Results

All the global minima that we have found are catalogued in Table 2.2 along with their energies, point groups, number of nearest neighbours, strain energies and the range of $\rho_0$ for which they are the lowest in energy. These minima are illustrated in the figures that appear throughout this chapter.

 

$$E_{\rm strain} \rho_0=3 \rho_0=6 \rho_0=10 \rho_0=14 \rho_{min} \rho_{max}$$ PG n_nn

5A D_3h 9 0.000 -9.2995 -9.0449 -9.0036 -9.0003

6A O_h 12 0.000 -13.5442 -12.4878 -12.0949 -12.0182

7A D_5h 16 0.062 -17.5530 -16.2076 -15.9565 -15.8831

8A D_2d 18 0.006 -22.0429 -19.1619 -18.2751 -18.0762 5.28

8B C_s 19 0.062 -19.3274 -18.9646 -18.8837 5.28

9A D_3h 21 0.002 -26.7784 -22.3308 -21.2135 -21.0380 3.42

9B C_2v 23 0.186 -26.6077 -23.4172 -22.8508 -22.6449 3.42

10A D_4d 24 0.002 -31.5198 -25.5039 -24.2050 -24.0320 2.28

10B C_3v 27 0.694 -31.8886 -27.4733 -26.5839 -26.1327 2.28

11A D_4d 34 10.374 -37.9308 -28.7952 -23.6661 3.40

11B C_2v 32 -37.8917 3.40 3.67

11C C_2v 31 0.792 -31.5219 -30.2652 -29.5881 3.67 13.57

11D C_s 30 -29.5961 13.57 15.29

11E C₂ 30 0.248 -36.6132 -30.6989 -29.8090 -29.5244 15.29 20.60

11F C_2v 29 0.001 -36.6978 -30.4317 -29.2155 -29.0379 20.60

12A C_2v 38 -43.9713 -35.1999 2.63

12B C_5v 36 1.704 -44.0979 -36.4003 -34.3668 -33.1159 2.63 12.15

12C C_s 34 -33.1995 12.15 13.03

12D D_2d 34 0.346 -41.8164 -34.8388 -33.7242 -33.3323 13.03 17.08

12E D_3h 33 0.001 -42.1214 -34.5680 -33.2223 -33.0383 17.08

13A I_h 42 2.425 -51.7370 -42.4399 -39.6630 -37.2589 14.76

13B D_5h 37 0.141 -49.9981 -39.3607 -37.2080 -36.7905 14.76

14A C_2v 46 4.373 -56.7547 -44.8275 -41.7170 3.23

14B C_3v 45 2.425 -56.6605 -45.6193 -42.6752 -40.2598 3.23 13.06

14C C_2v 41 0.141 -55.9716 -43.6340 -41.2493 -40.7983 13.06

15A D_6d 50 9.527 -63.1621 -47.5706 -40.5692 -35.7589 3.72

15B C_2v 49 2.573 -62.5939 -49.7484 -46.5414 -44.0866 3.72 12.71

15C C_2v 45 0.141 -62.6314 -47.9526 -45.2938 -44.8064 12.71

16A D_3h 54 11.222 -69.1406 -50.8342 -42.8876 3.39

16B C_s 53 2.868 -68.7572 -53.8458 -50.2619 -47.8320 3.39 11.99

16C C_2v 49 0.142 -68.5757 -52.2653 -49.3382 -48.8145 11.99

17A D_3h 58 -75.6624 -53.1560 3.42

17B C_3v 57 3.372 -75.1474 -57.8845 -53.7722 -51.3296 3.42 4.88

17C C_s 57 3.281 -75.0914 -57.9130 -53.8620 -51.4406 4.88 4.91

17D C₂ 57 3.163 -75.0054 -57.9414 -53.9836 4.91 11.30

17E C_2v 53 0.142 -74.8689 -56.5736 -53.3823 -52.8226 11.30

18A C₂ 65 -82.5793 -59.8814 2.14 3.03

18B C_5v 62 4.500 -82.5489 -62.6892 -57.6571 -54.0597 3.03 10.35

18C D_5h 57 0.142 -81.4902 -60.9265 -57.4297 -56.8309 10.35

19A D_5h 68 6.001 -90.6475 -68.4923 -62.1668 -56.6767 10.70

19B C_2v 61 0.151 -87.4857 -65.0648 -61.4271 -60.8124 10.70

20A C_2v 72 6.507 -97.4174 -72.5078 -65.6791 -61.3272 2.02 10.24

20B C_2v 65 0.162 -94.2224 -69.2027 -65.4237 -64.7920 10.24

21A C_2v 76 7.133 -104.337 -76.4873 -69.0687 -65.1796 5.40

21B C₁ 76 6.922 -104.004 -76.5291 -69.2763 -65.7789 5.40 9.84

21C C_s 69 0.169 -73.5770 -69.4499 -68.7836 9.84

22A C_s 81 8.198 -112.041 -81.1367 -73.0143 -68.5809 9.58

22B C₁ 73 0.169 -77.8879 -73.4943 -72.7917 9.58

23A D_3h 87 10.597 -120.787 -86.7355 -76.6306 -70.8161 8.35

23B D_3h 84 6.097 -116.279 -84.9406 -78.1439 -74.4424 8.35 9.71

23C C_s 78 0.433 -83.5049 -78.3254 -77.3025 9.71 22.45

23D C₁ 76 0.007 -82.2527 -76.8988 -76.1575 22.45

24A C_s 91 10.947 -127.885 -90.6854 -80.2955 -74.6137 2.79 8.55

24B C₁ 82 0.434 -87.8204 -82.3702 -81.3095 8.55 15.34

24C C_2v 81 0.007 -87.6268 -81.9414 -81.1642 15.34

25A C_s 96 12.090 -136.073 -95.1279 -84.1688 2.62 7.76

25B C_3v 87 0.893 -93.3428 -86.9897 -85.4774 7.76 15.40

25C C_s 85 0.010 -92.2415 -86.0154 -85.1768 15.40

26A T_d 102 15.393 -145.322 -100.550 -86.8823 7.89

26B C_2v 91 0.491 -138.941 -97.3632 -91.3703 -90.2108 7.89 14.20

26C D_3h 90 0.008 -97.6487 -91.0851 -90.1893 14.20

27A C_s 106 -152.514 -104.489 4.72

27B C_2v 106 -151.735 -104.745 4.72 7.71

27C C_s 97 3.076 -102.750 -94.7362 -91.2366 7.71 8.53

27D C_s 95 0.491 -146.171 -101.723 -95.4195 -94.2198 8.53 14.22

27E C_s 94 0.008 -101.920 -95.1213 -94.1956 14.22

28A C_s 111 -160.773 -108.855 4.82

28B C_s 111 -160.385 -108.998 4.82 6.70

28C C_3v 102 3.369 -108.186 -99.5240 -95.6921 6.70 8.88

28D C_s 100 0.983 -107.214 -100.008 -98.3317 8.88 14.57

28E C_2v 98 0.008 -106.239 -99.1611 -98.2021 14.57

29A D_3h 117 -170.116 -114.146 6.90

29B C₁ 106 3.370 -112.656 -103.586 -99.7027 6.90 8.95

29C C₁ 104 0.988 -111.544 -104.051 -102.333 8.95 11.07

29D C_2v 103 0.246 -111.509 -103.946 -102.744 11.07 11.17

29E D_5h 103 0.225 -164.721 -111.354 -103.927 -102.775 11.17 21.90

29F C_3v 102 0.010 -111.135 -103.299 -102.227 21.90

30A C_2v 121 -177.579 -118.116 4.47

30B C_2v 121 8.279 -176.941 -118.433 -103.408 4.47 6.86

30C C_2v 109 1.538 -117.011 -108.571 -106.426 6.86 12.61

30D C_s 107 0.238 -115.941 -107.985 -106.765 12.61 21.33

30E C_s 106 0.012 -115.625 -107.368 -106.239 21.33

31A C₁ 126 -185.984 -122.440 4.60

31B C_s 126 -185.299 -122.858 4.60 6.35

31C C_s 115 3.490 -122.342 -112.543 -108.723 6.35 8.95

31D C_s 112 0.276 -121.535 -113.057 -111.711 8.95 10.42

31E C_2v 112 0.246 -181.901 -121.367 -113.048 -111.761 10.42 20.90

31F C_s 111 0.011 -120.805 -112.376 -111.240 20.90

32A C_2v 132 -195.468 -127.644 5.93

32B C_2v 120 3.83 -127.771 -117.284 5.93 10.27

32C C_s 116 0.269 -125.954 -117.117 -115.733 10.27 11.33

32D C₁ 116 0.260 -125.950 -117.106 -115.748 11.33 20.32

32E C_s 115 0.011 -125.126 -116.416 -115.246 20.32

33A C_s 137 -204.209 -131.704 5.35

33B C_5v 137 -203.575 -131.774 5.35 5.74

33C C_s 124 3.832 -132.287 -121.352 5.74 8.46

33D C_2v 121 0.313 -131.556 -122.160 -120.666 8.46 9.71

33E C_2v 121 0.269 -131.379 -122.167 -120.741 9.71 19.88

33F C_s 120 0.011 -130.467 -121.455 -120.253 19.88

34A D_5h 143 -214.068 -136.468 5.87

34B C_2v 128 3.832 -209.947 -136.798 -125.420 5.87 8.52

34C C_s 125 0.313 -135.989 -126.212 -124.675 8.52 9.79

34D C_s 125 0.293 -135.973 -126.214 -124.711 9.79 19.06

34E C_4v 124 0.013 -135.486 -125.616 -124.282 19.06

35A C_s 147 -221.771 -140.503 4.05

35B C_3v 147 -221.294 -141.106 4.05 5.62

35C C₁ 133 3.975 -141.957 -130.282 -126.120 5.62 8.08

35D C_2v 130 0.304 -141.403 -131.274 -129.700 8.08 18.65

35E C_3v 129 0.011 -140.132 -130.533 -129.265 18.65

36A C_s 152 -230.508 -144.827 4.68

36B C₁ 152 -229.689 -145.274 4.68 5.07

36C C_s 138 4.35 -147.382 -134.989 5.07 9.51

36D C₁ 134 0.334 -145.985 -135.313 -133.656 9.51 10.50

36E C_s 134 0.314 -145.748 -135.305 -133.691 10.50 18.42

36F C_s 133 0.013 -145.279 -134.701 -133.296 18.42

37A C_s 158 -240.008 -149.766 5.25

37B C₁ 142 4.35 -151.891 -139.056 -134.765 5.25 7.62

37C C_2v 139 0.357 -151.436 -140.360 -138.622 7.62 10.57

37D C_s 139 0.318 -151.078 -140.343 -138.690 10.57 18.29

37E C_s 138 0.014 -150.885 -139.776 -138.308 18.29

38A D₂ 164 30.138 -249.159 -153.209 -134.320 3.15

38B C₁ 163 27.962 -249.088 -154.165 -135.519 -129.339 3.15 4.70

38C C_s 163 -248.600 -154.042 4.70 4.76

38D O_h 144 0.013 -246.415 -157.407 -145.850 -144.321 4.76 5.40 6.95

38E C_5v 147 3.714 -157.477 -144.756 5.40 6.95

39A C_2v 169 -258.945 -158.266 4.48

39B C_5v 153 5.039 -163.482 -149.456 4.48 9.46

39C C_4v 148 0.014 -161.702 -149.887 -148.327 9.46

40A C_s 174 -268.395 -163.957 4.51

40B C_s 157 5.040 -263.925 -167.993 -153.524 -148.493 4.51 9.51

40C D_4h 152 0.014 -261.116 -165.996 -153.925 -152.334 9.51

41A C_2v 180 -278.406 4.60

41B C_s 161 5.040 -273.700 -172.527 -157.593 4.60 8.75

41C C_2v 157 0.435 -171.125 -158.506 -156.512 8.75 9.60

41D C_s 157 0.368 -269.035 -170.808 -158.521 -156.631 9.60 11.06

41E C_2v 157 0.366 -267.688 -170.759 -158.518 -156.633 11.06 16.83

41F C_3v 156 0.015 -170.898 -158.063 -156.359 16.83

42A C_3v 186 -288.335 -172.252 4.54

42B C_s 166 5.275 -282.296 -177.680 -162.366 -157.125 4.54 9.77

42C C_s 161 0.368 -175.184 -162.566 -160.638 9.77 11.10

42D C_s 161 0.367 -175.136 -162.564 -160.641 11.10 16.82

42E C_s 160 0.017 -175.405 -162.134 -160.371 16.82

43A C_2v 192 -298.172 -175.540 4.40

43B C_s 171 -290.632 -183.093 4.40 9.34

43C C_2v 166 0.443 -180.838 -167.586 -165.511 9.34 9.57

43D C_2v 166 0.374 -288.302 -180.515 -167.603 -165.635 9.57 16.67

43E C_s 165 0.016 -289.533 -180.590 -167.142 -165.371 16.67

44A C_5v 198 -308.277 -179.032 4.38

44B C₁ 175 5.722 -302.004 -187.626 -171.071 -165.690 4.38 9.38

44C C_s 170 0.443 -185.214 -171.631 -169.519 9.38 9.65

44D C_s 170 0.374 -300.114 -184.841 -171.645 -169.642 9.65 16.67

44E C₁ 169 0.016 -184.937 -171.182 -169.378 16.67

45A I_h 204 -318.661 -182.301 4.24

45B C₁ 180 5.965 -310.750 -192.955 -175.389 -169.713 4.24 8.75

45C C_2v 175 0.452 -190.548 -176.666 -174.510 8.75 11.79

45D C_2v 175 0.451 -303.604 -190.500 -176.663 -174.512 11.79 15.10

45E C_s 174 0.016 -190.275 -176.222 -174.384 15.10

46A C_s 207 -327.033 3.96

46B C_2v 186 6.646 -320.119 -199.178 -181.236 -174.605 3.96 10.46

46C C_s 179 0.452 -194.924 -180.711 -178.518 10.46 11.79

46D C_s 179 0.451 -194.877 -180.709 -178.519 11.79 15.11

46E C_s 178 0.016 -194.622 -180.262 -178.391 15.11

47A C₃ 210 40.212 -336.666 -194.242 -170.368 4.03

47B C₁ 190 6.648 -331.591 -203.704 -185.302 -178.621 4.03 9.60

47C C_2v 184 0.461 -323.011 -200.257 -185.746 -183.508 9.60 14.94

47D C_3v 183 0.016 -199.964 -185.302 -183.397 14.94

48A C_2v 213 -346.663 4.01

48B C_s 195 7.464 -340.396 -209.044 -189.557 -182.618 4.01 9.81

48C C_s 188 0.461 -204.632 -189.791 -187.516 9.81 14.72

48D C_s 187 0.020 -205.419 -189.535 -187.440 14.72

49A C_s 219 -356.413 3.91

49B C_3v 201 7.770 -350.540 -215.254 -195.275 -187.212 3.91 10.32

49C D_5h 193 0.471 -337.433 -210.010 -194.828 -192.504 10.32 21.18

49D C_s 191 0.023 -210.023 -193.595 -191.449 21.18

50A C_5v 224 -366.636 4.04

50B C_s 205 7.770 -219.820 -199.345 -191.240 4.04 10.12

50C C_s 197 0.455 -216.105 -199.188 -196.612 10.12 15.27

50D C₁ 196 0.023 -215.519 -198.666 -196.462 15.27

51A C₂ 232 -376.673 -217.628 3.74

51B C_2v 210 9.360 -373.933 -225.391 -202.912 3.74 9.46

51C C_s 202 0.863 -221.114 -203.757 -200.829 9.46 10.17

51D C_s 201 0.022 -220.986 -203.737 -201.474 10.17

52A C_s 238 -387.587 3.71

52B C_3v 216 9.618 -384.521 -231.615 -208.673 -199.353 3.71 9.36

52C D_2h 207 0.021 -227.346 -209.769 -207.479 9.36

53A D_6d 244 -398.783 3.70

53B C_2v 222 9.902 -395.150 -237.835 -214.410 -203.181 3.70 10.32

53C C_s 211 0.021 -231.694 -213.809 -211.486 10.32

54A C_s 248 -407.966 -232.987 3.41

54B C_5v 228 10.215 -405.832 -244.058 -220.119 -208.330 3.41 10.32

54C C_2v 217 0.465 -238.115 -219.427 -216.637 10.32 15.17

54D C_2v 216 0.021 -237.038 -218.849 -216.492 15.17

55A C₁ 252 -417.919 3.25

55B I_h 234 10.543 -416.626 -250.287 -225.814 -213.524 3.25 11.15

55C C_2v 221 0.465 -242.622 -223.482 -220.646 11.15 15.18

55D C_s 220 0.021 -241.385 -222.889 -220.498 15.18

56A C_s 258 -428.611 3.60

56B C_3v 237 10.545 -425.709 -253.923 -228.900 -216.538 3.60 10.18

56C C_2v 226 0.481 -248.168 -228.553 -225.633 10.18 11.38

56D C_s 226 0.465 -248.051 -228.534 -225.655 11.38 15.20

56E D_2h 225 0.021 -410.499 -246.729 -227.929 -225.505 15.20

57A D_3h 264 -439.960 4.13

57B C_s 241 10.563 -436.125 -258.042 -232.878 -220.503 4.13 9.61

57C C_2v 231 0.465 -426.762 -253.482 -233.586 -230.664 9.61 21.83

57D C_s 229 0.028 -252.749 -232.227 -229.557 21.83

58A C_s 267 -449.432 -253.759 4.05

58B C_3v 246 10.585 -442.501 -263.411 -237.898 -225.469 4.05 10.13

58C C_s 235 0.481 -258.106 -237.660 -234.652 10.13 11.76

58D C_s 235 0.476 -258.102 -237.649 -234.661 11.76 11.85

58E C₁ 235 0.465 -257.876 -237.631 -234.671 11.85 14.94

58F C_3v 234 0.027 -258.241 -237.298 -234.570 14.94

59A C_3v 267 -459.509 -260.795 3.77

59B C_2v 272 -459.250 -261.475 3.77 4.26

59C C₁ 250 10.832 -267.858 -241.706 -229.054 4.26 4.86

7.93 9.09

59D C_2v 250 10.939 -453.110 -267.945 -241.627 -228.849 4.86 7.93

59E T_d 240 0.027 -264.710 -243.331 -240.572 9.09

60A C_s 278 -470.448 4.51

60B C_s 255 11.042 -273.341 -246.579 -233.702 4.51 9.34

60C C_s 245 0.540 -269.408 -247.786 -244.566 9.34 11.00

60D C_2v 245 0.536 -269.292 -247.782 -244.572 11.00 13.95

60E C_s 244 0.027 -269.050 -247.371 -244.579 13.95

61A T_d 284 -482.026 -271.693 4.65

61B C_2v 260 11.174 -278.727 -251.502 -238.499 4.65 9.54

61C C_3v 249 0.027 -274.135 -252.397 -249.587 9.54

62A C_s 293 -491.052 -272.268 4.00

62B C_s 264 11.707 -485.426 -283.183 -255.038 -241.549 4.00 6.33

62C C_2v 264 11.174 -283.103 -255.559 -242.510 6.33 9.24

62D C₁ 254 0.623 -279.852 -256.900 -253.442 9.24 9.78

62E C_s 254 0.560 -279.460 -256.908 -253.555 9.78 10.47

62F C₁ 254 0.526 -279.358 -256.897 -253.613 10.47 14.15

62G C₁ 253 0.027 -476.610 -278.482 -256.437 -253.593 14.15

63A C_s 301 -501.732 3.83

63B C₁ 269 11.880 -497.182 -288.561 -259.921 -246.279 3.83 8.87

63C C_s 259 0.533 495.357 -285.142 -262.030 -258.621 8.87 14.16

63D C_s 258 0.027 -283.826 -261.477 -258.600 14.16

64A D_3d 307 -512.832 3.85

64B C_s 274 12.066 -503.235 -293.932 -264.790 -250.989 3.85 8.22

64C C_2v 265 0.552 -501.818 -291.412 -268.024 -264.587 8.22 24.31

64D C_s 262 0.031 -289.324 -265.715 -262.643 24.31

65A C_s 310 -523.446 3.86

65B C₁ 278 12.494 -298.322 -268.407 -254.407 3.86 4.74

65C C₂ 278 12.574 -515.415 -298.392 -268.352 -254.301 4.74 6.32

65D C₁ 278 12.081 -298.318 -268.837 -254.984 6.32 8.12

65E C_2v 269 0.603 -296.318 -272.099 -268.514 8.12 10.53

65F C₁ 269 0.565 -296.028 -272.084 -268.578 10.53 11.30

65G C₁ 269 0.552 -295.807 -272.070 -268.595 11.30 19.72

65H C_s 267 0.027 -293.515 -270.557 -267.613 19.72

66A C_s 314 -534.464 3.82

66B C₁ 284 15.128 -530.517 -303.513 -271.712 -256.391 3.82 4.83

66C C₁ 283 12.758 -528.937 -303.763 -273.223 -259.079 4.83 7.83

66D C_s 274 0.640 -301.866 -277.135 -273.454 7.83 9.74

66E C_s 274 0.573 -301.465 -277.145 -273.574 9.74 11.50

66F C_s 274 0.552 -301.193 -277.116 -273.602 11.50 24.31

66G C_s 271 0.033 -299.454 -274.861 -271.668 24.31

67A C_s 318 -544.754 -297.548 3.80

67B C_s 289 15.345 -539.667 -308.980 -276.565 3.80 5.46

67C C₂ 288 12.945 -538.179 -309.130 -278.090 -263.853 5.46 7.72

67D C_2v 279 0.677 -307.349 -282.172 -278.396 7.72 11.36

67E C_2v 279 0.674 -307.234 -282.166 -278.401 11.36 21.85

67F C_3v 276 0.032 -304.942 -279.933 -276.681 21.85

68A C_s 324 -555.582 3.68

68B C₁ 294 15.980 -551.451 -314.375 -281.011 -265.754 3.68 7.29

68C C_s 293 12.938 -549.986 -314.147 -281.905 -267.101 7.29 7.78

68D C₁ 283 0.666 -311.913 -286.250 -282.430 7.78 9.67

68E C_2v 283 0.597 -311.516 -286.263 -282.554 9.67 11.24

68F C_s 283 0.573 -311.247 -286.237 -282.588 11.24 11.51

68G C_2v 283 0.552 -310.974 -286.208 -282.618 11.51 13.47

68H C_3v 282 0.031 -311.260 -285.949 -282.682 13.47

69A C₁ 329 -566.364 -307.246 3.65

69B C₁ 308 -560.639 -318.206 3.65 4.22

69C C_5v 298 14.909 -559.783 -319.688 -286.265 4.22 5.22

69D C_s 299 16.223 -319.820 -285.837 5.22 7.64

69E C_s 288 0.705 -317.396 -291.284 -287.367 7.64 9.66

69F C₁ 288 0.653 -317.293 -291.294 -287.462 9.66 18.06

69G C₁ 286 0.033 -315.850 -290.023 -286.694 18.06

70A C_2v 335 -577.740 3.22

70B C₁ 332 -577.287 -313.363 3.22 3.67

70C C₅ 314 -571.815 -324.212 3.67 4.16

70D C_5v 304 15.324 -570.747 -325.888 -291.872 4.16 7.89

70E C_s 293 0.676 -323.082 -296.412 -292.439 7.89 17.69

70F C_s 291 0.035 -321.460 -295.097 -291.707 17.69

71A C₁ 330 -588.397 -316.491 3.42

71B C₅ 320 -583.203 -330.363 3.42 4.28

71C C_5v 310 17.608 -581.945 -331.589 -295.612 4.28 7.29

71D C_2v 299 0.695 -329.356 -302.405 -298.405 7.29 21.50

71E C_2v 296 0.034 -575.523 -326.901 -300.164 -296.719 21.50

72A C₁ 335 -599.690 -320.666 3.40

72B C₁ 314 17.960 -595.065 -335.877 -299.301 -283.195 3.40 4.77

72C C_s 313 16.818 -592.641 -336.122 -299.750 -283.664 4.77 7.29

72D C₁ 303 0.713 -333.968 -306.460 -302.387 7.29 10.53

72E C₁ 303 0.695 -333.804 -306.454 -302.413 10.53 21.49

72F C_s 300 0.037 -331.605 -304.245 -300.732 21.49

73A C₃ 339 -610.937 -324.743 3.50

73B C_s 319 18.316 -603.874 -341.266 -304.027 -287.806 3.50 6.97

73C C_s 318 16.192 -602.727 -340.979 -305.190 -289.451 6.97 7.03

73D C_s 308 0.725 -339.405 -311.518 -307.374 7.03 10.72

73E C_s 308 0.695 -339.242 -311.503 -307.421 10.72 21.49

73F C_s 305 0.037 -597.775 -336.825 -309.251 -305.732 21.49

74A C_3v 357 -622.680 3.92

74B C_s 324 18.513 -614.020 -346.611 -308.886 3.92 6.83

74C C_5v 313 0.699 -610.492 -345.200 -316.651 -312.441 6.83 21.38

74D C_s 310 0.035 -342.317 -314.324 -310.745 21.38

75A C_s 364 -633.513 3.70

75B C₁ 328 18.559 -630.521 -351.177 -312.987 -296.599 3.70 5.81

75C D_5h 319 0.718 -351.472 -322.644 -318.407 5.81 21.14

75D C_4v 316 0.035 -348.841 -320.398 -316.758 21.14

76A C_s 370 -644.952 3.56

76B C₁ 334 21.847 -642.129 -355.971 -315.525 -298.049 3.56 4.16

76C C₁ 333 19.297 -641.559 -356.250 -317.225 -300.489 4.16 6.10

76D C_s 323 0.737 -356.085 -326.697 -322.387 6.10 10.93

76E C_2v 323 0.718 -355.772 -326.683 -322.414 10.93 21.14

76F C_s 320 0.035 -353.246 -324.440 -320.764 21.14

77A C₁ 368 -656.080 -349.696 3.68

77B C_s 339 22.562 -650.116 -361.293 -319.898 3.68 5.01

77C C₁ 339 22.076 -361.416 -320.372 -302.897 5.01 5.24

77D C_s 338 19.503 -647.494 -361.591 -322.075 5.24 6.04

77E C_2v 328 0.749 -361.521 -331.754 -327.372 6.04 20.66

77F C_s 325 0.035 -358.645 -329.483 -325.771 20.66

78A C₃ 384 -667.576 3.54

78B C₁ 344 22.724 -664.006 -366.727 -324.815 -307.287 3.54 5.18

78C C₁ 344 22.651 -366.761 -324.883 -307.076 5.18 6.04

78D C_s 338 11.849 -366.723 -330.163 -315.401 6.04 6.53

78E C₁ 332 0.769 -366.132 -335.807 -331.349 6.53 10.71

78F C_s 332 0.749 -650.436 -365.872 -335.796 -331.379 10.71 16.80

78G C_s 330 0.036 -364.252 -334.558 -330.784 16.80

79A C₁ 385 -678.940 -358.580 3.59

79B C_2v 348 19.381 -673.565 -372.832 -332.365 3.59 6.53

79C C_2v 343 11.973 -372.466 -335.139 -320.223 6.53 6.67

79D C_2v 337 0.783 -663.444 -371.568 -340.862 -336.332 6.67 11.06

79E C_s 337 0.752 -659.186 -371.199 -340.834 -336.380 11.06 11.55

79F O_h 336 0.036 -370.784 -340.632 -336.796 11.55

80A C₃ 387 -690.578 3.53

80B C_s 354 21.858 -683.221 -378.333 -335.898 3.53 6.36

80C C₁ 348 12.120 -377.972 -340.072 -324.944 6.36 7.34

80D C₁ 341 0.806 -674.292 -376.176 -344.911 -340.304 7.34 10.44

80E C_s 341 0.783 -375.919 -344.904 -340.340 10.44 11.02

80F C_s 341 0.752 -668.582 -375.572 -344.878 -340.387 11.02 11.57

80G C_4v 340 0.036 -375.075 -344.669 -340.803 11.57

  

Figure 2.5: Zero temperature `phase diagram' showing the variation of the lowest energy structure with N and $\rho_0$.The data points are the values of $\rho_0$ at which the global minimum changes. The lines joining the data points divide the phase diagram into regions where the global minima have similar structures. The solid lines denote the boundaries between the four main structural types--icosahedral, decahedral, close-packed and those associated with low $\rho_0$ (L)--and the dashed lines internal boundaries within a structural type, e.g. between icosahedral structures with Mackay and anti-Mackay overlayers, or between decahedral structures with different length decahedral axes.

  

Figure 2.6: Plots of $\Delta_2 E$ as a function of N for $\rho_0$=3, 6, 10 and 14 as marked.

The results are summarized in Figure 2.5 which provides a zero temperature `phase diagram', showing how the global minimum depends upon N and $\rho_0$. The structural behaviour of Morse clusters with less than eight atoms is rather uninteresting because the global minimum is independent of $\rho_0$.For all $N\ge 8$, however, the global minimum changes at least once as a function of $\rho_0$.For $N\ge 13$, icosahedral, decahedral and fcc structures all exist, and the form of the phase diagram is in very good agreement with the predictions made in §2.2.1. For most sizes the structure changes from icosahedral to decahedral to close-packed as the range of the potential is decreased. For N<23, however, the transition from decahedral to close-packed occurs at larger values than we consider in this study. There are also a number of sizes (N=38-40, 52, 53, 59 and 61) at which there is a transition directly from an icosahedral to a close-packed structure; this occurs when n_nn for the lowest energy close-packed structure is greater than or equal to that for the lowest energy decahedral structure. It can be confirmed by examining Table 2.2 that the causes of these transitions are the relative values of n_nn and $E_{\rm strain}$, as we suggested in §2.2.1.

  

Figure 2.7: Comparison of the energies of icosahedral (solid line with diamonds), decahedral (dashed line with crosses) and close-packed (dotted line with squares) N clusters at $\rho_0$=6. The energy zero is E_MI, the interpolated energy of Mackay icosahedra. E_MI=-3.0354+0.2624 N^1/3+8.8400 N^2/3-6.8381N and was obtained by fitting to the first four Mackay icosahedra (N=13, 55, 147 and 309).

The boundaries between the different morphologies are sensitive functions of N. Such size-dependence is observed for many properties of clusters, and gradually lessens as the size increases (because the addition of a single atom becomes a smaller perturbation) until the bulk limit is reached where properties are continuous functions of N. The decahedral to close-packed boundary is particularly sensitive, because the range of $\rho_0$ for which the decahedral structure is most stable changes dramatically even when the difference in n_nn between the decahedral and close-packed structures changes by one.

Sizes for which a morphology is the lowest in energy for a particularly large range of $\rho_0$indicate that the structure is especially stable. The optimal structures shown in Figure 2.1 are particularly good examples of this. Another indicator of sizes of especial stability is provided by $\Delta_2 E(N)=E(N+1)+E(N-1)-2 E(N)$.Peaks in $\Delta_2 E$ have been found to correlate well with the magic numbers (sizes at which clusters are particularly abundant) observed in mass spectroscopic studies of clusters formed in molecular beams[94]. Plots of $\Delta_2 E$ are shown in Figure 2.6 for a number of values of $\rho_0$. Unsurprisingly, the plot for $\rho_0$=6 is very similar to that for LJ clusters with peaks due to especially stable icosahedral clusters. At higher values of $\rho_0$ peaks corresponding to especially stable close-packed and decahedral clusters begin to occur. The plot at $\rho_0$=14 is very similar to that recently obtained for C₆₀ clusters using the Girifalco potential[95]. Similarly, if the energy is normalized by subtracting a suitable function of N, again particularly stable sizes stand out (Figure 2.7).

In the following subsections we will look at the growth sequences for each structural class in more detail. As well as the three ordered morphologies, we also examine the unusual structures that occur for the larger clusters at low $\rho_0$, which, as we will see, involve a mixture of order and disorder.

  

Figure 2.8: Global minima for $N\le 13$ which are formed by growth on the 7-atom pentagonal bipyramid and which lead to the 13-atom icosahedron.