Use of spherical coordinates is nicely applicable to symmetrical systems with molecules positioned on nested concentric spherical surfaces, such as the icosahedral water cluster. In place of the orthogonal x, y and z coordinates, r, θ (theta), and φ (phi) are used. The symbol r defines the distance from a central origin while θ and φ combine to define the direction. φ is the polar angle (from the z axis, latitude) and varies from 0° to 180°. θ is the azimuthal angle (in the xy plane, longitude) and varies between 0° to 360°.*

**Figure 1.** Spherical coordinates of the water oxygen atoms in icosahedral
water clusters (ES). Radial
values (r) are given on another page.

**Figure 2.** Spherical coordinates
of the water oxygen atoms in a super
cluster of 13 icosahedral water clusters (ES). Radial values (r) are given on
another page.

**Figure 3.** The hydrogen
bonding angles of the water molecules in icosahedral water
clusters (ES), given as spherical
coordinates about their oxygen atoms.

* Conversion between (x, y, z) and (r, θ, φ) coordinate systems:

r = √(x^{2}+y^{2}+z^{2}), θ= arctan(y/x ), φ = arccos(z/r)

if x is positive
and y is positive, 0° ≤ θ ≤ 90°

if x is negative
and y is positive, 90° ≤ θ ≤ 180°

if x is negative
and y is negative, 180° ≤ θ ≤ 270°

if x is positive
and y is negative, 270° ≤ θ ≤ 360°

x = r.cos(θ).sin(φ),
y = r.sin(θ).sin(φ),
z = r.cos(φ)

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This page was established in 2003 and last updated by Martin Chaplin on 21 July, 2015

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