Spectroscopic notation
Spectroscopic notation is a standardized system in atomic and molecular physics for labeling the quantum mechanical states of electrons in atoms, ions, and molecules, particularly emphasizing the coupling of their orbital and spin angular momenta.[1] In the most common form, known as term symbols, it uses the notation ^{2S+1}L_J, where L (denoted by letters S, P, D, F, etc., for L = 0, 1, 2, 3, \ldots) represents the total orbital angular momentum quantum number, S is the total spin angular momentum quantum number, the superscript $2S+1 indicates the multiplicity of the spin state, and the subscript J denotes the total angular momentum quantum number (ranging from |L - S| to L + S).[2] This notation succinctly captures the energy levels and spectroscopic properties of multi-electron systems under Russell-Saunders coupling (also called LS coupling), which is applicable to lighter atoms where spin-orbit interactions are relatively weak.[1] The origins of spectroscopic notation trace back to the late 19th and early 20th centuries, evolving from empirical classifications of spectral lines observed in atomic emission spectra.[2] Early spectroscopists, such as Johann Balmer in 1885, identified patterns in hydrogen lines, leading to formulas that categorized series as "sharp," "principal," "diffuse," and "fundamental," with corresponding letters s, p, d, and f initially denoting these line types rather than quantum numbers.[2] By the 1920s, the discovery of electron spin and the need to explain fine structure prompted Henry Norris Russell and Frederick Albert Saunders to formalize the LS coupling scheme in their 1925 paper, introducing the modern term symbol format to describe multiplet structures in spectra like those of alkaline-earth elements.[3] This development integrated quantum theory with spectroscopic observations, providing a framework that linked atomic energy levels to observable transitions.[4] In practice, spectroscopic notation is essential for predicting and interpreting atomic spectra, determining selection rules for allowed transitions (e.g., \Delta L = 0, \pm 1, \Delta S = 0, \Delta J = 0, \pm 1), and analyzing electron configurations in multi-electron atoms.[2] For example, the ground state of carbon is denoted as 2p^2 \, ^3P_0, indicating two electrons in the 2p orbital with total L=1, S=1, and J=0.[5] While LS coupling dominates for lighter elements (typically up to atomic number Z ≈ 50–60), alternative schemes like jj coupling are used for heavier atoms where relativistic effects strengthen spin-orbit interactions.[1] Today, this notation underpins applications in astrophysics, laser physics, and quantum computing, enabling precise modeling of atomic interactions and energy level diagrams.Basic Principles
Quantum Numbers
In atomic physics, the spectroscopic notation for atomic states relies on a set of fundamental quantum numbers that describe the quantum mechanical state of electrons in atoms. These numbers arise from the solutions to the Schrödinger equation for the hydrogen atom and its multi-electron extensions, providing essential labels for energy levels, orbital shapes, spatial orientations, and intrinsic spins. The four primary quantum numbers—principal, orbital angular momentum, magnetic, and spin—uniquely specify each electron's state within an atom, forming the foundation for more complex notations used in spectroscopy.[6][7] The principal quantum number n determines the energy level and average distance of the electron from the nucleus, taking positive integer values n = 1, 2, 3, \dots. Higher values of n correspond to higher energy states and larger orbital sizes. The orbital angular momentum quantum number l specifies the shape of the orbital and ranges from $0ton-1for a givenn, influencing the electron's angular momentum magnitude. The [magnetic quantum number](/page/Magnetic_quantum_number) m_ldescribes the orbital's orientation in space relative to an external [magnetic field](/page/Magnetic_field), with possible values from-lto+lin [integer](/page/Integer) steps. Finally, the [spin quantum number](/page/Spin_quantum_number)m_saccounts for the electron's intrinsic [spin](/page/Spin), which can be either+\frac{1}{2}or-\frac{1}{2}$, representing the two possible spin projections along a quantization axis. These numbers collectively define the possible states available to electrons in an atom.[8][6] The concept of these quantum numbers emerged in the early 20th century through the development of atomic models. In 1913, Niels Bohr introduced the principal quantum number n in his model of the hydrogen atom to quantize electron orbits and explain spectral line series, postulating discrete energy levels. Arnold Sommerfeld extended this in 1916 by incorporating relativistic effects and elliptical orbits, introducing the orbital angular momentum quantum number l (initially as a secondary quantum number) to account for fine structure in spectra, along with the magnetic quantum number for Zeeman splitting. The spin quantum number m_s was later proposed by George Uhlenbeck and Samuel Goudsmit in 1925 to explain spin-orbit coupling and anomalous Zeeman effects.[9] The Pauli exclusion principle, formulated by Wolfgang Pauli in 1925, states that no two electrons in an atom can occupy the same quantum state, meaning they cannot share identical values for all four quantum numbers n, l, m_l, and m_s. This principle governs the filling of orbitals by ensuring that each orbital (defined by n, l, and m_l) holds at most two electrons with opposite spins, leading to the structured buildup of atomic electron configurations and the periodic table's organization. These quantum numbers provide the basis for constructing term symbols in spectroscopic notation, which combine individual electron states to describe total atomic angular momenta.Angular Momentum Notation
In spectroscopic notation, the orbital angular momentum quantum number l for a single electron is represented by letters derived from early classifications of spectral line series in alkali metal atoms. These include s for l = 0 (sharp series), p for l = 1 (principal series), d for l = 2 (diffuse series), f for l = 3 (fundamental series), g for l = 4, and subsequent letters of the alphabet for higher values. This lettering system originated in the late 19th century from observations by spectroscopists such as George Liveing and James Dewar, who categorized line sharpness in alkali spectra, with Johannes Rydberg expanding the series descriptions around 1890 and Friedrich Hund formalizing the notation in 1927 to align with quantum mechanical subshells. For multi-electron atoms, the total orbital angular momentum quantum number L employs uppercase letters following the same sequence: S for L = 0, P for L = 1, D for L = 2, F for L = 3, and so forth.[10] The magnitude of L arises from the vector sum of the individual orbital angular momenta of equivalent electrons, expressed as \mathbf{L} = \sum_i \mathbf{l}_i, where the possible values of L range from the maximum sum down to 0 in steps of 1, depending on the electron configuration.[11] The total spin angular momentum quantum number S is denoted by the multiplicity $2S + 1, which serves as a left superscript in spectroscopic descriptions to indicate the number of possible spin states.[8] This convention, part of the broader Russell-Saunders coupling scheme, reflects the degeneracy due to spin orientation and was standardized in early 20th-century atomic spectroscopy to classify energy levels based on experimental spectra.[8]Atomic Notation
Electron Configurations and Orbitals
Electron configurations in spectroscopic notation describe the distribution of electrons among atomic orbitals for atoms in their ground or excited states. This notation specifies the principal quantum number n, the azimuthal quantum number \ell (represented by letters s, p, d, f for \ell = 0, 1, 2, 3), and the number of electrons in each subshell as a superscript. For example, the ground state configuration of neon is $1s^2 2s^2 2p^6, indicating two electrons in the 1s orbital, two in 2s, and six in 2p.[12][13] The arrangement follows the Aufbau principle, which states that electrons occupy orbitals starting from the lowest energy levels, ordered by increasing n + \ell, and for equal n + \ell, by increasing n. This building-up process, combined with the Pauli exclusion principle (limiting each orbital to two electrons of opposite spin), and Hund's rule (maximizing unpaired electrons by filling degenerate orbitals singly with parallel spins before pairing), determines the ground state configuration. Hund's rule minimizes electron-electron repulsion and exchange energy, leading to higher total spin and orbital angular momentum for stability./Quantum_Mechanics/10:_Multi-electron_Atoms/Electron_Configuration)[12][14] Atomic orbitals vary in shape and electron capacity based on \ell: s orbitals (\ell=0) are spherical and hold up to 2 electrons; p orbitals (\ell=1) are dumbbell-shaped along the x, y, or z axes and accommodate 6 electrons; d orbitals (\ell=2) have more complex cloverleaf or double-dumbbell shapes and hold 10 electrons. These shapes arise from the angular part of the wave function and influence electron probability density.[6][15] For the first 20 elements, ground state configurations follow the Aufbau order, filling 1s, then 2s and 2p, 3s and 3p, and 4s. Anomalies occur due to the stability of half-filled or fully filled subshells, as seen in chromium (Z=24), which adopts [Ar] 4s^1 3d^5 instead of [Ar] 4s^2 3d^4, prioritizing the half-filled 3d subshell for lower energy. Copper (Z=29) similarly shows [Ar] 4s^1 3d^{10} over [Ar] 4s^2 3d^9. The table below lists configurations for hydrogen through calcium:| Element | Atomic Number | Ground State Configuration |
|---|---|---|
| H | 1 | $1s^1 |
| He | 2 | $1s^2 |
| Li | 3 | $1s^2 2s^1 |
| Be | 4 | $1s^2 2s^2 |
| B | 5 | $1s^2 2s^2 2p^1 |
| C | 6 | $1s^2 2s^2 2p^2 |
| N | 7 | $1s^2 2s^2 2p^3 |
| O | 8 | $1s^2 2s^2 2p^4 |
| F | 9 | $1s^2 2s^2 2p^5 |
| Ne | 10 | $1s^2 2s^2 2p^6 |
| Na | 11 | $1s^2 2s^2 2p^6 3s^1 |
| Mg | 12 | $1s^2 2s^2 2p^6 3s^2 |
| Al | 13 | $1s^2 2s^2 2p^6 3s^2 3p^1 |
| Si | 14 | $1s^2 2s^2 2p^6 3s^2 3p^2 |
| P | 15 | $1s^2 2s^2 2p^6 3s^2 3p^3 |
| S | 16 | $1s^2 2s^2 2p^6 3s^2 3p^4 |
| Cl | 17 | $1s^2 2s^2 2p^6 3s^2 3p^5 |
| Ar | 18 | $1s^2 2s^2 2p^6 3s^2 3p^6 |
| K | 19 | $1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 |
| Ca | 20 | $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 |