The L-Uncoupling Operator

The L-uncoupling operator, —(l/2//ii2)(J+L-+J-L+), which is responsible for the evolution as J increases from Hund s case (a) to case (d), causes numerous perturbations between states that differ by AO = AA = 1 and with AS = 0. This specific type of rotational perturbation is often called a gyroscopic perturbation. 

Since J+L = J+ 17) the L-uncoupling operator is a one-electron operator, and consequently, in the single-configuration approximation, the configurations describing the two interacting states can differ by no more than one spin-orbital. The electronic part of the perturbation matrix element is then proportional to the same (7r+ l+ r) or b parameter that appeared in the spin-electronic perturbation. However, owing to the presence of the J+ operator, the total matrix element of the B(.R)J+L operator between (fi — 1 and ft) is proportional to [J(J + 1) — 0(0 — 1)]1 2- 

Therefore, owing to the effects of the /-mixing electrostatic perturbation, the L-uncoupling matrix element is slightly different from zero between states of different nominal /, since for a nonspherical system / is no longer a good 

If the energy interval between the A-components of a Rydberg /-complex is similar to the magnitude of the vibrational interval, degeneracy can occur, for example, between a II(np7r) v = 0 level and a E(npcr) v = 1 level. Equation (3.5.5) shows that a nonzero interaction can occur between these two levels. 

Consider now a perturbation between a cr7r47r ni state and a r2n3n Eq state. Such an example appears in the spectrum of CO (see Fig. 3.19 Field, et al., 1972). It is easy to see from the method presented in Section 3.2.3 that the 7t37t configuration gives rise to two E states from 7r+(7r-)27r,+ and (7r+)27r-7r subshell occupancies. The 1E- and 1E+ wavefunctions, properly symmetrized with respect to the crv operator acting only on the spatial part of the wavefunction, must be constructed as linear combinations of the four possible A = 0, E = 0 Slater determinants  

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When the S-uncoupling operator acts between two components of a multiplet state that belong to the same vibrational quantum number, then the vibrational part of the B(R)L S matrix element is... 

However, if by chance there is a near degeneracy between the O spin-component of the nth level and the fi = fi 1 components of the (v + l)th level of the same electronic state, then the S-uncoupling operator can cause a perturbation between these levels. In the harmonic approximation and using the phase choice that all vibrational wavefunctions are positive at the inner turning point,... 

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

Effantin, et al., (1982), Amiot and Islami (1986), and Wada and Kanamori (1999), in their analysis of CO a3n a 3E+ perturbations, found it necessary to include in their Heff fit model, in addition to the spin-orbit ( o) and L-uncoupling (/ o) off-diagonal interaction parameters, a direct spin-spin interaction parameter (e) as well as a second-order Hso x HROT interaction term (p3). The p3 parameter is not a centrifugal correction to either o or (3o. It is a cross term between the implicit iZ-dependence of the spin-orbit coupling constant of the c3n state and the explicit iZ-dependence of the B(R)J-h perturbation operator. 

Figure 6.22 displays the rotational plus electronic fine structure of the NO 15/ <— A2E+(v = 1) transition from the N = 3, Ms = —1 /2 Zeeman component of the intermediate level (Guizard, et at, 1991). The parity selection rule permits transitions from the — parity TV = 3 initial level of the A2E+ state to the three + parity N+ = 1,3,5 rotational clusters (separated by 10B+ and 18B+) of an nf complex. The structure of an nf <— A2E+IV = 3, Ms = +1/2 transition (not shown) is identical to that originating from the Ms = —1/2 Zeeman component. The electric dipole transition operator operates exclusively on the spatial coordinates of the electron, thus AMs = 0 is a rigorous selection rule. Since the d-character of the A2E+ state is exclusively responsible for making the nf <— A2E+ transition allowed, one expects Z-polarized transitions that terminate on mi = —2, —1,0, +l,+2 Zeeman components in each N+ cluster. The observed intensity patterns in Fig. 6.22b are in excellent agreement with those calculated for an uncoupled case (d) <— case (b) pure / d transition (Guizard, et al., 1991). 

We have assumed that no angular momentum contribution assists. Then the basis set of spin functions consists of the uncoupled set Sa,Msa) Sb, Msb), or the coupled set SA,SB,S,Ms), its size is N = (2Sa + l)(2Ss + 1). Additionally, the orbital angular momentum can be added and then the basis set becomes a direct product of all orbital and spin functions. In a special case, spin delocalisation (double exchange) operates. 

The result follows from spin orthogonality. It is perfectly clear from experiments on atoms and molecules (Zeeman effect) that singlet-triplet and other apparently spin-forbidden transitions do occur, so we are led to assume that the spin and orbital motions of an electron are not uncoupled. In the above example, the transition moment will never vanish identically if the state with spin-orbit coupling operator ffjo- Assume that the state with index n and spin function a-j can interact with another state, say tn, with spin perturbation theory, the corrected state xi l is given by... 

Here f denotes the transition operator and E the energy transferred to the system (e.g. the photon energy in optical absorption). The initial state l kj) (with energy Ei) can, but need not, belong to the set ) of vibronic states (with energies E ) which constitute the set of final states given by the solutions of the vibronic Hamiltonian. In the applications to be discussed below, l i) will be the zero vibrational level of the electronic ground state which is assumed to be vibronicaUy uncoupled from the excited states. 

The uncoupled basis is formed by the determinants la o l, a, pa and The result of acting with the isotropic part of Hamiltonian on these determinants can directly be written down with the help of Eqs. 3.79, but the anisotropic part requires a little more work. Based on the relations given in Eqs. 1.16a and 1.20a, the following is easily derived for the products of one-electron operators... 

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