Tensor Correlation Methods

Tensor correlation methods refer to the selection of two anisotropic interactions so that the pattern of the 2D correlation spectrum can reveal the magnitudes and relative orientation of the two tensorial interactions. Below we will discuss many different SSNMR techniques suggested for the determination of peptide backbone 

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Before one can compare different correlated methods and their results for the rotational g tensor, one should discuss the quality of the employed basis set. Therefore, I have performed also calculations with rotational London orbitals for all the molecules at the SCF and the level of theory and compare... 

Each coupled spin level in the zero-field or correlation diagram had energy E(Si2, S34, S) for each wave function Si2, S34, S, M). The coupling scheme adopted was S12 = Sj + S2 S34 = S3 + S4 S = S12 + S34. Since the matrix of Eq. (14) cannot be diagonalized, matrix elements were worked out by tensor operator methods (50, 68). 

With quantum-mechanical methods, the second derivatives of the energy could be used directly for the FF and atomic polar tensors (APT) for the dipole derivatives. Both are standardly computed in most quantum-chemistry programs but for accurate results, moderately large basis sets and/or some accommodation for correlation interaction is needed. Until recently, this has restricted most ab initio studies to modest-sized molecules. 

This approach yields spectral densities. Although it does not require assumptions about the correlation function and therefore is not subjected to the limitations intrinsic to the model-free approach, obtaining information about protein dynamics by this method is no more straightforward, because it involves a similar problem of the physical (protein-relevant) interpretation of the information encoded in the form of SD, and is complicated by the lack of separation of overall and local motions. To characterize protein dynamics in terms of more palpable parameters, the spectral densities will then have to be analyzed in terms of model-free parameters or specific motional models derived e.g. from molecular dynamics simulations. The SD method can be extremely helpful in situations when no assumption about correlation function of the overall motion can be made (e.g. protein interaction and association, anisotropic overall motion, etc. see e.g. Ref. [39] or, for the determination of the 15N CSA tensor from relaxation data, Ref. [27]). 

It is also possible to employ highly correlated reference states as an alternative to methods that employ Hartree-Fock orbitals. Multiconfigu-rational, spin-tensor, electron propagator theory adopts multiconfigura-tional, self-consistent-field reference states [37], Perturbative corrections to these reference states have been introduced recently [38],... 

Different equilibrium, hydrodynamic, and dynamic properties are subsequently obtained. Thus, the time-correlation function of the stress tensor (corresponding to any crossed-coordinates component of the stress tensor) is obtained as a sum over all the exponential decays of the Rouse modes. Similarly, M[rj] is shown to be proportional to the sum of all the Rouse relaxation times. In the ZK formulation [83], the connectivity matrix A is built to describe a uniform star chain. An (f-l)-fold degeneration is found in this case for the f-inde-pendent odd modes. Viscosity results from the ZK method have been described already in the present text. 

The first observation one can make is that the correlation effects for the rotational g tensor in HF, H2O, and NH3 are in general small, 1.5-3.5%, and negative, i.e., correlation reduces the values of the rotational g tensor and therefore the amount of coupling between the electronic and rotational motion. Methane is an exception in that respect, because correlation increases the value of the g factor and because some methods, MPn and CCSD, predict much larger correlation corrections. 

Comparing the SOPPA and SOPPA(CCSD) results one can see that both methods predict correlation corrections which are comparable to the results obtained at the CCSD or RAS iSD level. However, it is clear that SOPPA(CCSD) is in better agreement with these more expensive methods. The SOPPA(CCSD) results for HF, for gip and goop in H2O are close to the respective MP3 results. In case of NH3 and the parallel component of the g tensor of H2O, where the correlation corrections are somewhat smaller, 1.5%, SOPPA(CCSD) predicts values close to the CCSD numbers. 

The components of the rotational g tensor of hydrogen fluoride, water, ammonia and methane have been calculated at their equilibrium geometries with different correlated ab initio methods and two large basis sets. 

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