Order-parameter matrix

Another principal difficulty is that the precise effect of local dynamics on the NOE intensity cannot be determined from the data. The dynamic correction factor [85] describes the ratio of the effects of distance and angular fluctuations. Theoretical studies based on NOE intensities extracted from molecular dynamics trajectories [86,87] are helpful to understand the detailed relationship between NMR parameters and local dynamics and may lead to structure-dependent corrections. In an implicit way, an estimate of the dynamic correction factor has been used in an ensemble relaxation matrix refinement by including order parameters for proton-proton vectors derived from molecular dynamics calculations [72]. One remaining challenge is to incorporate data describing the local dynamics of the molecule directly into the refinement, in such a way that an order parameter calculated from the calculated ensemble is similar to the measured order parameter. 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

Among the methods discussed in this book, FEP is the most commonly used to carry out alchemical transformations described in Sect. 2.8 of Chap. 2. Probability distribution and TI methods, in conjunction with MD, are favored if there is an order parameter in the system, defined as a dynamical variable. Among these methods, ABF, derived in Chap. 4, appears to be nearly optimal. Its accuracy, however, has not been tested critically for systems that relax slowly along the degrees of freedom perpendicular to the order parameter. Adaptive histogram approaches, primarily used in Monte Carlo simulations - e.g., multicanonical, WL and, in particular, the transition matrix method - yield superior results in applications to phase transitions,... 

One of the characteristic features of mesogenic molecules in uniaxial mesophases is their orientational order specified by a set of order parameters, which forms a symmetric and traceless order matrix S. For rigid molecules, the elements of order matrix Sa/Y defined by Saupe,52 are given by... 

Let us also define a it x 1 matrix of first-order parameter estimates, f ... 

Finally, we define a k x k matrix of second-order parameter estimates, S ... 

The single-factor second-order parameter estimates lie along the diagonal of the S matrix, and the two-factor interaction parameter estimates are divided in half on either side of the diagonal. 

Norden et al. have examined the CD spectral changes of dyes oriented in a liquid crystal matrix, on variing the order parameter q, which is estimated by NMR observation 262). Kuball et al. presented a theoretical description of the optical activity of oriented molecules, and they found that the C D of the transition A of a given oriented molecule Ae (v) is described by ... 

The super-CI method now implies solving the corresponding secular problem and using tpq as the exponential parameters for the orbital rotations. Alternatively we can construct the first order density matrix corresponding to the wave function (4 55), diagonalize it, and use the natural orbitals as the new trial orbitals in I0>. Both methods incorporate the effects of lpq> into I0> to second order in tpq. We can therefore expect tpq to decrease in the next iteration. At convergence all t will vanish, which is equivalent to the condition ... 

The full machinery of ESPS is customizable in a number of respects, notably the choice of reference configurations, of order parameter and of transformation matrix. These issues are best explored in the context of specific examples. 

Vra / ft ) is the quadrupole coupling constant. The matrix of S values represents the order parameters, and they give the alignment of the compound with respect to the applied magnetic field. They can be, and usually are, defined in terms of a molecular-fixed coordinate system. S is a symmetrical 3x3 matrix, and the sum of the diagonal elements of S is zero, so that in a molecular-fixed coordinate system, the number of components of the S matrix varies from 5 for compounds with no elements of symmetry, such as chiral species, to 1 for entities with a C3 or higher axis of symmetry. 

The average <(poz)3>=, or order parameter, is calculated over the equilibrium density matrix, p=e H/T/Tr[e H/T], where Tr denotes the trace (sum over diagonal elements). The Ising Hamiltonian can be expressed as ... 

We have used that the second-order perturbed density matrix elements (see Eq. (70)) can be separated into components due to first- and second-order parameters, respectively,... 

Here, we have also separated the perturbed density matrix into terms that depend on first- and second-order, and third-order parameters, respectively, as... 

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