Approximations no-pair

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

A fully relativistic extension of the scheme put forward in [12] has been introduced in [19], including the transverse electron-electron interaction (Breit +. .. ) and vacuum corrections. Restricting the discussion to the no-pair approximation [28] for simplicity, we here compare this perturbative approach to orbital-dependent Exc to the relativistic variant of the adiabatic connection formalism [29], demonstrating that the latter allows for a direct extraction of an RPA-like orbital-dependent functional for Exc- In addition, we provide some first numerical results for atomic Ec. 

In the electronic sector the presence of the potential leads to an inhomogeneous reference system. Within the no-pair approximation. 

As all quantities discussed in this publication are understood within the no-pair approximation, we will omit the index np in the following for brevity). In Eqs. (2.21, 2.22) bk and b are the annihilation and creation operators for positive energy KS states, which allow to write the electronic ground state as... 

A more appropriate spin-orbit coupling Hamiltonian can be derived if electron-positron pair creation processes are excluded right from the beginning (no-pair approximation). After projection on the positive energy states, a variationally stable Hamiltonian is obtained if one avoids expansion in reciprocal powers of c. Instead the Hamiltonian is transformed by properly chosen... 

Here, hD ) and gf"uIomb are one- and two-electron operators, respectively, a and P are Pauli matrices, c is the speed of light, Nelec is number of electrons, and Vnuc(A) is the nuclear attraction potential. The electron-electron repulsion is assumed to be the Coulomb interaction and electron-positron interactions are disregarded with no pair approximation. 

The most important, and at the same time most legitimate, simplification if one aims at electronic structure calculations in quantum chemistry and condensed matter physics is the no-sea (or alternatively no-pair) approximation. In this approximation all radiative contributions to the four current and are neglected,... 

In addition the vacuum contributions in the functional dependence of E c on j are dropped , so that one is led to the RDFT analogue of the no-pair approximation applied in conventional relativistic many-body approaches (see e-g- [7]). An a posteriori perturbative evaluation of these corrections is possible and should be adequate, except in special circumstances as for instance the calculation of the structure of super-heavy atoms (with Z 137 [33]). The resulting RKS-equations are then still given by Eqs. (3.15-3.17), but f(x) and Tj are determined by the simpler expressions (3.7) and (3.11). 

The definition of the no-sea approximation for is not completely unambiguous. As discussed in Appendix B we define it through neglect of all vacuum fermion loops in the derivation of an approximate [/]. Alternatively, one could project out all negative energy states, thus generating a direct equivalent of the standard no-pair approximation. As one would expect the differences between these two schemes to be small, we do not differentiate between these approximations here. 

For the case of the longitudinal no-pair approximation and a purely electrostatic external potential F" = (F°, 0), to which we restrict further discussion of the ROPM, Eq. (3.32) reduces to (summation over the spinor indices a, h = 1,..., 4 is implicitly understood)... 

In fact, (4.13) is also satisfied by the x-only limit of g, i.e. its lowest order contribution in e. In the relativistic case only this limit of the pair correlation function of the RHEG, g ikplr — Ikp), specified in Eq. (B.68), is known (within the no-pair approximation [19,102]), so that we restrict the subsequent discussion to the x-only limit. 

The NpPolMe basis sets were developed recently (10) for the investigation of relativistic effects using the DK transformed hamiltonian (13, 18-20). This is the spin-averaged no-pair approximation which reduces the 4-component relativistic one-electron hamiltonian to a 1-component form without introducing strongly singular operators. NpPolMe basis sets indirectly incorporate some relativistic effects on the wave function. Let us note that both PolMe and NpPolMe contracted sets share the same exponents of primitive Gaussians. Contraction coefficients are, however,... 

At this point it is convenient, though not necessary, to define the exchange component Ex of Exc. As in the nonrelativistic context (Langreth and Mehl 1983 Sahni et al. 1982 Sham 1985) we identify Ex with the first-order contribution to Exc resulting from perturbation theory on the basis of the KS auxiliary Hamiltonian (Engel et al. 1998a). Within the no-pair approximation this leads to... 

In principle, not only low-order perturbative Ec can be obtained in this way, but also resummed forms like the RPA (Engel and Facco Bonetti 2000). In practice, however, the resulting functionals are computationally much more demanding than the exact Ex, so that until now only the lowest-order contribution has been applied. Within the no-pair approximation and neglecting the transverse interaction, this second-order term reads... 

Qvac is the total charge of the vacuum, which vanishes for free electrons, but is finite in the presence of an external field (the phenomenon of vacuum polarization). Note that whilst Q is conserved for all processes, the total number of particles need not be it is always possible to add virtual states incorporating electron-positron pairs without changing Q. The neglect of such terms in the total wavefunction of an n-electron system is called the no-pair approximation. 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

A formal definition of is thus necessary to apply the no-pair approximation... 

The previous section considered the derivation of second quantized Hamiltonians that can be used in post-DHF calculations. From now on we will regard the matrix elements of h and g as (complex) numbers and direct the attention to the associated operators. By applying the no-pair approximation we retained only particle conserving operators in the Hamiltonian. Such operators can concisely be expressed using the replacement operators Eq = a p Q and... 

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. 

The curve G corresponds to the first order Coulomb interelectron interaction, the curve C++ corresponds to the no pair approximation for the second-order Coulomb box interelectron interaction (Fig.7a). The curve B corresponds to the first order Breit interaction, the curve BC corresponds to the second-order Coulomb-Breit box interaction (exchange of the one Coulomb and one transverse photons) Fig.7c,d. The curve denoted by ( ) includes the contributions ( )= GC, BB, X, where GG is the negative-energy contribution to the Coulomb - Coulomb box interaction Fig.7a, BB is the Breit-Breit box interaction Fig.7g, X denotes all cross interactions Fig.7b,e,f,h. The order of magnitude of all ( ) corrections is defined by the high-energy intermediate electron state contributions. This means that the corresponding effective interelectron interaction potential does not depend on the ex-... 

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