Hamiltonian in atomic units

For H2, let us write down the zeroth-order electronic Hamiltonian (in atomic unit) ... 

These transformations result in the internal Hamiltonian (in atomic units) ... 

For simplicity, let us illustrate the calculations of entanglement for two spin- particles. The general Hamiltonian, in atomic units, for such a system is given by [57]... 

N stands here for the number of elementary cells and yja(r) = Xa(r—Rj—Ra) is an AO centred in the cell j at Rj+Ra. Applying the non-relativistic electronic Hamiltonian (in atomic units)... 

In the context of attempts to use only units based on the SI system an alternative definition of atomic units was proposed. One starts from the SI system (with 4 basic units), and measures then mass, electric charge, action and quantities of the dimension of the dielectric constant of the vacuum in units of m, e, h, and 47reo respectively. On this way one arrives at the same Hamiltonian in atomic units, as following Hartree - as long as no magnetic quantities are involved. We must therefore reconsider the system of units, when we come to electrons in the presence of magnetic fields (section 2,10). 

When formulating the Hamiltonian in atomic units, we prefer to have the choice between the original Hartree system (with 6 = 1/c) and an SI-based system or a system based on esu (with 6=1). The latter two lead formally to the same Hamiltonian, and for all three choices the potential of a point nucleus is —Z/r. 

The effective N-electron Hamiltonian (in atomic units) for the development of our MR-MP algorithm is taken to be the relativistic no-pair Dirac-... 

Treating the radiation in the semiclassical dipole approximation, the Hamiltonian operator in the presence of an electric field e(t) may be written in atomic units as ... 

The Hamiltonian of helium, in the center of mass frame and under the action of an electromagnetic field polarized along the x axis, with field amplitude F and frequency w, reads, in atomic units,... 

Treating the electronic Schrodinger equation in the usual clamped nuclei (Bom-Oppenheimer) approximation, [20] we have (in atomic units) the Hamiltonian, H, and the spectrum of eigenvalues and eigenvectors, and fife,... 

Within the Born-Oppenheimer approximation, the non-relativistic electronic Hamiltonian of an A-electron molecular system in the presence of an external potential can be written (in atomic units) as... 

External fields are introduced in the relativistic free-particle operator hy the minimal substitutions (17). One should at this point carefully note that the principle of minimal electromagnetic coupling requires the specification of particle charge. This becomes particularly important for the Dirac equation which describes not only the electron, but also its antiparticle, the positron. We are interested in electrons and therefore choose q = — 1 in atomic units which gives the Hamiltonian... 

Fig. 1. BLYP/uncDZ mean dipole polarizability of the mercury atom as a function of frequency. All values in atomic units. SR+SO refers to calculations based on the Dirac-Coulomb Hamiltonians, whereas SR refers to calculations in which all spin-orbit interaction has been eliminated.

One of the purposes of this work is to make contact between relativistic corrections in quantum mechanics and the weakly relativistic limit of QED for this problem. In particular, we will check how performing plane-wave expectation values of the Breit hamiltonian in the Pauli approximation (only terms depending on c in atomic units) we obtain the proper semi-relativistic functional consistent in order ppl mc ), with the possibility of analyzing the separate contributions of terms with different physical meaning. Also the role of these terms compared to next order ones will be studied. 

The direct variational solution of the Schrddinger equation after separation of the center of mass motion is in general possible and can be performed very accurately for three- and four- body systems such as (Kolos, 1969) and H2 (Kolos and Wolniewicz, 1963 Bishop and Cheung, 1978). For larger systems it is unlikely to perform such calculations in the near future. Therefore the usual way in quantum chemistry is to introduce the adiabatic approximation. The nonrelativistic hamiltonian for a diatomic N-electron molecule in the center of mass system has the following form (in atomic units). 

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. 

In this equation H, the Hamiltonian, is defined by precise quantum mechanical rules and can be written in atomic units (Appendix A. 10-1) as... 

Another interesting example is the chaotic autoionization of molecular Rydberg states caused by the interaction of the electron with the degrees of freedom of the core. We consider the model in which the core consists of a positive Coulomb charge plus a rotating dipole that lies in the same plane of the electron orbit (m = l). The Hamiltonian reads (atomic units)... 

If the relativistic effects are sufficiently large and therefore cannot be accounted for as corrections, then as a rule one has to utilize relativistic wave functions and the relativistic Hamiltonian, usually in the form of the so-called relativistic Breit operator. In the case of an N-electron atom the latter may be written as follows (in atomic units, in which the absolute value of electron charge e, its mass m and Planck constant h are equal to one, whereas the unit of length is equal to the radius of the first Bohr orbit of the hydrogen atom) ... 

The Hamiltonian operator for a confined quantum system is written in atomic units as... 

The total energies and wavefunctions of the Hamiltonian (1) have been calculated as the eigenvalues and eigenvectors of a Cl matrix. Full Cl has been used for all calculations of quasi-one-dimensional quantum dots and for quasi-two-dimensional quantum dots with N = 2, while multi-reference Cl has been used for quasi-two-dimensional quantum dots with N = 3 and 4. The results are presented in atomic units. They can be scaled by the effective Bohr radius of 9.79 nm and the effective Hartree energy of 11.9 meV for GaAs semiconductor quantum dots [25,26]. 

The basis for studies of the electronic structure of materials is the (non-relativistic) time independent Schrodinger equation, HQ> = 0, where the Hamiltonian, H, in atomic units is ... 

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