Adaptation spin-adapted operators

While previous variational 2-RDM calculations for electronic systems have employed the above formulation [20-31], the size of the largest block diagonal matrices in the 2-RDMs may be further reduced by using spin-adapted operators Ci in Eq. (9). Spin-adapted operators are defined to satisfy the following mathematical relations [54, 55] ... 

These operators satisfy the formal definition for spin-adapted operators in Eqs. (79) and (78). Inserting these four operators into Eq. (8), we can generate four... 

By comparing the calculations of the 2j-order EOM method I with Simons Nj results (using the same orbital basis set), columns one and three of Table I, respectively, the effect of spin-symmetry adapting the operators basis can be assessed, since this is the only difference between the calculations. The results exhibit a shift in the IPs of 0.30 to 0.44 V when the symmetry-adapted operators are employed. This quite large shift is really a reflection of the restricted accuracy of the diagonal approximation... 

In the actual calculations, spin-symmetry-adapted operators and... 

Finally, let us emphasize that in contrast to a spin-independent case, the U(n) irreps in the U(2n) D U(n) SU(2) bases are generally changed or shifted by U(2 ) operators. Thus, one-body spin-dependent operators can change the total spin [or U(n) irrep label] by A5 = 0, 1 (or Ah = 0, 2) and two-body ones by A5 = 0, 1, 2 (or Ah = 0, 2, 4) and thus take us out of the U(n) framework. Nonetheless, as we have indicated above, the U( )-adapted creation (C ) and annihilation (C) type operators— that represent very useful tensors serving as fundamental building blocks for various U(n) tensors—are also useful in the spin-dependent U(2n) case. Indeed, since xj (X, ) are vector (contragredient vector) operators when acting on the irrep modules of U(n), their MEs in the U(n) basis are clearly related to those of (Cf) operators. In view of this fact, the MEs of one-body operators must be related to those of cfCj. The latter were carefully examined in [36] and briefly reviewed above. 

With the core potentials and spin-orbit operators given in the forms of Eqs. (2.1) and (2.2), existing programs for nonrelativistic calculations can be adapted to include relativistic effects. In the literature, the coefficients of the spin-orbit potentials are not always defined in the same manner. In our implementation of the spin-orbit interaction in NWChem, the spin-orbit potential is defined as the A in (2.3) scaled by... 

The spin-orbit coupling is incorporated into the calculations at the post-Hartree-Fock level, particularly with CL This is generally referred to as spin-orbit CL The core potential and associated basis sets for Li—Kr, including the spin-orbit operator, have been published." ° Implementation of the spin-orbit Cl method has been discussed by Pitzer and co-workers. It has also been reported that the spin-orbit Cl code has been adapted to a parallel computing enviroment. " ... 

To summarize it is often convenient to assume that the basis functions in an expansion such as (3.1.4) are all of the same pure symmetry species with respect to both space and spin symmetry operations. The symmetry-adapted functions are no longer necessarily single Slater determinants, and may often be linear combinations of a considerable number. Such functions, constructed within the individual configurations, are often called configurational functions (CFs), a terminology that we frequently adopt.t... 

Spin-adapted rotations The spin-free nonrelativistic Hamiltonian commutes with the total and projected spin operators. We are therefore usually interested only in wave functions with well-defined spin quantum numbers. Such functions may be generated from spin tensor operators that are totally symmetric in spin space. For optimizations, we need consider only singlet opeiatois since these are the only ones that conserve the spin of the wave function. Spin poturbations, on the other hand, may mix spin eigenstates and require the inclusion also of triplet rotations. 

Chapters 1-3 introduce second quantization, emphasizing those aspects of the theory that are useful for molecular electronic-structure theory. In Chapter 1, second quantization is introduced in the spin-orbital basis, and we show how first-quantization operators and states are represented in the language of second quantization. Next, in Chapter 2, we make spin adaptations of such operators and states, introducing spin tensor operators and configuration state functions. Finally, in Chapter 3, we discuss unitary transformations and, in particular, their nonredundant formulation in terms of exponentials of matrices and operators. Of particular importance is the exponential parametrization of unitary orbital transformations, used in the subsequent chapters of the book. 

In summary, proper spin eigenfunetions must be eonstmeted from antisymmetrie (i.e., determinental) wavefunetions as demonstrated above beeause the total and total Sz remain valid symmetry operators for many-eleetron systems. Doing so results in the spin-adapted wavefunetions being expressed as eombinations of determinants with eoeffieients determined via spin angular momentum teehniques as demonstrated above. In... 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

Various optical detection methods have been used to measure pH in vivo. Fluorescence ratio imaging microscopy using an inverted microscope was used to determine intracellular pH in tumor cells [5], NMR spectroscopy was used to continuously monitor temperature-induced pH changes in fish to study the role of intracellular pH in the maintenance of protein function [27], Additionally, NMR spectroscopy was used to map in-vivo extracellular pH in rat brain gliomas [3], Electron spin resonance (ESR), which is operated at a lower resonance, has been adapted for in-vivo pH measurements because it provides a sufficient RF penetration for deep body organs [28], The non-destructive determination of tissue pH using near-infrared diffuse reflectance spectroscopy (NIRS) has been employed for pH measurements in the muscle during... 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

To generate the spin-adapted matrix, we spin-adapt the products of one creation operator and one annihilation operator ... 

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