som rdil Gussin-typ function, nd obtind th nonrltivistic D 0 = cm with n stimtd convrgnc rror of 0.28 cm. Approximt rltivistic ( 0.54 cm ) nd rditiv ( 0.22 cm ) corrctions compltd th dissocition nrgy to th vlu displyd in Tbl 2. A mor ccurt rltivistic dissocition nrgy of th HD molcul ws first obtind by Wolniwicz 27 in 983, nd ltr by Ko"os nd coworkrs. 28,29 In 995 Wolniwicz hs mrkdly improvd his lctronic wv functions nd rfind th finl dissocition nrgy to gt cm shown in Tbl 2. This vlu diffrs from ours by fw thousnds of wv numbr in ccord with th uncrtinty stimtd by Wolniwicz. Concrning th QED corrction to th ground Tbl 2 Comprison of thorticl nd xprimntl rsults for D 0 (in cm ) of th v =0,J = 0 stt of HD. d is diffrnc from our rsult Componnt D 0 d This work (0) Thory Stnk t l. (2009) Wolniwicz (995) Ko"os nd Rychlwski (993) Ko"os, Szlwicz, Monkhorst (986) Wolniwicz (983) Bishop nd Chung (978) Exprimnt Zhng t l. (2004) (6) Blkrishnn t l. (993) (0) 0.05 Eylr nd Mlikchi (993) (0) 0.0 Hrzbrg (970) 33, (4) 0.4 Th originl D 0 = cm from rf. 9 hs bn ugmntd by sum of our 3 nd 4 QED corrctions qul to cm. stt D 0 w mntion th old but vry good stimtion 0.97 cm by Ldik. 30 It grs surprisingly wll with th currnt rigorous rsult, s Tbl. Lst yr, Stnk t l. 9 prformd nw vritionl nondibtic clcultion mploying xplicitly corrltd Gussin bsis functions. Thir nonrltivistic totl nrgy of (20) E h, whn subtrctd from th sum of th tomic nondibtic nrgis, qn (34), yilds D 0 = cm in good grmnt with our nonrltivistic subtotl vlu in Tbl (th diffrnc is (2) cm ). Thir rltivistic corrction computd with th nondibtic wv function is E h. Bcus th corrsponding tomic limit ( 2 /4 E h ) is known to high ccurcy (th lding ordr rcoil trm vnishs), th rltivistic D 0 cn b infrrd from this dt s qul to cm. W not hr tht now th discrpncy incrss to 0.002(5) cm in comprison with our rltivistic rsult. If this diffrnc wr ttributd to th rltivistic rcoil contribution, it would b lmost 3 tims lrgr thn th consrvtiv stimt of this ffct discussd bov. Tbl 2 lso collcts dissocition nrgis dtrmind xprimntlly. Th first msurmnt of D 0 for HD ws prformd by Hrzbrg nd Monfils in yilding cm. Motivtd by discrpncy with th fmous thorticl rsults by Ko"os nd Wolniwicz, 32 Hrzbrg rptd his xprimnt 33,34 using n improvd pprtus nd stblishd D 0 = (4) cm shown in Tbl 2. Tbl 3 Comprison of thorticl nd xprimntl rsults for th nrgy diffrnc DE (in cm ) btwn v = 0 nd v = rottionlss stts of HD. d is diffrnc from our rsult Sourc DE d This work (5) Thory Stnk t l. (2009) Wolniwicz (995) Ko"os nd Rychlwski (993) Exprimnt Stnk t l. (2009) (7) b Rich t l. (982) (6) c 0.00 McKllr t l. (976) (9) c Th originl DE = cm from rf. 9 hs bn ugmntd by sum of our 3 nd 4 QED corrctions qul to cm. b s uncrtinty. c 3s uncrtinty. Tbl 4 Componnts of thorticlly prdictd trnsition nrgy DE btwn J = 0 nd J =, nd btwn J = 0 nd J = 2 rottionl lvls of th ground vibrtionl stt (v = 0) of HD. All ntris in cm Componnt DE(0 - ) DE(0-2) BO Adibtic corrction Nondibtic corrction (6) (9) 0 subtotl (6) (9) 2 corrction (2) (5) subtotl (6) (20) 3 corrction () (2) 4 corrction (4) (9) Totl (8) (22) Exprimnt 46, (5) (0) This journl is c th Ownr Socitis 200 Phys.Chm.Chm.Phys., 200, 2,

8 Tbl 5 Dissocition nrgy (in cm ) of ll 400 bound stts of HD. v nd J r th vibrtionl nd rottionl quntum numbrs, rspctivly v/j v/j v/j Phys. Chm. Chm. Phys., 200, 2, This journl is c th Ownr Socitis 200

9 This vlu, howvr, is in fct n rithmtic mn of two indpndnt msurmnts: cm nd cm, th formr bing vry clos to our vlu. In 993, Eylr nd Mlikchi 35 dtrmind th dissocition thrshold from th EF S + g stt nd, in combintion with th spctr msurd by Dick, 36 obtind D 0 = (0) cm. At th sm tim, Blkrishnn t l. 37 prformd dlyd dtction of th fluorscnc spctrum of photodissocitd hydrogn nd rrivd t D 0 = (0) cm. Ths rsults, lthough systmticlly lrgr, r in grmnt within thir uncrtintis with currnt thorticl prdictions. An ordr of mgnitud mor ccurt msurmnts wr rportd by th Eylr group in In thr-stp xprimnt iming t dtrmintion of th scond dissocition thrshold thy obtind D 0 = (6) cm. This rsult is 3s wy from our thorticl vlu. In viw of n incrsd prcision on both th xprimntl nd thorticl sid it must b sttd tht currntly thr is discrpncy of c cm in th dtrmintion of D 0 for HD. Accurcy of th prsnt rsults cn lso b ssssd by comprison of th nrgy diffrnc corrsponding to th lowst rottionlss vibrtionl trnsition with th vilbl litrtur dt (s Tbl 3). Th most ccurt thorticl prdictions by Wolniwicz nd by Ko"os nd Rychlwski s wll s th xprimntl dt r in vry good grmnt with th prsnt rsult (5) cm. Hr, w stimtd th uncrtinty in th sm wy s for th dissocition nrgy (s bov) i.. ssuming tht th rror componnts r proportionl to corrsponding corrctions. In contrst to th homonuclr isotopomrs, th lctric dipol trnsitions btwn th lowst rottionl stts of HD r llowd nd th trnsition nrgy cn, in principl, b msurd dirctly. In Tbl 4 w prsnt vlus of ll significnt contributions to th lowst J = 0-, 2 trnsition nrgis nd compr with th vilbl xprimntl dt w not 2s diffrnc btwn th thory nd msurmnts. Th ioniztion potntil (IP) of HD cn b rltd to its dissocition nrgy by IP = D 0 (HD) E(H) D 0 (HD + ). (52) Sinc th dissocition nrgy of HD +, s wll s th totl nrgy of th hydrogn tom, is known vry ccurtly, w cn vlut IP with n ccurcy dqut to tht of D 0 (HD). Up-to-dt vlus of E(H) = cm nd D 0 (HD + ) = cm hv bn compild by Liu t l. 3 on th bsis of currnt fundmntl constnts 7 nd clcultions by Korobov. 39,40 IP computd for HD from th bov formul mounts to (0) cm with th uncrtinty trnsfrrd dirctly from D 0. IX. Conclusion Th high ccurcy of 0.00 cm for th thorticlly prdictd dissocition nrgy of H 2 nd isotopomrs hs bn chivd du to th rcnt progrss md in two dirctions. Th first on, nbld complt trtmnt of th lding QED ffcts. In prticulr, th pproch to ffctivly clcult th mny lctron Bth logrithm nd mn vlus of singulr oprtors, lik th Arki Suchr trm, hs bn dvlopd.,4,42 Th scond dirction, indispnsbl for rching this ccurcy, is th nondibtic prturbtion thory, 0,,24 which nbls rigorous pproch to th finit nuclr mss ffcts byond th dibtic pproximtion. Howvr, n ccurt nondibtic corrction to rltivistic contribution still rmins to b vlutd. In comprison of thorticl prdictions with rcnt xprimntl rsults w obsrv vry good grmnt for dissocition nrgis of H 2 nd D 2, nd smll discrpncy of 0.045(6) cm for HD. Thrfor, nw msurmnt with n incrsd prcision of dissocition nd trnsition nrgis of HD molcul would b vry dsirbl. Not ddd in proof Aftr submitting this ppr w bcm wr of nw msurmnts of HD dissocition nrgy [D. Sprchr, J. Liu, C. Jungn, W. Ubchs, F. Mrkt, 200, to b publishd]. Th nw vlu of D 0 = (36) cm is in vry good grmnt with our thorticl prdiction. Acknowldgmnts KP cknowldgs support by NIST through Prcision Msurmnt Grnt PMG 60NANB7D653. JK cknowldgs support by th Polish Ministry of Scinc nd Highr Eduction Grnt No. N N nd by computing grnt from Poznn Suprcomputing nd Ntworking Cntr. Rfrncs K. Piszcztowski, G. Lch, M. Przybytk, J. Koms, K. Pchucki nd B. Jziorski, J. Chm. Thory Comput., 2009, 5, J. Liu, E. J. Slumbids, U. Hollnstin, J. C. J. 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