# Meaning of atomic positional parameters

I am trying to model some materials with DFT, so I intended to start out with the structure. However, some texts that present XRD data, use a term called atomic positional parameter. I can't seem to figure out what this means. Here's an image from a paper:

Can someone help breakdown what format the coordinates are represented in ?

• +1. Is this not just the xyz file? Meaning that these are the (x,y,z) coordinates of the corresponding atoms in the molecule, most likely in Angstrom units but possibly in a.u. (Bohr radii)? The numbers in parentheses are likely the uncertainties in the last digits shown. Please provide in your question, a reference to the paper where this screenshot has been acquired from. – Nike Dattani Jun 24 '20 at 18:14
• @NikeDattani I added the link to the paper. The problem is, when these coordinates are fractions, you know that they can only be the fractional coordinates, which are easy to work with. But when they list coordinates like this (some other XRD papers do this as well), it becomes hard for me to decrypt.. Also, I think you're definitely right about the uncertainty part .. I've seen that plenty.. listing errors in closed parantheses.. But what I dont understand are the numbers themselves.. – livars98 Jun 24 '20 at 18:51
• What exactly do you mean by coordinates being fractions? They are never fractions like 1/3 or 1/2. They are usually numbers like 1.203 Angstroms. In this case the table's caption says that the coordinates are in 10^4 Angstroms. So when it says for C(6) a value of 1383(3), that means (1.383 +/- 3) Angstroms. You are also given the geometry in a clearer way in Table 2. If you can let me know exactly what you're looking for, I can turn my comments into an answer. – Nike Dattani Jun 24 '20 at 19:33
• What I meant by that was, say for some crystals that crystallize in cubic, it's common to find fractional coordinates like 0.25, 0.75 etc. Your answer makes total sense, please turn it into an official answer. Thanks. – livars98 Jun 24 '20 at 19:53

Above the table that you provided in the screenshot, is a caption that says what these numbers mean:

I don't blame you for coming here to ask this, because the notation ($$\times 10^4$$) can be confusing for someone that's not used to it (and I don't think it's taught in school, generally). But what it means is that numbers like 1383, which is the $$x$$-coordinate (along the $$a$$ axis of the unit cell) given for the 6th carbon atom, C(6), actually mean 0.1383. The number 1383(3) means (0.1383 +/- 0.0003). Similarly, the $$y$$ and $$z$$ coordinates are actually fractional lengths along the $$b$$ and $$c$$ axes of the crystal unit cell.

I also don't blame you if you're confused by what these numbers mean, because the table's caption says "atomic positional parameters" which seems (at least to me) as an unconventional way of saying "atomic coordinates". At least a different part of the exact same paper describes this table as listing atomic coordinates though:

Table 2 also has the "internal coordinates" that you would use if your modeling program requires input in ZMAT format.

In summary: The table in your question corresponds to the fractional coordinates for the atoms with respect to the unit cell (although it also has uncertainties on each coordinate value). You mentioned in the comments that you're used to seeing exact fractions like 1/4 and 3/4, but in this case the molecule is not so simple like textbook lattice structures such as NaCl. So, when they are measured experimentally, they won't correspond to simple fractions and will have an associated uncertainty.

In the above case the fraction would be approximately 1383/10000, but remember that there's an uncertainty, so in fractional form the $$x$$-coordinate would actually be:

$$\frac{1383 \pm 3}{10000}.$$

• My friends Tyberius and Cody Aladaz have told me that the coordinates might not be in Angstroms, but instead they could be "fractional coordinates" like you say: meaning that they are a fraction of the unit cell. The rest of my answer would be correct, except for the unit of length. They might offer an answer or edit. – Nike Dattani Jun 25 '20 at 0:13
• I have made some small edits. The substance of the answer is mostly the same, but I clarified units and that these values are tied to the unit cell dimensions. – Tyberius Jun 25 '20 at 1:23
• @Tyberius Teamwork makes the dream work – Cody Aldaz Jun 25 '20 at 3:06

These are in fractional coordinates, which are coordinates with respect to the lattice axis commonly referred to as the a, b, and c axis. More specifically, a, b and c are the lengths of the cell edges, and the axis have angles $$\alpha$$, $$\beta$$ and $$\gamma$$ between them.

Many matter modeling software can use either fractional coordinates or Cartesian coordinates but the conversion between them is straightforward. .

Here is a snippet adapted from https://pymolwiki.org/index.php/Cart_to_frac to convert Cartesians to fractional coordinates

# (scaled) volume of the cell
v = sqrt(1 -cos(alpha)*cos(alpha) - cos(beta)*cos(beta) - cos(gamma)*cos(gamma) + 2*cos(alpha)*cos(beta)*cos(gamma))

tmat = numpy.matrix( [
[ 1.0 / a, -cos(gamma)/(a*sin(gamma)), (cos(alpha)*cos(gamma)-cos(beta)) / (a*v*sin(gamma))  ],
[ 0.0,     1.0 / (b*sin(gamma)),         (cos(beta) *cos(gamma)-cos(alpha))/ (b*v*sin(gamma))  ],
[ 0.0,     0.0,                        sin(gamma) / (c*v)                                    ] ])

frac_coord = cart_coord * tmat.T    # CRA renamed for MMSE



Here is a snippet adapted from I https://gist.github.com/Bismarrck/a68da01f19b39320f78a to convert fractional coordinates to Cartesians:

cosa = np.cos(alpha)
sina = np.sin(alpha)
cosb = np.cos(beta)
sinb = np.sin(beta)
cosg = np.cos(gamma)
sing = np.sin(gamma)
volume = 1.0 - cosa**2.0 - cosb**2.0 - cosg**2.0 + 2.0 * cosa * cosb * cosg
volume = np.sqrt(volume)
r = np.zeros((3, 3))
r[0, 0] = a
r[0, 1] = b * cosg
r[0, 2] = c * cosb
r[1, 1] = b * sing
r[1, 2] = c * (cosa - cosb * cosg) / sing
r[2, 2] = c * volume / sing

cart_coord = np.dot(r,frac_coord.T)   # CRA added this for MMSE