Physical Measurement Laboratory

Triatomic Spectral Database, List of Symbols

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4. List of Symbols

a. Quantum Numbers
J	Resultant total angular momentum quantum number, excluding nuclear spins.
N	Rotational angular momentum quantum number, excluding electron and nuclear spins, in the case where electron spin is present.
K	Projection of J (or N) on the symmetry axis in the limiting prolate or oblate symmetric top.
F₁	Resultant angular momentum quantum number including nuclear spin for one nucleus.
F	Resultant total angular momentum quantum number.
λ or ε	Quantum number employed when F₁ is not a good quantum number. This value simply numbers the levels from lowest to highest energy for the same F quantum number.
ν₁,ν₂,ν₃ v₁v₂v₃	Vibrational modes (ν) and quantum numbers (v). ν₁ is the highest energy symmetric stretching mode, ν₂ is the bending mode, and ν₃ is the asymmetric stretch for XY₂ molecules or lowest energy stretch for XYZ molecules.
U or L	Upper or lower energy level or transition frequency.
′ or ″	Prime or double prime is used to distinguish the upper (′) and lower (″) levels in a transition. They occur as superscripts on the quantum numbers.
ℓ	Quantum number for vibrational angular momentum.
I or (I_i)	Angular momentum quantum number of nuclear spin for one (or i^th) nucleus.
S	Resultant angular momentum number of electron spins. Σ is the projection of S on the molecular axis.
Λ	Absolute values of the projection of the resultant orbital angular momentum on the molecular axis.
Ω	Absolute value of the projection of the total electronic angular momentum on the molecular axis.
Σ, Π, Δ	Electronic state designation for which Λ = 0,1,2, respectively.
b. Molecular Constants
A, B, C	Rotational constants (MHz). These are related to the principal moments of inertia: A = h/8π²I_a, etc.
${\bar B}$	B-bar equals (B + C)/2.
τ, Δ, δ, D	Quartic centrifugal distortion constants (MHz or kHz).
H, h	Sextic centrifugal distortion constants (MHz or kHz).
L, ℓ, G, g	Octic centrifugal distortion constants (MHz).
P, p, F	Dectic centrifugal distortion constants (MHz).
eQq_aa(X), ... χ_aa(X), ...	Nuclear electric quadrupole coupling constant along principal axis indicated for nucleus X (MHz).
eQq η	Product of the nuclear quadrupole coupling constant and the asymmetry parameter for the bending vibrational state.
µ_a,b	Components of the electric dipole moment along the a- or b-principal axes.
α_v, γ_vv	Rotation-vibration coefficients in the power series representing B_v (eq 3).
q_v	ℓ-doubling constant (MHz).
σ_\|\| , σ_⊥	Components of the magnetic shielding tensor which are parallel and perpendicular to the molecular axis, respectively.
Q	Molecular quadrupole moment relative to the center of mass (esu · cm²).
α_\|\| - α_⊥	Electric polarizability anisotropy (cm³).
χ_⊥, χ_\|\|	Components of the magnetic susceptibility tensor which are, respectively, perpendicular and parallel to the molecular axis (erg/G² · mol).
χ_xx - χ_yy	Magnetic susceptibility anisotropy.
g_⊥ , g_\|\|	Components of the molecular G tensor which are, respectively, perpendicular and parallel to the molecular axis. g_⊥ is sometimes denoted g or g_J for linear molecules in the ground state (µ_N).
g_xx - g_yy	Anisotropy of the molecular G tensor perpendicular to the molecular axis (µ_N).
c_X or M	Spin rotation constant related to nucleus X (kHz).
S_XY	Spin-spin interaction constant between nucleus X and nucleus Y (kHz).
α_p , β_p	Ω-type doubling parameters, $\alpha_p = 4\Sigma(-1)^S \times~ \frac{\langle \Pi\|\, (A+2B) L_y\, \|\Sigma\rangle \, \langle\Sigma\|\, BL_y\, \|\Pi\rangle} {E_\Sigma - E_\Pi}$ (eq 14) $\beta_p = 4\Sigma(-1)^S \times ~ \frac {\| \langle\Pi\|\, BL_y\, \|\Sigma\rangle \|^2} {E_\Sigma - E_\Pi}$ (eq 15)
p_eff	Λ-type doubling constant in the ²Π_1/2 state (MHz).
a, b, c, d	Magnetic hyperfine coupling constants (MHz) where, $a = 2\mu_{\rm B} g_{\rm N}\mu_{\rm N} \langle 1/r^3\rangle$ (eq 16) $b = - \mu_{\rm B}g_{\rm N}\mu_{\rm N} ~\left\langle {\displaystyle \frac{3\cos^2\chi-1}{r^3}}\right\rangle + {\displaystyle \frac{16}{3}} ~ \pi \mu_{\rm B} g_{\rm N}\mu_{\rm N} ~ \Psi^2(0)$ (eq 17) $c = 3\mu_{\rm B}g_{\rm N}\mu_{\rm N} ~\left\langle \frac{3\cos^2\chi-1}{r^3}\right\rangle$ (eq 18) $d = 3\mu_{\rm B}g_{\rm N}\mu_{\rm N} ~\left\langle \frac{\sin^2\chi}{r^3}\right\rangle$ (eq 19) Here µ_B is the Bohr magneton, µ_N is the nuclear magneton, and g_N is the nuclear g-value.
A	Spin-orbit coupling constants defined by the power series expansion, A = A_e + A₍₁₎ξ + A₍₂₎ξ² + ... .
(O)_S	Spin-rotation interaction constant (ref [21]). Coefficient in the Hamiltonian term (O)_SN · S. This is related to ε_ii's in Lin's notation [19] as (O)_S =1/3(ε_xx + ε_yy + ε_zz).
(ij)_S,T_ij	Spin-rotation interaction constants where i and j are a, b, or c and the Hamiltonian term is Σ(ij)_SN_iS_j. See Curl and Kinsey [21] and Bowater et al. [22] (for T_ij notation) for correspondence to the notation of other workers.
(O)_I,a_I	Fermi interaction constant and coefficient of the I · S operator. (O)_I =(16π/3)g_I µ_Bµ_N [Ψ(O)]².
(ij)_I	Magnetic dipole-dipole interaction constant in the Hamiltonian term (ij)_IS_iI_j where i and j are a, b, or c. (ij)_S =g_S g_I µ_Bµ_N [δ_ij - (3r_ir_j / r³]_Av .
(ij)_Q	Nuclear electric quadrupole interaction constant in the term (ij)_QI_iI_j where i and j are a, b, or c and (ij)_Q =[eQ / 2I(2I - l)] (∂²V / ∂i∂j).
A_S, B_S, C_S	Combination of spin-rotation constants where A_S =(aa)_S + (O)_I,B_S =(bb)_S + (O)_I, and C_S =(cc)_S + (O)_I.
c. Other
*	Asterisks in the uncertainty column indicate that the transition frequency is calculated rather than measured.
(...)	Parentheses in the numerical listings contain measured or estimated uncertainties. For example, the value 1.407(83) should be interpreted as 1.407 ± 0.083. Thus the value in parentheses refers to the last significant digits given.
a,b,c	Designate principal axes corresponding to A, B, and C, respectively.
r(X-Y)	Distance between centers of mass of atoms X and Y (Å).
∠XYZ	Angle formed by atoms X, Y, and Z (degrees).

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Created May 10, 2018, Updated February 19, 2025