Pseudo-diagonal

Fig. 4. 2Q-HoMQC of the DNA [d(AAATATAGCTATATTT)]2 at 500 MHz and 25°C without (top) and with (bottom) aliasing. The 2Q pseudo-diagonal is denoted by dot-dashed lines. Regions for direct correlations and remote correlations are highlighted by ellipsoids and rectangles, respectively. For more details see ref. [9]. (Copyright Academic Press, with...

Fig. 5. Multiple aliasing for 2Q-HoMQC of a 24-mer RNA hairpin 7], The spectrum was acquired in D2O with presaturation of the residual water signal. Relevant sections of the 2Q pseudo-diagonal are denoted with solid line (no aliasing), dashed line (single aliasing),...

Pairwise symmetry of cross peaks to a single pseudo-diagonal applies only to the direct correlation peaks in OQ- and 2Q-HoMQC spectra. All other peaks, and those in correlations of higher number of quanta, occupy systematic and predictable positions, of course, but their relative location can not be expressed in terms of mirror image symmetry of some kind. 

Figure 5.76. The 2D INADEQUATE (magnitude mode) and conventional carbon spectrum of n-butanol 5.10. Ibe dotted line indicates the double-quantum pseudo diagonal.

Wolinski K, Haacke R, Hinton JF, Pulay P (1997) Methods for parallel computation of SCF NMR chemical shifts by GIAO method Efficient integral calculation, multi-Fock algorithm, and pseudo-diagonalization. J Comp Chem 18 816-825... 

The equations may be simplified by choosing a unitary transformation (Chapter 13) which makes the matrix of Lagrange multipliers diagonal, i.e. Ay 0 and A This special set of molecular orbitals (f> ) are called canonical MOs, and they transform eq. (3.40) mto a set of pseudo-eigenvalue equations. 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

The structure was refined with block diagonal least squares. In cases of pseudo-symmetry, least squares refinement is usually troublesome due to the high correlations between atoms related by false symmetry operations. Because of the poor quality of the data, only those reflections not suffering from the effects of decomposition were used in the refinement. With all non-hydrogen atoms refined with isotropic thermal parameters and hydrogen atoms included at fixed positions, the final R and R values were 0.142 and 0.190, respectively. Refinement with anisotropic thermal parameters resulted in slightly more attractive R values, but the much lower data to parameter ratio did not justify it. 

At each stage of the refinement of a new set of parameters, the hat matrix diagonal elements were calculated in order to detect the influential observations following the criterium of Velleman and Welsh [8,9]. The inspection of the residues of such reflections revealed those which are aberrant but progressively, these aberrations disappeared when the pseudo-atoms model was used (introduction of multipoler coefficients). This fact confirms that the determination of the phases in acentric structures is improved by sophisticated models like the multipole density model. 

Eu3+-doped LaCl3 (C3h or D3h), LaA103 (D3) and La203 (C3v) have also been studied (145). In LaCl3, there are no off-diagonal elements between/-orbitals (in the pseudo-symmetry D3h) in the other compounds, two off-diagonal elements arise. 

The pseudo-inverse for the calculation of the shift vector in equation (4.67) has been computed traditionally as J+= (J Jp1 J. Adding a certain number, the Marquardt parameter mp, to the diagonal elements of the square matrix J J prior to its inversion, has two consequences (a) it shortens the shift vector 8p and (b) it turns its direction towards steepest descent. The larger the Marquardt parameter, the larger is the effect. In matrix formulation, we can write ... 

A normal matrix is one that commutes with its adjoint, AA = A A. Normal matrices include diagonal, real symmetric, orthogonal, unitary, Hermitian (self-adjoint), permutation, and pseudo-permutation matrices. 

A parameterization method of the Hamiltonian for two electronic states which couple via nuclear distortions (vibronic coupling), based on density functional theory (DFT) and Slaters transition state method, is presented and applied to the pseudo-Jahn-Teller coupling problem in molecules with an s2-lone pair. The diagonal and off-diagonal energies of the 2X2 Hamiltonian matrix have been calculated as a function of the symmetry breaking angular distortion modes and r (Td)] of molecules with the coordination number CN = 3... 

In the principal magnetic axes system, the magnetic susceptibility tensor is diagonal and the three last terms of eq. (27) vanish, thus leading to eq. (28) for the pseudo-contact shifts... 

These additional pseudo- and quasitorques produce the pseudo- and quasirotations of the hybridization tetrahedron of the boundary atom R. In the linear response approximation, it corresponds to the treatment of the corresponding pseudo- and quasitorques by the fV7/0 1 matrix which is simple (diagonal in the basis of the and SAi variables) in the case of symmetric hydride ... 

In the pseudo-natural orbital (PSNO) method,188 a natural orbital calculation is performed on selected pairs of electrons in the Hartree-Fock field of the n-2 electron core. These orbitals are then used as a basis for a Cl calculation. In their work on the HeH+ system, for which the orbital occupancy is 1 first-order density matrix, only 45 configurations were found to have significant occupation numbers. The results of the application of this method to Hs184 and HeH 185 will be discussed below. 

An alternative, and more recent, idea is the pseudo-spectral expansion,67 in which ffo is diagonalized in some large basis of convenient functions and the summation in the second-order energy is evaluated exactly over the resulting eigenfunctions. This method has been used most recently by Bukta and Meath, who use it to obtain third-... 

Plot the feed composition on the diagonal (expressed as the appropriate binary equivalent or pseudo-light-component composition as per item 1 above),... 

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