The No-Pair Approximation

When deriving mean-field methods, we had to consider the role of the negative-energy states in the variation of the wave function. Now that we are considering correlation, we must examine the role of these functions in the fV-particle wave function. 

The most general approach to variational calculations is represented by the MCSCF wave function, with the simultaneous optimization of both the one-particle and the iV-particle functions. The MCSCF wave function may be written (Jprgensen and Simons 1981) 

Here and in the following, we will use indices i, j, k. to denote one-particle functions occupied in the reference determinant, and a,b,c. to denote virtual one-particle functions. In the nonrelativistic case, these one-particle functions are spin-orbitals, normally derived as eigenvalues of the Hartree-Fock operator. For the relativistic case, the 

In this chapter we use the opposite sign convention for ic to that used in chapter 8. 

While this no-pair approximation is chemically reasonable and convenient to work with, it does have consequences for our use of the one-particle variational space. For the nonrelativistic case, the orbital rotation space is the electronic spin-orbitals, and we can indicate this explicitly in the expression for the MCSCF wave function as 

---

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... 

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). 

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. 

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-... 

In the case of the many-body terms the neglect of vacuum corrections is no longer uniquely defined. Two possible approaches can be distinguished, both set up within the KS Furry picture in order to be consistent with (70). In the no-pair approximation the contribution of the negative energy solutions to all intermediate sums over states are ignored. For instance, the DPT analog of the... 

Quite generally, the no-pair approximation within the KS scheme corresponds to the reduction of the complete KS propagator G to... 

Even after the introduction of the no-pair approximation the physical background of the RDFT formalism outlined in this Section is not yet identical with that of the standard methods based on the no-pair DC- or DCB-Hamiltonian. In RDFT the complete electron-electron interaction via the photon propagator is taken into account both in the Hartree energy (58) and in the construction of approximations for Exc (see Appendix C). On the other hand, there is also no fundamental problem in restricting RDFT to the Coulomb or Coulomb-Breit level. Choosing Feynman gauge, the full D y, Eqs.(201),(202), reduces to... 

Even within the no-pair approximation the RKS equations (62) are more involved than the nonrelativistic KS equations due to the four vector structure of the RKS potential. Thus the question arises whether one can find simplified forms in which the RKS potential reduces to one or two components. Fortunately, in most applications the external magnetic field... 

---