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Hamiltonian equations approximation

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... [Pg.334]

Oppenheimer approximation, hydrogen molecule, Hamiltonian equation, 514— 516... [Pg.86]

We first construct an adiabatic approximation for the classical Hamiltonian equations of motion. We start from an unconstrained model with a classical... [Pg.174]

Now, the two-state Hamiltonian with approximations made can be simulated near Rx by the model Hamiltonian equation (26) with parameters cos 6 - h a - , Ae - Aeeft. The transition probability P can be obtained from equations (27) and (22) provided interference is neglected ... [Pg.343]

Conditions in Strong Fields.—It will be necessary to discuss this case, though only our method of approach is new, while the results were known from Bohr s and Kramers work. If the electric perturbation dominates over the relativistic, our process of successive approximations must start from considering the electric terms. As we have pointed out, the most important one among them is the degenerate term (11). Our second step will be, therefore, the consideration of the Hamiltonian equation... [Pg.4]

The construction of approximate electronic wave functions is a difficult many-body problem. The source of the difficulties is the presence of the two-body electron-electron repulsion term in the Hamiltonian equation [Eq. (3)]. In the absence of this term, there would be no interactions among the electrons and it would be sufficient to consider one electron at the time, independently of the others. Indeed, in this case, the... [Pg.59]

The delightful thing about one-electron operators is that we can exactly solve the Schrodinger equation if the Hamiltonian is approximated by its one-electron part V = S, h(i) = Hq) since a separable wavefunction can be construaed as a produa of one-particle funaions, ,(ry). [Pg.68]

Table 3.1 Effective Nuclear Charge (Scaling Parameter) Z ff for Approximate Spin-Orbit Interaction Calculations Using the One-Electron Term in the Breit-Pauli Hamiltonian (Equation 3.8, developed by Koseki et... Table 3.1 Effective Nuclear Charge (Scaling Parameter) Z ff for Approximate Spin-Orbit Interaction Calculations Using the One-Electron Term in the Breit-Pauli Hamiltonian (Equation 3.8, developed by Koseki et...
Let us assume the Bom-Oppenheimer approximation (p. 269). Thus, the nuclei occupy some fixed positions in space, and in the electronic Hamiltonian equation (12.59), we have the electronic charges qj = —e and masses mj = nto = m (we skip the subscript 0 for the rest mass of the electron). Now, let us refine the Hamiltonian by adding the interaction of the particle magnetic moments (of the electrons and nuclei the moments result from the orbital motion of the electrons, as well as from the spin of each particle) with themselves and with the external magnetic field. We have, therefore, a refined Hamiltonian of the system, the particular terms of the Hamiltonian corresponds to the relevant terms of the Breit HamiltonianS (p. 147)... [Pg.763]

Desclaus has developed a computer code to solve the many-electron Dirac-Rock equation for atoms in a numerical self-consistent method. In this method, the relativistic Hamiltonian is approximated within the Dirac-Fock method, ignoring the two-electron Breit interaction. The Breit interaction is introduced as a first-order perturbation to energy after self-consistency is achieved. Relativistic wavefunctions and energies calculated this way are available for a number of atoms. ... [Pg.292]


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See also in sourсe #XX -- [ Pg.276 , Pg.277 ]




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