Complex energy formalism

In this chapter, discussion will be restricted to intensity effects arising from bound bound interactions. Bound free predissociation and autoionization interactions are discussed in Chapters 7 and 8. Interactions between broadened states (or resonances) are dealt with by a complex-energy formalism in Section 9.3. 

This question was answered some time ago in Ref. [101], where we showed how the corresponding complex eigenvalue Schrodinger equation (CESE) is derived via the appropriate consideration of boundary condifions. The concomitant results justify the computation of resonance sfafes in ferms of non-Hermitian, complex-energy formalism via fhe use of superposifions of square-integrable real and complex funcfions. 

Chu S I 1991 Complex quasivibrational energy formalism for intense-field multiphoton and above-threshold dissociation—complex-scaling Fourier-grid Hamiltonian method J. Chem. Phys. 94 7901... 

Group 2 complexes are formally electron deficient and conformationally floppy only small energies (often only 1-2 kcal mol-1) are required to alter their geometries by large amounts (e.g., bond angles by 20° or more). In such cases, the inclusion of electron-correlation effects becomes critical to an accurate description of the molecules structures. Both HF/MP2 and density functional theory (DFT) methods have been applied to organoalkaline earth compounds. DFT approaches, which implicitly incorporate electron correlation in a computationally efficient form, are generally the more widely used. Molecular orbital calculations that successfully reproduce bent... 

It must be emphasised that the relatively low activation energy of the insertion reaction in this complex is characteristic of complexes having formally a d° 16-electron configuration, which is just adopted by the cationic group 4 metallocene species complexed with a coordinating olefin as the two-electron donor [136]. 

To summarize, for our model Hamiltonian, resonances appear after a bound-virtual and a virtual-virtual resonance transition. There is no method to obtain virtual energies using a square-integrable basis set, even in the complex-rotated formalism. Then, at this point we can ask if FSS is a useful method to study this kind of resonance. As we will show in the next subsection, the answer is yes FSS is a method to obtain near-threshold properties, and with FSS we can characterize the near-threshold resonances by solving the Hermitian (not complex-rotated) Hamiltonian using a real square-integrable basis-set expansion. Moreover, the critical point of the virtual resonance-resonance transition, Xr, could also be obtained using FSS. 

Combining with the non-hermltlan Floquet theory (also called the complex quasi-energy formalism) (37. 38), the CCCC method can be extended to the study of photodlssoclatlon or multiphoton dissociation of vdW molecules. Work in this direction is in progress. 

Resonances are defined formally as poles of the scattering matrix in the complex energy or momentum plane (98-100). The pole location in the complex energy plane may be written as... 

Multiphoton processes taking place in atoms in strong laser fields can be investigated by the non-Hermitian Floquet formalism (69-71,12). This time-independent theory is based on the equivalence of the time-dependent Schrodin-ger description to a time-independent field-dressed-atom picture, under assumption of monochromaticity, periodicity and adiabaticity (69,72). Implementation of complex coordinates within the Floquet formalism allows direct determination of the complex energy associated with the decaying state. The... 

The two previous secfions were devoted to modeling quantum resonances by means of effective Hamiltonians. From the mathematical point of view we have used two principal tools projection operators that permit to focus on a few states of interest and analytic continuation that allows to uncover the complex energies. Because the time-dependent Schrodinger equation is formally equivalent to the Liouville equation, it is attractive to try to solve the Liouville equation using the same tools and thus establishing a link between the dynamics and the nonequilibrium thermodynamics. For that purpose we will briefly recall the definition of the correlation functions which are similar to the survival and transition amplitudes of quantum mechanics. Then two models of regression of a fluctuation and of a chemical kinetic equation including a transition state will be presented. 

In the following sections, I discuss basic elements of formalisms for these three cases, i.e., the description involves either real energy, or complex energy, or time. Their proper consideration provides insight into the formal aspects of the QM of resonances and into the problem of how to compute them efficiently. 

---