Hydrogenic functions expectation value

Because particles have wavelike properties, we cannot expect them to behave like pointlike objects moving along precise trajectories. Schrodinger s approach was to replace the precise trajectory of a particle by a wavefunction, i]i (the Greek letter psi), a mathematical function with values that vary with position. Some wavefunctions are very simple shortly we shall meet one that is simply sin x when we get to the hydrogen atom, we shall meet one that is like e x. 

J. B. Mann, Atomic Structure Calculations, Los Alamos Scientific Laboratory, Univ. California, Los Alamos, NM, Part I Hartree—Fock Energy Results for the Elements Hydrogen to Lawrencium, 1967 Part II Hartree-Fock Wave Functions and Radial Expectation Values, 1968. 

To get some idea of the use of trial wave functions and the variation principle, evaluate the expectation value of the energy using the Hydrogen atom Hamiltonian, and normalized Is orbitals with variable Z. That is, evaluate ... 

As a last open shell system, we have chosen the H2C0 radical, which has been well characterized both at the theoretical and experimental level [77-80]. In this radical the symmetry of the singly occupied molecular orbital (a rt-in plane orbital located mainly on the oxygen center) [79] determines that only spin polarization effects contribute to isotropic hcc s of C and O, whereas spin densities at hydrogens have also a direct delocalization contribution. The results of table 5 show that, as expected, isotropic hcc s at the H atoms are well reproduced by aU the functionals, whereas the results for heavy atoms are much more scattered. In particular, the hyperfine constant of the carbon atom is -35 G at the PBEO level, in better agreement with the experimental value (-39 G)[80] than the B3LYP prediction (-34 G). It is noteworthy that the PBEO functional provides values for the H and C atoms which are sufficiently accurate. Unfortunately, no experimental value is available for the oxygen atom. 

Next let us turn to the wave functions of the hydrogen atom in the well. For the states with either = 0 or = 0 the expectation values of the y- and z-coordinate are zero. For n = 0 the wave function is centered near the minimum of the well. With increasing quantum number the center of the wave function... 

Figure 1.14 Expectation value of the electric field strength for the lowest-lying states of a hydrogen-like atom in the range Z = 1-92. Electron wave functions for extended nuclear-charge distributions are employed.

As you see, the main difference is the need to calculate an extra potential term to represent the attraction between the electron and a second nucleus. We can calculate the expectation values for any approximate trial function exactly as in the case of the hydrogen atom, then optimize the result using the variation theorem using any available disposable parameter available. 

In elementary textbooks, one often sees plots of the hydrogen radial distribution function, B Rni(R) P. If we wish to find the expectation value of some function of R, we multiply that function by the... 

Then it turned out that a promising approach (the test was for the hydrogen molecule) is to start with an accurate solution to the Schrodinger equation and go directly toward the expectation value of the Breit-Pauli Hamiltonian with this wave function (i.e., to abandon the Dirac equation), and then to the QED corrections. This Breit-Pauli... 

With Hpar in [97], a different problem is encountered in variational calculations. The occurrence of the delta function renders the expectation value of Hoar proportional to the density at the nucleus—that is, it can be zero, but it can never become negative. In other words, the Darwin term will never lower the energy of the system, and the best we can do in a variational calculation is to make it as small as possible. By choosing a wavefunction for hydrogen of the form... 

For atomic systems, theory can determine rather accurately the absolute shielding constants. First, fairly large nonelectronic contributions such as the zero-point vibrational corrections are absent. Secondly, because of symmetry only the diamagnetic term - an expectation value, which is much easier to compute than an accurate value of the response function - contributes. For instance, for the hydrogen atom the nonrelativistic shielding constant can be calculated as ... 

Our hydrogen atom orbitals correspond to stationary states, as do all wave functions that are products of a coordinate factor and a time factor. The expectation value of any time-independent variable in a stationary state is time-independent, and can be calculated from the coordinate wave function, as shown in Eq. (16.4-4). 

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