Potential acting on an electron

On the other hand, it still remained a mystery how it is that the mutual action of the electrons does not completely destroy the free mobility of the electron. For this mutual action is not, as might be expected, a feeble one, but is of the same order as that between the atoms and the electrons, and certainly cannot be completely explained by a screening effect. If in the total potential acting on an electron. 

We now make a further approximation beyond assuming the wave function to be a product of one-electron orbitals. We assume that the effective potential acting on an electron in an atom can be adequately approximated by a function of r only. This central-field approximation can be shown to be generally accurate. We therefore average Vi(ri, 01, < i) over the angles to arrive at a potential energy that depends only on ri. 

The auxiliary Hamiltonian is chosen to have the usual kinetic operator and an effective local potential acting on an electron of spin o at point r. 

Pi is the projector on the /-space of spherical harmonics. V r) is a function of r describing the potential acting on an electron locaUy of / symmetry. The matrix elements of semi-local pseudopotentials can be expressed in terms of almost analytical expressions, the computational time of which is negligible with respect to that of the two-electron matrix elements. The most general pseudopotentials can be written in the non-local form... 

Zhao and Yang [7, 8] define the potential acting on an electron in a molecule (PAEM) as the interaction energy of a local electron that belongs to the molecule (say electron 1) with all the nuclei and the remaining electrons. The instantaneous Coulomb interaction energy of the first electron at r with the rest of particles is... 

Since p r)/N measures the probability of finding the first electron at r and )] P2 r,r2) the probability of finding the first electron at r while the second electron is at r2, the potential acting on an electron at r is expressed as... 

The function is thus a local quantity, which has different values at different points in the species, N is the total number of electrons, p is the chemical potential and V is the potential acting on an electron due to all nuclei present. Since p r) as a function of N has slope discontinuities, Eq. (4) provides the following three reaction indices [14] ... 

CCSD(T)/aug-cc-pVDZ calculations and molecular face theory have been applied to the 5 2 reaction between F and CH3CI. The calculations indicate that the molecular intrinsic characteristic contour (MICC) of F contracts (the electron density on the Mice increases) slowly as the reactant complex forms. Then, the MICC of the fluoride ion increases (the electron density decreases) rapidly as one goes to the transition state and to the product complex. The MICC contracts and the electron density at chlorine increases throughout the reaction. The potential acting on an electron in a molecule (the PAEM) decreases between F and C and increases between C and Cl. 

Sommerfeld suggested that the potential in a metal crystal could be assumed constant. This assumption implies that the forces acting on an electron cancel to zero and that the electrons in a metal can be described like a non-interacting gas of electrons, confined to a box that represents the metal. The only restriction on electronic motion would be the Pauli principle. The electronic energy in a three-dimensional rectangular box is known as... 

In variance with the hydrogen-like and Slater functions the potential employed to formally construct the gaussian basis states has nothing to do with the real potential acting upon an electron in an atom. On the other hand the solutions of this (actually three-dimensional harmonic oscillator problem) form a complete discrete basis in the space of orbitals in contrast to the hydrogen-like orbitals. 

In the following sections, particularly for the calculation of phonon spectra, the correct small-wave-vector limit of the pseudopotential is required for the internal consistency of the calculations. This limit is easily calculated here. The ionic contribution for —> 0, is the Fourier transform of the Coulomb potential of an ion with charge —Ze acting on an electron with charge e. Therefore ... 

Ojiq, acting on an electron near the interface is given by Equation 2, while in the vapor above the liquid surface the potential is given by Equation 1. The image potential of an electron in a liquid with r = 2 and in the vapor with 1 is shown in Figure 13. 

We follow Thompson and Mead [13] to discuss the behavior of the electronic Hamiltonian, potential energy, and derivative coupling between adiabatic states in the vicinity of the D31, conical intersection. Let A be an operator that transforms only the nuclear coordinates, and A be one that acts on the electronic degrees of freedom alone. Clearly, the electronic Hamiltonian satisfies... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

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