Schrodinger equation adiabatic approximation

The familiar BO approximation is obtained by ignoring the operators A completely. This results in the picture of the nuclei moving over the PES provided by the electrons, which are moving so as to instantaneously follow the nuclear motion. Another common level of approximation is to exclude the off-diagonal elements of this operator matrix. This is known as the Bom-Huang, or simply the adiabatic, approximation (see [250] for further details of the possible approximations and nomenclature associated with the nuclear Schrodinger equation). 

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) which is a solution to the electronic Schrodinger equation. The PES is independent of the nuclear masses (i.e. it is the same for isotopic molecules), this is not the case when working in the adiabatic approximation since the diagonal correction (and mass polarization) depends on the nuclear masses. Solution of (3.16) for the nuclear wave function leads to energy levels for molecular vibrations (Section 13.1) and rotations, which in turn are the fundamentals for many forms of spectroscopy, such as IR, Raman, microwave etc. 

When the non-adiabatic coupling terms x and x 2 are considered negligibly small and dropped from Eq. (B.15), we get the uncoupled approximate Schrodinger equation... 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrodinger equation are derived by employing the time-dependent variational principle (TDVP). 

As a starting point, we consider the Schrodinger equation (30) in the adiabatic classical-path approximation. This equation can be recast in a density-matrix... 

The adiabatic approximation ignores the action of the slow variable kinetic energy operator, T", on the X /jV) When this approximation is made, then the wavefrmction for the entire system can be written as a product of fast and slow factors, T = and the wavefrmction for the slow variable subsystem satisfies the slow variable Schrodinger equation... 

The fundamental approximation used for describing the electron and nuclear motion in molecules and in condensed media is the well-known adiabatic approximation. Let us recall its essence. It is based upon the large difference in the masses of electrons and nuclei. Due to this difference the electron motion is fast in comparison with the nuclear motion, and thus electrons have time to adjust themselves to the nuclear motion and at every moment they can be in a state very close to the one they would be in if nuclei were immobile. Within this picture, as the first step in the construction of the complete wave function of the system, it proves useful to find wave functions describing electron motion with fixed positions of the nuclei, i.e. to resolve the Schrodinger equation... 

Within the adiabatic and one-electron approximations, when electron spin is neglected, electron states in crystals are described by the eigenfunctions and their corresponding eigenvalues, which are the solutions of the Schrodinger equation,... 

The vibrationally adiabatic approximation should be made for the fast HCl vibrational coordinate r. Solving the one-dimensional Schrodinger equation with (R,9) fixed leads to the parameterized form

rotationally averaged potential is then used to compute the one-dimensional potential... 

In the adiabatic approximation one neglects the coupling elements on the right-hand side of (3.33) so that each wavefunction n is determined by a one-dimensional Schrodinger equation t... 

The adiabatic approximation means the neglect of the nuclear motion in the Schrodinger equation. The electronic structure is thus calculated for a set of fixed nuclear coordinates. This approach can in principle be exact if one uses the set of wave functions for fixed nuclear coordinates as a basis set for the full Schrodinger equation, and solves the nuclear motion on this basis. The adiabatic approximation stops at the step before. (The Born-Oppenheimer approximation assumes a specific classical behavior of the nuclei and hence it is more approximate than the adiabatic approximation.)... 

From this formal, but exact statement of Schrodinger s equation, which corresponds to the non-adiabatic approximation referred to above, one sees that the approximation made in obtaining equation (7) is to assume that... 

In the adiabatic approximation, the electron motion is diagonalized under the condition that the atoms are fixed at a value g of the reaction coordinate, as a solution to a Schrodinger equation... 

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