Mark E. Tuckerman Dept. of Chemistry Courant Institute of Mathematical Sciences New York University New York, NY 10003

Transcription

1 Two (unrelated) topcs n quantum transport: Evaluatng centrod and rngpolymer molecular dynamcs and calculatng STM mages for molecules on semconductor surfaces Mark E. Tuckerman Dept. of Chemstry Courant Insttute of Mathematcal Scences New York Unversty New York, NY 10003

2 Ensemble average: Quantum Statstcal Mechancs á la Feynman Energy egenvalues: Ĥ E k E k E k Canoncal partton functon: Ĥ E Q Tr e e 1 1 A Tr Ae e E A E ˆ ˆ Ĥ E k ˆ k Q Q k k k k Ths law s the summt of statstcal mechancs, and the entre subject s ether the slde-down from ths summt, as the prncple s appled to varous cases, or the clmb-up to where the fundamental law s derved and the concepts of thermal equlbrum and temperature T clarfed -- Statstcal Mechancs: A Set of Lectures

3 ath Integrals n Quantum Statstcal Mechancs Carry out trace n coordnate bass: Trotter theorem: H ˆ (Kˆ V) ˆ Q Tr e dx x e x (Kˆ V) ˆ V ˆ / K ˆ / lm lm ˆ e e e Insert completeness of poston egenvectors: Q lm dx x ˆ x ˆ ˆ ˆ ˆ lm dx1 dx dx3 dx x1 x x x3 x3 x x x1 Matrx elements: 1/ ˆ m exp m 1 1 ( ) x x x x V x

4 Dscrete ath Integral Q lm Q / m m 1 1 exp 1 ( ) 1 Q dx dx x x V x x x 1 1 k m Classcal somorphsm: D. Chandler and. G. Wolynes J. Chem. hys. 74, 4078 (1981)

5 ath ntegral molecular dynamcs M. arrnello and A. Rahman, J. Chem. hys. 80, 860 (1984); R. W. Hall and B. J. Berne, J. Chem. hys. 81, 3641 (1984). Add Gaussan ntegrals to partton functon: p 1 1 Q dp1 dpdx1 dx exp m x x 1 V ( x ) 1 m x x 1 1 artton functon can be sampled usng the Hamltonan: p H m x x V x ( ) 1 m x x rmtve equatons of moton: p x p m x x x 1 1, 1 1 Heat bath m x 1 V

6 Stagng transformaton: u Varable Transformatons n path ntegrals x 1 1 ( k 1) x x k1 1 uk xk, k,..., Normal mode transformaton: MET, G. J. Martyna, M. L. Klen and B. J. Berne J. Chem. hys. 99, 796 (1993) x k k l1 a e l ( k 1)( l1)/ u a, u a, u Re( a ), u Im( a ) 1 1 ( )/ k k k 1 k Dagonalzaton of quantum knetc energy: x x u k k1 k k k1 k Stagng: k k Normal modes: k 1 k k 1 ( k 1) 1 cos

7 Transformed Hamltonan: ath ntegral molecular dynamcs MET, G. J. Martyna, M. L. Klen and B. J. Berne J. Chem. hys. 99, 796 (1993) p H m u V x u 1 1 ( ( )) 1 m x x 1 1 Fcttous masses: m 1 m, m m,,..., Equatons of moton: u p m 1 V x j p m u Heat bath( ) x u j1 j

8 Results for harmonc oscllator p 1 m, 15.8, 0.03, 400 m 1 1 V Evr V ( x) x 1 x H m x M. F. Herman, E. J. Bruskn and B. J. Berne J. Chem. hys. 76, 5150 (198). rmtve rmtve rmtve Stagng IMD Stagng IMD Stagng IMD Stagng IMC Stagng IMC Stagng IMC

9 A. érez, M. Müser, MET J. Chem. hys. 130, (009) 4 x V( x) 4 8, 3 σ Stagng + NHCs rmtve + velocty scalng Error n vral estmator Lqud para-hydrogen Slvera-Goldman potental T 14 K, A N 56, 64 Tme = 6 ps -3 rmtve Stagng Error n vral estmator I. F. Slvera and V. V. Goldman, JC 69, 409 (1978)

10 Non-symmetrc correlaton functon: Quantum tme correlaton functons G. Krlov, E. Sm and B. J. Berne J. Chem. hys. 114, 1075 (001) 1 C ˆ ˆ ˆ AB( t) A(0)B( t) Tr e Ae Be Q Hˆ H ˆ t / H ˆ t / Assume Aˆ A( ˆ xˆ), Bˆ B( ˆ xˆ), trace n coordnate bass 1 Hˆ H ˆ / H ˆ / AB( ) ' '' ' ( ') ' t t C t dx dx dx x e x a x x e x '' b( x '') x '' e x Q x Real tme (backward) Imag Imagnary tme x t Real x Real tme (forward)

11 Quantum tme correlaton functons Symmetrc complex-tme correlaton functon: 1 G ˆ AB( t) Tr Ae B e, t Q H ˆ*/ H ˆ / Assume Aˆ A( ˆ xˆ), Bˆ B( ˆ xˆ), trace n coordnate bass 1 H ˆ */ H ˆ / AB( ) ' ( ) G t dx dx a x x e x ' b( x ') x ' e x Q Complex tme (forward) Imag x x Complex tme (backward) / t Real C ( ) e G ( ) Fourer transforms: AB AB

12 Quantum tme correlaton functons Kubo-transformed correlaton functon: 1 H ˆ */ H ˆ ˆ / KAB( t) d Tr Ae B e, t Q 0 Assume Aˆ A( ˆ xˆ), Bˆ B( ˆ xˆ), trace n coordnate bass 1 H ˆ */ H ˆ / KAB( t) ddx dx ' a( x) x e x ' b( x ') x ' e x Q 0 Imag Fourer transforms: C AB ( ) KAB( ) 1 e λ t Real

13 Centrod molecular dynamcs J. Cao and G. A. Voth J. Chem. hys. 99, (1993) J. Cao and G. A. Voth J. Chem. hys. 100, 5106 (1994) J. Cao and G. A. Voth J. Chem. hys. 101, 6168 (1994) ath Centrod x c 1 1 x x c s the unbound mode u 1 n a normal-mode transformaton. Centrod potental of mean force: (Feynman and Klenert hys. Rev. A 34, 5080 (1986)) 1 W x W u kt du du m u V x u ( c) ( 1) ln exp k k ( k ( )) k k1 Dscrete parttonfuncton: Q m / dx e c W( x ) c

14 Centrod molecular dynamcs Approxmate quantum dynamcs as dynamcs on the centrod MF x c p c m, p c dw dx Approxmaton to Kubo-transformed tme correlaton functon: 1 p c K AB ( t) dpcdxca( xc (0)) b( xc ( t))exp W ( xc ) Q m Generate centrod MF on the fly by performng magnary-tme IMD wth m 1 m, m k mk, 1 called partally adabatc CMD. (J. Cao and G. J. Martyna, J. Chem. hys. 104, 08 (1996)) Short-tme accuracy (B. J. Braams and D. E. Manolopoulos J. Chem. hys. 17, (007)) K t x x t O t K t v v t O t 6 4 xx ( ) c(0) c( ) ( ), vv( ) c(0) c( ) ( ) Exact n harmonc and classcal lmts. c

15 Classcal rng-polymer Hamltonan: Rng-polymer molecular dynamcs p 1 H x p m x x V x cl(, ) 1 ( ) 1 m x x Equatons of moton: 1 1 p V x, p m ( x x 1 x 1) m x Approxmate correlaton functon: I. R. Crag and D. E. Manolopoulos J. Chem. hys. 11, 3368 (004) (Can also be wrtten n terms of normal-mode and stagng varables) 1 K AB ( t) d pd x a(0) b ( t)exp Hcl( x, p) / Q 1 1 a( t) a( x ( t)), b ( t) b( x ( t)) 1 1 Short-tme accuracy: (B. J. Braams and D. E. Manolopoulos J. Chem. hys. 17, (007)) K t x x t O t K t v v t O t 8 6 xx ( ) (0) ( ) ( ), vv( ) (0) ( ) ( )

16 Some numercal examples Recommended algorthm: x V ( x) 0.1x 0.01x 3, CMD x V( x) 4 3, CMD

17 Some numercal examples T N Lqud para-hydrogen 14 K, A 56, 64, Tme = 6 ps -3 CMD ( x.) ( x1.5) CMD V ( x) 0.1 x 4 e 0.5e 3, D D D exp CMD RMD 0.4 A / ps 0.31 A / ps 0.6 A / ps Dffuson constants agree wth Hone et al. JC 14, (006) and Mller and Manolopoulos JC 1, (005)

18 Self-consstent qualty measure of CMD and RMD correlaton functons Imagnary-tme mean-square dsplacement: 1 G( ) xˆ(0) xˆ( ) ( ) free partcle m N 1 G( ) rˆ(0) ˆ r ( ) ( ) free partcles N m 1 Relaton to real-tme poston and velocty correlaton functons: 1 G( ) de K xx( ) cosh cosh 1 Kvv ( ) de cosh cosh

19 Error Measure A. érez, M. Müser, MET J. Chem. hys. 130, (009) (est) 1 G ( ) G( ) d 0 G()

20 D D D exp CMD RMD 0.4 A / ps 0.31 A / ps 0.6 A / ps T N Lqud para-hydrogen 14 K, A 56, 64, Tme = 6 ps -3 CMD Consderatons of CMD and RMD n calculatng vbratonal spectra: A. Wtt et al. JC 130, (009) S. D. Ivanov et al. JC 13, (010)

21 Organc-Semconductor Interfaces Organc chem. Nano-lthography Surface physcs Example: - assvate S(100) surface wth benzene - Create nm wde patterns wth STM tp - React wth vnyl ferrocene Kruse and Wolkow Appl. hys. Lett. 81 (00) 44. Chemo-/bo-sensors Self-assembled nanowres and other nanostructures Example: - styrene forms lnes on H-S(100) - precursor to molecular electroncs DLabo, va, Kruse, and Wolkow JACS 16 (004) Monolayers Example: - monolayer of 1,5-cyclooctadene absorbed on S(100) - -bond on surface avalable for further rxns - precursor to molecular sensor DLabo, va, Kruse, and Wolkow JACS 16 (004)

22 Non-concerted mechansm of Dels-Alder adduct from ab nto MD. Mnary and MET J. Am. Chem. Soc. 16, 1390 (004). Mnary and MET J. Am. Chem. Soc. 17, 1110 (005) R. L. Hayes and MET J. Am. Chem. Soc. 19, 117 (007) R. Iftme,. Mnary, MET roc. Natl. Acad. Sc. 10, 6654 (005) 1,3-cyclohexadene on S(100)-x1

23 Calculaton of Empty-State STM Images R. L. Hayes and MET (to be submtted) L. C. Teague and J. J. Boland JCB 107, 380 (003) Tersoff-Hamann Bardeen Tersoff-Hamann: I r E E ( tp) F Bardeen: I f ( E ) 1 f ( E ev ) M E E,

24 Calculaton of STM Images Naydenov et al. Nano Lett. 6, 175 (006) Need to perform self-consstent conductance calculatons (NEGF) on a full grd!

25 Acknowledgments Students eter Mnary (now at Stanford) Alejandro érez Yanl Zhang ostdocs Robn Hayes (now AAAS Fellow) External Martn Müser (Saarbrücken)