Schrodinger equation Coulomb potential

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

We consider an /V-electron system where the electrons experience the mutual Coulomb interaction along with an external potential vv( r) due to the nuclei. The system is then subjected to an additional TD scalar potential (r, t) and a TD vector potential A(r, t). The many-electron wavefunction [Pg.74]

Consequently, from the density the Hamiltonian can be readily obtained, and then every property of the system can be determined by solving the Schrodinger equation to obtain the wave function. One has to emphasize, however, that this argument holds only for Coulomb systems. By contrast, the density functional theory formulated by Hohenberg and Kohn is valid for any external potential. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density n, of the f-th excited state is known, the Hamiltonian H is also known in principle and its eigenvalue problem ... 

The singularity in the interelectron Coulomb potential r 1 creates a cusp in the exact solution of the Schrodinger equation,... 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

It is interesting to note that the Coulomb matrix and the matrix of the nuclear potential present in Vc are opposite in sign. This means that an underestimation, or complete neglect, of the Coulomb matrix will lead to a larger Vc and thus to an overestimation of the relativistic effect. If Vc is negligable compared to 2c the ZORA equation reduces to the non relativistic Schrodinger equation. 

The method ofmany-electron Sturmian basis functions is applied to molecules. The basis potential is chosen to be the attractive Coulomb potential of the nuclei in the molecule. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nuclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

No assumptions concerning potentials u r,r ) and v(r) (like Coulombic character), unless they lead to bounded solutions of the Schrodinger equation [see Eqs. (10)-(12)]... 

Momentum-space methods, pioneered by McWeeny, Fock, Shibuya, Wulfman, Judd, Koga, Aquilanti and others [4,17-26] provide us with an easy and accurate method for constructing solutions to the Schrodinger equation of a single electron moving in a many-center Coulomb potential... 

For large r, G(f, Fo) must vanish, which requires that A = 0. For small distances, where kf < < 1, it should be identical to the Coulomb potential, which requires that 5=1. Finally, we find that the Green s function of the Schrodinger equation in vacuum is the Yukawa potential. 

There are both bound and continuum solutions to the radial Schrodinger equation for the attractive coulomb potential because, at energies below the asymptote the potential confines the particle between r=0 and an outer turning point, whereas at energies above the asymptote, the particle is no longer confined by an outer turning point (see the figure below). 

In principle, it should be possible to obtain the electronic energy levels of the molecules as a solution of the Schrodinger equation, if inter-electronic and internuclear cross-coulombic terms are included in the potential energy for the Hamiltonian. But the equation can be solved only if it can be broken up into equations which are functions of one variable at a time. A simplifying feature is that because of the much larger mass of the nucleus the motion of the electrons can be treated as independent of that of the nucleus. This is known as the Bom-Oppen-heimer approximation. Even with this simplification, the exact solution has been possible for the simplest of molecules, that is, the hydrogen molecule ion, H + only, and with some approximations for the H2 molecule. 

Many problems in nuclear physics and chemistry involve potentials, such as the Coulomb potential, that are spherically symmetric. In these cases, it is advantageous to express the time-independent Schrodinger equation in spherical coordinates (Fig. E.6). The familiar transformations from a Cartesian coordinate system (x, y, z) to spherical coordinates (r, 0, tp) are (Fig. E.6)... 

To find the energy levels of an electron in a hydrogen atom, we have to solve the appropriate Schrodinger equation. An electron in an atom is like a particle in a box, in the sense that it is confined within the atom by the pull of the nucleus, so we can expect quantized energy levels. However, the Coulomb potential energy of the electron, % varies with distance, r, from the nucleus ... 

On the other hand, we sometimes discuss the Schrodinger equation without an absorption potential, for example, the Coulomb few-body Schrodinger equation with the Hamiltonian (3), assuming the energy to take complex values. This is analytic continuation of the quantal problem into fake, complex energies. This way, we depart from what actually occurs in nature for real energies. By regarding the nature as a special... 

The absorption potential — z(1,3Vabs) is quite weak compared with the Coulomb potential, as is noted below Eq. (109). Therefore,1,3in Eq. (115) can normally be replaced by the solution of the spin-independent Coulomb Schrodinger equation... 

Schrodinger s equation for a single electron and a nucleus with Z protons is an extension into three dimensions (x,y,z) of Equation 6.8, with the potential replaced by the Coulomb potential U(r) = Ze2/Ajt8or. This problem is exactly solvable, but it requires multivariate calculus and some very subtle mathematical manipulations which are beyond the scope of this book. 

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