Operators No-pair

In summary, conventional relativistic ECP s provide an efficient mean to calculate molecular properties up to and including the third row transition elements in cases where the spin-orbit coupling is weak. ECP s can also be used together with explicit relativistic no-pair operators. Such ECP s are somewhat more precise at at the atomic level, but of essentially the same quality as conventional relativistic ECP s in molecular applications. It should also be possible to combine the ECP formalism with full Fock-Dirac methods, but this has yet not been done. 

Moreover, electron-positron pair creation and other proce.s.ses ( radiative corrections ) de.scribed by quantum electrodynamics which has quantized degrees of freedom for both the fermions and the electromagnetic field are usually not included in the theory, although the chaige-conjugated degrees of freedom are still there. Therefore the literature often refers to the no virtual pair or, in short, no pair approximatioa Very few calculations go beyond this approximation. " Nevertheless. the no-pair operator based on the DCB Hamiltonian provides an exellent approximation to the full theory, generally sufficient for the determination of relativistic effects in the electronic structure of neutral atoms and molecules. 

We see that we can attach a definite physical meaning both to the existence of a neutral molecule in solution, and to the dissociation of this molecule into a pair of ions. Consider points near P and near Q in Fig. 27c. A point on the curve near P corresponds to the situation where the distance between the nuclei of the two ions has, say, the value OA, while a point on the curve near Q corresponds to the separation OB. If the separation of the nuclei is increased from OA to OB, a considerable amount of work is done against the short-range forces of attraction, in order to go from P to Q. But at Q the short-range forces are no longer operative and the neutral molecule has been dissociated into a pair of ions, between which there is the usual electrostatic attraction. 

Hess, B.A. (1986) Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators. Physical Review A, 33, 3742-3748. 

Employing a 2-Component No-Pair Formalism With External-Field Projection Operators. 

The definition of the pair part or, equivalently the no-pair part of Hmat is not unique. The precise meaning of no-pair implicitly depends on the choice of external potential, so that the operator Hm t depends implicitly on the external potential, whereas the sum Hmat Hm t + Hm t is independent of the choice of external potential. Since the no-pair part conserves the number of particles (electrons, positrons and photons) we can look for eigenstates of Hm j in the sector of Fock space with N fermions and no photons or positrons. Following Sucher [18,26,28], the resulting no-pair Hamiltonian in configuration space can be written as... 

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... 

Let us now consider bosonizing [4] the pair operators, based on the reduced matrix elements that appear on the rhs of (1). We then have an expansion which contains no s bosons in it. This BET, which we may call SR+BET, is exact (assuming that the boson expansion is carried out to a desired order). 

As may be seen by comparing Eqs. [103] and [105], the no-pair spin-orbit Hamiltonian has exactly the same structure as the Breit-Pauli spin-orbit Hamiltonian. It differs from the Breit-Pauli operator only by kinematical factors that damp the 1/rfj and l/r singularities. 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

Since the relativistic many-body Hamiltonian cannot be expressed in closed potential form, which means it is unbound, projection one- and two-electron operators are used to solve this problem [39], The operator projects onto the space spanned by the positive-energy spectrum of the Dirac-Fock-Coulomb (DFC) operator. In this form, the no-pair Hamiltonian [40] is restricted then to contributions from the positive-energy spectrum and puts Coulomb and Breit interactions on the same footing in the SCF calculations. 

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. 

Employing a Two-Component No-Pair Eormalism with External Eield Proj ection Operators. 

How can one reduce the number of conformations that have to be checked Here the concept of construction becomes useful. One constructs the conformations in a stepwise fashion, starting with an initial aggregate and adding a second aggregate at a given torsional increment for the torsional variable T that is applied to the rotatable bond connecting the two. If any pair of atoms overlaps for that increment, then one can terminate the construction because no addition operation will relieve that steric overlap. In effect, one has truncated the combinatorial possibilities that would have included that subconformation that is, one has pruned the combinatorial tree. 

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,... 

Having defined our starting point, the second quantized no-pair Hamiltonian, we may now take a closer look at the relations between the matrix elements. For future convenience we will also change the notation of these matrix elements slightly. Due to hermiticity of the Dirac Hamiltonian and the Coulomb-Breit operator we have... 

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. 

In principle problems of relativistic electronic structure calculations arise from the fact that the Dirac-Hamiltonian is not bounded from below and an energy-variation without additional precautions could lead to a variational collapse of the desired electronic solution into the positronic states. In addition, at the many-electron level an infinite number of unbound states with one electron in the positive and one in the negative continuuum are degenerate with the desired bound solution. A mixing-in of these unphysical states is possible without changing the energy and might lead to the so-called continuum dissolution or Brown-Ravenhall disease. Both problems are avoided if the Hamiltonian is, at least formally, projected onto the electronic states by means of suitable operators (no-pair Hamiltonian) ... 

It is easy to verify that the DKl operator, Eq. (Ill), is equivalent to the Fock operator derived from the no-pair or free-particle FW Hamiltonian. Likewise, the higher order DK operators are also derived straightforwardly by repeating the DK transformations, though their formulae are omitted only because of their lengthy forms. 

As molecular applications of the extended DK approach, we have calculated the spectroscopic constants for At2 equilibrium bond lengths (RJ, harmonic frequencies (rotational constants (B ), and dissociation energies (Dg). A strong spin-orbit effect is expected for these properties because the outer p orbital participates in their molecular bonds. Electron correlation effects were treated by the hybrid DFT approach with the B3LYP functional. Since several approximations to both the one-electron and two-electron parts of the DK Hamiltonian are available, we dehne that the DKnl -f DKn2 Hamiltonian ( 1, 2= 1-3) denotes the DK Hamiltonian with DKnl and DKn2 transformations for the one-electron and two-electron parts, respectively. The DKwl -I- DKl Hamiltonian is equivalent to the no-pair DKwl Hamiltonian. For the two-electron part the electron-electron Coulomb operator in the non-relativistic form can also be adopted. The DKwl Hamiltonian with the non-relativistic Coulomb operator is denoted by the DKwl - - NR Hamiltonian. 

It should be noted that the projection operator A+ and, consequently, the no-pair Hamiltonian depends on the background potential U. One finds however that energies obtained from the no-pair Hamiltonian are only weakly dependent on the potential and that small differences between calculations starting from different potentials can be accounted for in terms of omitted negative-energy corrections. We elaborate on this point in Sec. 4. 

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