בעיתון הטוב הכללי של ברוסקי et. al. , ^[1] אלגוריתם של אופן סימטריה של מולקולה "כמעט סימטרית" על ידי סימטריה של מטריצת המרחק ניתן לדוגמא למתאן.

הרעיון העיקרי הוא שלאטומים מקבילים סימטריים יש אותה רשימת מרחקים מחויבת בעמודה ובשורה של מטריצת המרחק. לפיכך על ידי ממוצע והתקבצות של מרחקים אלה ניתן לקבל את מטריצת המרחק של המולקולה הסימטרית הקרובה ביותר. בהתאם ל"כמות הממוצע "המולקולה נאלצת לסימטריות גבוהות וגבוהות יותר.

אולם כשאני מנסה להחיל אלגוריתם זה על מולקולה מורכבת יותר (נגיד Hexanitrocuprate (II)) הממוצע של מטריצת המרחק החדשה הממוצעת על פני האטומים המקבילים הסימטריים שלה (SEA) עם הסובלנות של 25.0 בערב, הופכת ללא אוקלידית ואילו מטריצת המרחק המקורית היא אוקלידית. כדי לקבל את מטריצות המרחק המיקומים של המולקולה הוסבו מאנגסטרום ל- pm ונעזרו בסדר היסודות כקבוצות של O, N ו- Cu.

להסבר לבעיה זו הנקראת אוקלידית. בעיית מרחק, ראה למשל עבודת הגמר של אל-הומידן ^[2] . לחפור מעט בתיאוריה ניתן ליישם שיטת השלכה אשר הופכת את מטריצת המרחק הלא-אוקלידית בחזרה למאכלת אוקלידית. האלגוריתם המוזכר ב- Beruski et. אל. בונה מחדש את מיקומי האטום במולקולה ישירות באמצעות אופטימיזציה כמעט של ניוטון של פונקציית ניקוד על מטריצת המרחק.

עם זאת, שתי הגישות "מורידות" את מטריצת המרחק חזרה למצב שקרוב מטריצת מרחק של המולקולה המקורית (לפחות לדוגמא Hexanitrocuprate (II)). אני מניח שהבעיה העיקרית היא האילוץ למרחקים שמחזיר את המולקולה לצורתה המקורית.

כמובן שיש אלגוריתמים שקובעים תחילה את הסימטריה (קבוצת הנקודות) של המולקולה (ראה למשל את התשובה ל זה ו זה שאלות מכימיה חישובית), ואז סימטריה המולקולה עם מידע זה. עם זאת ברצוני להשתמש בסימטריה המקומית של המולקולה ללא שימוש באלגוריתם מורכב למדי של זיהוי קבוצות נקודות.

יש לך מושג כיצד לפתור בעיה זו?

הפניות

1. O. Beruski, L. N. Vidal. J. מחשב. כימיה. 2014, 35, 290–299.
2. סולימאן סאלח אל-הומידן. שיטות היברידיות לבעיות אופטימיזציה עם אילוצי מטריקס חיוביים למחצה. עבודת דוקטורט, אוניברסיטת דנדי, דנדי, דצמבר 2003.

תוכן הקובץ המוזכר

Hexanitrocuprate (II ) (קובץ mol)

  OpenBabel10241500433D 19 18 0 0 0 0 0 0 0 0999 V2000 6.2190 -0.6777 -1.0333 O 0 5 0 0 0 0 0 0 0 0 0 0 5.5672 -1.4099 -0.2754 N 0 3 0 0 0 0 0 0 0 0 0 6.0808 -2.3487 0.3568 O 0 0 0 0 0 0 0 0 0 0 0 0 3.7257 -1.2558 0.0053 Cu 0 7 0 0 0 0 0 0 0 0 0 3.5134 -2.5214 1.3633 N 0 3 0 0 0 0 0 0 0 0 0 4.2665 -2.2644 2.3200 O 0 5 0 0 0 0 0 0 0 0 0 2.7186 -3.4711 1.3724 O 0 0 0 0 0 0 0 0 0 0 0 2.1847 -0.7314 0.9234 N 0 3 0 0 0 0 0 0 0 0 0 0 2.1383 -0.1356 2.0089 O 0 5 0 0 0 0 0 0 0 0 0 0 1.1629 -1.0996 0.3148 O 0 0 0 0 0 0 0 0 0 0 0 0 4.4060 0.3341 0.7046 N 0 3 0 0 0 0 0 0 0 0 0 0 4.2381 1.3976 0.0898 O 0 5 0 0 0 0 0 0 0 0 0 4.9585 0.2682 1.8160 O 0 0 0 0 0 0 0 0 0 0 0 3.4004 -0.4833 -1.6643 N 0 3 0 0 0 0 0 0 0 0 0 0
2.3776 0.2229 -1.6723 O 0 5 0 0 0 0 0 0 0 0 0 4.1301 -0.6158 -2.6565 O 0 0 0 0 0 0 0 0 0 0 0 3.3488 -2.7126 -1.0984 N 0 3 0 0 0 0 0 0 0 0 0 0 2.1566 -2.9077 -1.3889 O 0 5 0 0 0 0 0 0 0 0 0 0 4.2641 -3.4679 -1.4571 O 0 0 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 0 2 3 2 0 0 0 0 2 4 1 0 0 0 0 4 5 1 0 0 0 0 4 8 1 0 0 0 0 4 11 1 0 0 0 0 4 14 1 0 0 0 0 4 17 1 0 0 0 0 5 6 1 0 0 0 0 5 7 2 0 0 0 0 8 9 1 0 0 0 0 8 10 2 0 0 0 0 11 12 1 0 0 0 11 13 2 0 0 0 0 14 15 1 0 0 0 0 14 16 2 0 0 0 0 17 18 1 0 0 0 0 17 19 2 0 0 0 0 M CHG 8 1 -1 2 1 4 -3 5 1 6 -1 8 1 9 -1 11 1 M CHG 5 12 -1 14 1 15 -1 17 1 18 -1 M END 

מטריצת מרחק מקורי (קובץ csv)

  0., 285.286949929365,365.22 30816911768,231.30761855157303,273.544384149995,270.1046991446095,123.91014849478631,124.29027476033676,302.6233946673654,388.3080251552883,431.26815544391866,445.4489939375776,312.76083674270984,274.96343866048807,384.58956837127033,289.23154098403586,388.7863095583486,270.7369295090716,186.91339037104862 285.286949929365,0., 227.223583723169,306.3413243426358,365.1435658751226,247.45944091911304,405.7485996278977,276.3043665959697, 124.43807697003355,123.8434471419461,282.8381340979324,294.0360137806251,418.39638311056177,317.41328674143426,424.68919800249216,449.11493284013613,309.2691185682787,306.8235470103297,186.84160805345257 365.2230816911768,227.223583723169,0., 247.33251666531837,286.97859484637524,306.0711925353316,448.4098021899164,425.63235379843957,293.8447174954826,282.7119788760285,123.91290691449379,124.5006200787771,307.3106168032598,308.05683501587816, 277.2283785978629,407.5981909920602,317.55003684458933,418.0803374950801,186.88456249781575
231.30761855157303,306.3413243426358,247.33251666531837,0.,270.01965354395963,369.4707259039612,270.7582805751285,318.1718988220048,306.2871146489842,421.57825655505525,265.7871304258353,356.7231046624258,123.9839384759171,124.29034636688402,312.6727620052632,350.3630121745159,446.674488302164,437.58755123974913,186.53714348622367  273.544384149995,365.1435658751226,286.97859484637524,270.01965354395963,0.,230.05833738423826,289.4902126152109,384.045655228646,444.93924203198793,431.431883036013,389.95115001753743,305.0093236279835,270.4883160138345,388.6504335517973,124.29409800951929,123.87409656582768,273.8722139976964,311.3959969235315,186.81917728113459  270.1046991446095,247.45944091911304,306.0711925353316,369.4707259039612,230.05833738423826,0.,351.89634922800775,311.670567266144,356.77044566499615,266.0324591097861,421.4445697835007,306.2198891319765,436.99448245944706,446.8783960989835,314.4797433858022,272.66601438389785,124.24955130703692,123.9724917068299,186.61386711603188  123.91014849478631,405.7485996278977,448.4098021899164,270.7582805751285,289.4902126152109,351.89634922800775,0.,217.80078627038978,419.21944170088295,508.3628252537749,511.8687892419306,524.9716223759146,308.0939549877602,325.6086231966224,399.6968266073675,264.61506117377365,464.78430610337955,343.31409073907815,276.2143091876306  124.29027476033676,276.3043665959697,425.63235379843957,318.1718988220048,384.045655228646,311.670567266144,217.80078627038978,0.,267.449905963715,368.7223068923279,481.3579984793023,507.42245929797,418.3490884416984,319.9529268501853,494.40938108009243,398.5994203457903,433.1200195096044,280.05786437806034,262.00150133157643
302.6233946673654,124.43807697003355,293.8447174954826,306.2871146489842,444.93924203198793,356.77044566499615,419.21944170088295,267.449905963715,0.,217.94643975068732,302.61870216495214,387.42636771391795,428.77549603492974,267.33766588342917,506.8832586503523,524.4238664477429,431.52580583320855,396.42026209567035,258.21647294469807  388.3080251552883,123.8434471419461,282.7119788760285,421.57825655505525,431.431883036013,266.0324591097861,508.3628252537749,368.7223068923279,217.94643975068732,0.,344.49151789267614,302.7000578130107,525.9104743204873,438.13579470296645,479.9178584924716,513.5864635093102,287.36807842904193,322.40736250898493,279.11945310207244  431.26815544391866,282.8381340979324,123.91290691449379,265.7871304258353,389.95115001753743,421.4445697835007,511.8687892419306,481.3579984793023,302.61870216495214,344.49151789267614,0.,217.96045444070813,323.15332413577306,285.548470316337,370.6348631739869,509.54701883143224,438.51930561835024,525.7033852848962,279.08958705046666  445.4489939375776,294.0360137806251,124.5006200787771,356.7231046624258,305.0093236279835,306.2198891319765,524.9716223759146,507.42245929797,387.42636771391795,302.7000578130107,217.96045444070813,0.,396.7806431770582,430.47714968392916,267.8258529343275,422.69329270760846,267.5679164249705,428.5547332605253,258.61424032717144  312.76083674270984,418.39638311056177,307.3106168032598,123.9839384759171,270.4883160138345,436.99448245944706,308.0939549877602,418.3490884416984,428.77549603492974,525.9104743204873,323.15332413577306,396.7806431770582,0.,218.50141784437005,281.8931845575554,340.6993872903208,500.54774028058506,510.5552453946586,270.3742881636491
274.96343866048807,317.41328674143426,308.05683501587816,124.29034636688402,388.6504335517973,446.8783960989835,325.6086231966224,319.9529268501853,267.33766588342917,438.13579470296645,285.548470316337,430.47714968392916,218.50141784437005,0.,433.95084733181466,463.36765974763495,531.1155093762561,501.53582304756657,266.85213752188685  384.58956837127033,424.68919800249216,277.2283785978629,312.6727620052632,124.29409800951929,314.4797433858022,399.6968266073675,494.40938108009243,506.8832586503523,479.9178584924716,370.6348631739869,267.8258529343275,281.8931845575554,433.95084733181466,0.,217.79172573814643,315.11605671561716,415.056597706867,261.1181544818361  289.23154098403586,449.11493284013613,407.5981909920602,350.3630121745159,123.87409656582768,272.66601438389785,264.61506117377365,398.5994203457903,524.4238664477429,513.5864635093102,509.54701883143224,422.69329270760846,340.6993872903208,463.36765974763495,217.79172573814643,0.,327.93776269286224,309.6932477468632,276.7366726691639  388.7863095583486,309.2691185682787,317.55003684458933,446.674488302164,273.8722139976964,124.24955130703692,464.78430610337955,433.1200195096044,431.52580583320855,287.36807842904193,438.51930561835024,267.5679164249705,500.54774028058506,531.1155093762561,315.11605671561716,327.93776269286224,0.,218.174964879108,267.10750756951785  270.7369295090716,306.8235470103297,418.0803374950801,437.58755123974913,311.3959969235315,123.9724917068299,343.31409073907815,280.05786437806034,396.42026209567035,322.40736250898493,525.7033852848962,428.5547332605253,510.5552453946586,501.53582304756657,415.056597706867,309.6932477468632,218.174964879108,0.,270.58962895868723
186.91339037104862,186.84160805345257,186.88456249781575,186.53714348622367,186.81917728113459,186.61386711603188,276.2143091876306,262.00150133157643,258.21647294469807,279.11945310207244,279.08958705046666,258.61424032717144,270.3742881636491,266.85213752188685,261.1181544818361,276.7366726691639,267.10750756951785,270.58962895868723,0.  

המרחק חדש מטריקס (קובץ CSV)

  0., 284.90364177745516,365.1833237831497,239.03947838006067,277.35510979538634,269.22803953053614,123.88514977926349,124.29218638492802,301.66740166629756,404.11562988556216,434.849076029974,431.45426939883123,312.7048608574015, 273.827178600421,376.9980984165578,277.0514027606322,388.71837155507296,274.0233685968097,186.76829146761784 284.90364177745516,0., 237.30978125769235,310.23931151726777,381.83722368114843,239.03947838006067,404.11562988556216,271.73490110511295,124.46934852440532,123.88514977926349,277.0514027606322,301.66740166629756,410.5138853378226,317.61308082818346,429.3 481046605877,434.849076029974,317.61308082818346,312.7048608574015,186.76829146761784 365.1833237831497,237.30978125769235,0., 239.03947838006067,277.35510979538634,310.23931151726777,434.849076029974,429.3481046605877,301.66740166629756,277.0514027606322,123.88514977926349,124.46934852440532,312.7048608574015,317.61308082818346,271.73490110511295,404.11562988556216,317.61308082818346,410.5138853378226,186.76829146761784 239.03947838006067,310.23931151726777 , 239.03947838006067,0., 277.35510979538634,359.11572725631135,277.0514027606322,315.3439925075028,301.66740166629756,404.11562988556216,277.0514027606322,356.74677516371094,123.9782150913735,124.26994883696048,315.3439925075028,351.8266306604487,437.4102258851276,432.97806574866286,186.76829146761784
277.35510979538634,381.83722368114843,277.35510979538634,277.35510979538634,0.,239.03947838006067,277.0514027606322,376.9980984165578,431.45426939883123,434.849076029974,404.11562988556216,301.66740166629756,274.0233685968097,388.71837155507296,124.29218638492802,123.88514977926349,273.827178600421,312.7048608574015,186.76829146761784  269.22803953053614,239.03947838006067,310.23931151726777,359.11572725631135,239.03947838006067,0.,351.8266306604487,315.3439925075028,356.74677516371094,277.0514027606322,404.11562988556216,301.66740166629756,432.97806574866286,437.4102258851276,315.3439925075028,277.0514027606322,124.26994883696048,123.9782150913735,186.76829146761784  123.88514977926349,404.11562988556216,434.849076029974,277.0514027606322,277.0514027606322,351.8266306604487,0.,217.79625600426812,431.45426939883123,515.6449618418281,515.6449618418281,515.9253016929949,312.7048608574015,317.61308082818346,407.9254832754308,277.0514027606322,464.0759829255072,342.00673901469946,268.8361627756964  124.29218638492802,271.73490110511295,429.3481046605877,315.3439925075028,376.9980984165578,315.3439925075028,217.79625600426812,0.,264.5020090797186,351.8266306604487,472.3569557056971,515.9253016929949,410.5138853378226,317.61308082818346,487.5236547829897,404.11562988556216,437.4102258851276,274.0233685968097,268.8361627756964  301.66740166629756,124.46934852440532,301.66740166629756,301.66740166629756,431.45426939883123,356.74677516371094,431.45426939883123,264.5020090797186,0.,217.8748515499829,315.27662940585355,392.0134101751411,432.97806574866286,273.827178600421,507.1528589741612,515.6449618418281,437.4102258851276,410.5138853378226,268.8361627756964
404.11562988556216,123.88514977926349,277.0514027606322,404.11562988556216,434.849076029974,277.0514027606322,515.6449618418281,351.8266306604487,217.8748515499829,0.,351.8266306604487,301.66740166629756,512.4679856204754,437.4102258851276,487.5236547829897,515.6449618418281,273.827178600421,312.7048608574015,268.8361627756964  434.849076029974,277.0514027606322,123.88514977926349,277.0514027606322,404.11562988556216,404.11562988556216,515.6449618418281,472.3569557056971,315.27662940585355,351.8266306604487,0.,217.95344709569773,312.7048608574015,273.827178600421,376.9980984165578,515.6449618418281,437.4102258851276,512.4679856204754,268.8361627756964  431.45426939883123,301.66740166629756,124.46934852440532,356.74677516371094,301.66740166629756,301.66740166629756,515.9253016929949,515.9253016929949,392.0134101751411,301.66740166629756,217.95344709569773,0.,410.5138853378226,437.4102258851276,271.73490110511295,434.849076029974,273.827178600421,432.97806574866286,268.8361627756964  312.7048608574015,410.5138853378226,312.7048608574015,123.9782150913735,274.0233685968097,432.97806574866286,312.7048608574015,410.5138853378226,432.97806574866286,512.4679856204754,312.7048608574015,410.5138853378226,0.,218.338191361739,271.73490110511295,351.8266306604487,501.0417816640758,512.4679856204754,268.8361627756964  273.827178600421,317.61308082818346,317.61308082818346,124.26994883696048,388.71837155507296,437.4102258851276,317.61308082818346,317.61308082818346,273.827178600421,437.4102258851276,273.827178600421,437.4102258851276,218.338191361739,0.,429.3481046605877,472.3569557056971,531.1155093762561,512.4679856204754,268.8361627756964  376.9980984165578,429.3481046605877,271.73490110511295,315.3439925075028,124.29218638492802,315.3439925075028,407.9254832754308,487.5236547829897,507.1528589741612,487.5236547829897,376.9980984165578,271.73490110511295,271.73490110511295,429.3481046605877,0.,217.8748515499829,317.61308082818346,410.5138853378226,268.8361627756964
277.0514027606322,434.849076029974,404.11562988556216,351.8266306604487,123.88514977926349,277.0514027606322,277.0514027606322,404.11562988556216,515.6449618418281,515.6449618418281,515.6449618418281,434.849076029974,351.8266306604487,472.3569557056971,217.8748515499829,0.,317.61308082818346,312.7048608574015,268.8361627756964  388.71837155507296,317.61308082818346,317.61308082818346,437.4102258851276,273.827178600421,124.26994883696048,464.0759829255072,437.4102258851276,437.4102258851276,273.827178600421,437.4102258851276,273.827178600421,501.0417816640758,531.1155093762561,317.61308082818346,317.61308082818346,0.,218.338191361739,268.8361627756964  274.0233685968097,312.7048608574015,410.5138853378226,432.97806574866286,312.7048608574015,123.9782150913735,342.00673901469946,274.0233685968097,410.5138853378226,312.7048608574015,512.4679856204754,432.97806574866286,512.4679856204754,512.4679856204754,410.5138853378226,312.7048608574015,218.338191361739,0.,268.8361627756964  186.76829146761784,186.76829146761784,186.76829146761784,186.76829146761784,186.76829146761784,186.76829146761784,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,268.8361627756964,0.