Potential nucleus

Tranquillizing drugs based on the phenothiazine nucleus potentiate the action of narcotic analgesics [157] although there are some reports to the contrary [158] and certain derivatives, notably methotrimeprazine (levomepromazine, Levo-prome, Veractil) (—)-(LV), are claimed to have intrinsic analgesic activity. In mice, (-)-(LV) is more potent than morphine in a variety of tests [159, 160] for example hot-plate EDso value of 1.02 mg/kg, that of morphine 2.1 mg/kg. 

Some of these quantities are experimentally measurable from scattering experiments [1,2] or relate to physically interesting quantities which appear in different problems. For example p ) gives the kinetic energy of any nonrelativistic system, (r ) is related to the diamagnetic susceptibility, (r ) relates to the electron-nucleus potential energy,... 

Figure 6.1 Schematic representations of a general neutron-nucleus potential and a proton...

Figure 7.12 Proton-nucleus potential for the semiclassical calculation of the 151Lu partial proton half-life. [From S. Hofmann, In D. N. Poenaru (Ed.), Nuclear Decay Modes, Copyright 1996 by IOP Publishing. Reprinted by permission of IOP Publishing.]...

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... 

The electron—nucleus potential is not quite trivial. For larger atoms the radial functions that are large at the nucleus are affected by the finite charge distribution of the nucleus. It is sufficient to use the potential for a uniform charge distribution of radius R... 

The appropriately chosen electron-nucleus potential is denoted as Vnuc(0- Usually, we shift the energy scale with /T = /3 — 1 to yield energy expectation values which are directly comparable with those obtained from nonrelativistic calculations. 

A general central electron-nucleus potential may be written... 

The integral representation in Eq. (32) shows that V r) is, in general, a continuous function, even when p(r) is not continuous. As a consequence, the electron-nucleus potential V (r) itself will be continuous and continuously differentiable, in general. In addition, we understand immediately... 

Radial functions and energy eigenvalues for hydrogen-like atoms with this electron-nucleus potential are well-known in closed from, both in the non-relativistic and in the relativistic case. They can be found in every good textbook on quantum mechanics, for a compact reference see [48]. 

Due to the discontinuity of p r) at r = R, the second and higher derivar tives of V r) do not exist at this point. Only in the non-relativistic case the radial functions for hydrogen-like atoms with this electron-nucleus potential are known analytically in closed form. The corresponding energy eigenvalues must be determined iteratively [57-59]. 

The latter always contains a centrifugal part and the modified electron-nucleus potential The additional term U r) is optional,... 

The application of a modified electron-nucleus potential together with analytical basis functions requires the evaluation of appropriate matrix elements (nuclear attraction integrals) ... 

Consequently, the difficult problem for a system interacting electrons has been mapped onto that of a system of noninteracting electrons moving in an effective potential. It is essential that the electron feels the summarized field of all other electrons. The effective potential for an electron in a point is a sum of the nucleus potential and potentials of all other electrons. 

The radius R has to be fixed empirically and may be understood as the "size" of the nucleus. This charge density distribution leads through Eq. (6.150) and multiplication by qe = to the homogeneous electron-nucleus potential energy operator... 

We have already seen above that the choice of a point-like atomic nucleus limits the Dirac theory to atoms with a nuclear charge number Z < c, i.e., Zmax 137. A nonsingular electron-nucleus potential energy operator allows us to overcome this limit if an atomic nucleus of finite size is used. In relativistic electronic structure calculations on atoms — and thus also for calculations on molecules — it turned out that the effect of different finite-nucleus models on the total energy is comparable but distinct from the energy of a point-like nucleus (compare also section 9.8.4). 

We approach the effect of finite nuclear charge models from a formal perspective and introduce a general electron-nucleus potential energy Vnuc, which may be expanded in terms of a Taylor series around the origin. 

As can be seen from the last section, this expression holds for closed- and open-shell systems. If we add the electron-nucleus potential, this is just the potential of the ion, which the electron leaves behind, i.e., —(Z — N + l)/r. 

As was pointed out in section 9.6 the Eqs. (9.105) with a singular electron-nucleus potential of Coulomb type yield nonanalytic solution functions. To obtain regular functions for the point-like nuclei the operator identity... 

As can be seen from the table, the effect on the total energy is notable when switching from the singular potential of a point nucleus to a finite-nucleus potential. The finite-nucleus potentials, however, not differ very much. Most important is the effect on relative energies. Here we may note that the effect... 

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