Second-order expansion

In the chemically important region below lOkcal/mol above the bottom, a second-order expansion is analogous to normally sufficient, as illustrated in Figure 2.4. 

We casually ignore the possibility of a more accurate second order expansion. That s because the higher order terms are nonlinear, and we need a linear approximation. Needless to say that with a first order expansion, it is acceptable only if h is sufficiently close to hs. 

At this point it might be helpful to summarize what has been done so far in terms of effective potentials. To obtain the QFH correction, we started with an exact path integral expression and obtained the effective potential by making a first-order cumulant expansion of the Boltzmann factor and analytically performing all of the Gaussian kinetic energy integrals. Once the first-order cumulant approximation is made, the rest of the derivation is exact up to (11.26). A second-order expansion of the potential then leads to the QFH approximation. 

Typically, in the MM framework, the increment from the bending is considered a quadratic function of valence angles. The formula for bending eq. (3.151) can be rewritten in this form. This is obtained by substituting eq. (3.140) to the second order expansion eq. (3.151) and significant simplifications based on vector algebra. After that we see that the bending force field constant can be written as ... 

We would like to point out that the general second-order expansion for a change in the system energy due to d N,... 

L.X. 0.2 -0.1 0.0 0.1 0.2 AR(A) approximation. In the chemically important region below lOkcal/mol above the bottom, a second-order expansion is analogous to str normally sufficient, as illustrated in Figure 2.4. Angles where the central atom is di- or bivalent (ethers, alcohols, sulfides, amines and... 

J the wave function is usually done by a second-order expansion of the energy in terms of... 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

The second-order expansion given in eqn (6.96) recovers all of the physical quantities needed to describe a quantum system and determine its properties the charge density p and its gradient vector field Vp define atoms and determine many of their properties in a stationary state the current density determines the system s magnetic properties and the change in p in a time-dependent system and, finally, the stress tensor determines the local and average mechanical properties of the system. Thus, one does not need all the... 

The quadratic response to a perturbation requires the solution of from the second-order expansion of Eq. (68),... 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

In this work, a recently developed semi-empirical method, SCC-DFTB method, is employed to account for the electronic structure of QM part. The details of this method and its implementation to CHARMM have been summarized elsewhere [6, 22-24]. Here we just give a short description. This method is derived by a second order expansion of the DFT total energy functional with respect to the charge density fluctuation around a given reference density. The total energy can be expressed as following [22] ... 

Figure P.2 iiAX S]/kT as a function of pAAB according to the first- and second-order expansion in pAAB at one composition xa = xb=0.5.

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

The second-order expansion of Eq. [11] leads to a macroscopic momentum conservation equation that differs from Navier-Stokes equations only in irrelevant terms of higher order provided that the mean velocity u is small. Thus lattice-gases may be used as models for fluids. 

These values correspond to a second-order expansion with respect to c, and thus arc consistent with the approximation which is used here. 

R - result of a renormalization calculation obtained by starting from second-order expansions in e = 4 — d ... 

As we saw previously, we can deduce from it a second-order expansion of the ratio N(z) = 6Rq/R2 in powers of z. This ratio can be written in the form of a Pade approximant. 

As mentioned (Section 21.3.2), the MNDO-type methods attempt to incorporate the effects of Pauli exchange repulsion in an empirical manner, through an effective atom-pair potential that is added to the core-core repulsion. It would clearly be better to include the underlying orthogonahzation corrections explicitly in the electronic calculation and to remove the effective atom-pair potential from the core-core repulsion. In a semiempirical context, the dominant one-electron orthogonahzation correchons can be represented by parametric funchons that reflect the second-order expansions of the Lowdin orthogonahzation transformation in terms of overlap. These corrections can then be adjusted during the parametrization process. 

Intuitively designed damped gradient corrections have been also used to improve the EDA for correlation. The first attempt of this kind was made by Ma and Brueckner [121] in their paper on the exact second-order expansion of Ec[p]. where they also propose the functional... 

The analytic form of Eq. (142) is sufficiently flexible to parametrize the exact second-order expansion of the exchange hole for any many-electron system, if it is generalized as... 

FORTRAN source code in which the maximum likelihood is evaluated with one of two different first-order expansions (FO or FOCE) and a second-order expansion about the conditional estimates of the random effects (Laplacian) S-PLUS algorithm utilizing a generalized least-squares (GLS) procedure and Taylor series expansion about the conditional estimates of the interindividual random effects... 

A second-order expansion of the FMPES at a stationary point on the zero-F PES yields... 

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