Effective Hamiltonian Born-Oppenheimer approximation

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

In both cases we can introduce a similar picture in terms of an effective Hamiltonian giving rise to an effective Schrodinger equation for the solvated solute. Introducing the standard Born-Oppenheimer approximation, the solute electronic wavefunction ) will satisfy the following equation ... 

To determine an effective dressed Hamiltonian characterizing a molecule excited by strong laser fields, we have to apply the standard construction of the free effective Hamiltonian (such as the Born-Oppenheimer approximation), taking into account the interaction with the field nonperturbatively (if resonances occur). This leads to four different time scales in general (i) for the motion of the electrons, (ii) for the vibrations of the nuclei, (iii) for the rotation of the nuclei, and (iv) for the frequency of the interacting field. It is well known that it is a good strategy to take into account the time scales from the fastest to the slowest one. 

In the clamped-nucleus Born-Oppenheimer approximation, with neglect of relativistic effects, the molecular Hamiltonian operator in atomic units takes the form... 

Bunker, P.R., Moss, R.E. Breakdown of Born-Oppenheimer approximation—effective vibration-rotation Hamiltonian for a diatomic molecule. Mol. Phys. 1977,33,417-24 ... 

The connection between the full molecular N-particle Hamiltonian of Equation 12.3 including rotational excitations and dressing fields, and the effective Hamiltonian of (Equation 12.1) can be made using a Born-Oppenheimer approximation. The diagonalization of the Hamiltonian [92] Hbo = - d,Ej - -... 

With the Born-Oppenheimer approximation and assuming that there is no charge transfer between regions, the effective Hamiltonian of the system can be separated into three terms ... 

The Born-Oppenheimer approximation separates the electronic and nuclear coordinates. For Hj, we figured out there are nine coordinates overall. The three coordinates of the electron are separated into the effective Hamiltonian. Once we have obtained the electronic energy as a function of R, we can add and solve the Schrodinger equation along the remaining six coordinates for motion of the nuclei ... 

Applying the Born-Oppenheimer approximation groups the electron-dependent terms into an effective Hamiltonian ... 

The Born-Oppenheimer approximation allowed us to isolate the electronic energy using an effective Hamiltonian for the molecule with fixed, non-rotating nuclei, solving the Schrodinger equation at each value of R (Eq. 5.9) ... 

Nevertheless, progress in these directions has been extremely promising based on approximate three particle variational calculations with suitably modified potentials and effective non-Born Oppenheimer kinetic energy terms in the Hamiltonian. Indeed, predictions based on this ah initio procedure now reproduce existing experimental data for Hg to nearly spectroscopic levels of precision, i.e., within a few hundredths of a cm By virtue of the additional non-Born-Oppenheimer terms in the Hamiltonian, a particularly stringent test is provided by the asymmetrically substituted isotopomers, which prior to these studies had not been observed beyond the fundamental manifold. 

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

There has been considerable controversy in the literature as to whether the nonradiative decay rates kn T are to be evaluated using the adiabatic Bom-Oppen-heimer (ABO) or the crude Born-Oppenheimer (CBO) approximation mo,31,39-43) In Sect. 5 it was noted that the complimentary principle of quantum mechanics requires that the rates exactly calculated within these two schemes be the same provided that 4>s in both schemes is, as expected, a reasonable approximation to the true physical state. As noted also, in both the ABO and the CBO approximations. we have the same mechanistic schemes of (f>s coupled to the effective quasicontinuum 0,. Thus, both cases represent differing, yet reasonable representations of s and 10,). In the present discussion it is necessary to consider the remainder of the vibronic states of the moleculae, 0C in addition to f coupling matrix elements are no longer the effective values, but are actual matrix elements of the perturbing Hamiltonian. 

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