Chemistry · Ch 2 — Quantum Mechanical Model of Atom

Shapes of atomic orbitals

2.5.1

Shapes of atomic orbitals

Solving the Schrödinger equation for a one-electron system like hydrogen, written in spherical polar coordinates $r$ (distance from nucleus), $\theta$ and $\varphi$ (the two angles fixing direction), factorises neatly into a product of three separate one-variable functions:

\Psi(r,\theta,\varphi)=R(r)\cdot f(\theta)\cdot g(\varphi)\qquad\text{— eq. (2.15)}

$R(r)$ , depending only on the distance from the nucleus, is called the radial wave function; $f(\theta)$ and $g(\varphi)$ together, depending only on direction, are called the angular wave functions. Because $\Psi$ itself is not physically meaningful but $|\Psi|^2$ is (as established in section 2.4.1), this factorisation lets you study two separate questions independently: how does the probability of finding the electron vary with distance from the nucleus (the radial distribution), and how does it vary with direction from the nucleus (the angular distribution)?

Radial distribution function. Take the simplest case: the single electron of a hydrogen atom in its ground state, where $n=1$ and $l=0$ (a 1s orbital). Plotting $R(r)^2$ (the squared radial wavefunction, ignoring direction) against $r$ shows this quantity is largest exactly at the nucleus ( $r=0$ ) and falls off smoothly as $r$ increases (Figure 2.3). Taken at face value that would suggest the electron is "most likely" to be found sitting right on top of the nucleus - but that overlooks an important geometric fact: the raw function $R(r)^2$ is only a probability density, and the actual probability of finding the electron also depends on how much three-dimensional space is available at each distance $r$ .

To account for this properly, consider the thin spherical shell of volume $dV$ trapped between two spheres of radius $r$ and $r+dr$ (Figure 2.4). Since the volume of a sphere is $V=\tfrac{4}{3}\pi r^3$ ,

\frac{dV}{dr}=4\pi r^2\qquad\Rightarrow\qquad dV=4\pi r^2\,dr

so the actual probability of finding the electron somewhere within that thin shell is

\Psi^2\,dV=4\pi r^2\,\Psi^2\,dr\qquad\text{— eq. (2.16)}

Plotting this shell-weighted quantity, $4\pi r^2\cdot R(r)^2$ , against $r$ for the 1s orbital gives a very different-looking curve (Figure 2.5): it starts at zero right at the nucleus (because the shell volume itself shrinks to zero as $r\to0$ , even though $R(r)^2$ was largest there), rises to a single peak at $r=0.52$ Å - matching the Bohr radius exactly - and then falls back toward zero at larger distances. This peak is the genuine, most probable distance of the electron from the nucleus, even though there remains some (smaller) probability of finding it at other distances too.

Radial nodes. For orbitals beyond 1s, the shape of this radial-distribution curve gets more interesting. For a 2s orbital, as $r$ increases from the nucleus, the probability density first rises, reaches a small local maximum, then drops sharply all the way to zero before rising again to a second, larger maximum, and finally decaying to zero at large $r$ (Figure 2.6). The particular radius at which the curve touches zero in between its humps is called a nodal surface or radial node - literally, a spherical shell around the nucleus where the electron will essentially never be found. The general counting rule, illustrated by comparing 2s, 3s, 3p and 3d in Figure 2.6, is that an ns orbital has $(n-1)$ radial nodes: 2s has one radial node, 3s has two. More generally, across any orbital,

\text{number of radial nodes}=n-l-1

which correctly reduces to $n-1$ for any s orbital (where $l=0$ ), and predicts fewer nodes as $l$ grows for a fixed $n$ - exactly the pattern visible when comparing 3s (2 nodes), 3p (1 node) and 3d (0 nodes) in Figure 2.6.

Angular distribution function and orbital shapes. The other half of the story - which direction from the nucleus the electron is likely to be found in - is governed by the azimuthal quantum number $l$ of the orbital.

s orbitals ( $l=0$ ). For an s orbital, $m=0$ as well, and the angular functions work out to constants: $f(\theta)=1/\sqrt{2}$ and $g(\varphi)=1/\sqrt{2\pi}$ , so the overall angular distribution is the constant $1/(2\sqrt{\pi})$ - completely independent of both angles $\theta$ and $\varphi$ . In plain terms: an s orbital has exactly equal probability of finding the electron in any direction from the nucleus, which is precisely why its shape is a perfect sphere (Figure 2.7), regardless of how many radial nodes it has (1s has none, 2s has one concentric spherical node, 3s has two).
p orbitals ( $l=1$ ). Here $m=-1,0,+1$ , giving three distinct spatial orientations rather than one spherically symmetric shape. The angular functions themselves are more mathematically involved and are not worked out explicitly, but the resulting shape (Figure 2.8) is a "dumbbell": two lobes of electron density pointing in opposite directions along a single axis. The three $m$ values correspond to three orbitals - $p_x$ , $p_y$ and $p_z$ - whose lobes point along the x, y and z axes respectively, and each 2p orbital has exactly one nodal plane passing through the nucleus, perpendicular to its own axis (for instance the nodal plane of $p_x$ is the yz-plane, since that is the plane of zero density separating the two lobes). …

Figure 2.3Plot of $R(r)^2$ versus r for the 1s orbital of hydrogen

What this figure shows. A simple curve starting at its highest value when r = 0 (right at the nucleus) and falling off smoothly and monotonically to zero as r increases past about 15 Bohr radii, with no bumps or secondary peaks. It shows that the raw radial wavefunction squared, $R(r)^2$ , taken purely as a function of distance from the nucleus without accounting for how much space exists at each distance, is deceptively largest exactly at the nucleus itself …

Figure 2.4Volume element between two spheres of radii r and r + dr

What this figure shows. A cutaway sketch of two concentric spheres, an inner one of radius r and an outer one of radius r + dr, with the thin spherical shell of volume dV trapped between them shaded to highlight it. This is the geometric picture behind the formula $dV=4\pi r^2\,dr$ used immediately afterward: it shows visually why the amount of available space in a thin shell grows with $r^2$ even while the wavefunction itself may be shrinking, which is exactly why the true probability of finding the electron is not maximum at the nucleus once this g …

Figure 2.5Plot of $4\pi r^2\cdot R(r)^2$ versus r for the 1s orbital of hydrogen

What this figure shows. A curve that starts at zero at the nucleus (r = 0), rises to a single rounded peak at r = 0.52 Å, and then falls back toward zero at larger r. Unlike figure 2.3, this curve is zero exactly at the nucleus because it has been weighted by the growing shell volume $4\pi r^2$ , and its peak at 0.52 Å - matching the Bohr radius - is the genuine most-probable distance of the electron from the nucleus in the 1s orbital, even though the raw wavefunction itself was largest a …

Figure 2.6Radial distribution plots of $4\pi r^2\cdot R(r)^2$ versus r for the 2s, 3s, 3p and 3d orbitals of hydrogen

What this figure shows. Four separate curves, one per orbital, each plotted against r in units of the Bohr radius (out to about 25). The 2s curve rises to a small hump, drops all the way to zero at one particular radius (a node), then rises again to a second, larger and broader hump before finally decaying to zero. The 3s curve shows the same rise-fall-rise pattern but with two zero-crossings (two nodes) before its final broad hump. The 3p curve shows one node before its outermost hump, and the 3d curve shows a single smooth hump with no interior node at all. Together the four curves are the visual evidence for the radial-node counting rule stated in the text: an ns orbital has (n−1) radial nodes, and in general any orbital has (n−l−1) radial nodes, so 2s has 1, 3s has 2, …

Figure 2.7Shapes of 1s, 2s and 3s orbitals

What this figure shows. Three separate three-dimensional sketches, each drawn on x, y, z axes, showing the boundary surface within which the electron is most likely to be found. The 1s orbital is drawn as a single solid sphere centred on the nucleus with no internal structure. The 2s orbital is drawn as two concentric spherical surfaces - an inner small sphere and an outer larger shell - separated by a thin spherical gap labelled 'Node' where the electron probability drops to zero. The 3s orbital is drawn with three concentric spherical regions, separated by two labelled spherical nodal gaps. All three shapes are perfectly round in every direction, which is the picture behind the text's statement that the s orbital's angular part is completely independent of direction …

Figure 2.8Shapes of 2p orbitals (cartoon representation)

What this figure shows. Three separate dumbbell-shaped sketches on x, y, z axes, one for each of $p_x$ , $p_y$ and $p_z$ . Each drawing shows two balloon-like lobes of electron density pointing in opposite directions along one single axis - $p_x$ 's two lobes point along the x-axis, $p_y$ 's along the y-axis, and $p_z$ 's along the z-axis - with a flat plane of exactly zero electron density (labelled the nodal plane) passing through the nucleus perpendicular to that axis: the yz-plane for $p_x$ , the xz-plane for $p_y$ , and the xy-plane for $p_z$ . The three drawings side by side make clear that the three m values of the p subshell correspond to three physically distinct orientations i …

Figure 2.9Shapes of the five 3d orbitals

What this figure shows. Five separate sketches on x, y, z axes: $3d_{xy}$ shows four clover-leaf lobes lying in the xy-plane, pointing between the x and y axes, with two nodal planes (the xz-plane and the yz-plane); $3d_{yz}$ and $3d_{xz}$ show the same four-lobed clover shape but rotated into the yz-plane and xz-plane respectively, each again with its own pair of nodal planes; $3d_{x^2-y^2}$ shows four lobes lying along the x and y axes themselves (rather than between them), again with two nodal planes (the diagonal planes bisecting the x and y axes); and $3d_{z^2}$ looks visually different from the other four - a single dumbbell-shaped lobe along the z-axis plus a donut-shaped ring of density around its middle in the xy-plane. All five orbitals are drawn to show they each contain exactly two nodal surfaces, matching the gene …

Figure 2.10Shapes of the seven f-orbitals

What this figure shows. Seven separate, visually intricate multi-lobed sketches on x, y, z axes, one for each of the seven f orbitals $f_{z^3}$ , $f_{xz^2}$ , $f_{yz^2}$ , $f_{xyz}$ , $f_{z(x^2-y^2)}$ , $f_{x(x^2-3y^2)}$ and $f_{y(3x^2-y^2)}$ , each considerably more complex-looking than the d orbitals with six or eight separate lobes arranged symmetrically in three dimensions. Every one of the seven shapes contains exactly three nodal planes, consistent with an …

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