Band Structure for Diamond (Back)(Zip)

Diamond is both a very simple solid, consisting only of carbon atoms, and also a complicated solid, in that the band-structure is very complicated.  As such, it is ideal for illustrating features of the Brillouin zone.

A good starting point for investigating the Brillouin zone of diamond is the simple data-set shown in Figure 1.  Here, the geometry consists of the two carbon atoms of the primitive unit cell and the three translation vectors that define where the primitive unit cell is moved to by the
Figure 1: Data-set for Diamond for use with MAKPOL    
*
* Data set for diamond, for use with BZ
*
 MERS=(6,6,6) BCC let symmetry INT
 Diamond

 C
 C  1.545 1 0 0 0 0 1
Tv  1.784 0 54.73561 0 0 0 1 2 
Tv  1.784 0 54.73561 0 120 0 1 2 3
Tv  1.784 0 54.73561 0 240 0 1 2 3
 
 translation operations.  This data-set would be run using the program MAKPOL to create a much larger data-set that could then be used by MOPAC. The MAKPOL data-set uses five keywords. In order, these are:

MERS=(6,6,6) Six unit cells in each of the three translational directions is sufficient to adequately represent the solid-state structure.

BCC Only unit cells where the sum of the cell indices is even are used.

LET For Brillouin zone work, atoms are not rearranged into the most aesthetically attractive form.

Figure E1: Diamond Electronic Band Structure
Example of a "Walk" in k-space, from Program BZ

SYMMETRY Diamond has very high symmetry; keyword SYMMETRY can be used to both simplify the geometry optimization and to preserve the precise octahedral symmetry of the lattice.  

INT The resulting geometry is to be in internal coordinates (these are better than Cartesian coordinates for high-symmetry solids.)

Names for data sets for MAKPOL need to be in the form MAKE_<name>.dat, so for diamond a suitable name would be MAKE_Diamond.dat.  When MAKE_Diamond.dat is run using MAKPOL, a much larger data-set, Diamond.mop, is generated.  This data-set contains some symmetry, but still has several geometric parameters flagged for optimization.

Addition of extra symmetry to the MOPAC data-set

All angles and dihedral angles in diamond are, of course, fixed, so without loss of generality all angle and dihedral angle flags that were "1" can be changed to "0".
Figure 2: Symmetrized data-set for Diamond for use with MOPAC    
  MERS=(6,6,6) BCC LET SYMMETRY 
 Diamond

  C    0.000000 +0   0.000000 +0    0.000000 +0                   
  C    1.545000 +1   0.000000 +0    0.000000 +0     1             
  C    2.958427 +0 100.024785 +0    0.000000 +0     2     1       
  C    1.545000 +0 100.024785 +0  180.000000 +0     3     2     1 
  C    2.958427 +0 100.024785 +0  180.000000 +0     4     3     2 
  C    1.545000 +0 100.024785 +0  180.000000 +0     5     4     3 
  C    1.544986 +0 109.470848 +0  -60.000000 +0     2     1     3 
  C    1.545000 +0 109.470848 +0  180.000000 +0     7     2     1 
  C    1.544986 +0 109.470848 +0  120.000000 +0     4     3     2 
  C    1.545000 +0 109.470848 +0  180.000000 +0     9     4     3 
  C    1.544986 +0 109.470848 +0  120.000000 +0     6     5     4 
  C    1.545000 +0 109.470848 +0  180.000000 +0    11     6     5 
  C    2.522957 +0  90.000000 +0  -90.000000 +0     7     2     1 
  C    1.545000 +0  90.000000 +0 -109.470848 +0    13     7     2 
  C    1.544986 +0 109.470848 +0  180.000000 +0     8     7     2 
  C    1.545000 +0 109.470848 +0  180.000000 +0    15     8     7 
  C    1.544986 +0 109.470848 +0  180.000000 +0    10     9     4 
  C    1.545000 +0 109.470848 +0  180.000000 +0    17    10     9 
  C    1.544986 +0 109.470848 +0  -90.000000 +0    14    13     7 
  C    1.545000 +0 109.470848 +0  180.000000 +0    19    14    13 
  C    1.544986 +0 109.470848 +0  180.000000 +0    16    15     8 
  C    1.545000 +0 109.470848 +0  180.000000 +0    21    16    15 
  C    1.544986 +0 109.470848 +0  180.000000 +0    18    17    10 
  C    1.545000 +0 109.470848 +0  180.000000 +0    23    18    17 
  C    2.522957 +0  90.000000 +0  -90.000000 +0    19    14    13 
  C    1.545000 +0  90.000000 +0 -109.470848 +0    25    19    14 
  C    1.544986 +0 109.470848 +0  180.000000 +0    20    19    14 
  C    1.545000 +0 109.470848 +0  180.000000 +0    27    20    19 
  C    1.544986 +0 109.470848 +0  180.000000 +0    22    21    16 
  C    1.545000 +0 109.470848 +0  180.000000 +0    29    22    21 
  C    1.544986 +0 109.470848 +0  -90.000000 +0    26    25    19 
  C    1.545000 +0 109.470848 +0  180.000000 +0    31    26    25 
  C    1.544986 +0 109.470848 +0  180.000000 +0    28    27    20 
  C    1.545000 +0 109.470848 +0  180.000000 +0    33    28    27 
  C    1.544986 +0 109.470848 +0  180.000000 +0    30    29    22 
  C    1.545000 +0 109.470848 +0  180.000000 +0    35    30    29 
  C    1.544986 +0 109.470848 +0   60.000000 +0     2     1     3 
  C    1.545000 +0 109.470848 +0  180.000000 +0    37     2     1 
  C    1.544986 +0 109.470848 +0 -120.000000 +0     4     3     2 
  C    1.545000 +0 109.470848 +0  180.000000 +0    39     4     3 
  C    1.544986 +0 109.470848 +0 -120.000000 +0     6     5     4 
  C    1.545000 +0 109.470848 +0  180.000000 +0    41     6     5 
  C    1.544986 +0 109.470848 +0  180.000000 +0     2     1     3 
  C    1.545000 +0 109.470848 +0  180.000000 +0    43     2     1 
  C    1.544986 +0 109.470848 +0  -60.000000 +0     8     7     2 
  C    1.545000 +0 109.470848 +0  180.000000 +0    45     8     7 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    10     9     4 
  C    1.545000 +0 109.470848 +0  180.000000 +0    47    10     9 
  C    1.544986 +0 109.470848 +0   30.000000 +0    14    13     7 
  C    1.545000 +0 109.470848 +0  180.000000 +0    49    14    13 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    16    15     8 
  C    1.545000 +0 109.470848 +0  180.000000 +0    51    16    15 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    18    17    10 
  C    1.545000 +0 109.470848 +0  180.000000 +0    53    18    17 
  C    1.544986 +0 109.470848 +0  150.000000 +0    14    13     7 
  C    1.545000 +0 109.470848 +0  180.000000 +0    55    14    13 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    20    19    14 
  C    1.545000 +0 109.470848 +0  180.000000 +0    57    20    19 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    22    21    16 
  C    1.545000 +0 109.470848 +0  180.000000 +0    59    22    21 
  C    1.544986 +0 109.470848 +0   30.000000 +0    26    25    19 
  C    1.545000 +0 109.470848 +0  180.000000 +0    61    26    25 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    28    27    20 
  C    1.545000 +0 109.470848 +0  180.000000 +0    63    28    27 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    30    29    22 
  C    1.545000 +0 109.470848 +0  180.000000 +0    65    30    29 
  C    1.544986 +0 109.470848 +0  150.000000 +0    26    25    19 
  C    1.545000 +0 109.470848 +0  180.000000 +0    67    26    25 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    32    31    26 
  C    1.545000 +0 109.470848 +0  180.000000 +0    69    32    31 
  C    1.544986 +0 109.470848 +0  -60.000000 +0    34    33    28 
  C    1.545000 +0 109.470848 +0  180.000000 +0    71    34    33 
  C    2.522957 +0  90.000000 +0  -90.000000 +0    43     2     1 
  C    1.545000 +0  90.000000 +0 -109.470848 +0    73    43     2 
  C    1.544986 +0 109.470848 +0  180.000000 +0    38    37     2 
  C    1.545000 +0 109.470848 +0  180.000000 +0    75    38    37 
  C    1.544986 +0 109.470848 +0  180.000000 +0    40    39     4 
  C    1.545000 +0 109.470848 +0  180.000000 +0    77    40    39 
  C    1.544986 +0 109.470848 +0   60.000000 +0    44    43     2 
  C    1.545000 +0 109.470848 +0  180.000000 +0    79    44    43 
  C    1.544986 +0 109.470848 +0  180.000000 +0    46    45     8 
  C    1.545000 +0 109.470848 +0  180.000000 +0    81    46    45 
  C    1.544986 +0 109.470848 +0  180.000000 +0    48    47    10 
  C    1.545000 +0 109.470848 +0  180.000000 +0    83    48    47 
  C    1.544986 +0 109.470848 +0  180.000000 +0    44    43     2 
  C    1.545000 +0 109.470848 +0  180.000000 +0    85    44    43 
  C    1.544986 +0 109.470848 +0  180.000000 +0    50    49    14 
  C    1.545000 +0 109.470848 +0  180.000000 +0    87    50    49 
  C    1.544986 +0 109.470848 +0  180.000000 +0    52    51    16 
  C    1.545000 +0 109.470848 +0  180.000000 +0    89    52    51 
  C    1.544986 +0 109.470848 +0   60.000000 +0    56    55    14 
  C    1.545000 +0 109.470848 +0  180.000000 +0    91    56    55 
  C    1.544986 +0 109.470848 +0  180.000000 +0    58    57    20 
  C    1.545000 +0 109.470848 +0  180.000000 +0    93    58    57 
  C    1.544986 +0 109.470848 +0  180.000000 +0    60    59    22 
  C    1.545000 +0 109.470848 +0  180.000000 +0    95    60    59 
  C    1.544986 +0 109.470848 +0  180.000000 +0    56    55    14 
  C    1.545000 +0 109.470848 +0  180.000000 +0    97    56    55 
  C    1.544986 +0 109.470848 +0  180.000000 +0    62    61    26 
  C    1.545000 +0 109.470848 +0  180.000000 +0    99    62    61 
  C    1.544986 +0 109.470848 +0  180.000000 +0    64    63    28 
  C    1.545000 +0 109.470848 +0  180.000000 +0   101    64    63 
  C    1.544986 +0 109.470848 +0   60.000000 +0    68    67    26 
  C    1.545000 +0 109.470848 +0  180.000000 +0   103    68    67 
  C    1.544986 +0 109.470848 +0  180.000000 +0    70    69    32 
  C    1.545000 +0 109.470848 +0  180.000000 +0   105    70    69 
  C    1.544986 +0 109.470848 +0  180.000000 +0    72    71    34 
  C    1.545000 +0 109.470848 +0  180.000000 +0   107    72    71 
  C    1.544986 +0 109.470848 +0  150.000000 +0    74    73    43 
  C    1.545000 +0 109.470848 +0  180.000000 +0   109    74    73 
  C    1.544986 +0 109.470848 +0  180.000000 +0    76    75    38 
  C    1.545000 +0 109.470848 +0  180.000000 +0   111    76    75 
  C    1.544986 +0 109.470848 +0  180.000000 +0    78    77    40 
  C    1.545000 +0 109.470848 +0  180.000000 +0   113    78    77 
  C    1.544986 +0 109.470848 +0  -90.000000 +0    74    73    43 
  C    1.545000 +0 109.470848 +0  180.000000 +0   115    74    73 
  C    1.544986 +0 109.470848 +0  180.000000 +0    80    79    44 
  C    1.545000 +0 109.470848 +0  180.000000 +0   117    80    79 
  C    1.544986 +0 109.470848 +0  180.000000 +0    82    81    46 
  C    1.545000 +0 109.470848 +0  180.000000 +0   119    82    81 
  C    1.544986 +0 109.470848 +0   60.000000 +0    86    85    44 
  C    1.545000 +0 109.470848 +0  180.000000 +0   121    86    85 
  C    1.544986 +0 109.470848 +0  180.000000 +0    88    87    50 
  C    1.545000 +0 109.470848 +0  180.000000 +0   123    88    87 
  C    1.544986 +0 109.470848 +0  180.000000 +0    90    89    52 
  C    1.545000 +0 109.470848 +0  180.000000 +0   125    90    89 
  C    1.544986 +0 109.470848 +0  180.000000 +0    86    85    44 
  C    1.545000 +0 109.470848 +0  180.000000 +0   127    86    85 
  C    1.544986 +0 109.470848 +0  180.000000 +0    92    91    56 
  C    1.545000 +0 109.470848 +0  180.000000 +0   129    92    91 
  C    1.544986 +0 109.470848 +0  180.000000 +0    94    93    58 
  C    1.545000 +0 109.470848 +0  180.000000 +0   131    94    93 
  C    1.544986 +0 109.470848 +0   60.000000 +0    98    97    56 
  C    1.545000 +0 109.470848 +0  180.000000 +0   133    98    97 
  C    1.544986 +0 109.470848 +0  180.000000 +0   100    99    62 
  C    1.545000 +0 109.470848 +0  180.000000 +0   135   100    99 
  C    1.544986 +0 109.470848 +0  180.000000 +0   102   101    64 
  C    1.545000 +0 109.470848 +0  180.000000 +0   137   102   101 
  C    1.544986 +0 109.470848 +0  180.000000 +0    98    97    56 
  C    1.545000 +0 109.470848 +0  180.000000 +0   139    98    97 
  C    1.544986 +0 109.470848 +0  180.000000 +0   104   103    68 
  C    1.545000 +0 109.470848 +0  180.000000 +0   141   104   103 
  C    1.544986 +0 109.470848 +0  180.000000 +0   106   105    70 
  C    1.545000 +0 109.470848 +0  180.000000 +0   143   106   105 
  C    2.522957 +0  90.000000 +0  -90.000000 +0   115    74    73 
  C    1.545000 +0  90.000000 +0 -109.470848 +0   145   115    74 
  C    1.544986 +0 109.470848 +0  180.000000 +0   110   109    74 
  C    1.545000 +0 109.470848 +0  180.000000 +0   147   110   109 
  C    1.544986 +0 109.470848 +0  180.000000 +0   112   111    76 
  C    1.545000 +0 109.470848 +0  180.000000 +0   149   112   111 
  C    1.544986 +0 109.470848 +0   60.000000 +0   116   115    74 
  C    1.545000 +0 109.470848 +0  180.000000 +0   151   116   115 
  C    1.544986 +0 109.470848 +0  180.000000 +0   118   117    80 
  C    1.545000 +0 109.470848 +0  180.000000 +0   153   118   117 
  C    1.544986 +0 109.470848 +0  180.000000 +0   120   119    82 
  C    1.545000 +0 109.470848 +0  180.000000 +0   155   120   119 
  C    1.544986 +0 109.470848 +0  180.000000 +0   116   115    74 
  C    1.545000 +0 109.470848 +0  180.000000 +0   157   116   115 
  C    1.544986 +0 109.470848 +0  180.000000 +0   122   121    86 
  C    1.545000 +0 109.470848 +0  180.000000 +0   159   122   121 
  C    1.544986 +0 109.470848 +0  180.000000 +0   124   123    88 
  C    1.545000 +0 109.470848 +0  180.000000 +0   161   124   123 
  C    1.544986 +0 109.470848 +0   60.000000 +0   128   127    86 
  C    1.545000 +0 109.470848 +0  180.000000 +0   163   128   127 
  C    1.544986 +0 109.470848 +0  180.000000 +0   130   129    92 
  C    1.545000 +0 109.470848 +0  180.000000 +0   165   130   129 
  C    1.544986 +0 109.470848 +0  180.000000 +0   132   131    94 
  C    1.545000 +0 109.470848 +0  180.000000 +0   167   132   131 
  C    1.544986 +0 109.470848 +0  180.000000 +0   128   127    86 
  C    1.545000 +0 109.470848 +0  180.000000 +0   169   128   127 
  C    1.544986 +0 109.470848 +0  180.000000 +0   134   133    98 
  C    1.545000 +0 109.470848 +0  180.000000 +0   171   134   133 
  C    1.544986 +0 109.470848 +0  180.000000 +0   136   135   100 
  C    1.545000 +0 109.470848 +0  180.000000 +0   173   136   135 
  C    1.544986 +0 109.470848 +0   60.000000 +0   140   139    98 
  C    1.545000 +0 109.470848 +0  180.000000 +0   175   140   139 
  C    1.544986 +0 109.470848 +0  180.000000 +0   142   141   104 
  C    1.545000 +0 109.470848 +0  180.000000 +0   177   142   141 
  C    1.544986 +0 109.470848 +0  180.000000 +0   144   143   106 
  C    1.545000 +0 109.470848 +0  180.000000 +0   179   144   143 
  C    1.544986 +0 109.470848 +0  150.000000 +0   146   145   115 
  C    1.545000 +0 109.470848 +0  180.000000 +0   181   146   145 
  C    1.544986 +0 109.470848 +0  180.000000 +0   148   147   110 
  C    1.545000 +0 109.470848 +0  180.000000 +0   183   148   147 
  C    1.544986 +0 109.470848 +0  180.000000 +0   150   149   112 
  C    1.545000 +0 109.470848 +0  180.000000 +0   185   150   149 
  C    1.544986 +0 109.470848 +0  -90.000000 +0   146   145   115 
  C    1.545000 +0 109.470848 +0  180.000000 +0   187   146   145 
  C    1.544986 +0 109.470848 +0  180.000000 +0   152   151   116 
  C    1.545000 +0 109.470848 +0  180.000000 +0   189   152   151 
  C    1.544986 +0 109.470848 +0  180.000000 +0   154   153   118 
  C    1.545000 +0 109.470848 +0  180.000000 +0   191   154   153 
  C    1.544986 +0 109.470848 +0   60.000000 +0   158   157   116 
  C    1.545000 +0 109.470848 +0  180.000000 +0   193   158   157 
  C    1.544986 +0 109.470848 +0  180.000000 +0   160   159   122 
  C    1.545000 +0 109.470848 +0  180.000000 +0   195   160   159 
  C    1.544986 +0 109.470848 +0  180.000000 +0   162   161   124 
  C    1.545000 +0 109.470848 +0  180.000000 +0   197   162   161 
  C    1.544986 +0 109.470848 +0  180.000000 +0   158   157   116 
  C    1.545000 +0 109.470848 +0  180.000000 +0   199   158   157 
  C    1.544986 +0 109.470848 +0  180.000000 +0   164   163   128 
  C    1.545000 +0 109.470848 +0  180.000000 +0   201   164   163 
  C    1.544986 +0 109.470848 +0  180.000000 +0   166   165   130 
  C    1.545000 +0 109.470848 +0  180.000000 +0   203   166   165 
  C    1.544986 +0 109.470848 +0   60.000000 +0   170   169   128 
  C    1.545000 +0 109.470848 +0  180.000000 +0   205   170   169 
  C    1.544986 +0 109.470848 +0  180.000000 +0   172   171   134 
  C    1.545000 +0 109.470848 +0  180.000000 +0   207   172   171 
  C    1.544986 +0 109.470848 +0  180.000000 +0   174   173   136 
  C    1.545000 +0 109.470848 +0  180.000000 +0   209   174   173 
  C    1.544986 +0 109.470848 +0  180.000000 +0   170   169   128 
  C    1.545000 +0 109.470848 +0  180.000000 +0   211   170   169 
  C    1.544986 +0 109.470848 +0  180.000000 +0   176   175   140 
  C    1.545000 +0 109.470848 +0  180.000000 +0   213   176   175 
  C    1.544986 +0 109.470848 +0  180.000000 +0   178   177   142 
  C    1.545000 +0 109.470848 +0  180.000000 +0   215   178   177 
 XX    2.522957 +0  90.000000 +0   90.000000 +0    11     6     5 
 XX    2.522957 +0  90.000000 +0  -90.000000 +0    31    26    25 
 XX    2.522957 +0  90.000000 +0  -90.000000 +0   187   146   145 
 Tv 10.70400010.704000 +1   0.000000 +0    0.000000 +0     1   217     2 
 Tv 10.70400010.704000 +0   0.000000 +0    0.000000 +0     1   218     2 
 Tv 10.70400010.704000 +0   0.000000 +0    0.000000 +0     1   219     2 
 
2 19 1.914854   3
2 19 1.632993  13
2 19 6.928203 220
  2  1    4    6    7    8    9   10   11   12   14   15
   2  1   16   17   18   19   20   21   22   23   24   26
   2  1   27   28   29   30   31   32   33   34   35   36
   2  1   37   38   39   40   41   42   43   44   45   46
   2  1   47   48   49   50   51   52   53   54   55   56
   2  1   57   58   59   60   61   62   63   64   65   66
   2  1   67   68   69   70   71   72   74   75   76   77
   2  1   78   79   80   81   82   83   84   85   86   87
   2  1   88   89   90   91   92   93   94   95   96   97
   2  1   98   99  100  101  102  103  104  105  106  107
   2  1  108  109  110  111  112  113  114  115  116  117
   2  1  118  119  120  121  122  123  124  125  126  127
   2  1  128  129  130  131  132  133  134  135  136  137
   2  1  138  139  140  141  142  143  144  146  147  148
   2  1  149  150  151  152  153  154  155  156  157  158
   2  1  159  160  161  162  163  164  165  166  167  168
   2  1  169  170  171  172  173  174  175  176  177  178
   2  1  179  180  181  182  183  184  185  186  187  188
   2  1  189  190  191  192  193  194  195  196  197  198
   2  1  199  200  201  202  203  204  205  206  207  208
   2  1  209  210  211  212  213  214  215  216
   3  1    5
   3  2    4    5    6
   4  3    5    6    8   10   12   15   16   17   18   20
   4  3   21   22   23   24   27   28   29   30   32   33
   4  3   34   35   36   38   40   42   43   44   46   48
   4  3   50   52   54   56   58   60   62   64   66   68
   4  3   70   72   75   76   77   78   80   81   82   83
   4  3   84   85   86   87   88   89   90   92   93   94
   4  3   95   96   97   98   99  100  101  102  104  105
   4  3  106  107  108  110  111  112  113  114  116  117
   4  3  118  119  120  122  123  124  125  126  127  128
   4  3  129  130  131  132  134  135  136  137  138  139
   4  3  140  141  142  143  144  147  148  149  150  152
   4  3  153  154  155  156  157  158  159  160  161  162
   4  3  164  165  166  167  168  169  170  171  172  173
   4  3  174  176  177  178  179  180  182  183  184  185
   4  3  186  188  189  190  191  192  194  195  196  197
   4  3  198  199  200  201  202  203  204  206  207  208
   4  3  209  210  211  212  213  214  215  216
   7  2    8    9   10   11   12   15   16   17   18   19
   7  2   20   21   22   23   24   27   28   29   30   31
   7  2   32   33   34   35   36   37   38   39   40   41
   7  2   42   43   44   45   46   47   48   49   50   51
   7  2   52   53   54   55   56   57   58   59   60   61
   7  2   62   63   64   65   66   67   68   69   70   71
   7  2   72   75   76   77   78   79   80   81   82   83
   7  2   84   85   86   87   88   89   90   91   92   93
   7  2   94   95   96   97   98   99  100  101  102  103
   7  2  104  105  106  107  108  109  110  111  112  113
   7  2  114  115  116  117  118  119  120  121  122  123
   7  2  124  125  126  127  128  129  130  131  132  133
   7  2  134  135  136  137  138  139  140  141  142  143
   7  2  144  147  148  149  150  151  152  153  154  155
   7  2  156  157  158  159  160  161  162  163  164  165
   7  2  166  167  168  169  170  171  172  173  174  175
   7  2  176  177  178  179  180  181  182  183  184  185
   7  2  186  187  188  189  190  191  192  193  194  195
   7  2  196  197  198  199  200  201  202  203  204  205
   7  2  206  207  208  209  210  211  212  213  214  215
   7  2  216
   7  3   45   47   51   53   57   59   63   65   69   71
   7 14   37   79   91  103  121  133  151  163  175  193
   7 14  205
   9  3   11
   9 14   39   41
  13  1   25   73  145  217  218  219
  13  2   14   25   26   73   74  145  146  217  218  219
  13  3   19   25   31   73  115  145  187  218  219
  13 14  217
  14  3   26   74  146
  49  3   61
  55  3   67  109  181
 220  1  221  222
 

All distances are related by symmetry to the C1-C2 distance, here 1.545 Å.  This means that the three unique distances, 2.958427, 2.522957, and 10.704000 Å, are related to the C1-C2 distance.  If these three distances are defined using symmetry functions within MOPAC, then their optimization flags can also be set to zero.  This can be done using internal coordinate symmetry-function 19.  The resulting data set is shown in Figure 2.  Note the three function 19 definitions at the start of the symmetry data.  Within the automatically-generated symmetry data there are many references to symmetry functions 2 and 3 - these define angles and dihedral angles to be the same - but none of these angles are allowed to vary.  These symmetry functions can be deleted or left in place; their presence or absence will not affect the calculation.

Diamond.mop can the be run using MOPAC to generate a BRZ file, Diamond.br. For convenience, this file should be renamed "Diamond for electronics.br"  This prevents it accidentally getting overwritten in the event that Diamond.mop was re-run. 

Electronic Band Structure

Figure E2: Diamond Electronic Band Structure
Example of a "Walk" in k-space, from Excel
 
 

The Brillouin Zone for Diamond is a truncated octahedron. This structure has a high symmetry, so for ease of study, only the irreducible wedge need be examined.  This wedge is defined by the six vertices (0,0,0), X (0.5,0,0), L (0.25,0.25,0.25),
Point Location    Description Symmetry*
Γ    (0,0,0)    Center of BZ Oh
X    (1/2,0,0)    Center of square face ~D4h
L    (1/4,1/4.1/4)    Center of hexagonal face ~(D3h and S6)
K    (3/8,3/8,0)    Center of edge of hexagonal face C2v
W    (1/2,1/4,0)    Corner of a square face ~S4
U    (1/2,1/8,1/8)    Middle of edge of square face C2v
*: "~" indicates no equivalent point-group, the point-group(s)
 shown indicate some of the properties of the group.

Band-structure as a "Walk"

If "Diamond for electronics.brz" is opened using BZ, various walks in k-space can be generated.  One of the more informative walks is Γ-X-W-L-Γ-K-W-U-X, shown in Figure E1.  This figure, generated by BZ, is of some use (as shown here), but not easy to edit.  When BZ is run, in addition to the simple graphic, it also produces a file that ends in ".txt" that contains all the information that was used in making the graphic.  This information can then be using in Excel to make a chart that can easily be edited to add axes, text boxes, etc.  An example is shown in Figure E2.

Band-structure as a surface

BZ can generate surfaces, these are square cross-sections through k-space.  These cross-sections are defined by a point (three numbers) marking the center of the surface, a vector perpendicular to the surface (three numbers), and a distance (one number) from the point to the edge of the square.  A single graphic can display only one energy band at a time.  

An example of Band 1 is given in Figure E3, and of Band 2 in E4. As with a walk, these graphics, generated by the program BZ, are simple to make and to understand, but not easy to edit.

If the associated .txt file is used in Excel, other styles of display become available.  Two examples of Band 1 at Γ are shown in Figures E5 and E6.  In addition, as with the walks, these graphics can be annotated easily.

Symmetry of individual points in the irreducible wedge
Figure E3 Diamond Electronic Band Structure
(001) Surface 1 at Γ from BZ
Figure E5 Diamond Electronic Band Structure
(001) Surface 1 at Γ from Excel
Figure E4 Diamond Electronic Band Structure
(111) Surface 2 at L (from BZ)
Figure E6 Diamond Electronic Band Structure
3D (001) Surface 1 at G from Excel

When a file containing the symmetry operations of the lattice are supplied, the symmetry properties of the eigenfunctions for individual points can be examined.  An examination of the symmetry operations of the diamond lattice reveals that they map exactly onto the operations of the point-group Oh.  Some points in the irreducible wedge have a symmetry exactly isomorphous with a point-group symmetry.  For example the center of the Brillouin zone Γ has the same symmetry as point-group Oh.  Other points, X is a good example, are unrelated to any point-group, although in the case of X, there is a superficial similarity to D4h.   Other points, L being a good example, have a sub-group symmetry that maps to a point-group, here C3, but also has operations such as S6, C2 and I that look as if they belong to D6h, however the operation D6 is, of course, not present in the Oh point-group. 

Finally, some points, although superficially different, are in fact related by a space-group symmetry operation, therefore the eigenvalues of the two points are the same and their associated eigenvectors have the same symmetry. This is seen in K and U, where, in addition to having the same little-group symmetry, that symmetry maps to the point-group C2v.

Phonon Band Structure

The vibrational or phonon Brillouin zone is constructed from the mass-weighted Hessian matrix for the Large Unit Cell. Within BZ, the behavior of the electronic and phonon energy matrices are almost the same, the only difference being the sizes of the matrices; the size of the electronic energy matrix is equal to the number of atomic orbitals, but in the phonon energy matrix the size is three times the number of atoms.  Because of this similarity, working with the phonon Brillouin zone is exactly the same as working with the electronic Brillouin zone. 

One dramatic difference between the appearances of the electronic and phonon Brillouin zones is the presence in the phonon structure of sharp points at Γ where the equivalent structure in the electronic Brillouin zone is of a local maximum or minimum.  This is most dramatically seen in the Excel figures P3 and P4
Figure P1: Diamond Phonon Band Structure
Walk in k-space, from Program BZ
Figure P2: Diamond Phonon Band
Structure (001) Surface 1 at Γ from BZ
Figure P3: Diamond Phonon Band
Structure (001) Surface 1 at Γ from Excel
Figure P4: Close-up of Diamond Phonon
Band Structure (001) Surface 1 at Γ from BZ