3. RESULTADOS DEL ANÁLISIS DE DOCUMENTOS
3.1 FUNDAMENTOS DE LA ERE
turbations. Diabaticisation does not remove the effect of state mixing, but merely shifts its origin fromHvib to Hel. This persistence arises because the approximations made during the definition of Born-Oppenheimer wavefunctions are inherently insufficient.
Either adiabatic or diabatic electronic wavefunctions and potential-energy curves may be chosen to provide a complete description of a diatomic molecule, but in either case must be accompanied by a specification of the state mixing. The more appropriate basis is that which has smaller and less broadly distributed off-diagonal matrix elements, and minimises the set of basis functions necessary to reproduce experimental data. Several studies [145, 167, 174] consider alternative treatments of a coupled state using diabatic and adiabatic representations.
Frequently, a diabatic representation proves simpler to implement, and more efficient to compute. Additionally, physical insight and qualitative predictions may be more easily achieved in the single-configuration picture because off-diagonal mixing terms are less likely to be R-dependent.
Some selection rules for the off-diagonal matrix elements of Hel may be determined without complex calculation. First, the rigorous quantum-numbers; representing total angular-momentum, parity and exchange of nuclei; may not be degraded by state mixing. Then, an electronic perturbation mixing states iandj, are limited to the following cases:
Ji =Jj; (2.33)
e∼e, f ∼f ande≁f; (2.34)
u∼u, g∼g and u≁g. (2.35)
Additionally, for a molecule well-represented by Hund’s case (a) quantum numbers, some additional restrictions may be applied,
Λi= Λj, Ωi = Ωj, Si =Sj and Σi= Σj. (2.36)
As can be seen from Eq. (2.32), Helis composed of terms involving the coordinates of at most two electrons. Only one or two product functions of an electronic configuration in the form of Eq. (2.32) may then be modified by an individual term. This imposes a further restriction on the matrix elements ofhφdi|Hel|φdji if the diabatic states iandj are each represented by a single configuration. That is, all terms of this matrix element will certainly be zero if the configurations ofiand j differ by more than two spin-orbitals.
Theoretical calculations of nonadiabatic or electronic perturbations requires detailed knowledge of the electronic wavefunctions involved. Alternatively, their magnitude, and possibly R-dependence, may be deduced from observed spectra by finding empirical pa- rameters that correctly reproduce energy level shifts, transition intensity anomalies, or predissociation linewidths.
2.5
Rotational perturbations
Born-Oppenheimer wavefunctions, whether adiabatic or diabatic, are calculated assuming nonrotating nuclei, and will not be eigenfunctions of the total Hamiltonian once rotation is included. This results from the breaking of cylindrical symmetry, as discussed in Sec. 2.1. The off-diagonal mixing of electronic states may, however, be small enough to be treated perturbatively for sufficiently slow rotation.
tor; the rotational part of the Hamiltonian, defined in Eq. (2.16), may be written more succintly as Hrot(R) = 1 2µR2 |R| 2 = 1 2µR2 |J −L−S| 2. (2.37)
The second equality in Eq. (2.37) may be expanded by making use of the following relation,
|J −L−S|2=|J|2+|L|2+|S|2−Jz2−L2z−Sz2
−(J+L−+J−L+)−(J+S−+J−S+) + (L+S−+L−S+), (2.38) where
X± =Xx±iXy forX =J, L, orS. (2.39)
In deriving Eq. (2.38) it is necessary to recognise that nuclear rotation is necessarily orthogonal to the internuclear axis, so that Rz = 0.
In Eq. (2.39), raising and lowering operators are written X+ and X−, respectively, where ~pX(X+ 1) is any quantised angular-momentum and its projection along the
internuclear axis is given by~Xz. Pure|XXzistates are not eigenfunctions ofX±. Instead,
they are transformed according to [90, p. 73],
X±|XXzi= (
~[X(X+ 1)−Xz(Xz±1)]1/2|XXz±1i; normal
~[X(X+ 1)−Xz(Xz∓1)]1/2|XXz∓1i; anomalous. (2.40)
The normal case applies to X =L or S but J is anomalous. This occurs because J and
JZ are properly defined in the laboratory frame of reference but the raising and lowering operators in Eq. (2.39) act on Jz = Ω in the molecular coordinate system.
Assuming that a Hund’s case (a) basis is adopted for the electronic wavefunctions, and using Eqs. (2.37) to (2.40), then the diagonal elements of Hrot(R) are given by,
Erot(R) =hJMΩΛSΣ|Hrot(R)|JMΩΛSΣi =B(R) J(J+ 1)−Ω2+S(S+ 1)−Σ2+L(L+ 1)−Λ2 , (2.41) where B(R) = ~ 2 2µR2. (2.42)
Generally,L is not a good quantum number because of the nonsphericity encouraged by the internuclear axis. Thus, neither of the nuclear-rotational and electronic-orbital angular momenta are quantised according to a single value of their respective quantum numbers. The nonquantised term that replaces [L(L+1)−Λ2] in Eq. (2.41) may be formally transferred from Erot(R) to Eel(R). Alternatively, it may be treated as an arbitrary R- dependent fitting parameter which is distinct for each electronic state. TheR-independent expectation value of Erot(R) may then be found by integrating over the internuclear distance, which, assuming a Born-Oppenheimer separation of the wavefunction, leads to the form
Erot=Bv
J(J + 1)−Ω2+S(S+ 1)−Σ2
§2.5 Rotational perturbations 17
where the rotational constant is given by,
Bv= Z ∞ 0 χ†(R) ~ 2 2µR2χ(R)dR, (2.44)
and is sometimes replaced with an approximate value calculated at the equilibrium inter- nuclear distance,Re, according to
Bv ≃ ~2 2µR2 e . (2.45) 2.5.1 L-uncoupling
The three bracketed terms in Eq. (2.38) generate off-diagonal matrix elements ofHrot(R), and are responsible for perturbations between different Hund’s case (a) electronic states. TheL-uncoupling operator arising from nuclear rotation is given by−2µR1 2 (J+L−+J−L+),
and mixes states with Λ and Ω differing by±1 and with common J,M,S, Σ, parity, and
g/u symmetry.
Using the raising and lowering operators of Eq. (2.39), the effect of the second bracketed term of theL-uncoupling operator on a case (a) state is,
J−L+|JMΩΛSΣi=
[J(J + 1)−Ω(Ω + 1)]1/2[L(L+ 1)−Λ(Λ + 1)]1/2~2|JMΩ + 1Λ + 1SΣi, (2.46)
with a similar result for the first term. Here,L±is responsible for the conversion Λ→Λ±1, andJ∓generates Ω→Ω±1. Then, theR-dependent off-diagonal matrix elements mixing
pure case (a) electronic states are given by, JMΩ±1Λ±1SΣ − 1 2µR2J ±L∓ ΩΛSΣ =−B(R) [J(J + 1)−Ω(Ω±1)]1/2[L(L+ 1)−Λ(Λ±1)]1/2. (2.47) BecauseLis rarely a good quantum number, the factor [L(L+ 1)−Λ(Λ±1)]1/2 must be calculated ab initio or treated as an adjustable parameter which may even be R- dependent.
If each relevant electronic state is well defined by a single configuration of single- electron spin-orbitals, only those differing by exactly one orbital may be mixed by the
L-uncoupling operator. This is because, once written as a sum of single-electron operators,
L±=X i
l±i ,
it is clear that each term in the orbital angular-momentum raising and lowering operators will affect precisely one term of a configuration of single-electron spin-orbitals. Addition- ally,J± does not affect the electronic wavefunctions at all and soJ±L∓is a single-electron operator.
2.5.2 S-uncoupling
TheS-uncoupling operator comprises the part ofHrot(R) given by− 1
2µR2(J+S−+J−S+).
otherwise identical electronic states. The observed splitting of sublevels is similar to that caused by the spin-orbit operator discussed in Sec. 2.6, but is dependent on internuclear distance, R, and total angular-momentum, J.
Assuming the validity of the Born-Oppenheimer approximation, the off-diagonal ma- trix elements arising between Hund’s case (a) electronic wavefunctions due to the S- uncoupling operator are
JMΩ±1ΛSΣ±1− 1 2µR2J ±S∓ JMΩΛSΣ =−B(R) [J(J + 1)−Ω(Ω±1)]1/2 [S(S+ 1)−Σ(Σ±1)]1/2. (2.48) 2.5.3 Spin-electronic perturbations
The last parenthesised term in Eq. (2.38) is the spin electronic operator (L+S−+L−S+), which generates off-diagonal matrix elements mixing Hund’s case (a) stationary states with common Ω, according to
JMΩΛ±1SΣ∓1 1 2µR2L ±S∓ JMΩΛSΣ =B(R) [L(L+ 1)−Λ(Λ±1)]1/2 [S(S+ 1)−Σ(Σ∓1)]1/2. (2.49) Because these matrix elements have no J-dependence they may be absorbed into an ef- fective electronic-potential energy.