Triplet state
In quantum chemistry and spectroscopy, a triplet state refers to an electronic excited state of an atom or molecule where two unpaired electrons occupy distinct molecular orbitals with parallel spins, yielding a total spin quantum number S = 1 and a spin multiplicity of 3, which corresponds to three possible spin projections (M_S = +1, 0, -1) in a magnetic field. This configuration arises from the promotion of an electron from the ground-state singlet (paired spins, S = 0) to an excited orbital while aligning spins, often via intersystem crossing, and is characterized by lower energy than the corresponding singlet excited state due to reduced electron-electron repulsion governed by Hund's rule.[1] Triplet states are metastable, with lifetimes typically ranging from microseconds to seconds, enabling phosphorescence—a delayed emission of light upon return to the ground state—distinct from the rapid fluorescence of singlet states. The concept of the triplet state emerged in the early 20th century through studies of atomic and molecular spectroscopy, with key insights from G. N. Lewis, who in 1916 proposed "diradical" structures for certain molecules that prefigured the parallel-spin configuration, though without direct spectroscopic ties.[2] By the 1940s, Gilbert N. Lewis and Michael Kasha formalized the assignment of phosphorescence to triplet-singlet transitions, revolutionizing photophysics by explaining long-lived emissions in organic compounds and establishing spin conservation rules in electronic transitions.[3] In practice, triplet states play a pivotal role in photochemistry, facilitating energy transfer, sensitization processes, and reactions such as cis-trans isomerization or cycloadditions in molecules like ethylene or formaldehyde, where the singlet-triplet energy gap (\Delta E_{ST}) influences reactivity—often around 10 kcal/mol for n,π* transitions but larger (∼70 kcal/mol) for π,π* types.[1] Spin-orbit coupling, arising from relativistic effects, enables the forbidden intersystem crossing between singlet and triplet manifolds, which occurs on timescales typically ranging from 10⁻¹¹ to 10⁻⁶ s.[1] Notable examples include molecular oxygen (O₂), which exists in a triplet ground state due to its half-filled π* orbitals, making it paramagnetic and reactive in diradical fashion, and organic dyes where triplets mediate applications in photodynamic therapy or organic light-emitting diodes (OLEDs). Experimental detection often involves electron paramagnetic resonance (EPR) spectroscopy, which reveals the zero-field splitting from electron-spin interactions, or time-resolved phosphorescence at low temperatures to minimize non-radiative decay.[4] Overall, triplet states exemplify the interplay of quantum mechanics, spin statistics, and Pauli exclusion in dictating molecular behavior under light excitation.[1]Fundamentals
Definition
In quantum mechanics, the triplet state refers to a specific spin configuration of a system composed of two spin-1/2 particles, such as electrons, where the total spin quantum number S = 1.[5] This configuration arises from the parallel alignment of the individual spins, yielding three possible projections along the z-axis: m_s = -1, 0, +1.[6] The designation "triplet" originates from the three-fold degeneracy of these spin states, which, in the absence of an external magnetic field, results in identical energies for the three projections.[6] For fermions like electrons, the triplet spin state is symmetric under particle exchange, necessitating an antisymmetric spatial wavefunction to satisfy the Pauli exclusion principle and ensure the overall wavefunction remains antisymmetric.[7] The concept of the triplet state emerged in the early 20th century within atomic physics, particularly through Werner Heisenberg's 1926 analysis of the helium atom spectrum, which distinguished triplet (orthohelium) and singlet (parahelium) series, building on the hypothesis of electron spin by George Uhlenbeck and Samuel Goudsmit and Wolfgang Pauli's exclusion principle.[7][8][9] In contrast to the triplet, the singlet state features S = 0 with a single m_s = 0 projection and an antisymmetric spin wavefunction paired with a symmetric spatial one.[7]Relation to Total Spin
The total spin angular momentum \mathbf{S} for a system of two particles, each with spin s_1 = s_2 = 1/2, is defined as the vector sum \mathbf{S} = \mathbf{S}_1 + \mathbf{S}_2, where \mathbf{S}_1 and \mathbf{S}_2 are the individual spin operators.[10] The possible eigenvalues of the total spin quantum number S are 0 or 1, corresponding to the singlet and triplet states, respectively.[10] The triplet state arises when S = 1, and its three substates are characterized by the magnetic quantum numbers m_S = 1, 0, -1. These states are constructed using Clebsch-Gordan coefficients for combining two spin-1/2 angular momenta. Specifically, the basis states in the uncoupled representation are combined as follows: |S=1, m_S=1\rangle = |\uparrow \uparrow \rangle |S=1, m_S=0\rangle = \frac{1}{\sqrt{2}} \left( |\uparrow \downarrow \rangle + |\downarrow \uparrow \rangle \right) |S=1, m_S=-1\rangle = |\downarrow \downarrow \rangle where |\uparrow\rangle and |\downarrow\rangle denote the spin-up and spin-down states for each particle, and the coefficients ensure normalization and proper symmetry.[10][11] The spin multiplicity, given by $2S + 1, equals 3 for the triplet state (S=1), in contrast to 1 for the singlet state (S=0). This multiplicity reflects the degeneracy of the triplet substates.[10] In the triplet state, the spins are parallel on average—both aligned up for m_S=1, both down for m_S=-1, and with equal probability of alignment in the symmetric m_S=0 combination—leading to exchange energy differences relative to the singlet state.[10][12]Quantum Description
Spin Configurations for Two Particles
For two indistinguishable spin-1/2 particles, such as electrons, the triplet state arises when their spins couple to a total spin quantum number S = 1, resulting in three possible projections along the z-axis: m_S = 1, 0, -1.[6] The corresponding spin wavefunctions are symmetric under particle exchange, ensuring that the overall wavefunction remains antisymmetric as required for fermions by the Pauli exclusion principle.[13] The explicit triplet spin functions, using the single-particle spin functions \alpha (spin up, m_s = +1/2) and \beta (spin down, m_s = -1/2), are as follows:- For m_S = 1: \alpha(1)\alpha(2)
- For m_S = -1: \beta(1)\beta(2)
- For m_S = 0: \frac{1}{\sqrt{2}} \left[ \alpha(1)\beta(2) + \beta(1)\alpha(2) \right]