Mass-velocity terms

The quasirelativistic (QR) PP of Hay and Wadt [61] use two-component wave functions, but the Hamiltonian includes the Darwin and mass-velocity terms and omits the spin-orbit effects. The latter are then included via the perturbation operator after the wave functions have been obtained. The advantage of die method is the possibility to calculate quite economically rather large systems. The method is implemented in the commercial system Gaussian 98 It has extensively been applied to calculations of transition-element and actinide systems [62],... 

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

We can now turn to a discussion of how relativistic effects will modify the molecular energies, Eab (or AEab )r as well as their functional dependence on the interatomic distance, Rab-There are two non-zero Relativistic corrections from the first order Hamiltonian, Hi, of Eq. (5). One is the first order correction, T to the kinetic energy from the mass-velocity term, Hmv /and... 

The CASSCF wavefiinction is used as reference function in a second-order estimate of the remaining dynamical correlation effects. All valence electrons were correlated in this step and also the 3s and 3p shells on copper. Relativistic corrections (the Darwin and mass-velocity terms) were added to all CASPT2 energies. They were obtained at the CASSCF level using first-order perturbation theory. A level-shift (typically 0.3 Hartree) was added to the zeroth order Hamiltonian in order to remove intruder states [30]. Transition moments were conputed with the CAS state-interaction method [31] at the CASSCF level. They were... 

Spin-orbit (SO) coupling corrections were calculated for the Pt atom since the relativistic effects are essential for species containing heavy elements. Other scalar relativistic corrections like the Darwin and mass-velocity terms are supposed to be implicitly included in (quasi)relativistic pseudopotentials because they mostly affect the core region of the considered heavy element. Their secondary influence can be seen in the contraction of the outer s-orbitals and the expansion of the d-orbitals. This is considered in the construction of the pseudoorbitals. The effective SO operator can be written within pseudopotential (PS) treatment in the form71 75... 

Results for dipole moments are shown in Table 29. ° Clearly, coupled with good basis sets, correlated methods can do quite well for this property. Elsewhere, we have used relaxed density-based CC and MBPT methods to study spin densities and the related hyperfine coupling constants to evaluate relativistic corrections (Darwin and mass-velocity term) when impor-tant and to evaluate highly accurate electric field gradients to extract nuclear quadruple moments. 

Actually the mass-velocity term shows already up explicitly in (473), and the contributions involving Vi and V2 can be reduced to the Darwin and spin-orbit terms as in section 4.6. The only new terms are... 

Including the Darwin and mass-velocity terms from the Pauli equation. 

The also so-called mass-velocity term, is mainly related to the fourth power of momentum, and thus with the operator V. The adequate operator structure will be given by a diagonal operator V = Diag l -V )= Diag A) acting on the extended wavefunctions, in this way, using Green s first theorem ... 

In systems with heavier elements, relativistic effects must be included. In the medium range of atomic numbers (up to about 54) the so called scalar relativistic scheme is often used [21], It describes the main contraction or expansion of various orbitals (due to the Darwin s-shift or the mass-velocity term), but omits spin-orbit interaction. The latter becomes important for the heavy elements or when orbital magnetism plays a significant role. In the present version of WIEN2k the core states always are treated fully relativistically by numerically solving the radial Dirac equation. For all other states, the scalar relativistic approximation is used by default, but spin-orbit interaction (computed in a second-variational treatment [22]) can be included if needed [23]. 

Corrections for scalar-relativistic effects (one-electron Darwin and mass-velocity terms, MVD) were calculated at the ae-CCSD(T)/aug-cc-pwCVTZ level [101, 102]. For the C atom, the spin-orbit correction (SO) to the total electronic energy amounts to AEso = -0.35399 kJmol-Ml03]. 

Aev is a core-valence correction obtained as the difference between ae-CCSD(T)/cc-pCVQZ and fc-CCSD(T)/cc-pCVQZ energies. Azpve is the harmonic zero-point vibrational correction obtained at the ae-CCSD(T)/cc-pCVTZ level, AAnh. is the correction due to anharmonic effects, calculated at the fc-MP2/cc-pVDZ level. Amvd is the correction for scalar-relativistic effects (one electron Darwin and mass-velocity terms) obtained at the ae-CCSD(T)/cc-pCVTZ level [101, 102], Aso is a spin-orbit coupling correction, which may be non-zero only for open-shell species. For the C, O and F atoms, Aso amounts to —0.35599, —0.93278 and —1.61153 kJ/mol, respectively [103]. The remaining contributions take care of the correction to the full triple excitations and perturbative treatment of quadruples Ax = ccsdt/cc-pvtz - ccsd(t)/cc-pvtz, A(q) = E CCSDT(Q)/cc-pVDZ—-E ccsDT/cc-pVDz- The final atomization energies are obtained by adding all the incremental contributions... 

Recent results within the context of a benchmark ab initio study of atomic electron affinities of first and second row atoms [35], confirmed the quality of these hybrid functionals. The study enabled us to compare DFT calculated EA values with what we considered to be the best non relativistic values available until now. They were obtained by adding valence and core correlation terms to basis set extrapolated SCF results, finally correcting for differences with full Cl and basis set incompleteness. Adding relativistic corrections (spin orbit, Darwin and mass velocity terms) one obtains what we called the "best calculated values" (vide supra) which show a mean absolute difference of only 9 10" eV with experiment. In Table 1 these reference values are given, together with the most recent and accurate experimental values. 

Of the terms in H", mc is a trivial constant, and we recognize as the nonrelativistic Hamiltonian. The mass-velocity term is a correction to the kinetic energy due to the relativistic mass change, which could also have been obtained from a Taylor expansion of the free-particle energy (Eq. [45]) ... 

It is instructive to compare the 2-component Hamiltonian T-Cl with the Pauli Hamiltonian, which was derived in the previous section. The term Ep contains the relativistic free-particle energy, which is well-behaved for all values of the momentum. Ep, which is the kinetic energy operator for the positive-energy states, is a positive definite operator. In the Pauli Hamiltonian we see that this operator is expanded in powers of p/mc, which does not converge if p/mc > 1—a situation that will occur in any potential if the electron is sufficiently close to the nucleus. As mentioned above, the mass-velocity term is not bounded from below and so cannot be used variationally. 

The mass-velocity term is therefore the lowest-order term from the relativistic Hamiltonian that comes from the variation of the mass with the velocity. The second relativistic term in the Pauli Hamiltonian is called the Darwin operator, and has no classical analogue. Due to the presence of the Dirac delta function, the only contributions for an atom come from s functions. The third term is the spin-orbit term, resulting from the interaction of the spin of the electron with its orbital angular momentum around the nucleus. This operator is identical to the spin-orbit operator of the modified Dirac equation. 

This is the correction to the Hartree-Fock energy. Obviously, only the terms that are diagonal in the spin and are spatially totally symmetric will contribute. These terms are the scalar relativistic corrections the mass-velocity term, the one- and two-electron Darwin terms, and the orbit-orbit term. Other terms such as the spin-spin term and the z component of the spin-orbit interaction contribute for open-shell systems, where the spin is nonzero. 

The singular nature of the Pauli operators can in fact be traced to the normalization of the wave function. When we renormalized the large component, we introduced operators that canceled the energy-dependent term and produced the mass-velocity term. This connection between the singular operators and the normalization has been made in another way by Kutzelnigg (1999). 

Moreover, the negative sign of the mass-velocity term inevitably leads to a variational collapse in the sense that there is no repulsive term n H = f/nomei + hat could prevent the energy from going to —oo upon free variation on a domain that contains functions of arbitrarily high momentum. These undesirable features of the straightforward expansion get worse in higher orders. ... 

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