3 TABLE II: Matrix element values of Ω - doubling constant divided to 2 B $^{′}$( R ) and electronic dipole moments (1 D = 0.3934 a.u.) for the ground and the second excited states of PbF molecule H sr constants < Ψ | Δ / 2B $^{′}$| Ψ > State $^{2}$Π$_{1}$$_{/}$$_{2}$ A $^{2}$Σ + 1 / 2 Internuclear distance 3.8 3.9 4.0 4.1 4.2 3.8 3.9 4.0 4.1 4.2 [2] -0.347 [4] -0.303 1.49 This work -0.412 -0.342 -0.369 -0.391 -0.411 1.49 1.42 1.45 1.47 1.49 D(R) ( D ) State $^{2}$Π$_{1}$$_{/}$$_{2}$ A $^{2}$Σ + 1 / 2 Internuclear distance 3.8 3.9 4.0 4.1 4.2 3.8 3.9 4.0 4.1 4.2 [2] 4.62 5.5 This work 4.27 4.264 4.28 4.266 4.255 2.32 2.42 2.55 2.67 2.79 where Ψ e Ω = ˆ U ˜ Ψ e $_{Ω}$, ˆ U = exp ( iβJ e $_{2}$) · exp ( iαJ e $_{3}$) is the unitary operator. Then one can obtain that the inversion, ˆ ˜ P = ˆ U ˆ P ˆ U $^{+}$, in the body frame system of coordinates is same as in eq. (14) of [1]. In the experimental work [4] of the Shafer-Ray group the obtained A$_{⊥}$ values are nearly the same by absolute value but have the opposite sign for both states. How- ever, it was noted in [4] that the energy level structure is determined by the expression, in which A$_{⊥}$ is multi- plied by p , where p = ± 1 is the parity of the electronic- rotational state. Therefore, it was suggested that the parity rather than sign of A$_{⊥}$ has been incorrectly as- signed in previous works. However, we believe that the changes in parity or sign of the perpendicular component are not necessary. Some corrections for the energy level expression are required instead. As we found, ( − 1) J +1 / 2 should be replaced by ( − 1) J − 1 / 2 in eqs. (9) and (23) of [1]. This leads to change of sign in front of pA$_{⊥}$ in eq. (24) of [1] for the energy level structure. where i = √ − 1, N is the number of electrons. Note that the above transformation of the electronic wavefunction (corresponding to the reflection in the η − ζ plane) gives the electronic wavefunction with the opposite sign of Ω, which has the same adiabatic potential. In the case of Ω = 0 electronic wavefunction whether not changed or is multiplied by the − 1 factor. The rotational wavefunc- tions are also transformed into each other with changing sign of Ω [27]: Let us consider the symmetry properties of wavefunc- tion of a diatomic molecule at the space inversion opera- tion in more details. Let x, y, z be a laboratory axes, and the ξ, η, ζ body frame axes in which the molecule is fixed. The α and β ( γ = 0) Euler angles are defined in such a way that the ζ axis coincides with the molecule axis. The vibrational wavefunction is not important here, and electronic-rotational wavefunction within the adiabatic approximation in the Hund case c is written as The total electronic-rotational wavefunction (4), except the case Ω = 0, has no definite parity. In the case Ω = 0 two wavefunctions (4) with different signs of Ω are transformed into each other, and we can form from them positive ( p = 1) and negative ( p = − 1) wavefunctions. Let us construct the electronic-rotational wavefunction with definite parity for the case Ω = ± 1 / 2 of our in- terest, and, in particular, for the $^{2}$Π$_{1}$$_{/}$$_{2}$ and $^{2}$Σ+ states. Since the closed electronic shell is invariant under reflec- tion, consider transformation only of the valence spinor ϕ e $_{ω}$( ξ$_{a}$, η$_{a}$, ζ$_{a}$, ˜ σ ), where ω = ± 1 / 2 is projection of the to- tal angular momentum, j , of the valence electron on the ζ axis. One can verify that for phase factors accepted by eq. (14) of [1] where M$_{J}$ is projection of the total angular momen- tum, J , on the z axis, subscript a enumerate electrons, ˜ σ$_{a}$ = ± 1 is the spin variable which corresponds to the projection of the spin of the a -th electron on the ζ axis, R is the internuclear distance. θ J M J , $_{Ω}$( α, β ) = √ 2 J +1 4 π D J M J , $_{Ω}$( α, β, γ = 0) is rotational wavefunction. In the laboratory system of coordinates the inversion operation, ˆ P , is and hence, accounting for (6) and (7), the electronic- rotational wavefunction | J, M$_{J}$, p 〉 with parity p becomes iϕ e 1 / $_{2}$( − ξ$_{a}$, η$_{a}$, ζ$_{a}$, − ˜ σ ) = iϕ e − 1 / $_{2}$( ξ$_{a}$, η$_{a}$, ζ$_{a}$, ˜ σ ) (7) | J, M$_{J}$ , p 〉 = Ψ e 1 / $_{2}$θ J M J , 1 / 2 + ( − 1) J − 1 / $^{2}$p Ψ e − 1 / $_{2}$θ J M J , − 1 / 2 (8) ˆ P ˜ Ψ e $_{Ω}$( . . . , x$_{a}$, y$_{a}$, z$_{a}$, σ$_{a}$ . . . , R, α, β ) θ J M J , $_{Ω}$( α, β ) = ˜ Ψ e $_{Ω}$( . . . , − x$_{a}$, − y$_{a}$, − z$_{a}$, σ$_{a}$ . . . , R, α + π, π − β ) × θ J M J , $_{Ω}$( α + π, π β ) , Ψ e $_{Ω}$( . . . , ξ$_{a}$, η$_{a}$, ζ$_{a}$, ˜ σ$_{a}$, . . . , R ) θ J M J , $_{Ω}$( α, β ) , (4) θ J M J , $_{Ω}$( α + π, π − β ) = ( − 1) J +2Ω θ J M J , − $_{Ω}$( α, β ) (6)