Ring polymer molecular dynamics

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1 Ring polymr molcular dynamics David Manolopoulos Dpartmnt of Chmistry, Univrsity of Oxford Mariapfarr Workshop 29, Lctur II

2 <latxit sha_bas64="jkdzeiy9kilcxa5qssn6glub8=">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</latxit>. Quantum mchanical corrlation functions Many dynamical proprtis of condnsd phas systms can b rlatd to raltim corrlation functions of th form c AB (t) = h Q tr Ĥ Â() ˆB(t) i, whr Q =tr h Ĥ i, and ˆB(t) = +iĥt/~ ˆB iĥt/~.

3 <latxit sha_bas64="2kawxscs932nh+rcbfvlpsfb2ci=">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</latxit> For xampl, th di usion co cint of a molcul i in a liquid is givn by D(T )= 3 c vi v i (t) dt, chmical raction rat co cints can b calculatd from k(t )= Q r (T ) c ff (t) dt, and dipol absorption spctra from n(!) (!) =! 3~cV ( ~! )C µ µ (!), whr C µ µ (!) = 2 i!t c µ µ (t) dt.

4 <latxit sha_bas64="5sg+tsew2flutsc8kafttsin84=">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</latxit> Th standard ral-tim corrlation function is c AB (t) = h Q tr Ĥ Â() ˆB(t) i, whras th Kubo-transformd corrlation function is c AB (t) = Q d tr h ( )ĤÂ() i Ĥ ˆB(t). Thr ar a numbr of rasons why c AB (t) is th mor classical of th two objcts and it is c AB (t) that is approximatd in RPMD.

5 If and C AB (!) = 2 C AB (!) = 2 thn it is straightforward to show that whr i!t c AB (t) dt i!t c AB (t) dt, C AB (!) =D(!) C AB (!), D(!) = ~! ~!. So standard corrlation functions can asily b rconstructd from Kubo-transformd corrlation functions, and viv vrsa. <latxit sha_bas64="+9udtkmsmjdmf59vq/yi+rgmce=">aaad43icrvjlb9naehyshsw8ujhygeitvitxqejlkif9adiugr9shuautxvkq9a+2oewwrw4gbbx/hq3fgo3dpmglbbcggmn22vpn5dqiie4b6/+nuvkwvx274n2/dvno3vx7vwkhsc9znklp6kgigmyfxnwrlfrohmu4wfontxwylqi5r8t/mcrzmbsjeizsi6xuunx6fuqsyoyx+d+bsb/nbcvxhb4yqxwnbgudqbagyfdaicghqtk46toroxl9uqa4envfcscojovowgr3i7jktiuyskkcvydv/ugpfxuztbdfuprgiaqbgxpjiypjurpmi6bfjhuzybsrpcou/shx97qgn+wokyhds9oeeilewjppfo5ubvhvtnw/wddzdo9sp2tfcbxptvmi3+bpjtcaqocsubmrdahceejt7sjsmideqhzdlplp2tgnskdn+g2cbbbwyiimcwwfw8/xu9pv9rcgfgwahvxvj9o8wvrzm7a7wjblzhpqlglvmk+an5ygiwyp2utplqshznqfrsaa2prmd2rlq9kmdhptrsdyyr7lzmjjypodlsoskmjssiijjr8wsgpm/fvbuehflysvlfgnetbk/au6avvabjjwfusl4gdqcx73+m8hnnkznwvafq2/tc7yn3o73ytvz9j3/nd82pzc/nlcqfw9a35dnmy5vz3/vlwt9/a4+dpqy=</latxit>

6 <latxit sha_bas64="fhvpnzfw633xippajsp75r33kh8=">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</latxit> Proof (I): c AB (t) = h Q tr = h Q tr = X Q jk = X Q jk Ĥ Â() ˆB(t) i Ĥ Â +iĥt/~ ˆB iĥt/~i E j hj Â ki +ie kt/~ hk ˆB ji ie jt/~ E j A jk B kj +i(e k E j )t/~. ) C AB (!) = 2 = X Q = Q jk X jk i!t c AB (t) dt E j A jk B kj 2 E j A jk B kj (! [E k E j ]/~). i(! [E k E j ]/~)t dt

7 <latxit sha_bas64="lfjt9ge6wiguf3k9lybmrma/we=">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</latxit> Proof (II): c AB (t) = Q = Q d d tr X jk h ( )ĤÂ Ĥ +iĥt/~ ˆB iĥt/~i E j (E A jk B kj k E j ) +i(e k E j )t/~. ) CAB (!) = 2 = Q = i!t c AB (t) dt d ( ~! ) ~! X jk ~! d Q E j A jk B kj X jk C AB (!). (E k E j ) (! [E k E j ]/~) E j A jk B kj (! [E k E j ]/~)

8 Altrnativly (I): Not that c AB (t) = Q = Q d d tr tr h ( h )ĤÂ Ĥ Â +iĥ(t+i ~)/~ ˆB Ĥ +iĥt/~ ˆB iĥt/~i iĥ(t+i ~)/~i d c AB (t + i ~), whr c AB ( ) = Q tr h Ĥ Â +iĥ /~ ˆB iĥ /~i is an analytic function of in th strip appl Im( ) appl ~: β Im(τ) = λħ R(τ) = t

9 So C AB (!) = 2 = = = Altrnativly (II): d i!t c AB (t) dt 2!~ d!~ d = ( ~! ) ~! 2 2 C AB (!). i!t c AB (t + i ~) dt ++i ~ +i ~ + i! c AB ( ) d i! c AB ( ) d Im(τ) = λħ β R(τ) = t

10 It follows from this that dynamical obsrvabls can qually wll b writtn in trms of c AB (t). For xampl: and whr D(T )= 3 k(t )= Q r (T ) c vi v i (t) dt, n(!) (!) =!2 3cV Cµ µ (!), C µ µ (!) = 2 c ff (t) dt, i!t c µ µ (t) dt. Notic that non of ths quations involvs ~! <latxit sha_bas64="8q8mghv3pgyqyhy4sie8cah3q=">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</latxit>

11 <latxit sha_bas64="(null)">(null)</latxit> <latxit sha_bas64="(null)">(null)</latxit> <latxit sha_bas64="(null)">(null)</latxit> <latxit sha_bas64="(null)">(null)</latxit> <latxit sha_bas64="(null)">(null)</latxit> <latxit sha_bas64="(null)">(null)</latxit> <latxit sha_bas64="(null)">(null)</latxit> <latxit sha_bas64="(null)">(null)</latxit> 2. Ring polymr molcular dynamics 2 Rcall that: <latxit sha_bas64="vcoook6qskrbagoq9stipdvm3vg=">aaacd3icbvc7tsmwfhxko7wcjcwrfyipsgosikkscymgwpcaqrpxxwrh2jf9g6ii/gelv8lcaeksrgz8du6bgdrlb2dc6/tc+jucinb8olupqnufmf9zfpwvvw9tvwvupilruiwuvorbmmelayjhwa5szscjbbumb44l//kwacovvmbhyjojxev5xtqslvj5kkyx6t6j4zckl4oaa8iilxjdibsfrjfjdyu4qavj+h9jwjiqkxha9t6inqjyi+maoxphgknrwciryyiywkgn3dn2pksjjp5om8i3/bkj2/r7q9ev2x+njh8syyrlbyqrwyh57hfif86wf9jjuuwzjjohupnnrpyi3l8htmohhaalrz+fdkadrvthuul4o/jfqrxwracfavnvbloubjjtkiuyqkb6rbtsgparjk7skjsyvzopz5lw6b5prilpubjafcn6/anram9u=</latxit> Q =tr h Ĥ i q = (T )/ p 8 (T )=h/ p 2 mkt p = p mkt ) p q = ~ 2. Q = (2 ~) n dp dq nh n (p,q) H n (p, q) = nx j= " p 2 j 2m + 2 m!2 n(q j q j+ ) 2 + V (q j ) # ; n = /n;! n =/( n ~).

12 Path intgral molcular dynamics: PIMD uss th ring polymr trajctoris q n(p, ṗ n(p, as a sampling tool to calculat xact valus of static quilibrium proprtis such as hai = Q tr H ˆ A ˆ. Ring polymr molcular dynamics: RPMD uss th sam trajctoris to approximat Kubotransformd tim corrlation functions of th form c ˆ ˆ ˆ AB (t) = h i d tr ( )H A() H B(t) ˆ, Q whr B(t) ˆ = +iht/~ ˆ B ˆ iht/~ ˆ.

13 Ring polymr molcular dynamics: Th RPMD approximation to c AB (t) = Q d tr h ( )H ˆ A() ˆ i H ˆ B(t) ˆ is simply c AB (t) ' (2 ~) n Q dp dq nh n (p,q ) A n (q )B n (q t ), whr A n (q) = n nx A(q j ) and B n (q) = n nx B(q j ). j= j= Classical molcular dynamics in an xtndd phas spac!

14 In short, th RPMD approximation includs both: tunnling and zro point nrgy But it nglcts QM intrfrnc ffcts in th ral-tim dynamics.

15 3<latxit sha_bas64="p9cpfj+ayo5aslqngfqejxu7lpm=">aaab7hicbvbns8naej3urq/qh69llacp5kgh4lxjxwmg2hjwwz3brln5uwuxfk6g/w4kerr/4gb/4bn2ko2vpg4phddpz/jgzpw372yptbg5t75r3k3v7b4dhotroossahlih7jvo8v5uxqvzpnat+wfic+pz/dpv5vscqfyveg57havxrlcaeayn5nyfw/xkqfqzg3yote6cgtsgqgdu/rqoi5kevgjcsvidx46l2kpgfurkmisayzpcedgwvoktks/njf+jckgmurnkuchxf+kofrqhvqmm8r6qla9tpzpgyq6upfsjujeugwi4keix2h7hmpiszgyckurwrkayajnpfokz+vi66tybtqth3zdr7asijjkcwtlcggpxiy76ialbbg8wyu8wcj6sd6tj2vryspmtueprm8fgingq==</latxit> On can show that RPMD is:. Exact in th high tmpratur limit 2. Exact in th short tim limit 3. Exact in th harmonic limit (for linar Aˆ or B) ˆ 4. Exact for A ˆ = ˆ (th unit oprator) 5. Faithful to all QM symmtris 6. Consistnt with th QM quilibrium distribution 2 2

16 E.g.: whn Â = ˆ whav c B (t) = Q = Q = Q tr h d d tr tr h ( h Ĥ ˆB(t) i )Ĥˆ Ĥ ˆB(t) i Ĥ ˆB(t) i = Q tr h = Q tr h Ĥ +iĥt/~ ˆB iĥt/~i i Ĥ ˆB hbi,

17 And in RPMD w also hav c B (t) = (2 ~) n Q = (2 ~) n Q = (2 ~) n Q = (2 ~) n Q hbi, dp dp t dp t dp dq nh n (p,q ) B n (q t ) dq t nh n (p,q ) B n (q t ) dq t nh n (p t,q t ) B n (q t ) dq nh n (p,q ) B n (q ) whr w hav usd Liouvill s thorm (dp dq = dp t dq t ) and th fact that RPMD trajctoris consrv H n (p t, q t ).

18 <latxit sha_bas64="wgdecbm9vbjb5gemzwyb9frm=">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</latxit> Non-local oprators So far, w hav only considrd local oprators Â = A(ˆq) and ˆB = B(ˆq). But d 2 dt 2 Qc qq(t) = d2 dt 2 tr h = d dt tr appl Ĥ ˆq +iĥt/~ ˆq iĥt/~i Ĥ ˆq +iĥt/~ i iĥt/~ [Ĥ, ˆq] ~ = d dt tr h = d h dt tr appl = tr h = tr h = tr Ĥ ˆq +iĥt/~ ˆv iĥt/~i i Ĥ iĥt/~ ˆq +iĥt/~ ˆv Ĥ iĥt/~ i ~ [Ĥ, ˆq] +iĥt/~ ˆv Ĥ iĥt/~ˆv i +iĥt/~ ˆv Ĥ ˆv +iĥt/~ˆv iĥt/~i Qc vv (t). So c vv (t) = d2 dt 2 c qq(t) and (similarly) c vv (t) = d2 dt 2 c qq(t).

19 <latxit sha_bas64="5mq3jmsg9qy+y/hwrvdflfwlwcq=">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</latxit> Thus th vlocity autocorrlation function can b calculatd in RPMD as c vv (t) = d 2 dt 2 c qq(t), which givs (ntirly naturally!) (2 ~) n Qc vv (t) = d2 dt 2 dp dq nh n (p,q ) q q t = d dp dq nh n (p,q ) q v t dt = d dp t dq t nh n (p t,q t) q v t dt = d dp dq nh n (p,q ) q t v dt = dp dq nh n (p,q ) v t v = dp t dq t nh n (p t,q t) v v t = dp dq nh n (p,q ) v v t. That is, c vv (t) =h v v t i,whr q = n nx q j and v = d dt q = n j= nx j= p j m.

20 Th sam argumnt applis to corrlation functions involving othr non-local oprators. For xampl, chmical raction rat co cints can b calculatd from Q r (T )k(t )= d c ff (t) dt = lim c fs (t) = lim t! t! dt c fs(t), whr c ff (t) = d dt c fs(t) = d2 dt 2 c ss(t), both in QM and in RPMD. But I shall not discuss this any furthr hr, as it is th subjct of Lctur III.

21 3. Exampl applications A. Quantum diffusion in liquid para-hydrogn K RPMD T.75 F.5 G L S D (A /ps) 2 o.25. Classical ρ N -/3 D = 3 D(L) =D() c v v (t) dt k BT 6 L

22 Consistncy chck: Rcall that C vv (!) = ht i = Q tr h ~! ~! C vv (!). So (in d notation) = m 2 c vv() = m 2 = m 2 = m 4 = m 2 i Ĥ ˆT = m 2 d! d! appl c vv () + h Q tr d! C vv (!) ~! C ~! vv (!) dt dt i Ĥ ˆvˆv ~! ~! i!t c vv (t) 2 d c vv (t) 2 t/ ~ dt, in which th last lin is obtaind by valuating th intgral ovr!. 5 By comparing this with th xact ht i = ht CV (q)i, on has a way to chck th accuracy of th RPMD approximation to c vv (t). <latxit sha_bas64="w35kdvnklsovw2v27dhs/ndplvi=">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</latxit>

23 For th 25 K liquid para-hydrogn xampl givn abov, this consistncy chck givs th following kintic nrgis pr atom (in 3d): 4 Kintic nrgy (K) Exact RPMD Classical Not bad th RPMD approximation to c vv (t) ovrstimats th quantum contribution to th kintic nrgy by lss than %. But not prfct RPMD is just an approximation to ral-tim quantum dynamics, aftr all! This is actually quit a stringnt tst, bcaus th thrmal tim ~ at 25 K is.3 ps, which is comparabl to th dcay tim of th p-h 2 vlocity autocorrlation function. So it provids som rason to hav faith in RPMD for othr (lss quantum mchanical) problms: <latxit sha_bas64="h+zhiq9+9xx6o9jwb+dglar+vo=">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</latxit>

24 B. Compting quantum ffcts in liquid watr 6 q-tip4p/f DQM/Dcl =.

25 C. Th vibrational spctrum of liquid watr RPMD (3 K) RPMD (35 K) n(ω)α(ω) / cm - 5 Spurious rsonancs! ω / cm -! =(2n/ ~)sin( /n) ' (2 / ~) = 3 cm at 3 K.

26 D. Thrmostattd RPMD 8 Non of th stablishd proprtis of RPMD is a ctd whn a PILE thrmostat is attachd to th intrnal mods of th ring polymr during th dynamics (TRPMD), which sms to b a good ida for calculating vibrational spctra: Curvatur problm (a) 3K 2K K CMD (c) 436K 35K 9K RPMD Spurious rsonancs σ(ω) (b) TRPMD (d) TRPMD Lorntzian broadning wavnumbr (cm - ) Vibrational spctra of an anharmonic OH molcul (2! x = 7 cm ).

27 4. Rfrncs. R. Kubo, J. Phys. Soc. Japan 2, 57 (957). 2. I. R. Craig and D. E. Manolopoulos, J. Chm. Phys. 2, 3368 (24). 3. B. J. Braams and D. E. Manolopoulos, J. Chm. Phys. 25, 245 (26). 4. T. F. Millr III and D. E. Manolopoulos, J. Chm. Phys. 22, 8453 (25). 5. B. J. Braams, T. F. Millr III and D. E. Manolopoulos, Chm. Phys. Ltt. 48, 79 (26). 6. S. Habrshon, T. E. Markland and D. E. Manolopoulos, J. Chm. Phys. 3, 245 (29). 7. S. Habrshon, G. S. Fanourgakis and D. E. Manolopoulos, J. Chm. Phys. 29, 745 (28). 8. M. Rossi, M. Criotti and D. E. Manolopoulos, J. Chm. Phys. 4, 2346 (24). 9. And finally, for a rviw of RPMD that was currnt in 23, s: S. Habrshon, D. E. Manolopoulos, T. E. Markland and T. F. Millr III, Ann. Rv. Phys. Chm. 64, 387 (23). <latxit sha_bas64="mdgbxzsjphy8qthq8dt/rlqhldq=">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</latxit>

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