Electron-nucleus Hamiltonian

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

Quantum numbers and shapes of atomic orbitals Let us denote the one-electron hydrogenic Hamiltonian operator by h, to distinguish it from the many-electron H used elsewhere in this book. This operator contains terms to represent the electronic kinetic energy ( e) and potential energy of attraction to the nucleus (vne),... 

We have already seen in Sec. 3.1 that before the Born-Oppenheimer separation of nuclear and electronic motion is made, the Coulomb Hamiltonian has very high symmetry, but that the clamped-nucleus Hamiltonian has only the spatial symmetry of the nuclear framework. That is, the Hamiltonian... 

The electronic term which is the first term in the Hamiltonian written in Eq. (3.13) and used to derive the Solomon and Bloembergen equations (Eqs. (3.16), (3.17), (3.19), (3.20), (3.26), (3.27)) may be inappropriate in many cases, since the electron energy levels may be strongly affected by the presence of ZFS or hyperfine coupling with the metal nucleus. Therefore, the electron static Hamiltonian to be solved to find the cos values, i.e. all electron energy transitions, and their probabilities, will be, in general,... 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

Even when confining the variation of the trial wavefunction to the LCAO-MO coefficients c U, the respective approximate solution of the Schrodinger equation is still quite complex and may be computationally very demanding. The major reason is that the third term of the electronic Hamiltonian, Hel (Equation 6.12), the electron-electron repulsion, depends on the coordinates of two electrons at a time, and thus cannot be broken down into a sum of one-electron functions. This contrasts with both the kinetic energy and the electron-nucleus attraction, each of which are functions of the coordinates of single electrons (and thus are written as sums of n one-electron terms). At the same time, orbitals are one-electron functions, and molecular orbitals can be more easily generated as eigenfunctions of an operator that can also be separated into one-electron terms. 

We have already dealt with the calculation of the wave functions of the hydrogen atom. We now proceed to consider many-electron atoms, first dealing with the simplest such example, the helium atom which possesses two electrons. The Hamiltonian for a helium-like atom with an infinitely heavy nucleus can be obtained by selecting the appropriate terms from the master equation in chapter 3. The Hamiltonian we use is... 

In this section we derive a set of regularized equations of motion and a triple collision manifold (TCM) for the Coulomb three-body system. Three particles (electron, nucleus, and electron) have masses mi = mg, m2 = m and m3 = mg and charges —e, Ze, and —e. We consider the Coulomb three-body system whose Hamiltonian is... 

The geometrical and electronic structure for molecular systems in general will depend on the balance between the different terms in the Hamiltonian i.e. electron-nucleus, electron-electron and nucleus-nucleus interaction including the valence as well as the core electrons of the constituent atoms. The full Hamiltonian for the molecular system is normally separated into a Hamiltonian Hn for the nuclei and another one Hgi for the electrons with fixed positions for the nuclei according to Born Oppenheimer approximation [31]. 

We first consider the symmetric one-electron operator T, which is the sum of operators U, i = 0, N, for each electron. A useful example of t, is the bare nucleus Hamiltonian X, -I- Vi, where T, is the electron—nucleus potential. The second-quantised form for T is found by considering matrix elements for [N + l)-electron determinants p ), p) of orbitals selected... 

We shall use the principle of stationary action to obtain a variational definition of the force acting on an atom in a molecule. This derivation will illustrate the important point that the definition of an atomic property follows directly from the atomic statement of stationary action. To obtain Ehrenfest s second relationship as given in eqn (5.24) for the general time-dependent case, the operator G in eqn (6.3) and hence in eqn (6.2) is set equal to pi, the momentum operator of the electron whose coordinates are integrated over the basin of the subsystem 1. The Hamiltonian in the commutator is taken to be the many-electroii, fixed-nucleus Hamiltonian... 

Second, the Hamiltonian operator for a relativistic many-body system does not have the simple, well-known form of that for the non-relativistic formulation, i.e. a sum of a sum of one-electron operators, describing the electronic kinetic energy and the electron-nucleus interactions, and a sum of two-electron terms associated with the Coulomb repulsion between the electrons. The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics.46... 

The terms in this Hamiltonian are successively, the kinetic energy associated with the nuclei, the kinetic energy associated with the electrons, the Coulomb interaction between the electrons, the electron-nucleus Coulomb interaction and finally, the nuclear-nuclear Coulomb interaction. Note that since we have assumed only a single mass M, our attention is momentarily restricted to the case of a one-component system. 

The reduced-mass and mass-polarization terms arise on transforming the many-electron plus nucleus Hamiltonian to center of mass coordinates. 

R. Harris, L. Stodolski, The effect of the parity violating electron-nucleus interaction on the spin-spin coupling Hamiltonian of chiral molecules, J. Chem. Phys. 73 (1980) 3862-3863. 

The one-electron Hamiltonian only involves differences of geometrical parameters the nuclear attraction terms involve the electron-nucleus distance not the absolute position of either particle. Likewise the molecular integrals dependence on molecular geometry is only via inter-centre distance the fact that the basis functions are atom-centred does not induce any dependence of the integrals on absolute position of the integrals. 

In solid phase the magnetic energy of the electron-nucleus spin system can be described by the static Hamiltonian... 

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