Generalized Temporal Acceleration Scheme for ... - ACS Publications

Feb 14, 2017 - can be practically executed during runtime. □ GENERALIZED TEMPORAL ACCELERATION. METHOD. Overview of the Algorithm. At the most funda...
1 downloads 12 Views 1MB Size
Subscriber access provided by UB + Fachbibliothek Chemie | (FU-Bibliothekssystem)

Article

Generalized Temporal Acceleration Scheme for Kinetic Monte Carlo Simulations of Surface Catalytic Processes by Scaling the Rates of Fast Reactions Eric Christopher Dybeck, Craig Patrick Plaisance, and Matthew Neurock J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00859 • Publication Date (Web): 14 Feb 2017 Downloaded from http://pubs.acs.org on March 2, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Journal of Chemical Theory and Computation is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

‡•‡”ƒŽ‹œ‡† ‡•’‘”ƒŽ ……‡Ž‡”ƒ–‹‘• …Š‡•‡ ˆ‘” ‹•‡–‹… ‘•–‡ ƒ”Ž‘ ‹•—Žƒ–‹‘•• ‘ˆ —”ˆƒ…‡ ƒ–ƒŽ›–‹… ”‘…‡••‡• „› …ƒŽ‹•‰ –Š‡ ƒ–‡• ‘ˆ ƒ•– ‡ƒ…–‹‘•• Œ]

X

Ç

l • U Œ ]P WX Wo ]• v



‰ Œšu vš }( Z u]



‰ Œšu vš }(

D]vv

Z u]

o vP]v o

vP]v



U v D ššZ Á E µŒ} l • •

Œ]vPU hv]À Œ•]šÇ }( s]ŒP]v] U Z Œo}šš •À]oo U s Œ]vP

v

D š Œ] o• ^ ] v

U hv]À Œ•]šÇ }( D]vv •}š U

‰}o]•U DE

„•–”ƒ…–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

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 43

UHDOLVWLF V\VWHPV DQG WKH FRPSXWDWLRQDO FRVW LV QRW OLPLWHG E\ WKH PDJQLWXGH RI WKH WLPHVFDOH GLVSDULW\ LQ WKH V\VWHP SURFHVVHV )XUWKHUPRUH WKH DOJRULWKP KDV EHHQ GHVLJQHG LQ D JHQHULF IDVKLRQ DQG FDQ HDVLO\ EH DSSOLHG WR RWKHU VXUIDFH FDWDO\WLF SURFHVVHV RI LQWHUHVW

•–”‘†—…–‹‘• .LQHWLF 0RQWH &DUOR .0& VLPXODWLRQV KDYH EHHQ XVHG H[WHQVLYHO\ WR PRGHO WKH G\QDPLF EHKDYLRU RI D ZLGH UDQJH RI FRPSOH[ V\VWHPV LQ VFLHQFH DQG HQJLQHHULQJ 7KHVH PHWKRGV KDYH PDGH LPSRUWDQW FRQWULEXWLRQV WR VLPXODWLQJ D EURDG UDQJH RI GLIIHUHQW DUHDV LQFOXGLQJ WUDQVSRUW FU\VWDO JURZWK

±

DQG SKDVH WUDQVLWLRQV

FKHPLFDO FDWDO\VLV

±

$OO RI WKHVH V\VWHPV VKDUH WKH FRPPRQ IHDWXUH WKDW VWRFKDVWLF RU

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

±

.0& VLPXODWLRQV RI VXUIDFH FDWDO\WLF SURFHVVHV W\SLFDOO\ IXQFWLRQ E\ UHSUHVHQWLQJ WKH V\VWHP RI LQWHUHVW DV D ODWWLFH RI VLWHV HDFK RI ZKLFK FDQ HLWKHU EH YDFDQW RU RFFXSLHG E\ RQH RI PDQ\ VXUIDFH VSHFLHV 7KLV ODWWLFH PRGHO DSSURDFK LV FRPPRQO\ UHIHUUHG WR DV D ODWWLFH JDV PRGHO

±

(YROXWLRQ RI WKH V\VWHP LV PRGHOHG

WKURXJK D VHULHV RI GLVFUHWH WUDQVIRUPDWLRQV RI WKH VSHFLHV RQ WKH ODWWLFH VXEMHFW WR WKH XQGHUO\LQJ NLQHWLFV

ACS Paragon Plus Environment

Page 3 of 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

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

±

$Q RYHUYLHZ RI WKHVH

8OWLPDWHO\ WKH OLQHDU LQFUHDVHV LQ HIILFLHQF\ SURYLGHG E\ WKHVH DSSURDFKHV

DUH QRW VXIILFLHQW WR UHGXFH WKH FRPSXWDWLRQDO FRVW WR DQ DFFHSWDEOH OHYHO ZKLFK LQ PDQ\ V\VWHPV RI LQWHUHVW FDQ HDVLO\ EH RUGHUV RI PDJQLWXGH EH\RQG ZKDW LV WUDFWDEOH 7KH KLJK FRPSXWDWLRQDO FRVW RI .0& VLPXODWLRQV RI VXUIDFH FDWDO\WLF SURFHVVHV W\SLFDOO\ DULVHV IURP D GLVSDULW\ LQ WLPHVFDOHV EHWZHHQ WZR RU PRUH UHDFWLRQ FKDQQHOV

)RU H[DPSOH DGVRUSWLRQ DQG GHVRUSWLRQ

SURFHVVHV W\SLFDOO\ RFFXU ZLWK UDWH FRQVWDQWV PDQ\ RUGHUV RI PDJQLWXGH KLJKHU WKDQ WKRVH RI UDWH OLPLWLQJ VXUIDFH UHDFWLRQV

%HFDXVH SURFHVVHV LQ .0& VLPXODWLRQV DUH VHOHFWHG EDVHG RQ WKHLU UHODWLYH UDWH

FRQVWDQWV LW LV IDU PRUH OLNHO\ WKDW D IDVW SURFHVV ZLOO RFFXU DW HDFK VWHS ZLWK RQO\ LQIUHTXHQW VDPSOLQJ RI WKH VORZHU SURFHVVHV 6LPLODUO\ WKH VLPXODWLRQ WLPH DIWHU HDFK VWHS DGYDQFHV RQ WKH WLPHVFDOH RI WKH IDVWHVW

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 43

SURFHVVHV DQG WKXV DQ LQWUDFWDEOH QXPEHU RI VLPXODWLRQ VWHSV PXVW RFFXU WR DFKLHYH ORQJ WLPHVFDOH VLPXODWLRQV XVLQJ D WUDGLWLRQDO EUXWH IRUFH DSSURDFK 6SDUVH VDPSOLQJ RI VORZ UHDFWLRQ FKDQQHOV JHQHUDOO\ XQGHUPLQHV WKH XWLOLW\ RI .0& EHFDXVH LQ PRVW VXUIDFH FDWDO\WLF V\VWHPV WKH UDUH SURFHVVHV RFFXUULQJ RQ WKH ORQJHU WLPHVFDOHV DUH SUHFLVHO\ WKH SURFHVVHV RI LQWHUHVW LQ WKH VLPXODWLRQ L H WKH RQHV FRQWUROOLQJ WKH WXUQRYHU IUHTXHQFLHV DQG VHOHFWLYLWLHV )XUWKHUPRUH WKH IDVW SURFHVVHV RIWHQ FRQYHUJH UDSLGO\ WR D TXDVL HTXLOLEUDWHG VWDWLRQDU\ VWDWH :KHQ WKH V\VWHP UHDFKHV WKLV TXDVL HTXLOLEUDWHG VWDWH IXUWKHU VDPSOLQJ RI WKH IDVW SURFHVVHV SURYLGHV HVVHQWLDOO\ QR QHZ LQIRUPDWLRQ DERXW WKH PDFURVFRSLF HYROXWLRQ RI WKH V\VWHP (VFDSLQJ WKLV NLQHWLF WUDS LQ WKH IUHH HQHUJ\ ODQGVFDSH FDQ EH DFFRPSOLVKHG E\ DFFHOHUDWLQJ WKH VLPXODWLRQ WKURXJK DSSURSULDWH FRDUVH JUDLQLQJ 2QH FDXVH RI WLPHVFDOH GLVSDULW\ RFFXUV ZKHQ RQH VXUIDFH VSHFLHV KDV VLJQLILFDQWO\ ODUJHU FRQFHQWUDWLRQV WKDQ DQRWKHU 7KH 2 OHDS PHWKRG LV RIWHQ LPSOHPHQWHG WR RYHUFRPH WKLV W\SH RI GLVSDULW\ LQ WLPHVFDOHV E\ H[HFXWLQJ PXOWLSOH HYHQWV LQ D VLQJOH .0& VWHS

±

$ VHFRQG FDXVH RI

WLPHVFDOH GLVSDULW\ DULVHV IURP D ODUJH GLIIHUHQFH LQ UDWHV RU UDWH FRQVWDQWV EHWZHHQ GLIIHUHQW SURFHVVHV +DVHOWLQH DQG 5DZOLQJV GHYHORSHG RQH PHWKRG WR DGGUHVV WKLV W\SH RI GLVSDULW\ ZKLFK SDUWLWLRQV WKH V\VWHP SURFHVVHV LQWR WZR JURXSV ZLWK WKH IDVW SURFHVVHV DSSUR[LPDWHG GHWHUPLQLVWLFDOO\ RU DV /DQJHYLQ SURFHVVHV DQG WKH VORZ SURFHVVHV PRGHOHG VWRFKDVWLFDOO\ ZLWK WKH *LOOHVSLH DOJRULWKP

±

2WKHU K\EULG

GHWHUPLQLVWLF VWRFKDVWLF DOJRULWKPV KDYH EHHQ GHYHORSHG WR FDOFXODWH RU HOLPLQDWH WKH FRQFHQWUDWLRQV RI KLJKO\ UHDFWLYH ODWWLFH VSHFLHV DQDO\WLFDOO\ LVVXH RI D GLVSDULW\ LQ UHDFWLRQ UDWHV

$GGLWLRQDO DOJRULWKPV DOVR H[LVW WR JHQHUDOO\ DGGUHVV WKH

±

7KH FRDUVH JUDLQLQJ DSSURDFKHV WR UHVROYH GLVSDULWLHV LQ UHDFWLRQ UDWHV UHSUHVHQW YDULRXV LPSOHPHQWDWLRQV RI WKH VDPH XQGHUO\LQJ DVVXPSWLRQ 1DPHO\ WKDW WKH IDVW UHDFWLRQV DUH TXDVL HTXLOLEUDWHG DQG LW LV WKH VORZ UHDFWLRQV WKDW DUH NLQHWLFDOO\ UHOHYDQW ,Q WKLV DSSUR[LPDWLRQ WKHVH IDVW UHDFWLRQV DUH DFFHOHUDWHG WR ZLWKLQ D IHZ RUGHUV RI PDJQLWXGH RI WKH WLPHVFDOHV RI WKH VORZ UHDFWLRQV 7KH SUHFLVH LPSOHPHQWDWLRQ RI WKLV

ACS Paragon Plus Environment

Page 5 of 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

DSSURDFK KRZHYHU LV QRW VWUDLJKWIRUZDUG DQG KDV UHVXOWHG LQ WKH GHYHORSPHQW RI D P\ULDG RI GLIIHUHQW DOJRULWKPV LQ WKH OLWHUDWXUH

±

±

0DQ\ RI WKHVH DOJRULWKPV UHTXLUH WKH XVHU WR VSHFLI\ D SULRUL LQIRUPDWLRQ DERXW WKH IDVW VORZ SURFHVVHV RU LGHQWLI\ WKH TXDVL HTXLOLEUDWHG UHDFWLRQV

7KH QHHG WR SURYLGH WKLV NLQHWLF LQIRUPDWLRQ XSIURQW LV

XQGHVLUDEOH EHFDXVH WKH TXDVL HTXLOLEUDWHG UHDFWLRQV PD\ FKDQJH DV WKH V\VWHP HYROYHV RU WKH UHDFWLRQ QHWZRUN PD\ EH VXIILFLHQWO\ FRPSOH[ WKDW WKH XVHU VLPSO\ GRHV QRW NQRZ LQ DGYDQFH ZKLFK UHDFWLRQV DUH TXDVL HTXLOLEUDWHG DW VWHDG\ VWDWH 2WKHU DOJRULWKPV KDYH EHHQ GHVLJQHG WR RSHUDWH RQ YHU\ VSHFLILF FODVVHV RI UHDFWLRQ V\VWHPV 7KH /DQJHYLQ PHWKRG IRU H[DPSOH DVVXPHV WKDW WKH G\QDPLFV RI WKH IDVW UHDFWLRQV DUH FRQWUROOHG E\ VSHFLHV ZLWK ODUJH FRQFHQWUDWLRQV 6LPLODUO\ WKH 2 OHDS PHWKRG UHTXLUHV WKDW UHDFWLRQ SURSHQVLWLHV UHPDLQ FRQVWDQW RYHU PXOWLSOH WLPH VWHSV WR VDWLVI\ WKH µOHDS FRQGLWLRQ¶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

IRU H[DPSOH

FRDUVH JUDLQLQJ LV DFKLHYHG XVLQJ D QHVWHG VLPXODWLRQ VFKHPH ZKHUH D QXPEHU RI LQGHSHQGHQW FRSLHV RI WKH V\VWHP DUH HYROYHG VLPXOWDQHRXVO\ LQ WLPH LQ DQ LQQHU ORRS ZKLOH D SURFHVV RQ WKH VORZ WLPHVFDOH LV IROORZHG LQ WKH RXWHU ORRS 7KH DOJRULWKP GHYHORSHG LQ UHI

UHTXLUHV WKH VLPXODWLRQ WR VWRUH PXOWLSOH

V\VWHP FRQILJXUDWLRQV VR WKDW D VORZ SURFHVV FDQ EH H[HFXWHG IURP DQ\ RQH RI WKHP 2WKHU DOJRULWKPV KDYH UHOLHG XSRQ FRPSXWLQJ DQDO\WLFDO SUREDELOLW\ GHQVLW\ IXQFWLRQV DQG WKHLU FRUUHVSRQGLQJ PRPHQWV $OWKRXJK WKHVH FRPSOH[ DOJRULWKPV FDQ DFFHOHUDWH VSHFLILF V\VWHPV ZKHUH DQDO\WLFDO PRPHQWV DUH DYDLODEOH WKH H[DFW LPSOHPHQWDWLRQ WR D JHQHUDO VHW RI .0& VLPXODWLRQV LV QRW VWUDLJKWIRUZDUG

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 43

$ WHPSRUDO DFFHOHUDWLRQ VFKHPH ZDV UHFHQWO\ GHYHORSHG E\ &KDWWHUMHH DQG 9RWHU WKDW LV DSSOLFDEOH WR DQ\ W\SH RI V\VWHP DQG UHTXLUHV QR VSHFLDO LQSXW IURP WKH XVHU DERXW ZKLFK SURFHVVHV DUH IDVW DQG VORZ

7KLV

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

ACS Paragon Plus Environment

Page 7 of 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

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

‹•‡–‹…

‘•–‡ ƒ”Ž‘ ƒ…•‰”‘—•†

7KH PDWKHPDWLFDO IRXQGDWLRQ IRU NLQHWLF 0RQWH &DUOR VLPXODWLRQV LV GHULYHG IURP WKH ZHOO NQRZQ FKHPLFDO PDVWHU HTXDWLRQ &0(

$OWKRXJK WKH FKHPLFDO PDVWHU HTXDWLRQ SURYLGHV D FRPSOHWH GHVFULSWLRQ RI WKH

VWRFKDVWLF G\QDPLFV WKH HTXDWLRQ FDQQRW EH VROYHG DQDO\WLFDOO\ IRU HVVHQWLDOO\ DOO V\VWHPV RI SUDFWLFDO LQWHUHVW 7KLV OHG WR WKH GHYHORSPHQW RI WKH UHMHFWLRQ IUHH 6WRFKDVWLF 6LPXODWLRQ $OJRULWKP 66$ SLRQHHUHG E\ *LOOHVSLH

HTXLYDOHQW WR WKH Q IROG PHWKRG RI %RUW] .DORV DQG /HERZLW]

ACS Paragon Plus Environment

WR HVWLPDWH V\VWHP

Journal of Chemical Theory and Computation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

REVHUYDEOHV QXPHULFDOO\ 7KH 66$ JHQHUDWHV UDQGRP VDPSOHV RI V\VWHP FRQILJXUDWLRQV WKDW LQ WKH OLPLW RI LQILQLWH VDPSOLQJ UHSURGXFHV WKH HQVHPEOH IURP WKH H[DFW VROXWLRQ WR WKH &0( ,Q WKH 66$ DOO HYHQWV LQ D SDUWLFXODU FRQILJXUDWLRQ DUH JLYHQ DQ LQWULQVLF UDWH FRQVWDQW EDVHG RQ WKH XQGHUO\LQJ V\VWHP NLQHWLFV /HW Tá UHSUHVHQW WKH FRQILJXUDWLRQ RI WKH V\VWHP DW VWHS J ZLWKLQ WKH IXOO FRQILJXUDWLRQ VSDFH : VXFK WKDW Tá Ð : $W HDFK VWHS J DQ HYHQW E LV UDQGRPO\ VHOHFWHG ZLWK SUREDELOLW\

2Ü L

GÜ :Tá ; ÃÝ GÝ :Tá ;

ZKHUH GÝ :Tá ; LV WKH UDWH FRQVWDQW RI HYHQW F LQ FRQILJXUDWLRQ Tá $IWHU WKH HYHQW H[HFXWLRQ WKH V\VWHP FORFN LV LQFUHPHQWHG E\

¿Pá L F

ZKHUH 40 LV D XQLIRUP UDQGRP QXPEHU EHWZHHQ

HJ>40? ÃÝ GÝ :Tá ;

DQG

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± L H ZKDW RQH ZRXOG FDOO D µFKHPLFDO UHDFWLRQ¶ LQ D GHWHUPLQLVWLF PRGHO 7KHUH LV D GHJUHH RI IOH[LELOLW\ LQ DVVLJQLQJ HYHQWV WR UHDFWLRQ FKDQQHOV )RU H[DPSOH RQ D VXUIDFH ZLWK WZR GLIIHUHQW W\SHV RI DGVRUSWLRQ VLWHV , DQG ,, IRU D VSHFLHV $ RQH FRXOG WUHDW LGHQWLFDO FKHPLFDO WUDQVIRUPDWLRQV IURP HDFK RI WKH WZR W\SHV RI VLWHV DV GLVWLQFW UHDFWLRQ FKDQQHOV RU RQH FRXOG WUHDW WKHP DV

ACS Paragon Plus Environment

Page 8 of 43

Page 9 of 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

FRUUHVSRQGLQJ WR WKH VDPH UHDFWLRQ FKDQQHO ,Q WKLV ZRUN ZH WDNH LGHQWLFDO FKHPLFDO WUDQVIRUPDWLRQV WR UHSUHVHQW D VLQJOH UHDFWLRQ FKDQQHO UHJDUGOHVV RI VLWH W\SH +RZHYHU RQH FRXOG VWUDLJKWIRUZDUGO\ DVVLJQ WKHVH HYHQWV WR GLIIHUHQW UHDFWLRQV FKDQQHOV LQ V\VWHPV ZKHUH WKH NLQHWLFV IURP GLIIHUHQW VLWHV DUH VLJQLILFDQWO\ GLIIHUHQW :H GHILQH WKH DYHUDJH UDWH FRQVWDQW Gà RI D SDUWLFXODU UHDFWLRQ FKDQQHO I LQ VWHS J DV EHLQJ HTXDO WR WKH VXP RYHU DOO HYHQWV F EHORQJLQJ WR UHDFWLRQ FKDQQHO I WKDW FDQ RFFXU IURP FRQILJXUDWLRQ Tá

Gà :Tá ; L Í GÝ :Tá ; ÝÐà

7KH UDWH Nà RI D UHDFWLRQ FKDQQHO I UHSUHVHQWV DQ DYHUDJH RYHU WKH HQWLUH FRQILJXUDWLRQ VSDFH ,Q SUDFWLFH WKLV LV DSSUR[LPDWHG E\ D WLPH DYHUDJH RYHU WKH VLPXODWLRQ VWHSV LQ WKH OLPLW RI ORQJ WLPHVFDOHV XWLOL]LQJ WKH IXQGDPHQWDOO\ HUJRGLF QDWXUH RI WKH .0& VLPXODWLRQ

Nà L Í Gà :T;L:T; L ëÐÑ

s Í Gà :Tá ;¿Pá P á

ZKHUH Gà :T; LV WKH DYHUDJH UDWH FRQVWDQW RI UHDFWLRQ FKDQQHO I LQ FRQILJXUDWLRQ T L:T; LV WKH SUREDELOLW\ RI WKH V\VWHP EHLQJ LQ WKH FRQILJXUDWLRQ T JHQHUDOO\ IROORZLQJ %ROW]PDQQ VWDWLVWLFV

Pá LV WKH DPRXQW RI

WLPH WKH V\VWHP VSHQGV LQ FRQILJXUDWLRQ Tá DQG P LV WKH WRWDO VLPXODWLRQ WLPH 0DQ\ RWKHU PDFURVFRSLF SURSHUWLHV æ#ç FDQ DOVR EH FRPSXWHG LQ WKH VDPH ZD\ DV D VLPSOH DYHUDJH RYHU FRQILJXUDWLRQV

Ã#Ä L Í #:T;L:T; L ëÐÑ

s Í #:Tá ;¿Pá P á

ZKHUH #:T; LV WKH LQVWDQWDQHRXV YDOXH RI WKH REVHUYDEOH LQ FRQILJXUDWLRQ T

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

7R REWDLQ DQ DFFXUDWH YDOXH IRU WKH UDWH RI WKH UHDFWLRQ I XVLQJ HT VXIILFLHQWO\ ORQJ WLPH ,W FDQ EH HDVLO\ VHHQ IURP HT

Page 10 of 43

WKH VLPXODWLRQ PXVW EH UXQ RYHU D

WKDW WKH WLPH LQFUHPHQW DW HDFK VWHS LV FRQWUROOHG

E\ WKH VXP RI WKH UDWH FRQVWDQWV RI DOO HYHQWV RFFXUULQJ IURP WKH FXUUHQW FRQILJXUDWLRQ ZKLFK LV GRPLQDWHG E\ WKH IDVW UHDFWLRQV 6LPLODUO\ HT

VKRZV WKDW LW LV IDU PRUH OLNHO\ IRU DQ HYHQW IURP D IDVW UHDFWLRQ

FKDQQHO WR EH VHOHFWHG DW HDFK .0& VWHS ([DPLQHG PRUH TXDQWLWDWLYHO\ FRQVLGHU D V\VWHP FRQWDLQLQJ D VHW RI IDVW UHDFWLRQV RFFXUULQJ RQ D WLPHVFDOH RI RUGHU ìÙ DQG D VHW RI VORZ UHDFWLRQV RFFXUULQJ RQ D WLPHVFDOH RI RUGHU ìæ ZLWK ìæ ( ìÙ +HUH WKH UDWLR ìæ ìÙ UHSUHVHQWV WKH GLVSDULW\ LQ WLPH VFDOHV EHWZHHQ WKH IDVW DQG VORZ UHDFWLRQV ,Q WKLV V\VWHP WKH G\QDPLFV DW HDFK VWHS ZLOO HYROYH DW WKH WLPHVFDOH ìÙ DFFRUGLQJ WR WKH UDWH RI WKH IDVW UHDFWLRQV 7KH SUREDELOLW\ RI DQ HYHQW FRUUHVSRQGLQJ WR D VORZ UHDFWLRQ H[HFXWLQJ LQ DQ\ JLYHQ FRQILJXUDWLRQ LV URXJKO\ ìÙ ìæ UHTXLULQJ WKDW 0Ù L ìæ ìÙ .0& VWHSV EH H[HFXWHG RQ DYHUDJH WR DFKLHYH D VLQJOH H[HFXWLRQ RI VXFK DQ HYHQW 7KLV GLVSDULW\ FDQ EH TXLWH ODUJH LQ SUDFWLFDO V\VWHPV DQG ZLOO SUHYHQW GLUHFW VLPXODWLRQ LI 0Ù LV VLJQLILFDQWO\ ODUJHU WKDQ WKH WRWDO QXPEHU RI .0& VWHSV WKDW FDQ EH SUDFWLFDOO\ H[HFXWHG GXULQJ UXQWLPH

‡•‡”ƒŽ‹œ‡† ‡•’‘”ƒŽ ……‡Ž‡”ƒ–‹‘•

‡–Š‘†

˜‡”˜‹‡™ ‘ˆ –Š‡ Ž‰‘”‹–Š• $W WKH PRVW IXQGDPHQWDO OHYHO WKH FRPSXWDWLRQDO OLPLWDWLRQ LQWURGXFHG E\ GLVSDUDWH WLPH VFDOHV LQ .0& VLPXODWLRQV H[LVWV EHFDXVH WKH H[HFXWLRQ UDWLR RI WKH IDVWHVW SURFHVVHV WR VORZHVW SURFHVVHV LV SURKLELWLYHO\ KLJK 7KXV DOPRVW DOO RI WKH FRPSXWDWLRQDO HIIRUW ZLOO EH VSHQW VLPXODWLQJ IDVW SURFHVVHV ZLWK WKH VORZ NLQHWLFDOO\ UHOHYDQW SURFHVVHV UDUHO\ RFFXUULQJ 3UHYLRXV FRDUVH JUDLQLQJ DOJRULWKPV KDYH IRFXVHG SULPDULO\ RQ DUWLILFLDOO\ LQFUHDVLQJ WKH VHOHFWLRQ RI WKH VORZ SURFHVVHV E\ µIRUFLQJ¶ D VORZ SURFHVV WR H[HFXWH

±

7KH SUHVHQW DOJRULWKP GLIIHUV IURP WKHVH DSSURDFKHV E\ PDNLQJ XVH RI WKH IDFW WKDW WKH IDVW SURFHVVHV DUH RIWHQ TXDVL HTXLOLEUDWHG RQ WKH WLPHVFDOH RI WKH VORZ QRQ HTXLOLEUDWHG SURFHVVHV WKDW DUH NLQHWLFDOO\ UHOHYDQW 7KLV PHDQV WKDW WKH UDWHV RI WKHVH IDVW SURFHVVHV FDQ EH DUWLILFLDOO\ GHFUHDVHG E\ VFDOLQJ WKHLU UDWH FRQVWDQWV VR WKDW WKHLU UDWHV DUH FORVHU WR WKH UDWHV RI WKH VORZHU SURFHVVHV 7KLV ORZHUV WKH H[HFXWLRQ UDWLR EHWZHHQ

ACS Paragon Plus Environment

Page 11 of 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

IDVW DQG VORZ SURFHVVHV WKXV UHGXFLQJ WKH WRWDO QXPEHU RI VLPXODWLRQ VWHSV UHTXLUHG WR REVHUYH D VXIILFLHQW QXPEHU VORZ SURFHVV H[HFXWLRQV 7KH DSSOLFDWLRQ RI VFDOLQJ WKH UDWHV YLD WKH UDWH FRQVWDQWV RI IDVW SURFHVVHV LV LOOXVWUDWHG VFKHPDWLFDOO\ LQ )LJXUH

IRU D VLPSOH VHULDO QHWZRUN RI LVRPHUL]DWLRQ UHDFWLRQV $Q HTXDO VFDOLQJ LV DSSOLHG WR WKH IRUZDUG

DQG UHYHUVH GLUHFWLRQV RI HDFK IDVW SURFHVV WR SUHVHUYH WKH XQGHUO\LQJ WKHUPRG\QDPLFV GHWDLOHG EDODQFH RI WKH V\VWHP

)LJXUH

*UDSKLFDO LOOXVWUDWLRQ RI WKH D XQVFDOHG DQG E VFDOHG UDWHV IRU D VHW RI VLPSOH LVRPHUL]DWLRQ

UHDFWLRQV RFFXUULQJ RQ GLVSDUDWH WLPHVFDOHV 7KH UDWH OLPLWLQJ UHDFWLRQ LV WKH ILQDO LVRPHUL]DWLRQ ':( ZLWK D UDWH HTXDO WR N5 7KLV UHDFWLRQ LV UDUHO\ VDPSOHG LQ DQ XQDFFHOHUDWHG .0& VLPXODWLRQ GXH WR WKH IDVW UDWHV RI WKH SUHFHGLQJ UHDFWLRQV ,Q WKH DFFHOHUDWLRQ VFKHPH WKH UDWH FRQVWDQWV RI WKH TXDVL HTXLOLEUDWHG UHDFWLRQV DUH VFDOHG WR JLYH UDWHV RI N6 L 0Ù N5 VR WKDW WKH VORZ UHDFWLRQ ZLOO EH VDPSOHG PRUH IUHTXHQWO\ GXULQJ WKH VLPXODWLRQ :H DSSO\ WKH VFDOLQJ SURFHGXUH WR UHDFWLRQ FKDQQHOV UDWKHU WKDQ WR LQGLYLGXDO HYHQWV LQ FRQWUDVW WR WKH PHWKRG RI &KDWWHUMHH DQG 9RWHU

7R FDUU\ RXW WKLV VFDOLQJ SURFHGXUH WKH UHDFWLRQ FKDQQHOV DUH ILUVW

SDUWLWLRQHG LQWR TXDVL HTXLOLEUDWHG DQG QRQ HTXLOLEUDWHG UHDFWLRQ FKDQQHOV 7KH UDWH FRQVWDQWV RI WKH HYHQWV LQ WKH TXDVL HTXLOLEUDWHG UHDFWLRQ FKDQQHOV DUH WKHQ UHGXFHG VR WKDW WKH UDWHV RI WKHVH UHDFWLRQ FKDQQHOV DUH ZLWKLQ D IHZ RUGHUV RI PDJQLWXGH RI WKH HVFDSH UDWH IURP WKH FXUUHQW VXSHUEDVLQ DV VKRZQ LQ )LJXUH

ACS Paragon Plus Environment

7KLV

Journal of Chemical Theory and Computation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

VFDOLQJ DOORZV IRU VXIILFLHQW VDPSOLQJ RI WKH VXSHUEDVLQ YLD WKH TXDVL HTXLOLEUDWHG UHDFWLRQV ZKLOH HQDEOLQJ WKH V\VWHP WR H[LW WKH VXSHUEDVLQ E\ H[HFXWLQJ D VORZ QRQ HTXLOLEUDWHG SURFHVV ZLWKLQ D WUDFWDEOH QXPEHU RI .0& VWHSV 7KH VFDOLQJ SURFHVV LQFOXGLQJ WKH UHDFWLRQ FKDQQHO SDUWLWLRQLQJ LV UHSHDWHG DW LQWHUYDOV RI 0æ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

ACS Paragon Plus Environment

Page 12 of 43

Page 13 of 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

%HJLQ VLPXODWLRQ

3RSXODWH HYHQW WDEOH

&DOFXODWH VFDOHG UDWH FRQVWDQWV IRU DOO SRVVLEOH HYHQWV HT

8SGDWH UHDFWLRQ UDWHV HT

3LFN HYHQW DQG H[HFXWH HTV DQG ,V WKLV D QRQ HTXLOLEUDWHG UHDFWLRQ"