Pair NOs

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. 

The zinc blende type is unknown for truly ionic compounds because there exists no pair of ions having the appropriate radius ratio. However, it is well known for compounds with considerable covalent bonding even when the zinc blende type is not to be expected according to the relative sizes of the atoms in the sense of the above-mentioned considerations. Examples are CuCl, Agl, ZnS, SiC, and GaAs. We focus in more detail on this structure type in Chapter 12. 

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... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

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... 

Polymers from 1,2-disubstituted ethylenes (XXXVI) possess 1,3-interactions, but the steric strain is not as severe as in XXXV. Both XXXV and XXXVI possess the same number of 1,3-interactions but the distribution of the interactions is different. For XXXV, pairs of 1,3-carbons each have a pair of 1,3-interactions. No pair of 1,3-carbons in XXXVI has more than... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [10]. The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component... 

One should not conclude, however, that T yA/i =HVj = OifT) < Ty. Although true in some cases, we see that it does not hold true for the first and last of the tableaux above. No pair in a row of the last is in a column of the first. In fact, the nonstandard tableau... 

Analyze the stereochemical relationships between the methyl groups of hexamethyl (Dewar benzene). Identify at least one pair of methyl groups in which the methyl groups are homotopic, enantiotopic, diastereotopic, or heterotopic. Note If there are no pairs of a given kind, say so. 

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