Spin-free

Spin-paired octahedral (P ions and spin-free octahedral (P ions appear to react by largely dissociative (D ot reactions, for example. 

The complete Hamiltonian of the molecular system can be wrihen as H +H or H =H +H for the commutator being linear, where is the Hamiltonian corresponding to the spin contribution(s) such as, Fermi contact term, dipolar term, spin-orbit coupling, etc. (5). As a result, H ° would correspond to the spin free part of the Hamiltonian, which is usually employed in the electron propagator implementation. Accordingly, the k -th pole associated with the complete Hamiltonian H is , so that El is the A -th pole of the electron propagator for the spin free Hamiltonian H . 

The spin free electronic Hamiltonian of the stem,, is partitioned according to the usual Moller-Plesset form (129),... 

The spin free many-body Hamiltonian Operator can be written in compact form by employing the 2-RO... 

Eleig, T. and Visscher, L. (2005) Large-scale electron correlation calculations in the framework of the spin-free dirac formalism the Au2 molecule revisited. Chemical Physics, 311, 63. 

We restrict ourselves to the local valence part of the EFG tensor to illustrate the principle. Since the EFG operator is spin-free, there are no off-diagonal elements M M and an inspection of Table 5.6 reveals that there are also no off-diagonal components between different configurations I A J- Hence ... 

Table 6.2 Energies of the lowest spin-free states originating from the SH multiplet and the energies of the low-lying Kramers doublets of the DyZn3 complex.

Atomic multiplet Spin free states J multiplet Kramers Doublets... 

Table 1 Spin-Free Excitation Energies in Re2Cls (in eV) Calculated at the CASSCF (CAS) and CASPT2 (PT2) Level.

The above operators apply only to primitive basis functions that have the spin degree of freedom included. In the current work we follow the work of Matsen and use a spin-free Hamiltonian and spin-free basis functions. This approach is valid for systems wherein spin-orbit type perturbations are not considered. In this case we must come up with a different way of obtaining the Young tableaux, and thus the correct projection operators. 

For the illustrative calculations shown here, the spin-free wave functions, 4, for the H/ isotopomers were obtained as 50-term expansions in a basis of FSECG s gi(r) ... 

But this is not the full story. The Hamiltonian operator employed is a spin-free operator and does not work on the spin functions a and p. H commutes therefore with the spin operators Sz and S ... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

In the quaternion modified Dirac equation the spin-free equation is thereby obtained simply by deleting the quaternion imaginary parts. For further details, the reader is referred to Ref. [13]. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

Table 4. The isotropic indirect spin-spin coupling constant of calculated at various levels of theory. LL refers to the Levy-Leblond Hamiltonian, std refers to a full relativistic calculation using restricted (RKB) or unrestricted (UKB) kinetic balance, spf refers to calculations based on a spin-free relativistic Hamiltonian. Columns F, G and whether quaternion imaginary parts are deleted (0) or not (1) from the regular Fock matrix F prior to one-index transformation, from the two-electron Fock matrix G...

Table 5. The NMR shielding constant and shielding polarizabilities of the xenon atom calculated at the Hartree-Fock level using the Drrac-Coulomb Hamiltonian (SR + SO), its spin-free version (SR) as well as the non-relativistic Levy-Leblond Hamiltonian. The shielding constant is given in ppm and shielding polarizabilities in ppm/(au field2) (1 a.u. field = 5.14220642X 10" V...

The arguments just described for the construction of the 2-RDM were extended without difficulty to the higher-order RDMs. The algorithms for these high-order RDMs were originally reported by Colmenero et al. [20] in a spin-free basis and Valdemoro et al. [46] obtained later on the algorithms in a spin-orbital basis. For the 3-RDM, the algorithm in a spin-orbital basis is... 

C. Spin-Free Excitation Operators and -Particle Density Matrices... 

Most Hamiltonians of physical interest are spin-free. Then the matrix elements in Eq. (9) depend only on the space part of the spin orbitals and vanish for different spin by integration over the spin part. Then it is recommended to eliminate the spin and to deal with spin-free operators only. We start with a basis of spin-free orbitals cpp, from which we construct the spin orbitals excitation operators carry orbital labels (capital letters) and spin labels... 

Greek letters). We define spin-free excitation operators carrying only orbital labels, by summation over spin... 

It would be in the spirit of a systematic formulation with lowercase letters for spin-orbital labels and capital letters for labels of spin-free orbitals, to choose... 

The spin-free one-particle density matrix Fi = F is diagonal in the basis of the (spin-free) natural orbitals (NOs)... 

The spin-free two-particle excitation operators and density matrices are symmetric with respect to simultaneous exchange of the upper and lower indices, but neither symmetric nor antisymmetric with respect to exchange of either upper or lower indices separately ... 

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