1162:
651:
1157:{\displaystyle {\begin{aligned}\Delta F(A\rightarrow B)&=\int _{0}^{1}{\frac {\partial F(\lambda )}{\partial \lambda }}d\lambda \\&=-\int _{0}^{1}{\frac {k_{B}T}{Q}}{\frac {\partial Q}{\partial \lambda }}d\lambda \\&=\int _{0}^{1}{\frac {k_{B}T}{Q}}\sum _{s}{\frac {1}{k_{B}T}}\exp{\frac {\partial U_{s}(\lambda )}{\partial \lambda }}d\lambda \\&=\int _{0}^{1}\left\langle {\frac {\partial U(\lambda )}{\partial \lambda }}\right\rangle _{\lambda }d\lambda \\&=\int _{0}^{1}\left\langle U_{B}(\lambda )-U_{A}(\lambda )\right\rangle _{\lambda }d\lambda \end{aligned}}}
88:), the free energy difference between two states cannot be calculated directly from the potential energy of just two coordinate sets (for state A and B respectively). In thermodynamic integration, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over ensemble-averaged enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. An example alchemical process is the
154:. The potential energy in either system can be calculated as an ensemble average over configurations sampled from a molecular dynamics or Monte Carlo simulation with proper Boltzmann weighting. Now consider a new potential energy function defined as:
639:
232:
80:
have different dependences on the spatial coordinates. Because the free energy of a system is not simply a function of the phase space coordinates of the system, but is instead a function of the
656:
446:
485:
1167:
The change in free energy between states A and B can thus be computed from the integral of the ensemble averaged derivatives of potential energy over the coupling parameter
1263:
1214:
534:
327:
301:
1303:
1283:
1234:
1185:
275:
255:
152:
125:
78:
51:
1402:
J Kästner; et al. (2006). "QM/MM Free-Energy
Perturbation Compared to Thermodynamic Integration and Umbrella Sampling: Application to an Enzymatic Reaction".
1311:
is a related free energy method. It adds a bias to the potential energy. In the limit of an infinite strong bias it is equivalent to thermodynamic integration.
505:
645:
If we take the derivative of F with respect to λ, we will get that it equals the ensemble average of the derivative of potential energy with respect to λ.
85:
542:
1442:
1447:
160:
89:
339:
23:
257:
is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of
1325:
1320:
81:
1363:
454:
1330:
1239:
1190:
510:
330:
306:
280:
1419:
1308:
1288:
1268:
1219:
1170:
260:
240:
1411:
1371:
130:
103:
56:
29:
1367:
490:
1436:
1305:
value, and finally computing the integral over the ensemble-averaged derivatives.
1423:
1187:. In practice, this is performed by defining a potential energy function
1335:
1415:
1375:
1354:
Kirkwood, John G. (1935). "Statistical
Mechanics of Fluid Mixtures".
1216:, sampling the ensemble of equilibrium configurations at a series of
1390:
Understanding
Molecular Simulation: From Algorithms to Applications.
26:
between two given states (e.g., A and B) whose potential energies
536:
as defined above. The free energy of this system is defined as:
634:{\displaystyle F(N,V,T,\lambda )=-k_{B}T\ln Q(N,V,T,\lambda )}
333:, the partition function of the system can be written as:
1236:
values, calculating the ensemble-averaged derivative of
227:{\displaystyle U(\lambda )=U_{A}+\lambda (U_{B}-U_{A})}
100:
Consider two systems, A and B, with potential energies
1291:
1271:
1242:
1222:
1193:
1173:
654:
545:
513:
493:
457:
342:
309:
283:
263:
243:
163:
133:
106:
59:
32:
1297:
1277:
1257:
1228:
1208:
1179:
1156:
633:
528:
499:
479:
440:
321:
295:
269:
249:
226:
146:
119:
72:
45:
507:in the ensemble with potential energy function
441:{\displaystyle Q(N,V,T,\lambda )=\sum _{s}\exp}
22:is a method used to compare the difference in
8:
1404:Journal of Chemical Theory and Computation
1290:
1270:
1241:
1221:
1192:
1172:
1138:
1118:
1096:
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1075:
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954:
944:
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864:
845:
838:
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827:
787:
772:
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691:
655:
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586:
544:
512:
492:
462:
456:
426:
417:
402:
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341:
308:
282:
262:
242:
215:
202:
183:
162:
138:
132:
111:
105:
64:
58:
37:
31:
1346:
277:varies from the energy of system A for
7:
16:Transfer targets in English football
1036:
1019:
971:
947:
798:
790:
722:
705:
659:
14:
487:is the potential energy of state
1388:Frenkel, Daan and Smit, Berend.
84:integral over phase space (i.e.
1356:The Journal of Chemical Physics
480:{\displaystyle U_{s}(\lambda )}
1252:
1246:
1203:
1197:
1130:
1124:
1108:
1102:
1031:
1025:
966:
960:
941:
920:
914:
898:
717:
711:
677:
671:
665:
628:
604:
573:
549:
523:
517:
474:
468:
435:
414:
408:
392:
370:
346:
221:
195:
173:
167:
1:
90:Kirkwood's coupling parameter
1258:{\displaystyle U(\lambda )}
1209:{\displaystyle U(\lambda )}
529:{\displaystyle U(\lambda )}
1464:
322:{\displaystyle \lambda =1}
296:{\displaystyle \lambda =0}
20:Thermodynamic integration
1326:Bennett acceptance ratio
1321:Free energy perturbation
1298:{\displaystyle \lambda }
1278:{\displaystyle \lambda }
1229:{\displaystyle \lambda }
1180:{\displaystyle \lambda }
270:{\displaystyle \lambda }
250:{\displaystyle \lambda }
1443:Computational chemistry
1299:
1279:
1259:
1230:
1210:
1181:
1158:
635:
530:
501:
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148:
121:
74:
47:
1448:Statistical mechanics
1300:
1280:
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1211:
1182:
1159:
636:
531:
502:
482:
443:
324:
298:
272:
252:
229:
149:
147:{\displaystyle U_{B}}
122:
120:{\displaystyle U_{A}}
75:
73:{\displaystyle U_{B}}
48:
46:{\displaystyle U_{A}}
1392:Academic Press, 2007
1289:
1269:
1240:
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1171:
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104:
57:
30:
1368:1935JChPh...3..300K
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1010:
837:
764:
701:
1331:Parallel tempering
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1177:
1154:
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1071:
996:
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477:
451:In this notation,
438:
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331:canonical ensemble
319:
293:
267:
247:
224:
144:
117:
86:partition function
82:Boltzmann-weighted
70:
43:
1416:10.1021/ct050252w
1376:10.1063/1.1749657
1309:Umbrella sampling
1043:
978:
890:
860:
858:
805:
785:
729:
500:{\displaystyle s}
376:
303:and system B for
1455:
1428:
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1386:
1380:
1379:
1351:
1304:
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1301:
1296:
1284:
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1276:
1265:with respect to
1264:
1262:
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1233:
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1215:
1213:
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1207:
1186:
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1137:
1133:
1123:
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1084:
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1034:
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1009:
1004:
989:
979:
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945:
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936:
927:
913:
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871:
868:
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849:
839:
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591:
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328:
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256:
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233:
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188:
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153:
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582:
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338:
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305:
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279:
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211:
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159:
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134:
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107:
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60:
55:
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33:
28:
27:
17:
12:
11:
5:
1461:
1459:
1451:
1450:
1445:
1435:
1434:
1430:
1429:
1410:(2): 452–461.
1394:
1381:
1362:(5): 300–313.
1345:
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15:
13:
10:
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6:
4:
3:
2:
1460:
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1174:
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1134:
1127:
1119:
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1105:
1097:
1093:
1088:
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1066:
1058:
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1050:
1045:
1039:
1028:
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991:
983:
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963:
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938:
933:
929:
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917:
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905:
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865:
861:
855:
851:
846:
842:
833:
828:
824:
820:
818:
810:
807:
801:
793:
782:
778:
773:
769:
760:
755:
751:
747:
744:
742:
734:
731:
725:
714:
708:
697:
692:
688:
684:
682:
674:
668:
662:
648:
647:
646:
625:
622:
619:
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613:
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587:
583:
579:
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537:
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112:
108:
95:
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91:
87:
83:
65:
61:
38:
34:
25:
21:
1407:
1403:
1397:
1389:
1384:
1359:
1355:
1349:
1307:
1166:
644:
450:
236:
99:
19:
18:
24:free energy
1437:Categories
1342:References
96:Derivation
1293:λ
1273:λ
1250:λ
1224:λ
1201:λ
1175:λ
1148:λ
1140:λ
1128:λ
1112:−
1106:λ
1073:∫
1059:λ
1051:λ
1040:λ
1037:∂
1029:λ
1020:∂
998:∫
984:λ
975:λ
972:∂
964:λ
948:∂
918:λ
902:−
896:
862:∑
825:∫
811:λ
802:λ
799:∂
791:∂
752:∫
748:−
735:λ
726:λ
723:∂
715:λ
706:∂
689:∫
672:→
660:Δ
626:λ
599:
580:−
571:λ
521:λ
472:λ
412:λ
396:−
390:
378:∑
368:λ
329:. In the
311:λ
285:λ
265:λ
245:λ
209:−
193:λ
171:λ
1424:26626532
1315:See also
1285:at each
1135:⟩
1089:⟨
1046:⟩
1014:⟨
92:method.
1364:Bibcode
1336:Alchemy
1422:
237:Here,
1420:PMID
127:and
53:and
1412:doi
1372:doi
893:exp
387:exp
1439::
1418:.
1406:.
1370:.
1358:.
596:ln
1426:.
1414::
1408:2
1378:.
1374::
1366::
1360:3
1253:)
1247:(
1244:U
1204:)
1198:(
1195:U
1145:d
1131:)
1125:(
1120:A
1116:U
1109:)
1103:(
1098:B
1094:U
1082:1
1077:0
1069:=
1056:d
1032:)
1026:(
1023:U
1007:1
1002:0
994:=
981:d
967:)
961:(
956:s
952:U
942:]
939:T
934:B
930:k
925:/
921:)
915:(
910:s
906:U
899:[
887:T
882:B
878:k
873:1
866:s
856:Q
852:T
847:B
843:k
834:1
829:0
821:=
808:d
794:Q
783:Q
779:T
774:B
770:k
761:1
756:0
745:=
732:d
718:)
712:(
709:F
698:1
693:0
685:=
678:)
675:B
669:A
666:(
663:F
641:,
629:)
623:,
620:T
617:,
614:V
611:,
608:N
605:(
602:Q
593:T
588:B
584:k
577:=
574:)
568:,
565:T
562:,
559:V
556:,
553:N
550:(
547:F
524:)
518:(
515:U
495:s
475:)
469:(
464:s
460:U
436:]
433:T
428:B
424:k
419:/
415:)
409:(
404:s
400:U
393:[
382:s
374:=
371:)
365:,
362:T
359:,
356:V
353:,
350:N
347:(
344:Q
317:1
314:=
291:0
288:=
222:)
217:A
213:U
204:B
200:U
196:(
190:+
185:A
181:U
177:=
174:)
168:(
165:U
140:B
136:U
113:A
109:U
66:B
62:U
39:A
35:U
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.