Eliminated another initialization error and tweaked rdf-adf example

The first ADF value for a bin is calculated from the histogram count by

dividing by the total number of triples satisfying the criteria,

so that the integral of the ADF w.r.t. angle is one i.e. the ADF

so that the integral of the ADF w.r.t. angle is 1, i.e. the ADF

is a probability density function.

The second ADF value is reported as a cumulative sum of

whether the bins are of uniform angular size from zero

to 180 ({degree}), zero to Pi ({radian}), or the 

cosine of the angle uniform in the range \[-1,1\] ({cosine}).

Regardless of which is chosen, the first ADF will be normalized

so that the integral w.r.t. the ordinate is one.

{cosine} has the advantage of eliminating the {acos()} function

call, which speeds up the compute by 2-3x, and it is also preferred

on physical grounds, because the for uniformly distributed particles 

in 3D, the angular probability density w.r.t dtheta is

sin(theta)/2, while for d(cos(theta)), it is 1/2,  

Regardless of which ordinate is chosen, the first column of ADF 

values is normalized w.r.t. the range of that ordinate, so that 

the integral is 1.

The simplest way to output the results of the compute adf calculation

to a file is to use the "fix ave/time"_fix_ave_time.html command, for

subsequent pair of columns gives the first and second kinds of ADF

for a specific set of ({itypeN},{jtypeN},{ktypeN}). The values 

in the first ADF column are normalized numbers >= 0.0,

whose integral w.r.t. the ordinate is one,

whose integral w.r.t. the ordinate is 1,

i.e. the first ADF is a normalized probability distribution. 

The values in the second ADF column are also numbers >= 0.0.

They are the cumulative density distribution of angles per atom.

# Liquid water RDFs and ADFs (~12 O-O-O/atom, ~1 O-H...O/atom)

units		real	

atom_style	full

compute gofr all rdf 100  1 1  2 2  1 2  # O-O, H-H, O-H

compute gofa all adf  45  1 1 1  2.0 3.5  2.0 3.5 & 

                          1 2 1  0.0 1.2  1.5 2.5

                          2 1 1  0.0 1.2  1.5 2.5   # O-O-O, O-H...O

fix rdf all ave/time 100 10 1000 c_gofr[*] file spce-rdf.dat mode vector

fix adf all ave/time 10 100 1000 c_gofa[*] file spce-adf.dat mode vector

# TimeStep Number-of-rows

# Row c_gofa[1] c_gofa[2] c_gofa[3] c_gofa[4] c_gofa[5]

1000 45

1 2 0 0 0.570997 4.0364e-06

2 6 0 0 1.34922 1.33246e-05

3 10 0 0 0.844388 1.94267e-05

4 14 0 0 0.476996 2.34028e-05

5 18 0 0 0.161187 2.46115e-05

6 22 0 0 0.0477465 2.47703e-05

7 26 0 0 0.00511569 2.48353e-05

8 30 0 0 0 2.48353e-05

9 34 0 0 0 2.48353e-05

10 38 0 0 0.0179049 2.497e-05

11 42 0.00139303 0.00112 0 2.497e-05

12 46 0.0970866 0.0791067 0 2.497e-05

13 50 0.421047 0.417227 0 2.497e-05

14 54 0.503109 0.821333 0 2.497e-05

15 58 0.403317 1.14533 0 2.497e-05

16 62 0.384523 1.45431 0 2.497e-05

17 66 0.425895 1.79643 0 2.497e-05

18 70 0.494575 2.1937 0 2.497e-05

19 74 0.547429 2.63327 0.00716197 2.50218e-05

20 78 0.584521 3.10253 0.0119366 2.51428e-05

21 82 0.612285 3.59409 0.0966866 2.56506e-05

22 86 0.643119 4.1105 0.0868815 2.64003e-05

23 90 0.669958 4.64829 0.245468 2.77959e-05

24 94 0.685175 5.19837 0.318537 3.01911e-05

25 98 0.68985 5.75207 0.608171 3.48379e-05

26 102 0.681411 6.29909 0.825076 4.16049e-05

27 106 0.660197 6.82903 1.2987 5.06221e-05

28 110 0.644879 7.34685 1.58753 6.12867e-05

29 114 0.613281 7.83923 1.1185 6.98041e-05

30 118 0.572272 8.29867 0.961282 7.65158e-05

31 122 0.534702 8.72795 0.546058 8.08946e-05

32 126 0.496697 9.12669 0.64867 8.51981e-05

33 130 0.453138 9.49045 0.518561 8.86029e-05

34 134 0.41004 9.81964 0.561575 9.23251e-05

35 138 0.36701 10.1143 0.355285 9.51805e-05

36 142 0.328717 10.3782 0.297648 9.74977e-05

37 146 0.294236 10.6145 0.0749449 9.81639e-05

38 150 0.261134 10.8242 0.0856879 9.87562e-05

39 154 0.223727 11.0039 0.0537574 9.9216e-05

40 158 0.19037 11.1567 0.0827037 9.95624e-05

41 162 0.156173 11.2822 0.0149208 9.97358e-05

42 166 0.119999 11.3785 0.00895247 9.98317e-05

43 170 0.0853136 11.447 0.00596831 9.98799e-05

44 174 0.0502058 11.4873 0 9.98799e-05

45 178 0.0171616 11.5011 0 9.98799e-05

1 2 0 0 0 0

2 6 0 0 0 0

3 10 0 0 0 0

4 14 0 0 0 0

5 18 0 0 0 0

6 22 0 0 0 0

7 26 0 0 0 0

8 30 0 0 0 0

9 34 0 0 0 0

10 38 0 0 0 0

11 42 0.00152868 0.00123333 0 0

12 46 0.096401 0.07888 0 0

13 50 0.418453 0.41588 0 0

14 54 0.506477 0.82378 0 0

15 58 0.408265 1.15264 0 0

16 62 0.39048 1.46717 0 0

17 66 0.43199 1.81511 0 0

18 70 0.495001 2.21377 0 0

19 74 0.549314 2.65613 0 0

20 78 0.586929 3.12873 0.000290833 2e-05

21 82 0.619347 3.62739 0.00150483 0.000123333

22 86 0.644793 4.14651 0.00573476 0.000516667

23 90 0.664615 4.68151 0.00996515 0.0012

24 94 0.680251 5.22921 0.0147164 0.00221

25 98 0.679622 5.77635 0.0195919 0.00355333

26 102 0.675331 6.32011 0.0279862 0.00547333

27 106 0.659266 6.85083 0.0431972 0.00843667

28 110 0.640721 7.36672 0.0679453 0.0130967

29 114 0.609075 7.85709 0.0903345 0.0192933

30 118 0.578386 8.32278 0.12826 0.0280933

31 122 0.535658 8.75409 0.169594 0.0397267

32 126 0.494864 9.15256 0.208251 0.05401

33 130 0.4498 9.51475 0.275366 0.0728967

34 134 0.40817 9.8434 0.354045 0.09718

35 138 0.362856 10.1356 0.459385 0.12869

36 142 0.328968 10.4005 0.606056 0.170257

37 146 0.291793 10.6354 0.796038 0.224857

38 150 0.259831 10.8446 1.04618 0.296613

39 154 0.2281 11.0283 1.34162 0.38863

40 158 0.191749 11.1827 1.62681 0.500213

41 162 0.15663 11.3088 1.86309 0.627997

42 166 0.121865 11.407 1.89445 0.757927

43 170 0.0881334 11.4779 1.67713 0.872957

44 174 0.0523029 11.52 1.17909 0.953827

45 178 0.0169789 11.5337 0.417313 0.98245

  // zero the central atom counts

  for (i = 0; i < ntriples; i++)

    iatomcount[m] = 0;

    iatomcount[i] = 0;

  // tally the ADFs

  // all three atoms i, j, and k must be in fix group