Unit 1: Atoms - Quantum Mechanical Model and Electron Configurations
Quantum Mechanical Model of the Atom
3.4.1 The Schrödinger Equation
The wavefunction \psi is a mathematical function that describes the quantum state of an electron in an atom, specifically its probability amplitude. It's a complex-valued function.
It is a function of 3D-coordinates: \psi(r, \theta, \phi). This means it describes the spatial distribution and behavior of the electron's wave in three dimensions.
It is obtained by solving the time-independent Schrödinger equation (S.E.), which describes the specific stationary states of an electron.
The Schrödinger equation is represented as: H\psi = E\psi or more explicitly, (-\frac{\hbar^2}{2me}\nabla^2 - \frac{e^2}{4\pi\varepsilon0 r})\psi(r,\theta,\phi) = E\psi(r,\theta,\phi).
H represents the Hamiltonian operator, which corresponds to the total energy of the system. It includes both the kinetic (T) and potential (V) energy terms of the electron. Specifically, (T + V)\psi = E\psi. The Hamiltonian determines the evolution of the quantum state.
E represents the total, quantized energy of the electron in a specific quantum state. This energy is a specific eigenvalue corresponding to the eigenfunction \psi.
The term (-\frac{\hbar^2}{2me}\nabla^2) is the kinetic energy operator, where \hbar (h-bar) is the reduced Planck constant (Planck's constant divided by 2\pi), me is the mass of the electron, and \nabla^2 (Laplacian operator) describes the curvature of the wavefunction, which is associated with kinetic energy.
The term (-\frac{e^2}{4\pi\varepsilon0 r}) is the potential energy operator, representing the electrostatic attraction between the negatively charged electron (charge e) and the positively charged nucleus (charge Z{eff}e, for hydrogen Z=1). Here, \varepsilon_0 is the permittivity of free space, and r is the distance from the nucleus.
Complexity: The Schrödinger equation is mathematically very complex, especially for multi-electron atoms where electron-electron repulsion must be included, making analytical solutions impossible. Students are generally not responsible for knowing the underlying math or how to solve it but should understand its purpose.
Solution Structure: Each valid solution of the Schrödinger equation, \psi{n, \ell, m\ell}(r, \theta, \phi), which is an atomic orbital, can be mathematically factored into two independent parts:
A radial part, R(r)_{n, \ell}, which depends only on the distance from the nucleus (r). This part describes how the probability of finding the electron varies with distance from the nucleus.
An angular part, Y(\theta, \phi){\ell, m\ell}, which depends on the angles ( heta and \phi). This part dictates the shape and orientation of the orbital in three-dimensional space.
Orbitals: Wavefunctions for single electrons are also called orbitals by chemists. They are mathematical constructs that describe the probable spatial distribution and energy of an electron in an atom, not a definite path.
3.4.2 Electron Probability Density
Probability density (|\psi|^2): Because the wavefunction \psi can be a complex number, its square magnitude, |\psi|^2 (or \psi^2 if \psi is real), represents the probability of finding the electron at a particular point (r, \theta, \phi) in space. This quantity is always real and non-negative, as required for a probability.
Electron cloud depiction (A): This is an imaginary, time-averaged picture representing the high-speed movement of the electron, illustrating regions where the electron is most likely to be found over time.
It is depicted as a cloud of dots around the nucleus, where each dot represents a possible instantaneous position of the electron.
The greater the density of dots, the higher the probability of finding the electron in that specific region of space.
Probability of finding the electron far from the nucleus (B): While the probability density (|\psi|^2) often decreases as the distance from the nucleus (r) increases, the volume of each concentric spherical layer of space increases proportional to r^2. Therefore, the total probability of finding the electron within a thicker, outer spherical shell can still be significant, leading to a non-zero probability of finding the electron at substantial distances.
Radial probability distribution (C and D): This plot represents the total probability of finding the electron at a given distance (r) from the nucleus, regardless of direction. It is obtained by integrating |\psi|^2 over all angles (\theta and \phi) and multiplying by the volume of a spherical shell (4\pi r^2 dr).
It takes into account both the intrinsic probability density (dots per unit volume) and the ever-increasing volume of space available at greater distances from the nucleus.
The radial probability distribution often exhibits one or more peaks at certain distances near the nucleus, indicating the most probable distance for the electron to be found, which can be related to the Bohr radius for the hydrogen atom.
Probability contour (E): This is a 3D boundary surface that defines the volume around the nucleus where there is a certain high probability (typically 90\%) of finding the electron. The 90\% contour is chosen as a good balance to visualize the orbital shape while encompassing most of the electron's probability.
It helps to visualize the characteristic shape and spatial extent of an orbital. It shows where the electron spends most of its time.
We can visualize the atom with a 90\% probability contour to understand the spatial extent of an electron's presence and compare the shapes of different orbitals.
3.4.3 The s Orbitals of Hydrogen
Different representations of s orbitals for the hydrogen atom illustrate the concepts of wavefunction, probability density, and shape:
\psi (wavefunction itself) - shows positive and negative phases.
\psi^2 (probability density) - always positive, shows areas of electron presence.
Electron density (visual cloud) - a visual average of \psi^2.
Boundary surface (probability contour) - defines the 90\% probability volume.
Radial probability distribution plot - shows peak probabilities at certain distances.
Examples are shown for 1s, 2s, and 3s orbitals, demonstrating how the size and the number of radial nodes (n - \ell - 1) change with increasing principal quantum number (n).
For 1s (n=1, \ell=0): 0 radial nodes.
For 2s (n=2, \ell=0): 1 radial node, a spherical region where \psi=0. The orbital is larger.
For 3s (n=3, \ell=0): 2 radial nodes. The orbital is even larger.
3.4.4 Quantum Numbers
The state of an electron in an atom is uniquely specified by a set of four quantum numbers, which arise naturally from the solutions to the Schrödinger equation. These numbers describe the discrete energy levels, shapes, and orientations of atomic orbitals.
Principal Quantum Number (n)
Symbol: n
Allowed values: Positive integers (1, 2, 3, \ldots). Higher values indicate higher energy and larger orbitals.
Interpretation: Indicates the principal energy level (or shell) of the electron. It primarily determines the size and energy of the orbital.
It also determines the total number of nodes in the orbital (n-1).
For hydrogen-like atoms (species with only one electron, like H, He$^+$, Li$^{2+}$), the energy depends only on n: En = -RH \frac{Z^2}{n^2}, where R_H is the Rydberg constant (2.179 \times 10^{-18} J), and Z is the nuclear charge (atomic number). In multi-electron atoms, energy also depends on \ell due to electron-electron repulsions.
Angular Momentum Quantum Number (\ell)
Symbol: \ell
Allowed values: Integers from 0 to n-1. For a given n, there are n possible \ell values.
Interpretation: Determines the shape of the orbital and defines the subshell within a principal energy level. It also determines the number of angular nodes in an orbital, which is equal to \ell.
\ell is related to the magnitude of the orbital's angular momentum.
Correspondence:
\ell = 0 corresponds to an s orbital (sharp, spherical shape).
\ell = 1 corresponds to a p orbital (principal, dumbbell shape).
\ell = 2 corresponds to a d orbital (diffuse, cloverleaf or double-dumbbell shapes).
\ell = 3 corresponds to an f orbital (fundamental, more complex shapes).
This letter nomenclature originates from early spectroscopy based on the appearance of spectral lines.
Magnetic Quantum Number (m_\ell)
Symbol: m_\ell
Allowed values: Integers from -\ell to +\ell (including 0). For a given \ell value, there are 2\ell+1 possible m_\ell values.
Interpretation: Determines the orientation of the orbital in space around the nucleus. In the absence of an external magnetic field, orbitals with the same n and \ell but different m_\ell values are degenerate (have the same energy).
The total number of m_\ell values for a given \ell value is 2\ell+1. This is equal to the number of orbitals within a subshell.
Relationship between n, \ell, m_\ell and number of orbitals:
For n=1:
\ell = 0 (1s subshell)
m_\ell = 0 (1 orbital)
Number of orbitals = 1
For n=2:
\ell = 0 (2s subshell) \Rightarrow m_\ell = 0 (1 orbital)
\ell = 1 (2p subshell) \Rightarrow m\ell = -1, 0, +1 (3 orbitals: 2px, 2py, 2pz)
Total number of orbitals = 1+3 = 4
For n=3:
\ell = 0 (3s subshell) \Rightarrow m_\ell = 0 (1 orbital)
\ell = 1 (3p subshell) \Rightarrow m\ell = -1, 0, +1 (3 orbitals: 3px, 3py, 3pz)
\ell = 2 (3d subshell) \Rightarrow m\ell = -2, -1, 0, +1, +2 (5 orbitals: 3d{xy}, 3d{xz}, 3d{yz}, 3d{x^2-y^2}, 3d{z^2})
Total number of orbitals = 1+3+5 = 9
General Rule: The total number of orbitals in a given principal energy level (n) is n^2. This means an electron in a shell with quantum number n can occupy any of n^2 degenerate orbitals (in a hydrogen atom).
Convention: While there's no exact physical relationship between a specific axis and a specific m\ell value without an external field, we typically associate the m\ell = 0 value with orbitals whose lobes lie along the z-axis (e.g., pz, d{z^2}) for convenience in standard coordinate systems.
Orbital Notation: Atomic orbitals are identified using the notation: n\ell (e.g., 1s, 2p, 3d), where the \ell value is represented by its corresponding letter (s, p, d, f).
3.4.5 Orbital Shapes
Orbital shapes are visual representations of the 90\% probability contour, meaning they define the volume around the nucleus where the electron is found 90\% of the time. These shapes are derived from the angular part of the wavefunction.
s orbitals (\ell=0):
Are perfectly spherical in shape, meaning the probability of finding the electron is independent of direction.
Increase in size with increasing n (e.g., 2s is larger than 1s) and possess more radial nodes.
1s is a simple sphere with no radial nodes.
2s and 3s orbitals have one (2-0-1 = 1) and two (3-0-1 = 2) radial nodes, respectively, within their spherical shapes, meaning these are spherical regions of zero electron probability.
p orbitals (\ell=1):
There are three degenerate p orbitals (px, py, pz) for each n \geq 2 (e.g., 2px, 2py, 2pz).
Each p orbital has two lobes on opposite sides of the nucleus, giving a dumbbell shape. The electron density is concentrated along a specific axis.
They are oriented along the x, y, and z axes, respectively, in three-dimensional space.
They possess one angular node (a flat plane) that passes directly through the nucleus, where the probability of finding the electron is zero.
d orbitals (\ell=2):
There are five degenerate d orbitals (d{xy}, d{xz}, d{yz}, d{x^2-y^2}, d_{z^2}) for each n \geq 3 (e.g., 3d).
Four of the d orbitals (d{xy}, d{xz}, d{yz}, d{x^2-y^2}) have four lobes each, usually in a cloverleaf pattern. For instance, d{xy} has lobes between the x and y axes, while d{x^2-y^2} has lobes along the x and y axes.
The d_{z^2} orbital has a unique shape with two lobes along the z-axis and a donut-shaped ring in the xy-plane.
These orbitals have two angular nodes, which can be planar or conical surfaces.
Learning Requirement: It is important to learn the shapes of all hydrogen s, p, and d orbitals, indicating their relative phases (represented by different colors or shades, crucial for understanding molecular bonding) and locating radial or angular nodes. Understanding nodes is key to comprehending the wave-like nature of electrons.
It's possible to sketch them in 3D perspective using right-handed coordinate axes or draw their projections onto planes if 3D drawing is difficult.
Nodes within an Atom
A node is a region in an orbital where the probability of finding an electron is zero. It arises from the wave nature of the electron, analogous to nodes in a standing wave.
Radial Nodes:
Occur where the radial part of the wavefunction, R(r), equals zero or changes sign. These are distinct spherical regions around the nucleus where the electron cannot be found.
These are spherical regions of zero electron probability.
Number of radial nodes = n - \ell - 1. This formula effectively counts the number of times the radial wavefunction passes through zero.
Angular Nodes:
Are typically flat planes (or conical surfaces for higher \ell values) at fixed angles where the angular part of the wavefunction, Y(\theta, \phi), is zero. They define the shape of the orbital.
These are planar or conical regions of zero electron probability that pass through the nucleus.
Number of angular nodes = \ell. This means s orbitals have 0 angular nodes, p orbitals have 1, d orbitals have 2, and so on.
Total Number of Nodes:
The sum of angular and radial nodes within an orbital.
Total number of nodes = (n - \ell - 1) + \ell = n - 1. This total number of nodes depends only on the principal quantum number n.
Recap: Atomic Orbital Concepts
The atomic orbital (\psi, wavefunction) is a mathematical description of the electron's wavelike behavior, providing information about its probability amplitude and energy in an atom.
The Schrödinger equation is a fundamental equation that, when solved, yields the allowed wavefunctions (orbitals) and corresponding quantized energy states of the atom.
The probability density of finding the electron at a particular location is represented by |\psi|^2. This quantity allows for the visualization of electron distribution.
For a given energy level, an electron density diagram (electron cloud) and a radial probability distribution plot show how the electron occupies the space near the nucleus, highlighting regions of high probability.
An atomic orbital is primarily described by three quantum numbers (n, \ell, m_\ell):
size and energy (n)
shape (\ell)
orientation (m_\ell)
Relationship between quantum numbers:
n (principal quantum number) limits \ell to n possible integer values from 0 to n-1, hence defining the number of subshells in a shell.
\ell (angular momentum quantum number) limits m_\ell to 2\ell+1 possible integer values from -\ell to +\ell, hence defining the number of orbitals within a subshell.
Terminology:
A shell consists of all orbitals that share the same principal quantum number (n) and thus the same principal energy level.
A subshell consists of all orbitals that share the same principal quantum number (n) and angular momentum quantum number (\ell), meaning they have the same size and shape but potentially different spatial orientations.
Orbital Shapes by \ell value:
A subshell with \ell=0 is described by a single spherical (s) orbital.
A subshell with \ell=1 is described by three two-lobed (p) orbitals (oriented along x, y, and z axes).
A subshell with \ell=2 is described by five multi-lobed (d) orbitals (complex shapes and orientations).
Hydrogen Atom Special Case: In the special case of the hydrogen atom (or any one-electron species), the energies of orbitals depend only on the n value. This means that orbitals within the same shell (e.g., 2s and 2p orbitals) are degenerate (have exactly the same energy). This degeneracy is lifted (removed) in multi-electron atoms due to electron-electron repulsion and shielding effects.
3.4.6 The Fourth Quantum Number: Spin Quantum Number
Spin Quantum Number (m_s)
Symbol: m_s
Allowed values: +\frac{1}{2} (spin up) or -\frac{1}{2} (spin down).
Interpretation: Describes the inherent, intrinsic angular momentum of the electron, often visualized as the electron spinning on its own axis. This property is crucial for understanding the Pauli Exclusion Principle and magnetic properties of atoms. Every electron has this fundamental property.