Nonrelativistic kinetic energy

The left-hand side is the nonrelativistic kinetic energy of one particle. It can be seen that the Higgs mechanism changes the classical nonrelativistic expression... 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

The second term on the right-hand side represents the familiar nonrelativistic kinetic energy and the third term is the first relativistic correction higher-order corrections have been suppressed here. We could have equally well expanded the energy-momentum relation as given by Eq. (3.118) to arrive at... 

We first consider the option to set up a quantum mechanical equation of motion which obeys the correspondence principle. If we apply the correspondence principle to the classical nonrelativistic kinetic energy expression E = (2m) we arrive at the time-dependent Schrodinger equation, in which... 

We recover the rest energy -2 and the nonrelativistic contributions the leading terms of the Foldy-Wouthuysen series. The term fp] offers low-order relativistic corrections to them. The form of the operator in the first term of Eq. (11.84) resembles the classical expression derived in Eq. (3.122). We shall investigate the physical meaning of the second term on the right-hand side of Eq. (11.84) in section 11.5.2. Moreover, we will encounter these operators again in section 13.1 where we will discuss how the Pauli elimination produces relativistic corrections to the nonrelativistic kinetic energy operator of the Pauli Eq. (5.140). 

An important aspect of DKH theory is that only standard operator matrices for the nonrelativistic kinetic energy and external potential are required for the evaluation of the DKH Hamiltonian, which is after all remarkable. Only one non-standard matrix is needed, which can, however, be calculated with little additional effort. This is the matrix representation of pVp,... 

Using (4.15) to reduce the second term, we find that it is the nonrelativistic kinetic energy operator f. The above equation is therefore... 

From the Dirac relation (4.14), (or p)(or p) = p, and the first term in the series gives p /2w = f, the nonrelativistic kinetic energy operator. Making use of the Dirac relation in the second term gives... 

The Pauli approximation may be used in conjunction with this method by neglecting the small component spinors Q) of the Dirac equation, leading to RECPs expressed in terms of two-component spinors. The use of a nonrelativistic kinetic energy operator for the valence region, and two-component spinors leads to Hartree-Fock-like expressions for the pseudoorbitals. Note that the V s (effective potentials) in this expression are not the same for pseudo-orbitals of different symmetry. Thus the RECPs are expressed as products of angular projectors and radial functions. In the Dirac-Fock approximation, the orbitals with different total j quantum numbers, but which have the same / values are not degenerate, and thus the potentials derived from the Dirac-Fock calculations would be y-dependent. Consequently, the RECPs can be expressed in terms of the /y-dependent radial potentials by equa-... 

Neglecting Vcpp in equation 6.1 for a moment, it is usually assumed that all relativistic effects are described by a suitable parametrization of the ECPs Vcv i-e- it sufficient to apply the nonrelativistic kinetic energy operator as well as the nonrelativistic Coulomb interaction between the valence electrons [19]. Besides the relativistic contributions, the ECP accounts for all interactions of the valence electrons with the nucleus and the (removed) core electrons, and it is given by [19]... 

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