U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Physical Measurement Laboratory

Triatomic Spectral Database, List of Symbols

Share

4. List of Symbols

a. Quantum Numbers

JResultant total angular momentum quantum number, excluding nuclear spins.
NRotational angular momentum quantum number, excluding electron and nuclear spins, in the case where electron spin is present.
KProjection of J (or N) on the symmetry axis in the limiting prolate or oblate symmetric top.
F1Resultant angular momentum quantum number including nuclear spin for one nucleus.
FResultant total angular momentum quantum number.
λ or εQuantum 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 LUpper 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.
ℓQuantum number for vibrational angular momentum.
I or (Ii )Angular momentum quantum number of nuclear spin for one (or ith) nucleus.
SResultant angular momentum number of electron spins. Σ is the projection of S on the molecular axis.
ΛAbsolute values of the projection of the resultant orbital angular momentum on the molecular axis.
ΩAbsolute value of the projection of the total electronic angular momentum on the molecular axis.
Σ, Π, ΔElectronic state designation for which Λ = 0,1,2, respectively.

b. Molecular Constants

A, B, C
 

script a, b, c

 

Rotational constants (MHz). These are related to the principal moments of inertia: A = h/8π2Ia, etc.
$${\bar B}$$B-bar equals (B + C)/2.
τ, Δ, δ, DQuartic centrifugal distortion constants (MHz or kHz).
H, hSextic centrifugal distortion constants (MHz or kHz).
L, ℓ, G, gOctic centrifugal distortion constants (MHz).
P, p, FDectic centrifugal distortion constants (MHz).
eQqaa(X), ...
χaa(X), ...		Nuclear electric quadrupole coupling constant along principal axis indicated for nucleus X (MHz).
eQq ηProduct of the nuclear quadrupole coupling constant and the asymmetry parameter for the bending vibrational state.
µa,bComponents of the electric dipole moment along the a- or b-principal axes.
αv , γvvRotation-vibration coefficients in the power series representing Bv (eq 3).
qvℓ-doubling constant (MHz).
σ|| , σ⊥Components of the magnetic shielding tensor which are parallel and perpendicular to the molecular axis, respectively.
QMolecular 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).
χxx - χyyMagnetic 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 gJ for linear molecules in the ground state (µN).
gxx - gyyAnisotropy of the molecular G tensor perpendicular to the molecular axis (µN).
cX or MSpin rotation constant related to nucleus X (kHz).
SXYSpin-spin interaction constant between nucleus X and nucleus Y (kHz).
αp , βp

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)

 

peffΛ-type doubling constant in the 2Π1/2 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 µB is the Bohr magneton, µN is the nuclear magneton, and gN is the nuclear g-value.

ASpin-orbit coupling constants defined by the power series expansion, A = Ae + A(1)ξ + A(2)ξ2 + ... .
(O)SSpin-rotation interaction constant (ref [21]). Coefficient in the Hamiltonian term (O)SN · S. This is related to εii's in Lin's notation [19] as (O)S =1/3(εxx + εyy + εzz).
(ij)S,TijSpin-rotation interaction constants where i and j are a, b, or c and the Hamiltonian term is Σ(ij)SNiSj. See Curl and Kinsey [21] and Bowater et al. [22] (for Tij notation) for correspondence to the notation of other workers.
(O)I,aIFermi interaction constant and coefficient of the I · S operator. (O)I =(16π/3)gI µBµN [Ψ(O)]2.
(ij)IMagnetic dipole-dipole interaction constant in the Hamiltonian term (ij)ISiIj where i and j are a, b, or c. (ij)S =gS gI µBµN [δij - (3rirj / r3]Av .
(ij)QNuclear electric quadrupole interaction constant in the term (ij)QIiIj where i and j are a, b, or c and (ij)Q =[eQ / 2I(2I - l)] (∂2V / ∂i∂j).
AS, BS, CSCombination of spin-rotation constants where AS =(aa)S + (O)I ,BS =(bb)S + (O)I , and CS =(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,cDesignate principal axes corresponding to A, B, and C, respectively.
r(X-Y)Distance between centers of mass of atoms X and Y (Å).
∠XYZAngle formed by atoms X, Y, and Z (degrees).
Analytical chemistry, Atomic / molecular / quantum and Reference data

Contacts

Sensor Science Division

Created May 10, 2018, Updated February 19, 2025