Interatomic Distances and Angles

The first thing anybody asks of a solved structure: how far apart are the atoms, and at what angles. Give a cell, a space group and the asymmetric unit — the page expands them by the symmetry, finds every neighbour inside a radius you choose, and names each one by the operation that produced it.

You supply: A unit cell, a space group and one atom per line of the asymmetric unit. Any constant or coordinate may carry its standard uncertainty in brackets, as a CIF writes it, and those are what the uncertainties below are propagated from.
Reading it: Uncertainties are propagated as if the parameters were uncorrelated, because a CIF does not carry the refinement’s variance–covariance matrix. That is not what a refinement program prints: for two atoms of one rigid group the correlation is usually positive, which makes the figure here larger. It is not an upper limit either.

Worked examples: Quartz, with uncertainties supplied · NaCl · Cu · α-Fe · CsCl · ZnS · quartz · cristobalite · berlinite · rutile · aragonite · ZrO₂ · albite

Unit cell

Å

A constant may carry its standard uncertainty in brackets, exactly as a CIF writes it — 4.9137(2). That is what the uncertainties below are built from.

Space group

Number or symbol

Conditions belong to a setting, not to a space group number. Pnma, Pbnm and Pmcn are one space group with its axes labelled three ways, and the three tables differ: the mirror that empties hk0 in one of them empties 0kl in another. Every setting in the International Tables is here with its own operations, so P2₁/n and P2₁/a answer for themselves.

Atoms in the asymmetric unit

Label, x, y, z, and optionally occupancy and B

One atom per line, for example Si 0.4697(1) 0 1/6. This is the same notation the structure factor page reads, so a list works on either — and a coordinate may carry its uncertainty, which only this page uses. A coordinate written as a fraction is one the symmetry fixes, and carries no uncertainty by definition.

How far to look

Å

Every neighbour inside this radius is listed. The angle table grows as the square of the neighbour count, so a small increase here is a large one there.

These are distances and angles, not bonds. Every neighbour inside the radius is listed; whether two atoms are bonded is a chemical judgement this page does not make. Contacts are found through every symmetry operation of the space group and across cell boundaries, so the list is the full environment and not only what lies inside one cell.

The cell, filled

axesdrag to rotate · scroll to zoomLook along

Zr Z 40 · r_cov 1.75 Å O Z 8 · r_cov 0.66 Å

Hover an atom to name it. Click one to pin it and mark every copy of that site.

Every atom of the cell is drawn, including the symmetry-equivalent ones. An atom lying on a face, an edge or a corner belongs to each cell it touches and is drawn in all of them, so counting the spheres overcounts the contents of one cell. Colours are the standard CPK ones, and spheres are drawn at half the covalent radius (Cordero et al., 2008), reduced further only if that would make two of them run into each other — at full size two bonded atoms touch by definition. These are not ionic radii: in a salt the cation is drawn larger than the anion, which is the opposite of the ionic picture. No bonds are drawn, because which contacts are bonds is a chemical judgement this page does not make.

The symmetry the atoms obey

The atoms obey exactly the 4 operations of the space group given, and no others that this cell would permit.

Distances

Interatomic distances, in Å.
atom	neighbour	distance	symmetry of the neighbour
Zr	O1	2.0531	`x,y,z`
	O1	2.0592	`-x,y-1/2,-z+1/2`
	O2	2.1531	`x,y-1,z`
	O1	2.1652	`x,-y+1/2,z-1/2`
	O2	2.1915	`-x+1,y-1/2,-z+1/2`
	O2	2.2220	`x,-y+1/2,z-1/2`
	O2	2.2873	`-x+1,-y+1,-z+1`
O1	Zr	2.0531	`x,y,z`
	Zr	2.0592	`-x,y+1/2,-z+1/2`
	Zr	2.1652	`x,-y+1/2,z+1/2`
	O2	2.5834	`-x+1,-y+1,-z+1`
	O1	2.5918	`-x,-y+1,-z+1`
	O1	2.8054	`x,-y+1/2,z-1/2`
	O1	2.8054	`x,-y+1/2,z+1/2`
	O1	2.8318	`-x,y-1/2,-z+1/2`
	O1	2.8318	`-x,y+1/2,-z+1/2`
	O2	2.9203	`x,y,z`
	O2	2.9319	`-x,y-1/2,-z+1/2`
	O2	2.9880	`-x,-y+1,-z+1`
O2	Zr	2.1531	`x,y+1,z`
	Zr	2.1915	`-x+1,y+1/2,-z+1/2`
	Zr	2.2220	`x,-y+1/2,z+1/2`
	Zr	2.2873	`-x+1,-y+1,-z+1`
	O1	2.5834	`-x+1,-y+1,-z+1`
	O2	2.6244	`-x+1,-y+2,-z+1`
	O2	2.6591	`x,-y+3/2,z-1/2`
	O2	2.6591	`x,-y+3/2,z+1/2`
	O2	2.7239	`-x+1,-y+1,-z+1`
	O1	2.9203	`x,y,z`
	O1	2.9319	`-x,y+1/2,-z+1/2`
	O1	2.9880	`-x,-y+1,-z+1`

Angles

Angles at each atom, in degrees.
at	between	angle
Zr	O1 and O1	87.041
	O1 and O2	117.458
	O1 and O1	83.325
	O1 and O2	100.994
	O1 and O2	166.715
	O1 and O2	72.828
	O1 and O2	88.188
	O1 and O1	75.644
	O1 and O2	146.101
	O1 and O2	88.441
	O1 and O2	140.242
	O2 and O1	153.277
	O2 and O2	115.713
	O2 and O2	74.841
	O2 and O2	72.388
	O1 and O2	72.734
	O1 and O2	83.453
	O1 and O2	132.875
	O2 and O2	76.218
	O2 and O2	72.807
	O2 and O2	117.776
O1	Zr and Zr	145.984
	Zr and Zr	109.078
	Zr and O2	57.769
	Zr and O1	158.685
	Zr and O1	50.048
	Zr and O1	99.468
	Zr and O1	46.568
	Zr and O1	133.605
	Zr and O2	107.002
	Zr and O2	99.172
	Zr and O2	121.485
	Zr and Zr	104.356
	Zr and O2	155.771
	Zr and O1	54.030
	Zr and O1	100.249
	Zr and O1	98.835
	Zr and O1	105.430
	Zr and O1	46.391
	Zr and O2	99.389
	Zr and O2	47.225
	Zr and O2	48.017
	Zr and O2	54.103
	Zr and O1	50.326
	Zr and O1	151.130
	Zr and O1	46.627
	Zr and O1	129.858
	Zr and O1	95.510
	Zr and O2	49.105
	Zr and O2	146.108
	Zr and O2	94.938
	O2 and O1	103.573
	O2 and O1	98.390
	O2 and O1	74.762
	O2 and O1	97.936
	O2 and O1	118.371
	O2 and O2	58.951
	O2 and O2	156.919
	O2 and O2	135.044
	O1 and O1	150.802
	O1 and O1	63.148
	O1 and O1	138.312
	O1 and O1	62.110
	O1 and O2	65.335
	O1 and O2	98.853
	O1 and O2	62.643
	O1 and O1	142.776
	O1 and O1	54.742
	O1 and O1	90.579
	O1 and O2	112.183
	O1 and O2	62.728
	O1 and O2	112.841
	O1 and O1	89.421
	O1 and O1	125.258
	O1 and O2	95.724
	O1 and O2	111.488
	O1 and O2	60.708
	O1 and O1	133.908
	O1 and O2	153.560
	O1 and O2	60.856
	O1 and O2	76.859
	O1 and O2	61.265
	O1 and O2	77.172
	O1 and O2	93.676
	O2 and O2	138.334
	O2 and O2	127.978
	O2 and O2	53.373
O2	Zr and Zr	105.698
	Zr and Zr	134.458
	Zr and Zr	107.612
	Zr and O1	129.371
	Zr and O2	56.170
	Zr and O2	53.757
	Zr and O2	128.157
	Zr and O2	142.092
	Zr and O1	100.066
	Zr and O1	44.587
	Zr and O1	92.679
	Zr and Zr	103.782
	Zr and Zr	100.199
	Zr and O1	53.163
	Zr and O2	112.105
	Zr and O2	55.257
	Zr and O2	123.392
	Zr and O2	52.395
	Zr and O1	87.776
	Zr and O1	114.252
	Zr and O1	138.517
	Zr and Zr	100.334
	Zr and O1	96.115
	Zr and O2	136.967
	Zr and O2	126.669
	Zr and O2	51.402
	Zr and O2	51.387
	Zr and O1	47.442
	Zr and O1	91.547
	Zr and O1	43.542
	Zr and O1	49.404
	Zr and O2	51.442
	Zr and O2	129.376
	Zr and O2	51.936
	Zr and O2	106.785
	Zr and O1	147.675
	Zr and O1	139.652
	Zr and O1	109.297
	O1 and O2	87.323
	O1 and O2	102.245
	O1 and O2	77.342
	O1 and O2	66.706
	O1 and O1	121.049
	O1 and O1	166.582
	O1 and O1	135.044
	O2 and O2	93.891
	O2 and O2	88.254
	O2 and O2	154.028
	O2 and O1	151.627
	O2 and O1	94.584
	O2 and O1	109.048
	O2 and O2	177.799
	O2 and O2	91.641
	O2 and O1	80.740
	O2 and O1	64.395
	O2 and O1	117.374
	O2 and O2	86.210
	O2 and O1	97.596
	O2 and O1	115.957
	O2 and O1	62.232
	O2 and O1	54.344
	O2 and O1	110.639
	O2 and O1	90.655
	O1 and O1	57.879
	O1 and O1	52.022
	O1 and O1	56.564

About the uncertainties

No uncertainties are quoted, because none were given. Nothing in the cell or the coordinates above carries a bracket, so there is nothing to propagate. An absent uncertainty is not a zero one.

Write a constant as 4.9137(2) or a coordinate as 0.4697(1) and every distance and angle below will carry its own, split into the part that comes from the cell and the part that comes from the coordinates.

Contacts were searched out to 3.2 Å. A neighbour is named by the operation that produces it from the atom in the list above, written out in full rather than as a numbered code.

Where this comes from

Computer derivation of the symmetry elements implied in a structure description
Y. Le Page, J. Appl. Cryst. 1987, 20, 264–269 · doi:10.1107/S0021889887086710
The method behind the additional-symmetry panel: search the metric symmetry of the lattice, then test each candidate against the atoms.
MISSYM 1.1 — a flexible new release
Y. Le Page, J. Appl. Cryst. 1988, 21, 983–984 · doi:10.1107/S0021889888007022
The follow-up, and the source of the tolerance handling: how far an atom may sit from its image and still count.
Covalent radii revisited
B. Cordero et al., Dalton Trans. 2008, 2832 · doi:10.1039/b801115j
The radii the distance table draws its spheres from. It publishes standard deviations with them, which is why they can carry a stated tolerance rather than a hidden one.