. Author manuscript; available in PMC: 2017 Sep 25.

Published in final edited form as: J Phys Chem A. 2015 Dec 1;119(49):11862–11874. doi: 10.1021/acs.jpca.5b10067

Table 3.

Equivalence relations between the model property P and actual properties α, G′ and A. The electric (µ) and magnetic (m) dipoles, used in IR and VCD intensities, are also reported. M is the atomic axial tensor (AAT).⁹⁴

P	P₀	P_i	P_ji	P_jki	s₀	s₁	s₂	S
μ	μ^eq	$\frac{\partial μ}{\partial q_{i}}$	$\frac{\partial^{2} μ}{\partial q_{i} q_{j}}$	$\frac{\partial^{3} μ}{\partial q_{i} q_{j} q_{k}}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2 \sqrt{2}}$	$\frac{1}{6 \sqrt{2}}$	+1
m	0	M_i	$\frac{\partial M_{i}}{\partial q_{j}}$	$\frac{\partial^{2} M_{i}}{\partial q_{j} q_{k}}$	$\frac{𝚤 ħ}{\sqrt{2}}$	$\frac{𝚤 ħ}{\sqrt{2}}$	$\frac{𝚤 ħ}{\sqrt{2}}$	−1

α	α^eq	$\frac{\partial α}{\partial q_{i}}$	$\frac{\partial^{2} α}{\partial q_{i} q_{j}}$	$\frac{\partial^{3} α}{\partial q_{i} q_{j} q_{k}}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2 \sqrt{2}}$	$\frac{1}{6 \sqrt{2}}$	+1
G′	G′^eq	$\frac{\partial G^{'}}{\partial q_{i}}$	$\frac{\partial^{2} G^{'}}{\partial q_{i} q_{j}}$	$\frac{\partial^{3} G^{'}}{\partial q_{i} q_{j} q_{k}}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2 \sqrt{2}}$	$\frac{1}{6 \sqrt{2}}$	+1
A	A^eq	$\frac{\partial A}{\partial q_{i}}$	$\frac{\partial^{2} A}{\partial q_{i} q_{j}}$	$\frac{\partial^{3} A}{\partial q_{i} q_{j} q_{k}}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2 \sqrt{2}}$	$\frac{1}{6 \sqrt{2}}$	+1