Eigenfunctions of Hamiltonian

The exact s-state eigenfunctions of Hamiltonian Eq. (36) has the functional form... 

In order to obtain virtual and resonances states, we have to find eigenfunctions of Hamiltonian (101) which grows up exponentially when x —> oo. Using the fact that the potential goes to zero very fast, we can obtain... 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

Further, the time-independent electionic basis functions are taken to be the eigenfunctions of the electionic Hamiltonian,... 

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

We write them as i / (9) to shess that now we use the space-fixed coordinate frame. We shall call this basis diabatic, because the functions (26) are not the eigenfunction of the electronic Hamiltonian. The matrix elements of are... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

The vibrational part of the molecular wave function may be expanded in the basis consisting of products of the eigenfunctions of two 2D harmonic oscillators with the Hamiltonians ffj = 7 -I- 1 /2/coiPa atid 7/p = 7p - - 1 /2fcppp,... 

To obtain the Hamiltonian at zeroth-order of approximation, it is necessary not only to exclude the kinetic energy of the nuclei, but also to assume that the nuclear internal coordinates are frozen at R = Ro, where Ro is a certain reference nucleai configuration, for example, the absolute minimum or the conical intersection. Thus, as an initial basis, the states t / (r,s) = t / (r,s Ro) are the eigenfunctions of the Hamiltonian s, R ). Accordingly, instead of Eq. (3), one has... 

Note that only the polynomial factors have been given, since the exponential parts are identical for all wave functions. Of course, any linear combination of the wave functions in Eqs. (D.5)-(D.7) will still be an eigenfunction of the vibrational Hamiltonian, and hence a possible state. There are three such linearly independent combinations which assume special importance, namely,... 

The eigenfunctions of the zeroth-order Hamiltonian are written with energies. ground-state wavefunction is thus with energy Eg° To devise a scheme by Lch it is possible to gradually improve the eigenfunctions and eigenvalues of we write the true Hamiltonian as follows ... 

Two states /a and /b that are eigenfunctions of a Hamiltonian Hq in the absence of some external perturbation (e.g., electromagnetic field or static electric field or potential due to surrounding ligands) can be "coupled" by the perturbation V only if the symmetries of V and of the two wavefunctions obey a so-called selection rule. In particular, only if the coupling integral (see Appendix D which deals with time independent perturbation theory)... 

To consider the question in more detail, you need to consider spin eigenfunctions. If you have a Hamiltonian X and a many-electron spin operator A, then the wave function T for the system is ideally an eigenfunction of both operators ... 

Since j/[ and xpl are eigenfunctions of the same hamiltonian, namely that in Equation (1.65), they are orthogonal, which means that, when v f v". 

The Hartree-Fock determinant and all of the substituted determinants are eigenfunctions of Hg these are the solutions to the part of the divided Hamiltonian for which we have a solution. Thus ... 

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