Hamiltonian central field case

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

As alluded to previously, the reason we discuss dynamics for two-dimensional systems arises from the fact that all one-dimensional conservative Hamiltonians are integrable and therefore do not admit chaotic motion. However, two-dimensional systems are in general anharmonic and nonintegrable, except in certain special cases (a particle in a central field in two dimensions, a two-dimensional normal-mode oscillator, etc.). Thus, Hamiltonians of the type given previously represent the simplest conservative systems that can exhibit dynamical chaos. Note that any function of four variables that is equal to a constant must correspond to a three-dimensional surface embedded in the fourdimensional space. 

The electron mass should be replaced by the reduced mass, in the case of hydrogen equal to p = m(l - 1/1836), since the nucleus has a finite mass equal to 1836m. The right part of Equation 2.1, where atomic units are used, assumes that the nucleus is infinitely heavy. The attraction between the electron and the nucleus is given by V(r) = -Z/r. Since V only depends on the distance r to the nucleus (central field), a polar coordinate system is used (Figure 2.1). The complete expression in polar coordinates for the Hamiltonian operator is given by... 

The actual form of the Hamiltonian H assumes that the nucleus can be treated as a point charge. Since exact solutions, i.e. those based on an explicit form of the Coulomb interaction, are only known in the one-electron cases, some method of approximation must be utilized. This is usually the central-field approximation. Each electron is assumed to move independently in the field of the nucleus and an additional central field composed of the spherically averaged potential fields of each of the other electrons in the system. In other words, each electron is treated... 

The 1-electron 1-centre eigenfunctions, which satisfy an equation of the form (1.2.9), are atomic orbitals (AOs) they are hydrogen-like AOs if we use the Hamiltonian (1.2.2a), corresponding to one electron in the field of the bare nucleus, but may have a somewhat different functional form if they are eigenfunctions of a more general Hamiltonian (1.1.2b) in whidi the potential V may allow partly for the presence of other electrons. In any case the AOs are essentially those of a central field, in w ich V depends only on the radial distance of the electron from the nucleus, and have characteristic forms which are well described in elementary textbooks on valency. For completeness, the forms of the Is, 2s, 2p,..., 4f orbitals and the significance of the classification are reviewed in Appendix 1. 

In this case, then, the orbital degeneracy can be described by introducing into the spin Hamiltonian the operator for the unquenched part of the total orbital angular momentum. The term y4L S2 is the analogue of the commonly adopted free-atom form fL S, which is appropriate in a strictly central-field situation. Again, however, the effective Hamiltonian corresponds to a simple model, all the complexities of the real system... 

We can speak of h as the hamiltonian for the one-particle case, and avail ourselves of the familiar methods to study the solutions of Eq. (10-318). For a central electrostatic field h takes the form... 

In quadrupolar nuclei, the situation differs notably the quadrupolar interaction only affects spins with I>% and is created by electric field gradient resulting from the asymmetry of charge distribution around the nucleus of interest. The quadrupolar interaction is characterized by the nuclear quadrupolar coupling constant Cq (from 0 in symmetrical environments to tens or hundreds of MHz) and an asymmetry parameter T]q. NMR spectra are usually recorded when Cq Vl the Larmor frequency of the quadrupolar spin. In such a case, the NMR spectrum can easily be simulated First, the first-order quadrupolar Hamiltonian, which is the quadrupolar interaction Hamiltonian truncated by the Larmor frequency, has to be taken into account. The first-order quadrupolar interaction (or the zeroth-order term in perturbation theory) is an inhomogeneous interaction and is modulated by MAS and does not affect symmetrical transition —m +m. Therefore, in half-integer spins, the single-quantum central transition (CT, i.e., —1/2 +1/2) is not affected by the first-order quadrupolar inter-... 

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