Atomic Many-Electron Hamiltonian

After having transformed all operators of the many-electron atomic Hamiltonian in Eq. (9.1) to polar coordinates, we may write the Dirac-Coulomb Hamiltonian for an atom explicitly as 

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From the atomic many-electron Hamiltonian we calculate the total electronic energy of an electronic state A according to Eq. (8.146). Then, along the lines of what has been presented in chapter 8, we obtain for the expectation value of an atomic Dirac-Coulomb Hamiltonian [351]... 

Applying the permutation operator P12 is therefore equivalent to interchanging rows of the determinant in Eq. (2.15). Having devised a method for constructing many-electron wavefunctions as a product of MOs, the final problem concerns the form of the many-electron Hamiltonian which contains terms describing the interaction of a given electron with (a) the fixed atomic nuclei and (b) the remaining (N— 1) electrons. The first step is therefore to decompose H(l, 2, 3,..., N) into a sum of operators Hj and H2, where ... 

Because electrons interact pairwise, the many-electron Hamiltonian for any atom or molecule can be written... 

The orbitals of an atom are labeled by 1 and m quantum numbers the orbitals belonging to a given energy and 1 value are 21+1- fold degenerate. The many-electron Hamiltonian, H, of an atom and the antisymmetrizer operator A = (V l/N )Ep sp P commute with total Lz =Ej Lz (i), as in the linear-molecule case. The additional symmetry present in the spherical atom reflects itself in the fact that Lx, and Ly now also commute with H and A. However, since Lz does not commute with Lx or Ly, new quantum... 

The time-consuming calculation is that of the electron-electron repulsion parts in matrix elements of the many-electron Hamiltonian. This calculation is reduced to that of the two-electron integrals, which is expressed in atomic unit as. 

By an apphcation of the DK transformation to the relativistic many-electron Hamiltonian, recently, we have shown that the many-electron DK Hamiltonian also gives satisfactory results for a wide variety of atoms and molecules compared with... 

The Hamiltonian of the two-electron atom already features all pair-interaction operators that are required to describe a system with an arbitrary number of electrons and nuclei. Hence, the step from one to two electrons is much larger than from two to an arbitrary number of electrons. For the latter we are well advised to benefit from the development of nonrelativistic quantum chemistry, where the many-electron Hamiltonian is exactly known, i.e., where it is simply the sum of all kinetic energy operators according to Eq. (4.48) plus all electrostatic Coulombic pair interaction operators. 

Qualitatively speaking, effects of the core electrons on the valence orbitals due to relativistically contracted core shells must be exerted by the surrogate potential. If an appropriate analytical form for this potential has been chosen, its parameters can be adjusted in four-component atomic structure calculations. Hence, the most important step is the choice of the analytical representation of the ECP. Formally we replace the many-electron Hamiltonian Hg/ of... 

In closing this chapter, we should emphasize again a point frequently forgotten by chemist. In the orbital approach to many-electron systems we have a convenient approximation. This is an imperfect but useful way to describe atomic structure. There are more accurate ways to approximate eigenfunctions of many-electron hamiltonians, but this usually involves more difficulty in interpretation. The orbital representation of appears to be the best compromise between accuracy and convenience for most chemical purposes. 

At this point, it is appropriate to make some conmrents on the construction of approximate wavefiinctions for the many-electron problems associated with atoms and molecules. The Hamiltonian operator for a molecule is given by the general fonn... 

In a recent paper Ostrovsky has criticized my claiming that electrons cannot strictly have quantum numbers assigned to them in a many-electron system (Ostrovsky, 2001). His point is that the Hartree-Fock procedure assigns all the quantum numbers to all the electrons because of the permutation procedure. However this procedure still fails to overcome the basic fact that quantum numbers for individual electrons such as t in a many-electron system fail to commute with the Hamiltonian of the system. As aresult the assignment is approximate. In reality only the atom as a whole can be said to have associated quantum numbers, whereas individual electrons cannot. 

Now consider a d ion as an example of a so-called many-electron atom. Here, each electron possesses kinetic energy, is attracted to the (shielded) nucleus and is repelled by the other electron. We write the Hamiltonian operator for this as follows ... 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

Quantum numbers and shapes of atomic orbitals Let us denote the one-electron hydrogenic Hamiltonian operator by h, to distinguish it from the many-electron H used elsewhere in this book. This operator contains terms to represent the electronic kinetic energy ( e) and potential energy of attraction to the nucleus (vne),... 

We assume for simplicity that the two-electron atom is described by a Hamiltonian (29) in which //(I) and //(2) are the hydrogen-like Dirac Hamiltonians and h, 2) = 0. Apparently, after this simplification the problem is trivial since it becomes separable to two one-electron problems. Nevertheless we present this example because it sheds some light upon formulations of the minimax principle in a many-electron case. 

Neglecting spin-orbit contributions (smaller than other relativistic corrections for the ground state of atoms, and zero for closed-shell ones), the Breit hamiltonian in the Pauli approximation [25] (weak relativistic systems) can be written for a many electron system as ... 

Here, F is a many-electron wavefunction and H is the so-called Hamiltonian operator (or more simply the Hamiltonian), which in atomic units is given by. 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

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