Scalar mass velocity

In addition to correlation, for the transition metals one has to start worrying about relativity. The common wisdom is that relativistic corrections will have little effect on the properties of 3d systems, a small but non-negjigible influence for the 4d series and dramatic, even qualitative, consequences for 5d. For the atoms, Martin and Hay " have evaluated the scalar (mass-velocity and... 

As expected, Ap vanishes if the strength of the spin-orbit coupling is reduced to 0 by reducing (co/c) or respectively. Both sets "f model calculations give nearly the same results indicating that the so-called scalar relativistic effects due to the mass-velocity and Darwin-term, are of minor importance for the absolute value of Ap. 

The scalar relativistic contribution is computed as the first-order Darwin and mass-velocity corrections from the ACPF/MTsmall wave function, including inner-shell correlation. 

Here we propose a new reduced-cost variant of W1 theory which we shall denote Wlc (for cheap ), with Wlch theory being derived analogously from Wlh theory. Specifically, the core correlation and scalar relativistic steps are replaced by the approximations outlined in the previous two sections, i.e. the MSFT bond additivity model for inner-shell correlation and scaled B3LYP/cc-pVTZuc+l Darwin and mass-velocity corrections. Representative results (for the W2-1 set) can be seen in Table 2.1 complete data for the molecules in the G2-1 and G2-2 sets are available through the World Wide Web as supplementary material [63] to the present paper. 

The first two terms, the mass-velocity and the Darwin operators, are called scalar relativistic terms since they do not involve the electron spin. They are given by... 

Equations (9.130) and (9.134) suggest that the cross coefficients Zrq andZrS are related to the gradients concentration and temperature. This reflects the vectorial character of the coupling coefficients Lrq and Trs, as they relate the vectorial flows of heat and mass with the scalar reaction velocity. 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

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... 

Martin " was the first to estimate the effects of relativity on the spectroscopic constants of Cu2. The scalar relativistic (mass-velocity and Darwin) terms were evaluated perturbatively using Hartree-Fock or GVB (Two configuration SCF (ffg -mTu)) wavefunctions. At these levels the relativistic corrections for r, cu and D, were found to be — 0.05 A, 15 cm and -h0.06eV for SCF, and —0.05 A, + 14 cm and -l-0.07eV for GVB. The shrinking of the bond length is less than half of the estimate based on the contraction of the 4s atomic orbital. 

Scalar relativistic (mass-velocity and Darwin) effects for the valence electrons were incorporated by using the quasi-relativistic method (55), where the first-order scalar relativistic Pauli Hamiltonian was diagonalized in the space of the nonrelativistic basis sets. The Pauli Hamiltonian used was of the form... 

Scalar relativistic effects (e.g. mass-velocity and Darwin-type effects) can be incorporated into a calculation in two ways. One of these is simply to employ effective core potentials (ECPs), since the core potentials are obtained from calculations that include scalar relativistic terms [50]. This may not be adequate for the heavier elements. Scalar relativity can be variationally treated by the Douglas-Kroll (DK) [51] method, in which the full four-component relativistic ansatz is reduced to a single component equation. In gamess, the DK method is available through third order and may be used with any available type of wavefunction. 

Contributions originating in the Darwin, mass-velocity and spin-orbit corrections to the ground state wave function are obtained in agreement with previous works where the Darwin and mass-velocity scalar effects were included within the unperturbed molecular Hamiltonian. ... 

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... 

We have already discussed in chapters 12 and 13 that low-order scalar-relativistic operators such as DKH2 or ZORA provide very efficient variational schemes, which comprise all effects for which the (non-variational) Pauli Hamiltonian could account for (as is clear from the derivations in chapters 11 and 13). It is for this reason that historically important scalar relativistic corrections which can only be considered perturbatively (such as the mass-velocity and Darwin terms in the Pauli approximation in section 13.1), are no longer needed and their significance fades away. There is also no further need to develop new pseudo-relativistic one- and two-electron operators. This is very beneficial in view of the desired comparability of computational studies. In other words, if there were very many pseudo-relativistic Hamiltonians available, computational studies with different operators of this sort on similar molecular systems would hardly be comparable. 

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