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Hydrocarbon Spectral Database, List of Symbols

4. List of Symbols

In most cases, a uniform set of quantum state and molecular parameter symbols is employed. This common set is listed here with a brief description of the molecular quantity represented by the symbol. However, there are a few special cases for which the reader is required to consult the literature cited to obtain this information since the Hamiltonian employed is unique to those species and reanalysis was not carried out here. The species for which the above applies are: CH, CH2, CH4, C2H, CH3CD3, C3H, C4H, and CH3CCCD3. The spectra of these species were not refit in this work since the published analyses are considered to be accurate and not readily reproduced by the authors. There are several instances where excited vibrational states have been assigned and the type of vibrations identified, i.e., bend, torsion, etc., but the mode number is not identified. We have followed the notation in the literature and used the following characters for the type of vibration: B = bend, T = torsion, P = ring pucker.

4.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-, 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.
A, E Torsional symmetry substates representing irreducible representations of the symmetry group of the rotation-internal rotation Hamiltonian.
ν1,ν2,ν3 Vibrational modes (ν) and quantum numbers (υ)
U or L Upper or lower energy level or transition frequency.
′ or ″ Prime or double prime is used to distinguish the upper (′) and lower (″) energy states in a transition. They occur as superscripts on the quantum numbers.
Quantum number for vibrational angular momentum.
I or (Ii) Angular momentum quantum number of nuclear spin for one (or ith) nucleus.
4.b. Molecular Constants
A, B, C Rotational constants (MHz). These are related to the principal moments of inertia: A = h/8π2Iα, 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, , G, g Octic centrifugal distortion constants (MHz).
Iα Moment of inertia of the methyl top around internal rotation axis (u Å2)
ρ Internal rotation interaction constant ρ = [Σx(λxIα /Ix)2]1/2.
λa, λb, λc Direction cosines between the internal rotation axis and the principal axes a,b,c, respectively.
α Angle of rotation around internal rotation axis.
F Internal rotation dynamical constant (GHz) F = h/8π2r Iα.
V3 Threefold component of torsional barrier potential V = V3(1-cos 3α)/2.
s Reduced barrier height s = 4V3 / 9F.
r r = 1 - Σx(λx2Iα / Ix).
Θ Angle between CH3 symmetry axis and a-principal axis.
ω1(s) Fourier coefficient.

Internal rotation interaction constant

$$\Delta_0= 3F~\frac{a_1(s)}{2}=\frac{28}{8}~F\,\omega_1(s) ~ .$$

(eq 13)

µaµbµc Components of the electric dipole moment along the a- or b- or c-principal axes.
αv , γvv Rotation-vibration coefficients in the power series representing Bv (see text).
qv -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).
χxx - χyy 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 gJ for linear molecules in the ground state (µN).
gxx - gyy Anisotropy of the molecular G tensor perpendicular to the molecular axis (µN).
cx or M Spin rotation constant related to nucleus X (kHz).
SXY Spin-spin interaction constant between nucleus X and nucleus Y (kHz).
αP , βP

Ω-type doubling parameters

$$\alpha_{\rm P}=4\Sigma(-1)^s \times \frac{\langle\Pi| \,(A+2B) L_y \, 
|\Sigma\rangle \, \langle\Sigma| \,B\,L_y \,|\Pi\rangle} {E_\Sigma-E\Pi} $$

$$\beta_{\rm P}=4\Sigma(-1)^s \times \frac{|\langle\Pi| \,B\,L_y 
\,|\Sigma\rangle|^2}{E_\Sigma-E\Pi} ~ .$$

(eq 14)


(eq 15)

peff Λ-type doubling constant in the 2Π1/2 state (MHz).
a, b, c, d

Magnetic hyperfine coupling constants (MHz) where,

$$a=2\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N} (1/r^3)$$

$$b=-\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N} 
\left\langle \frac{3\cos^2\, \chi-1}{r^3}\right\rangle +\frac{16}{3}~
\pi\,\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N}~\Psi^2(0) $$

$$c=3\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N} 
\left\langle \frac{3\cos^2\, \chi-1}{r^3}\right\rangle$$

$$d=3\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N} 
\left\langle \frac{\sin^2\, \chi}{r^3}\right\rangle ~ .$$

(eq 16)

(eq 17)


(eq 18)


(eq 19)

Here µB is the Bohr magneton, µN is the nuclear magneton and gN is the nuclear g-value.

A Spin-orbit coupling constants defined by the power series expansion, A = Ac + A(1)ξ + A(12)ξ2 + ... .
4.c. Other
# A hash mark located in front of the uncertainty denotes that the frequency is the hyperfine free center of the frequency.
(...) 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.


Sensor Science Division

Created April 23, 2018, Updated November 15, 2019