87:
you convert
Cartesian coordinates to a Z matrix and back again. While the transform is conceptually straightforward, algorithms of doing the conversion vary significantly in speed, numerical precision and parallelism. These matter because macromolecular chains, such as polymers, proteins, and DNA, can have thousands of connected atoms and atoms consecutively distant along the chain that may be close in Cartesian space (and thus small round-off errors can accumulate to large force-field errors.) The optimally fastest and most numerically accurate algorithm for conversion from torsion-space to cartesian-space is the Natural Extension Reference Frame method. Back-conversion from Cartesian to torsion angles is simple trigonometry and has no risk of cumulative errors.
1257:
107:, and more natural internal coordinates are used rather than Cartesian coordinates. The Z-matrix representation is often preferred, because this allows symmetry to be enforced upon the molecule (or parts thereof) by setting certain angles as constant. The Z-matrix simply is a representation for placing atomic positions in a relative way with the obvious convenience that the vectors it uses easily correspond to bonds. A conceptual pitfall is to assume all bonds appear as a line in the Z-matrix which is not true. For example: in ringed molecules like
79:, although it is not always the case that a Z-matrix will give information regarding bonding since the matrix itself is based on a series of vectors describing atomic orientations in space. However, it is convenient to write a Z-matrix in terms of bond lengths, angles, and dihedrals since this will preserve the actual bonding characteristics. The name arises because the Z-matrix assigns the second atom along the Z axis from the first atom, which is at the origin.
86:
and back, as the structural information content is identical, the position and orientation in space, however is not meaning the
Cartesian coordinates recovered will be accurate in terms of relative positions of atoms, but will not necessarily be the same as an original set of Cartesian coordinates if
98:
programs. A skillful choice of internal coordinates can make the interpretation of results straightforward. Also, since Z-matrices can contain molecular connectivity information (but do not always contain this information), quantum chemical calculations such as geometry
137:
C 0.000000 0.000000 0.000000 H 0.628736 0.628736 0.628736 H -0.628736 -0.628736 0.628736 H -0.628736 0.628736 -0.628736 H 0.628736 -0.628736 -0.628736
130:
C 0.000000 0.000000 0.000000 H 0.000000 0.000000 1.089000 H 1.026719 0.000000 -0.363000 H -0.513360 -0.889165 -0.363000 H -0.513360 0.889165 -0.363000
173:
Parsons, Jerod; Holmes, J. Bradley; Rojas, J. Maurice; Tsai, Jerry; Strauss, Charlie E. M. (2005). "Practical conversion from torsion space to
Cartesian space for in silico protein synthesis".
265:
Parsons, Jerod; Holmes, J. Bradley; Rojas, J. Maurice; Tsai, Jerry; Strauss, Charlie E. M. (2005). "Practical conversion from torsion space to
Cartesian space forin silico protein synthesis".
134:
Reorienting the molecule leads to
Cartesian coordinates that make the symmetry more obvious. This removes the bond length of 1.089 from the explicit parameters.
111:, a z-matrix will not include all six bonds in the ring, because all of the atoms are uniquely positioned after just 5 bonds making the 6th redundant.
915:
226:
Gordon, M. S.; Pople, J. A. (1968). "Approximate Self-Consistent
Molecular-Orbital Theory. VI. INDO Calculated Equilibrium Geometries".
1129:
348:
1220:
144:
C H 1 1.089000 H 1 1.089000 2 109.4710 H 1 1.089000 2 109.4710 3 120.0000 H 1 1.089000 2 109.4710 3 -120.0000
1139:
905:
17:
306:
1298:
940:
487:
1293:
704:
341:
779:
83:
935:
457:
100:
1039:
910:
824:
1144:
1034:
742:
422:
187:
1179:
1108:
990:
850:
447:
334:
316:
301:
95:
1049:
632:
437:
182:
37:
995:
732:
582:
577:
412:
387:
382:
547:
377:
357:
235:
148:
8:
1210:
1184:
762:
567:
557:
91:
239:
1261:
1215:
1205:
1159:
1154:
1083:
1019:
885:
622:
617:
552:
542:
407:
290:
208:
1272:
1256:
1059:
1054:
1044:
1024:
985:
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809:
804:
789:
784:
775:
770:
717:
612:
562:
507:
477:
472:
452:
442:
402:
311:
282:
200:
141:
The corresponding Z-matrix, which starts from the carbon atom, could look like this:
1267:
1235:
1164:
1103:
1098:
1078:
1014:
920:
890:
875:
855:
794:
747:
722:
712:
683:
602:
597:
572:
502:
482:
392:
372:
294:
274:
243:
212:
192:
860:
965:
900:
880:
865:
845:
829:
727:
658:
648:
607:
492:
462:
1225:
1169:
1149:
1134:
1093:
970:
930:
895:
819:
758:
737:
678:
668:
653:
587:
532:
522:
517:
427:
104:
103:
may be performed faster, because an educated guess is available for an initial
72:
1287:
1230:
1088:
1029:
960:
950:
945:
870:
799:
673:
663:
592:
512:
497:
432:
124:
64:
60:
1113:
1070:
975:
688:
627:
537:
417:
286:
204:
90:
They are used for creating input geometries for molecular systems in many
955:
925:
693:
527:
397:
33:
1006:
467:
68:
59:. It provides a description of each atom in a molecule in terms of its
278:
247:
196:
1240:
814:
123:
molecule can be described by the following
Cartesian coordinates (in
44:
1174:
326:
120:
108:
52:
320:
323:
module to build a Z-matrix from
Cartesian coordinates.
264:
172:
302:
Java implementation of the NERF conversion algorithm
307:
C++ implementation of the NERF conversion algorithm
1285:
312:Z-Matrix to Cartesian Coordinate Conversion Page
342:
225:
916:Fundamental (linear differential equation)
349:
335:
186:
147:Only the 1.089000 value is not fixed by
51:is a way to represent a system built of
1221:Matrix representation of conic sections
14:
1286:
168:
166:
164:
330:
29:Molecular modeling tool in chemistry
161:
24:
356:
317:Chemistry::InternalCoords::Builder
267:Journal of Computational Chemistry
175:Journal of Computational Chemistry
57:internal coordinate representation
25:
1310:
258:
55:. A Z-matrix is also known as an
1255:
1123:Used in science and engineering
228:The Journal of Chemical Physics
82:Z-matrices can be converted to
366:Explicitly constrained entries
219:
13:
1:
1140:Fundamental (computer vision)
154:
7:
906:Duplication and elimination
705:eigenvalues or eigenvectors
10:
1315:
839:With specific applications
468:Discrete Fourier Transform
114:
31:
1249:
1198:
1130:Cabibbo–Kobayashi–Maskawa
1122:
1068:
1004:
838:
757:Satisfying conditions on
756:
702:
641:
365:
36:meaning of this term see
1299:Computational chemistry
488:Generalized permutation
96:computational chemistry
1262:Mathematics portal
38:Z-matrix (mathematics)
84:Cartesian coordinates
149:tetrahedral symmetry
77:internal coordinates
18:Internal coordinates
1294:Molecular modelling
1211:Linear independence
458:Diagonally dominant
240:1968JChPh..49.4643G
92:molecular modelling
1216:Matrix exponential
1206:Jordan normal form
1040:Fisher information
911:Euclidean distance
825:Totally unimodular
1281:
1280:
1273:Category:Matrices
1145:Fuzzy associative
1035:Doubly stochastic
743:Positive-definite
423:Block tridiagonal
279:10.1002/jcc.20237
273:(10): 1063–1068.
248:10.1063/1.1669925
234:(10): 4643–4650.
197:10.1002/jcc.20237
181:(10): 1063–1068.
16:(Redirected from
1306:
1268:List of matrices
1260:
1259:
1236:Row echelon form
1180:State transition
1109:Seidel adjacency
991:Totally positive
851:Alternating sign
448:Complex Hadamard
351:
344:
337:
328:
327:
298:
252:
251:
223:
217:
216:
190:
170:
75:, the so-called
21:
1314:
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1307:
1305:
1304:
1303:
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1283:
1282:
1277:
1254:
1245:
1194:
1118:
1064:
1000:
834:
752:
698:
637:
438:Centrosymmetric
361:
355:
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220:
171:
162:
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145:
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117:
41:
30:
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12:
11:
5:
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1250:
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1226:Perfect matrix
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1218:
1213:
1208:
1202:
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1192:
1187:
1182:
1177:
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1162:
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1152:
1147:
1142:
1137:
1132:
1126:
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1120:
1119:
1117:
1116:
1111:
1106:
1101:
1096:
1091:
1086:
1081:
1075:
1073:
1066:
1065:
1063:
1062:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1011:
1009:
1002:
1001:
999:
998:
996:Transformation
993:
988:
983:
978:
973:
968:
963:
958:
953:
948:
943:
938:
933:
928:
923:
918:
913:
908:
903:
898:
893:
888:
883:
878:
873:
868:
863:
858:
853:
848:
842:
840:
836:
835:
833:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
773:
767:
765:
754:
753:
751:
750:
745:
740:
735:
733:Diagonalizable
730:
725:
720:
715:
709:
707:
703:Conditions on
700:
699:
697:
696:
691:
686:
681:
676:
671:
666:
661:
656:
651:
645:
643:
639:
638:
636:
635:
630:
625:
620:
615:
610:
605:
600:
595:
590:
585:
583:Skew-symmetric
580:
578:Skew-Hermitian
575:
570:
565:
560:
555:
550:
545:
540:
535:
530:
525:
520:
515:
510:
505:
500:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
445:
440:
435:
430:
425:
420:
415:
413:Block-diagonal
410:
405:
400:
395:
390:
388:Anti-symmetric
385:
383:Anti-Hermitian
380:
375:
369:
367:
363:
362:
354:
353:
346:
339:
331:
325:
324:
314:
309:
304:
299:
260:
259:External links
257:
254:
253:
218:
188:10.1.1.83.8235
159:
158:
156:
153:
143:
136:
129:
116:
113:
105:Hessian matrix
73:dihedral angle
28:
9:
6:
4:
3:
2:
1311:
1300:
1297:
1295:
1292:
1291:
1289:
1274:
1271:
1269:
1266:
1264:
1263:
1258:
1252:
1251:
1248:
1242:
1239:
1237:
1234:
1232:
1231:Pseudoinverse
1229:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1203:
1201:
1199:Related terms
1197:
1191:
1190:Z (chemistry)
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1127:
1125:
1121:
1115:
1112:
1110:
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1097:
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1077:
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1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
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1010:
1008:
1003:
997:
994:
992:
989:
987:
984:
982:
979:
977:
974:
972:
969:
967:
964:
962:
959:
957:
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952:
949:
947:
944:
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739:
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731:
729:
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721:
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581:
579:
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574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
548:Pentadiagonal
546:
544:
541:
539:
536:
534:
531:
529:
526:
524:
521:
519:
516:
514:
511:
509:
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489:
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481:
479:
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469:
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451:
449:
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431:
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421:
419:
416:
414:
411:
409:
406:
404:
401:
399:
396:
394:
391:
389:
386:
384:
381:
379:
378:Anti-diagonal
376:
374:
371:
370:
368:
364:
359:
352:
347:
345:
340:
338:
333:
332:
329:
322:
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292:
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276:
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97:
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88:
85:
80:
78:
74:
70:
66:
62:
61:atomic number
58:
54:
50:
46:
39:
35:
27:
19:
1253:
1189:
1185:Substitution
1071:graph theory
568:Quaternionic
558:Persymmetric
270:
266:
231:
227:
221:
178:
174:
146:
140:
133:
118:
101:optimization
89:
81:
76:
56:
48:
42:
34:mathematical
26:
1160:Hamiltonian
1084:Biadjacency
1020:Correlation
936:Householder
886:Commutation
623:Vandermonde
618:Tridiagonal
553:Permutation
543:Nonnegative
528:Matrix unit
408:Bisymmetric
1288:Categories
1060:Transition
1055:Stochastic
1025:Covariance
1007:statistics
986:Symplectic
981:Similarity
810:Unimodular
805:Orthogonal
790:Involutory
785:Invertible
780:Projection
776:Idempotent
718:Convergent
613:Triangular
563:Polynomial
508:Hessenberg
478:Equivalent
473:Elementary
453:Copositive
443:Conference
403:Bidiagonal
155:References
69:bond angle
1241:Wronskian
1165:Irregular
1155:Gell-Mann
1104:Laplacian
1099:Incidence
1079:Adjacency
1050:Precision
1015:Centering
921:Generator
891:Confusion
876:Circulant
856:Augmented
815:Unipotent
795:Nilpotent
771:Congruent
748:Stieltjes
723:Defective
713:Companion
684:Redheffer
603:Symmetric
598:Sylvester
573:Signature
503:Hermitian
483:Frobenius
393:Arrowhead
373:Alternant
183:CiteSeerX
125:Ångströms
45:chemistry
1069:Used in
1005:Used in
966:Rotation
941:Jacobian
901:Distance
881:Cofactor
866:Carleman
846:Adjugate
830:Weighing
763:inverses
759:products
728:Definite
659:Identity
649:Exchange
642:Constant
608:Toeplitz
493:Hadamard
463:Diagonal
287:15898109
205:15898109
67:length,
49:Z-matrix
32:For the
1170:Overlap
1135:Density
1094:Edmonds
971:Seifert
931:Hessian
896:Coxeter
820:Unitary
738:Hurwitz
669:Of ones
654:Hilbert
588:Skyline
533:Metzler
523:Logical
518:Integer
428:Boolean
360:classes
295:2279574
236:Bibcode
213:2279574
121:methane
115:Example
109:benzene
1089:Degree
1030:Design
961:Random
951:Payoff
946:Moment
871:Cartan
861:BĂ©zout
800:Normal
674:Pascal
664:Lehmer
593:Sparse
513:Hollow
498:Hankel
433:Cauchy
358:Matrix
293:
285:
211:
203:
185:
71:, and
47:, the
1150:Gamma
1114:Tutte
976:Shear
689:Shift
679:Pauli
628:Walsh
538:Moore
418:Block
291:S2CID
209:S2CID
53:atoms
956:Pick
926:Gram
694:Zero
398:Band
321:Perl
283:PMID
201:PMID
119:The
94:and
65:bond
1045:Hat
778:or
761:or
275:doi
244:doi
193:doi
127:):
43:In
1290::
319:—
289:.
281:.
271:26
269:.
242:.
232:49
230:.
207:.
199:.
191:.
179:26
177:.
163:^
151:.
63:,
1175:S
633:Z
350:e
343:t
336:v
297:.
277::
250:.
246::
238::
215:.
195::
40:.
20:)
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