Relativistic reference state

Several wind models of analytical nature exist. They differ in their level of physical sophistication and in their way to parametrize the wind characteristics. In all cases, the wind is assumed to be spherically symmetric, which appears to be a reasonable first approximation even in two-dimensional simulations, at least late enough after core bounce. In addition, the wind is generally treated as a stationary flow, meaning no explicit time dependence of any physical quantity at a given radial position. Newtonian and post-Newtonian descriptions of a spherically symmetric stationary neutrino-driven (supersonic) wind or (subsonic) breeze emerging from the surface of a PNS have been developed. The reader is referred to [24] for the presentation of a Newtonian, adiabatic and steady-state model for the wind and breeze regimes, and for a general-relativistic steady-state wind solution. 

In a non-relativistic theory we would now continue by adding a second quantized operator for two-body interactions. In the relativistic case we need to step back and first consider the interpretation of the eigenvalues of the Hamiltonian. Dirac stated that positrons could be considered as holes in an infinite sea of electrons . In this interpretation the reference state for a system with neither positrons nor electrons is the state in which all negative energy levels are filled with electrons. This vacuum state... 

There is only one subtle point with regard to the no-pair approximation that deserves some attention. In the non-relativistic case the Fock space formalism without truncation of the T operators gives just an alternative parametrization of the foil Cl wave function. In the relativistic case the situation is more complex because the states of interest may contain a different number of electrons than the reference state. This means that the no-pair approximation is less appropriate as it is based on a mean-field potential due to a different number of electrons. Formally this problem might be tackled by lifting the no-pair restriction but it will be very hard to turn the resulting complicated formalism into an efficient algorithm. The corrections would probably be small since the difference in potential mainly affects the valence region where the potential is small relative to the rest mass term anyway. 

We predict that W2, after the inclusion of relativistic effects, should have a stronger metal-metal bond than M02. Even in the non-relativistic case, the bonding interaction is stronger in W2 than in M02 if the two metal atoms are referred to the same reference state. 

There is no fundamental change in the concept of correlation between relativistic and nonrelativistic quantum chemistry in both cases, correlation describes the difference between a mean-field description, which forms the reference state for the correlation method, and the exact description. We can also define dynamical and non-dynamical correlation in both cases. There is in fact no formal difference between a nonrelativistic spin-orbital-based formalism and a relativistic spinor-based formalism. Thus we should be able to transfer most of the schemes for post-Hartree-Fock calculations to a relativistic post-Dirac-Hartree-Fock model. Several such schemes have been implemented and applied in a range of calculations. The main technical differences to consider are those arising from having to deal with integrals that are complex, and the need to replace algorithms that exploit the nonrelativistic spin symmetry by schemes that use time-reversal and double-group symmetry. 

Thus, while the ground state of these elements can be represented as a single, high-spin determinant in nonrelativistic theory, they must be represented by at least two determinants in relativistic theory. So for the relativistic case we may find ourselves forced to use methods beyond simple DHF to obtain reference states for cases that nonrelativistically could be comfortably treated in a mean-field model. 

The first sum is zero for a closed-shell system because io2i is a one-electron operator and the correlation perturbation only connects the reference state to doubly excited configurations. None of the remaining sums is necessarily zero. The mixed relativistic-correlation correction to a property can therefore be nonzero even if the corresponding correction to the energy is zero. 

Simulations of this scenario have been performed by Lee (Lee 2001 and references therein) using Newtonian gravity and polytropic equations of state of varying stiffness. Ruffert, Janka and Eberl performed similar simulations but with a detailed microphysics input (nuclear equation of state and neutrino leakage ). In our own simulations of NS-BH mergers we used a relativistic mean field equation of state together with three-dimensional smoothed particle hydrodynamics and a detailed, multiflavour neutrino treatment. 

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