Relativistic kinetic energy operator

The different techniques utilized in the non-relativistic case were applied to this problem, becoming more involved (the presence of negative energy states is one of the reasons). The most popular procedures employed are the Kirznits operator conmutator expansion [16,17], or the h expansion of the Wigner-Kirkwood density matrix [18], which is performed starting from the Dirac hamiltonian for a mean field and does not include exchange. By means of these procedures the relativistic kinetic energy density results ... 

For a quasi-relativistic framework as relevant to chemistry (21), we may neglect the magnetic retardation between the electrons and the nuclei and therefore employ standard Coulombic interaction operators for the electrostatic interaction. The interaction between the electrons and the nuclei is not specified explicitly but we only describe the interactions by some external 4-potential. For the sake of brevity this 4-potential shall comprise all external contributions. Explicit expressions for the interaction between electrons and nuclei will be introduced at a later stage. Furthermore, we can neglect the relativistic nature of the kinetic energy of the nuclei and employ the non-relativistic kinetic energy operator denoted as hnuc(I),... 

Herein, Ei denotes the relativistic kinetic energy operator... 

The so-called mass-velocity term Hmv /which represents the first order (in a ) relativistic correction to the non-relativistic kinetic energy operator... 

The superscript (4) indicates the Dirac four-component picture of operators and wave functions. is the relativistic kinetic energy functional of the Dirac-Kohn-Sham (DKS) reference system of non-interacting electrons with ground state density yO [45] ... 

The p gives the usual (non-relativistic) kinetic energy operator. Since p = -iV, the p -A term gives eq. (8.30). 

In the above method, one could introduce the Pauli approximation by neglecting the small Q component spinors of the Dirac equation. This leads to RECPs expressed as two-component spinors. The use of non-relativistic kinetic energy operator for the... 

However, the nonrelativistic term in arises from the relativistic second-order perturbation expressions given above like the correspondence between the nonrelativistic and relativistic kinetic energy discussed in section 11.2, there is a projection operator involving the small component (or in this case, a resolvent). If the small component is poorly represented, the cancellation between the sums over positive- and negative-energy states could show considerable error. 

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

The Dirac operator incorporates relativistic effects for the kinetic energy. In order to describe atomic and molecular systems, the potential energy operator must also be modified. In non-relativistic theory the potential energy is given by the Coulomb operator. 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

By inserting the equations defining the kinetic energy operators and the pairwise interaction operators into Eq. (8) we obtain the Dirac-Coulomb-Breit Hamiltonian, which is in chemistry usually considered the fully relativistic reference Hamiltonian. 

The first term on the right is the operator for the electrons kinetic energy the second term is the operator for the potential energy of attraction between the electrons and the nucleus (r, being the distance between electron i and the nucleus) the third term is potential energy of repulsion between all pairs of electrons ru being the distance between electrons / and j) the last term is the spin-orbit interaction (discussed below). In addition, there are other relativistic terms besides spin-orbit interaction, which we neglect. 

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

It is clear that with a high density of states the theoretical investigations of electronic spectra must sometimes go beyond a traditional BO and non-relativistic analysis that only refers to energy criteria, and that in the description of spectroscopic properties smaller terms of the Hamiltonian must be accounted for. The major corrections to the BO electrostatic Hamiltonian is the non-adiabatic coupling induced by the nuclear kinetic energy operator, and the electronic SOC treated in the present review. 

The simplest example of a complex symmetric operator is the non-relativistic many-particle Hamiltonian H for an atomic, molecular, or solid-state system, which consists essentially of the kinetic energy of the particles and their mutual Coulomb interaction. Since such a Hamiltonian is both self-adjoint and real, one obtains... 

Next we can dispose of the matrix elements of the one-electron operator Hy,Eq. (3-3 b) the kinetic energy operator, the electron-nuclear attraction potential arising from the metal nucleus, and the spin-independent relativistic terms have spherical symmetry and can be treated through the definition of the basis orbitals ip, Eq. (3-11). The spin-... 

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