Shapes of s, p, and d Orbitals
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You should be comfortable with the definitions of the quantum numbers \(n\), \(l\), and \(m_l\) from the previous lecture, and understand that an orbital is a mathematical function describing probability density, not a physical path.
Why this matters
The shape of an orbital dictates how atoms bond and how electrons interact with one another.
While the quantum numbers tell us the energy and orientation of an electron state, they do not immediately reveal the spatial distribution of the electron cloud. Understanding the specific geometries of s, p, and d orbitals is essential for visualising molecular geometry, predicting hybridisation, and explaining the trends in atomic size and ionisation energy that define the periodic table. Without this spatial intuition, the periodic table remains a list of numbers rather than a map of chemical behaviour.
From Numbers to Shapes
Imagine a cloud of fog in a room. The quantum number \(l\) determines the "texture" of that fog. When \(l=0\), the fog is perfectly spherical; it looks the same from any direction. This is the s orbital. When \(l=1\), the fog splits into two distinct lobes, separated by a flat plane where there is no fog at all. This is the p orbital. When \(l=2\), the pattern becomes more complex, often resembling a four-leaf clover or a dumbbell with a donut-shaped ring around the middle. These are the d orbitals. The key insight is that these shapes are not arbitrary; they are the only patterns that satisfy the mathematical boundary conditions of the atom while maintaining the specific angular momentum defined by \(l\).
The Spherical Symmetry of s Orbitals
For any principal quantum number \(n\), the s orbital corresponds to \(l=0\). Because \(l=0\), the magnetic quantum number \(m_l\) can only be 0. This means there is only one s orbital per shell, and it has no preferred orientation in space. The probability density \(|\psi|^2\) depends only on the distance \(r\) from the nucleus, not on the angles \(\theta\) or \(\phi\). Consequently, the boundary surface of an s orbital is a sphere. As \(n\) increases, the sphere becomes larger and more diffuse, reflecting the higher energy and greater average distance of the electron from the nucleus.
The angular part \(Y_{0}^{0}\) is a constant, confirming that the shape is isotropic. Separately, s orbitals are the only ones with non-zero density at the nucleus, which is why s electrons are described as "penetrating"; they can spend a significant amount of time very close to the nucleus compared to p or d electrons of the same shell.
Dumbbells and Nodal Planes: The p Orbitals
When \(l=1\), there are three possible values for \(m_l\): \(-1, 0, +1\). These correspond to the three p orbitals, conventionally labelled \(p_x\), \(p_y\), and \(p_z\). Each p orbital consists of two lobes of electron density separated by a nodal plane that passes through the nucleus. In this nodal plane, the probability of finding the electron is exactly zero. The lobes have opposite signs in the wavefunction \(\psi\), which is crucial for bonding, as it allows for constructive and destructive interference when orbitals overlap.
The specific orientation is determined by \(m_l\). The \(p_z\) orbital is aligned along the z-axis, while \(p_x\) and \(p_y\) lie in the xy-plane. Together with the s orbital, the three p orbitals are the raw material for the hybrid orbitals (sp, sp\(^2\), sp\(^3\)) behind linear, trigonal planar and tetrahedral bonding in organic molecules.
Complex Patterns: The d Orbitals
For \(l=2\), there are five possible values of \(m_l\): \(-2, -1, 0, +1, +2\). These generate five d orbitals. Four of them, often labelled \(d_{xy}\), \(d_{xz}\), \(d_{yz}\), and \(d_{x^2-y^2}\), have four lobes arranged in a clover-like pattern. In \(d_{xy}\), \(d_{xz}\) and \(d_{yz}\) the lobes point between the axes of their plane; in \(d_{x^2-y^2}\) they point along the x and y axes. The fifth orbital, \(d_{z^2}\), is unique. It has two large lobes along the z-axis and a toroidal (doughnut-shaped) ring in the xy-plane. This distinct shape arises from the specific angular dependence of the wavefunction for \(m_l=0\) when \(l=2\).
The d orbitals are critical in transition metal chemistry, where they are partially filled and determine the colour, magnetism, and catalytic properties of the elements. The directional nature of the d orbitals explains why transition metals often form complexes with specific coordination numbers and geometries, such as octahedral or square planar.
The result
Quantum numbers and atomic orbitals
Solving the time-independent Schrödinger equation for an electron in a hydrogen-like atom, by separating variables in spherical coordinates, forces every stationary electron state to be labelled by three integers arising from the boundary conditions on the separated equation, plus a fourth from electron spin: the principal quantum number \(n=1,2,3,\dots\) (energy shell), the orbital angular momentum quantum number \(l=0,1,\dots,n-1\) (orbital shape), the magnetic quantum number \(m_l=-l,\dots,+l\) (orbital orientation), and the spin quantum number \(m_s=\pm\tfrac12\) (intrinsic electron spin). Each allowed \((n,l,m_l)\) triple defines one atomic orbital — a specific spatial probability-density pattern — and each orbital holds at most two electrons, one with each value of \(m_s\).
A common misconception is that the "lobes" of a p or d orbital represent regions where the electron is physically confined. In reality, the wavefunction \(\psi\) extends to infinity, and the lobes simply indicate regions of high probability density with a specific phase sign. The electron is not "inside" the lobe; it is a delocalised quantum entity whose presence is described by the probability distribution. C
Worked example
Problem. Consider a carbon atom in its ground state. Determine the number of orbitals in the second shell (\(n=2\)) and calculate the total number of nodal planes associated with the p orbitals in this shell.
- Identify the allowed values of \(l\) for \(n=2\). Since \(l\) ranges from \(0\) to \(n-1\), the allowed values are \(l=0\) and \(l=1\).
- Count the orbitals for each \(l\). For \(l=0\) (s orbital), there is \(2(0)+1 = 1\) orbital. For \(l=1\) (p orbitals), there are \(2(1)+1 = 3\) orbitals.
- Sum the orbitals: \(1 + 3 = 4\) orbitals in the \(n=2\) shell.
- Determine the nodal planes for the p orbitals. Each p orbital (\(p_x, p_y, p_z\)) has one nodal plane passing through the nucleus. Since there are three p orbitals, the total number of nodal planes associated with the p subshell is 3.
Units check. Orbitals and nodal planes are dimensionless counts of mathematical states and geometric planes, so no physical units are required.
Three quick tests
- How many orbitals are there in the \(n=3\) shell?
Answer
9 orbitals (1 s, 3 p, 5 d). - Which d orbital has a toroidal ring in the xy-plane?
Answer
\(d_{z^2}\). - Does an s orbital have any nodal planes?
Answer
No, it has only radial nodes (if \(n>1\)), but no angular nodal planes.
Back to CU-101 →
Every lecture and every result in this unit, with prerequisites and threads.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.