Hamiltonian No-pair

The definition of the pair part or, equivalently the no-pair part of Hmat is not unique. The precise meaning of no-pair implicitly depends on the choice of external potential, so that the operator Hm t depends implicitly on the external potential, whereas the sum Hmat Hm t + Hm t is independent of the choice of external potential. Since the no-pair part conserves the number of particles (electrons, positrons and photons) we can look for eigenstates of Hm j in the sector of Fock space with N fermions and no photons or positrons. Following Sucher [18,26,28], the resulting no-pair Hamiltonian in configuration space can be written as... 

Since the relativistic many-body Hamiltonian cannot be expressed in closed potential form, which means it is unbound, projection one- and two-electron operators are used to solve this problem [39], The operator projects onto the space spanned by the positive-energy spectrum of the Dirac-Fock-Coulomb (DFC) operator. In this form, the no-pair Hamiltonian [40] is restricted then to contributions from the positive-energy spectrum and puts Coulomb and Breit interactions on the same footing in the SCF calculations. 

Computer programs were MOLCAS 3 program system (25) for SCF, CASSCF, and CASPT2 calculations and the program TITAN for closed shell calculations (26). The new version of the COMENIUS program was used for open shell CCSD(T) calculations based on the spin adapted singly and doubly excited amplitudes (15, 27-29). These codes were supplemented by the generator of the no-pair hamiltonian written by B. A. Hess in all DK calculations. 

In ECP theory an effective Hamiltonian approximation for the all-electron no-pair Hamiltonian Hnp is derived which (formally) only acts on the electronic states formed by nv valence electrons in the field of N frozen closed-shell atomic-like cores ... 

Having defined our starting point, the second quantized no-pair Hamiltonian, we may now take a closer look at the relations between the matrix elements. For future convenience we will also change the notation of these matrix elements slightly. Due to hermiticity of the Dirac Hamiltonian and the Coulomb-Breit operator we have... 

In principle problems of relativistic electronic structure calculations arise from the fact that the Dirac-Hamiltonian is not bounded from below and an energy-variation without additional precautions could lead to a variational collapse of the desired electronic solution into the positronic states. In addition, at the many-electron level an infinite number of unbound states with one electron in the positive and one in the negative continuuum are degenerate with the desired bound solution. A mixing-in of these unphysical states is possible without changing the energy and might lead to the so-called continuum dissolution or Brown-Ravenhall disease. Both problems are avoided if the Hamiltonian is, at least formally, projected onto the electronic states by means of suitable operators (no-pair Hamiltonian) ... 

The Douglas-Kroll transformation [40] of the Dirac-Coulomb Hamiltonian in its implementation by HeB [41-45] leads to one of the currently most successful and popular forms of a relativistic no-pair Hamiltonian. The one-electron terms of the Douglas-Kroll-HeB (DKH) Hamiltonian have the form... 

Since the model potential approach yields valence orbitals which have the same nodal structure as the all-electron orbitals, it is possible to combine the approach with an explicit treatment of relativistic effects in the valence shell, e.g., in the framework of the DKH no-pair Hamiltonian [118,119]. Corresponding ab initio model potential parameters are available on the internet under http //www.thch.uni-bonn.de/tc/TCB.download.html. 

In the study of the vibronic spectrum of a doublet HCCS radical, Peric et al. calculated the spin-orbit coupling constant at the equilibrium geometry of the radical by using the two-component relativistic no-pair Hamiltonian derived by Samzow et al. In the calculation, truncated (8,8)MRDCI wave functions were used with orbitals optimized for the triplet state of the corresponding cation. The spin-orbit coupling constant of 261 cm agreed well with the experimental data. 

Later, in Sec. 4, we will give a detailed discussion of the need for the no-pair Hamiltonian in relativistic calculations, its limitations, and its relation to QED. To establish a foundation for our studies of few-electron systems, we start in Sec. 2 with a discussion of the one-electron central-field Dirac equation and radiative corrections to one-electron atoms. In Sec. 3 we describe many-body perturbation theory (MBPT) calculations of few-electron atoms, and finally, in Sec. 4 we turn to relativistic configuration-interaction (RCI) calculations. 

Later, when making comparisons with nonrelativistic calculations, we subtract the electron rest energy mc firom e. ) The choice of the potential U r) is more or less arbitrary one important choice being the (Dirac) Hartree-Fock potential. Eigenstates of Eq. (1) fall into three classes bound states with —mc < k < rn< , continuum states with > m( , and negative energy (positron) states ej < —me . Since contributions from virtual electron-positron pairs are projected out of the no-pair Hamiltonian, we will be concerned primarily with bound and continuum electron states. 

Now let us turn to the many-body perturbation theory treatment of atoms with more than one electron. As discussed in the introduction, our approach is the no-pair Hamiltonian, which is given by... 

It should be noted that the projection operator A+ and, consequently, the no-pair Hamiltonian depends on the background potential U. One finds however that energies obtained from the no-pair Hamiltonian are only weakly dependent on the potential and that small differences between calculations starting from different potentials can be accounted for in terms of omitted negative-energy corrections. We elaborate on this point in Sec. 4. 

To bypass the complicated issue of constructing A+ in configuration space, we work with the second-quantized version of the no-pair Hamiltonian,... 

The rules of perturbation theory associated with the relativistic no-pair Hamiltonian are identical to the well-known rules of nonrelativistic many-body perturbation theory, except for the restriction to positive-energy states. The nonrelativistic rules are explained in great detail, for example, in Lindgren and Morrison [30]. Let us start with a closed-shell system such as helium or beryllium in its ground state, and choose the background potential to be the Hartree-Fock potential. Expanding the energy in powers of V) as... 

In the above equations, we denote sums over occupied levels by letters (o, 6, ) at the beginning of the alphabet, virtual states by letters (m, n, ) in the middle of the alphabet. Later, we use indices i or j to designate sums over both occupied and virtual states. The restriction to positive-energy states in the no-pair Hamiltonian leads to the restriction that virtual states be bound states and positive-energy continuum states in the expressions for the second- and third-order energy. Owing to the relatively small size of the Breit interaction, only terms linear in 6y/t( are important for most applications. The second-order correction from one Breit and one Coulomb interaction is easily found to be... 

Following [30], we write the no-pair Hamiltonian in normal order with respect to the (Is) closed shell. Thus, p ) = Hq + VqVi + V2 with... 

Now we turn to lithium and three-electron lithium-like ions. Again we start with the normally-ordered no-pair Hamiltonian given in Eq. (132), and choose the starting potential to be the Hartree-Fock potential of the (Is) helium-like core. We expand the energy of an atomic state in powers of the interaction potential... 

For RCI calculations, it is more convenient to work in the configuration space. Our starting point is the no-pair Hamiltonian given before with the Coulomb interaction only in Eq. (71). Here, we rewrite it as... 

Three quasi-relativistic approaches that are variationally stable are the Doug-lass-Kroll-Hess transformation of the no-pair Hamiltonian (for example, see Ref. 11, 20, 23-29), the zeroth order regular approximation, ZORA, (for example, see Ref. 30-34), and the approach of Barysz and Sadlej (for example, see Ref. 36). The results of the first two approaches differ considerably even when used by the same authors,which led them to try the third approach. A calibration study suggests that relativistic effects on heavy atom shieldings are significantly underestimated by ZORA in comparison to the four-component relativistic treatment, but that the neighboring proton chemical shifts are closer to experi-... 

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