Relativistic atomic Hamiltonian

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)  

In these formulas the Dirac matrices ad) and ft are defined in the usual way  

The terms H5+H6 are often written in the form of the sum of magnetic (Hm) and retarding (Hr) interactions, sometimes also called the relativistic Breit operator 

The energy operator considered above is an approximation, in which only the lowest terms of the correction for the retardation of the interaction are taken into account. More general is the formal quantum-electrodynamical interaction energy operator in the approximation of the exchange of one virtual photon [58] 

Here coy is the difference in eigenvalues of the energy operator. Only the real part of Eq. (2.14) corresponds to the energy, therefore, the exponent may be replaced by coscoyry. The lowest terms of the expansion of cosooyry in powers of 1/c lead to the Breit approximation discussed above. 

---

Let us consider in a similar way the relativistic atomic Hamiltonian in Breit approximation (2.1). Terms Hl, H2 and H3 are one-electron operators. It is very easy to find their irreducible forms. Operators H2 and H3 are complete scalars and do not require any transformations, whereas for H we have to substitute (19.1) and (19.2) in (2.2). This gives... 

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... 

While using (4.14) and exact wave functions. This supports the conclusion of Drake [83] that for electric dipole transitions, by considering the commutator of with the atomic Hamiltonian in the Pauli approximation, we obtain Qwith relativistic corrections of order v2/c2 (see (4.18)-(4.20)). However, for many-electron atoms and ions, one has to use approximate (e.g., Hartree-Fock) wave functions, and then this term gives non-zero contribution, conditioned by the inaccuracy of the model adopted. 

While Ho has the familiar form of the sum of the non-relativistic atomic/molecular Hamiltonian, (55), based purely on Coulombic interactions, and the Hamiltonian for free radiation (48), H has the unfamiliar feature of involving the essentially arbitrary Green s function g(x,x ) because no gauge for the vector potential is specified. In particular the form (55) does not require the Coulomb gauge condition. Of course, overall H is gauge-invariant, and observables must be as well, so we need to consider gauge-invariant calculation. 

Here, a is the Dirac matrix, E is a certain energy parameter, and Ho is the Hamiltonian of an isolated atom whose eigenvalues are denoted by er Ho r) = er r). Equation (4) can be readily inferred (see paper of V. Yakhontov in this volume) by taking into account the relativistic atom-photon field interaction... 

In this note, we discuss different approximation schemes for the evaluation of the two-photon transition rate between discrete states. Non relativistic atomic hydrogen is used as a test of the reliability of the methods. We consider a one particle system described by a Hamiltonian Ho, whose eigenstates and eigenvalues are denoted by n> and En, respectively. In the gauge with divA = 0, the interaction of the particle with the electromagnetic field has the usual form... 

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 this section we briefly review the main properties of the Dirac equation that is the basic equation to start with to build a relativistic effective Hamiltonian for atomic and molecular calculations. This single particle equation, as already stated in the introduction, was established in 1928 by P.A.M Dirac [1] as the Lorentz invariant counterpart of the Schrodinger equation. On a note let us recall that the first attempts to replace the Schrodinger equation by an equation fulfilling the requirements of special relativity started just after quantum... 

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... 

According to the relativistic Breit Hamiltonian, the EH MV DF values are more accused as the velocity of the involved object, in this case an electron, increases. In this way, the zones where KE increases will present a high value of MV variation, which will be mostly localised surrounding the atoms of the molecule. Away from atomic centres, the value of MV DF decreases, and it turns most similar to eDF due to the KE in these points is low and more homogeneous in space. 

The atomic Hamiltonian (11.1) does not involve electron spin. In reality, the existence of spin adds an additional term, usually small, to the Hamiltonian. This term, called the spinr-orbit interaction, splits an atomic term into levels. Spin-orbit interaction is a relativistic effect and is properly derived using Dirac s relativistic treatment of the electron. This section gives a qualitative discussion of the origin of spin-orbit interaction. 

When a proper relativistic derivation of the spin-orbit-interaction term Hs.o. the atomic Hamiltonian is carried out, one finds that for a one-electron atom (see Bethe and Jackiw, Chapters 8 and 23)... 

---