Adiabatic ansatz

Inserting this adiabatic ansatz into the Schrodinger Eq. (6), followed by multiplication from the left with and integration over the electronic coordinates, immediately leads to the following equation for the nuclear motion... 

In which cases is the adiabatic ansatz (15) expected to be accurate As shall become clear in the next subsection and from the discussion in Sec. 5, the adiabatic ansatz is accurate for an electronic state which is well separated energetically from all other electronic states. Under the same electronic conditions, the ansatz improves with increasing nuclear mass. On the other hand, we can expect that the larger the energetical distance of... 

At what order of k do other electronic states mix into the description of the dynamics As can be seen from Eq. (46), this mixing depends on the nonadiabatic couplings Aji between our electronic state j and the other states, i j. For the rigid coordinates, these couplings scale as K , but their contribution to the energy starts at second order perturbation theory and hence as k . This mixing imphes that the total wavefunction ceases to be merely a product of an electronic and a nuclear wavefunction. The adiabatic ansatz (15) looses its validity and one has to resort to the expansion (5) of the total wavefunction. 

In complete analogy to the diabatic case, the equations of motion in the adiabatic representation are then obtained by inserting the ansatz (29) into the time-dependent Schrodinger equation for the adiabatic Hamiltonian (7)... 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

If the system under consideration possesses non-adiabatic electronic couplings within the excited-state vibronic manifold, the latter approach no longer is applicable. Recently, we have developed a simple model which allows for the explicit calculation of RF s for electronically nonadiabatic systems coupled to a heat bath [2]. The model is based on a phenomenological dissipation ansatz which describes the major bath-induced relaxation processes excited-state population decay, optical dephasing, and vibrational relaxation. The model has been applied for the calculation of the time and frequency gated spontaneous emission spectra for model nonadiabatic electron-transfer systems. The predictions of the model have been tested against more accurate calculations performed within the Redfield formalism [2]. It is natural, therefore, to extend this... 

The surface-hopping trajectories obtained in the adiabatic representation of the QCLE contain nonadiabatic transitions between potential surfaces including both single adiabatic potential surfaces and the mean of two adiabatic surfaces. This picture is qualitatively different from surface-hopping schemes [2,56] which make the ansatz that classical coordinates follow some trajectory, R(t), while the quantum subsystem wave function, expanded in the adiabatic basis, is evolved according to the time dependent Schrodinger equation. The potential surfaces that the classical trajectories evolve along correspond to one of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the subsystem evolution is carried out coherently and may develop into linear combinations of these states. In such schemes, the environment does not experience the force associated with the true quantum state of the subsystem and decoherence by the environment is not automatically taken into account. Nonetheless, these methods have provided com-... 

In the adiabatic approximation21,22 one assumes that the right hand side of Eq. (1-6) can be neglected. This approximately corresponds to the following ansatz for the total wave function,... 

In summary, the model allows for two types of interactions between the mirror spaces, the weak kinematical perturbation and the adiabatic and sudden limits equivalent to Eq. (17) or Eqs. (29)-(34). The overwhelming rate of particles over antiparticles in the Universe is inferred in this picture once the particular particle state has been selected. The Minkowski metric of the special theory of relativity is represented here by a non-positive definite metric, Eq. (8), bringing about a quantum model with a complex symmetric ansatz. Although the latter permits general symmetry violations, it is nevertheless surprising that fundamental transformations between complex symmetric representations and canonical forms come out unitary. 

Kutzelnigg, W. The adiabatic approximation 1. The physical background of the Born-Handy ansatz. Mol. Phys. 1997, 90,909-16. 

The LVC and QVC approaches avoid the explicit construction of diabatic states because they result in a parametrized form of the adiabatic PES this can be used to determine the coupling parameters by comparing the parametrized form of the adiabatic PES with results from electronic structure calculations for these surfaces (diabatization by ansatz [26]). On the other hand, the fixed functional form leads to a model shape of the PES which may not always be flexible enough to reproduce these data well. To overcome this limitation, a modified construction scheme for the diabatic potential matrix has been introduced [27,28] where the LVC approach is applied to the adiabatic-to-diabatic (ADT) mixing angle only . In this form it can... 

This is without doubt a significant improvement since the application of the Born-Handy ansatz as a full replacement of the COM separation is not restricted by the size of the system under investigation. However, its applicability is unfortunately limited to adiabatic systems only. The Born-Handy formulation yields only the adiabatic corrections to the B-0 results. Beyond this approach we enter the enigmatic region, which is usually denoted as the break-down of B-O approximation. Therefore our main goal is here to find an extension of the Born-Handy formula , which is valid both in the adiabatic limit as well as beyond. 

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