Helium atom Hamiltonian operator

The helium atom serves as a simple example for the application of this construction. If the nucleus (for which Z = 2) is considered to be fixed in space, the Hamiltonian operator H for the two electrons is... 

Using the result in equation 5.38 we can recast the calculation of the energy of the helium atom as the problem to determine the eigenfunctions of the one-electron Hartree-Fock-Slater Hamiltonian. This identifies universally the Fock operator in the equation... 

In the homogeneous metal, there are no real" charge fluctuations, just as in the Helium atom there is no real" dipole moment. Van der Waals attraction between two Helium atoms comes about as the result of the correlated quantum fluctuations of a virtual dipole moment induced in each atom by the other. Similarly, an attractive force comes about from correlated, virtual, charge fluctuations In the two planes. I eliminate H from the Hamiltonian to leading order in W. using the operator generalization of second-order perturbation theory for the energy levels. This yields ... 

Q Show that the wavefunction given in equation (7.15) is an eigenfunction of the Hamiltonian operator for the helium atom when the electron repulsion term is ignored. 

In 1928,2 years after the Schrodinger equation was published, Hartree proposed a method solving this equation for multiple-electron systems, based on fundamental physical principles the Hartree method (Hartree 1928). Let us consider the electronic motion of a helium atom (Fig. 2.1). The Hamiltonian operator of this... 

Let us consider the exchange of electrons by taking the helium atom as an example. The Hamiltonian operator in Eq. (2.1) does not change by exchanging the coordinates of two electrons ... 

The hydrogen molecule, like the He atom, poses a real problem the Hamiltonian operator contains a term representing the repulsion between the electrons, and the presence of this term makes an exact solution of the Schrodinger equation impossible. In the case of the helium atom we turned to the hydrogen atom for guidance in the choice of approximate wavefunctions. In the case of the hydrogen molecule we turn to the hJ ion and assume that the wavefunction may be approximated by the product of two molecular orbitals... 

In order to properly write the complete form of the Schrodinger equation for helium, it is important to understand the sources of the kinetic and potential energy in the atom. Assuming only electronic motion with respect to a motionless nucleus, kinetic energy comes from the motion of the two electrons. It is assumed that the kinetic energy part of the Hamiltonian operator is the same for the two electrons and that the total kinetic energy is the sum of the two individual parts. To simplify the Hamiltonian, we will use the symbol V, called del-squared, to indicate the three-dimensional second derivative operator ... 

The problem is with the last term lATre rii- It contains a term, ri2, that depends on the positions of both of the electrons. It does not belong only with the terms for just electron 1, nor does it belong only with the terms for just electron 2. Because this last term cannot be separated into parts involving only one electron at a time, the complete Hamiltonian operator is not separable and it cannot be solved by separation into smaller, one-electron pieces. In order for the Schrodinger equation for the helium atom to be solved analytically, it either must be solved completely or not at all. 

The treatment of other atoms in zero order is analogous to the helium and lithium treatments. For an atom with atomic number Z (Z protons in the nucleus and Z electrons), the stationary-nucleus Hamiltonian operator is... 

In the perturbation method the Hamiltonian is written as + H, where corresponds to a Schrodinger equation that can be solved. The perturbation term H is arbitrarily multiplied by a fictitious parameter k, so that A. = 1 corresponds to the actual case. The method is based on representations of energy eigenvalues and energy eigenfunctions as power series in A. and approximation of the series by partial sums. The method can be applied to excited states. In the helium atom treatment the electron-electron repulsive potential energy was treated as the perturbation term in the Hamiltonian operator. 

In order to gain some understanding of the nature of the many-body problem in atoms and molecules, let us consider an array of well-separated systems, a Unear array of helium atoms, for example. By weU-separated we mean that the systems are not interacting. For simplicity, let us begin by considering just two weU-separated systems. The total Hamiltonian operator for the supersystem may be written... 

The helium atom consists of a system with two electrons around a nucleus. This model can be applied to any two-electron system with an atomic number Z including H, Li", and Be. The Hamiltonian includes kinetic energy operators for the two electrons, the Coulombic repulsion potential between the electrons, and a Coulombic attraction between each electron and the nucleus. This is shown schematically in Figure 84. 

Although the proper point of departure for relativistic atomic structure calculations is quantum electrodynamics (QED), very few atomic structure calculations have been carried out entirely within the QED framework. Indeed, almost all relativistic calculations of the structure of many-electron atoms are based on some variant of the Hamiltonian introduced a half century ago by Brown and Ravenhall [1] to understand the helium fine structure. By decoupling the electron and radiation fields in QED to order a (the fine-structure constant) using a contact transformation. Brown and Ravenhall obtained a relativistic momentum-space Hamiltonian in which the electron-electron Coulomb interaction was surrounded by positive-energy projection operators. Owing to the fact that contributions from virtual electron-positron pairs are automatically projected out of... 

---