4. Vibronic Couplings

Throughout this chapter, the matrix \({{\varvec{H}}} ({{\varvec{q}}})\) will refer exclusively to the representation of the electronic Hamiltonian \({H}^{el} ({{\varvec{q}}})\). It should not be confused with the matrix representation of the molecular Hamiltonian. The two quantities differ by the nuclear KEO matrix (including the non-adiabatic couplings and corrections, i.e., \({T}^{nu} {{\varvec{1}}} + {\varvec{\Lambda }}\) (see Sect. 3.​2.​4 for the definition of \({\varvec{\Lambda }}\)).

Often, the equation defining the angle is simply given asHowever, some information is missing regarding the relative signs of \(D({{\varvec{q}}})\) and \(W({{\varvec{q}}})\) and the corresponding domains of variation of the angle. A constant value should be added according to the corresponding quadrants if a continuity condition is required. A more accurate formulation iswhere \(-\pi < \text {Arg} (x + i y) \le \pi \) is the principal value of the argument of the complex number \(x + i y\). The corresponding function is sometimes denoted \(\text {atan2} (y,x)\) and this is how it is known in computer programming.

Note that two hypersurfaces that depend on \(3N - 6\) degrees of freedom can, in principle, intersect along a \((3N - 7)\)-dimensional hyperline. In contrast, two adiabatic potential energy surfaces can only intersect along the \((3N - 8)\)-dimensional seam, which is a direct consequence of the possible coupling between the degenerate states along certain directions.

Here, ill-defined local derivatives must be replaced by well-defined directional derivatives. Let us consider a first-order, two-dimensional case: \(\Delta (x,y) = \sqrt{(ax)^2 + (bx)^2}\). This function satisfies \(\Delta (0,0) = 0\), \(\Delta (x,0) = \vert ax \vert \), and \(\Delta (0,y) = \vert by \vert \). It behaves as a two-dimensional absolute-value function and is represented by an elliptic cone in the (x, y)-frame, the apex of which is located at (0, 0). The left- and right-derivatives are well-defined and satisfy \(\lim _{\varepsilon \rightarrow 0^+} \frac{\Delta (\varepsilon ,0) - \Delta (0,0)}{\varepsilon } = -\lim _{\varepsilon \rightarrow 0^-} \frac{\Delta (\varepsilon ,0) - \Delta (0,0)}{\varepsilon } = \vert a \vert \) and \(\lim _{\varepsilon \rightarrow 0^+} \frac{\Delta (0,\varepsilon ) - \Delta (0,0)}{\varepsilon } = -\lim _{\varepsilon \rightarrow 0^-} \frac{\Delta (0,\varepsilon ) - \Delta (0,0)}{\varepsilon } = \vert b \vert \). They are opposite on both sides. However, \(\lim _{\varepsilon \rightarrow 0} \frac{\Delta (\varepsilon ,0) - \Delta (-\varepsilon ,0)}{2\varepsilon } = \lim _{\varepsilon \rightarrow 0} \frac{\Delta (0,\varepsilon ) - \Delta (0,-\varepsilon )}{2\varepsilon } = 0\), which shows that a two-point formula is inadequate here: the local derivatives are ill-defined because of the cusp at the origin.

This apparent double-valuedness issue is related to what is known as the geometrical Berry phase (see Refs. [7, 16, 23, 24] for further details).

In fact, the same result can be obtained with a less restrictive hypothesis. It is enough to consider that both \(\vert \overline{\Phi }_1; {{\varvec{q}}}\rangle \) and \(\vert \overline{\Phi }_2; {{\varvec{q}}}\rangle \) vary smoothly with \({{\varvec{q}}}\) around \({{\varvec{q}}}_{\text {X}}\) such that \(\langle \overline{\Phi }_\alpha ; {{\varvec{q}}}_{\text{ X }} \vert \partial _j\Phi _\beta ; {{\varvec{q}}}_{\text{ X }}\rangle \) take finite values (as opposed to \(\langle \Phi _1^{el/ad}; {{\varvec{q}}} \vert \partial _j\Phi _2^{el/ad}; {{\varvec{q}}}\rangle \) that diverges when \({{\varvec{q}}}\) tends to \( {{\varvec{q}}}_{\text{ X }}\)). Using a Hellmann-Feynman-like formula, we getwhere we used, first, the fact that \(\overline{\Phi }\)-type states coincide with the eigenstates at \({{\varvec{q}}}_{\text{ X }}\) and, second, that they are assumed to be orthonormal,In addition, it is worth noticing that, for \(\alpha , \beta = 1,2\) (restriction to two states), there is no explicit involvement of the other possible eigenstates (there are no couplings \(\langle \overline{\Phi }_1; {{\varvec{q}}}_{\text{ X }} \vert \partial _j\overline{\Phi }_\gamma ; {{\varvec{q}}}_{\text{ X }}\rangle \) or \(\langle \overline{\Phi }_2; {{\varvec{q}}}_{\text{ X }} \vert \partial _j\overline{\Phi }_\gamma ; {{\varvec{q}}}_{\text{ X }}\rangle \) for \(\gamma \ge 3\) in the above expressions). This holds to first order but a second-order expansion would require considering such terms.

The working basis set (with the bar) was chosen as a pair of crude adiabatic states for which \(\varphi _{\text {ref}} = 0\) corresponded by convention to a specific pair of degenerate adiabatic states obtained from an actual calculation. This arbitrary angle occurs to get fixed implicitly from symmetry considerations when both degenerate states potentially belong to different irreducible representations (see Chap. 7). However, in a general situation, the value of \(\varphi _{\text {ref}}\) can be fixed for convenience through an extra constraint according to context. For example, it can be used to make the two branching-space vectors orthogonal. Alternatively, it may be convenient to fix the value of \(\varphi _{\text {ref}}\) to change from the original \(\overline{{\varvec{H}}} ({{\varvec{q}}})\) to an equivalent \({{\varvec{H}}} ({{\varvec{q}}})\) according to a condition such that the off-diagonal term is zero at some reference geometry, \({{\varvec{q}}} = {{\varvec{q}}}_{\text {ref}}\), where the states are not degenerate (often, the Franck-Condon point). This ensures coincidence with the adiabatic representation at this point in addition to coincidence at the conical intersection \({{\varvec{q}}}_{\text{ X }}\). The entries of both equivalent Hamiltonian matrices are related throughSetting \(W({{\varvec{q}}}_{\text {ref}}) = 0\) with \(D({{\varvec{q}}}_{\text {ref}}) = \Delta ({{\varvec{q}}}_{\text {ref}}) > 0\) thus yieldsThis is also useful in the context of degenerate perturbation theory in order to identify \({{\varvec{V}}} ({{\varvec{q}}}_{\text{ X }} + \delta {{\varvec{q}}})\) to \({{\varvec{H}}} ({{\varvec{q}}}_{\text{ X }} + \delta {{\varvec{q}}})\) defined as the the result of the diagonalisation of \(\overline{{\varvec{H}}} ({{\varvec{q}}}_{\text{ X }} + \delta {{\varvec{q}}})\) upon setting \({{\varvec{q}}}_{\text {ref}} = {{\varvec{q}}}_{\text{ X }} + \delta {{\varvec{q}}}\).

This notation reflects that \(\partial _{q_1} q_1 = 1\) and \(\partial _{q_2} q_1 = 0\) while \(\partial _{q_1} q_2 = 0\) and \(\partial _{q_2} q_2 = 1\).

Note that, as already shown, a two-state problem gets explicit solutions for the eigenvalues but this is no longer the case for problems involving more states. If so, the eigenvalues are to be calculated numerically using some diagonalisation procedure (for example the Jacobi algorithm).

This situation looks similar to an avoided crossing with respect to the adiabatic curves, see Fig. 4.3, except that now the strongly-coupled diabatic states produce curves that do not cross but rather coincide with the adiabatic ones at the origin; a \(\frac{\pi }{4}\)-rotation, sometimes referred to as a Nikitin transformation, would generate alternative diabatic states that cross and are weakly coupled. This would swap the roles of \(q_1\) and \(q_2\) as tuning and coupling modes, respectively.