Approximation second-order

The last step is to find a symplectic, second order approximation st to exp StL ). In principle, we can use any symplectic integrator suitable for time-dependent Schrddinger equations (see, for example, [9]). Here we focus on the following three different possibilities corresponding to special properties of the spatially truncated operators H q) and V q). 

Again, we use (2) to construct a symplectic, second order approximation to exp((5t L ). The resulting integrator for QCMD is of second order. 

This is the second-order approximation to additivity rules. 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

Requiring the gradient of the second-order approximation (14.3) to be zero produces the... 

The advantage of the NR method is that the convergence is second-order near a stationary point. If the function only contains tenns up to second-order, the NR step will go to the stationary point in only one iteration. In general the function contains higher-order terms, but the second-order approximation becomes better and better as the stationary point is approached. Sufficiently close to tire stationary point, the gradient is reduced quadratically. This means tlrat if the gradient norm is reduced by a factor of 10 between two iterations, it will go down by a factor of 100 in the next iteration, and a factor of 10 000 in the next ... 

As charge-dipole interaction between the electron and the atom is small, the perturbation theory expansion may be used to estimate f. The odd terms of this expansion disappear after averaging over impact parameters due to isotropy of collisions. In the second order approximation only those elements of P that are bilinear in V are non-zero. Straightforward calculation showed [176] that all components of the Stark structure are broadened but only those for which m = 0 interfere with each other ... 

The boundary conditions are unchanged. The method of lines solution continues to use a second-order approximation for dajdr and merely adds a Vr term to the coefficients for the points at r Ar. 

This is a second-order approximation and can be used to obtain derivatives up to the second. Differentiate to obtain... 

The second derivative is constant (independent of a) for this second-order approximation. We consider it to be a central difference. ... 

Example 8.11 Apply the various second-order approximations to /= xexp(x). 

The marching equation for reverse shooting. Equation (9.24), was developed using a first-order, backward difference approximation for dajdz, even though a second-order approximation was necessary for (faldz. Since the locations j—l, j, and j+ are involved an5rway, would it not be better to use a second-order, central difference approximation for dajdz ... 

As can readily be observed, the difference scheme of second-order approximation acquires the form... 

As one might expect, the derivative kii ) should be replaced by ku + k u. As a first step towards the construction of a second-order approximation, it will be sensible to carry out the forthcoming substitutions... 

Conditions for the second-order approximation (18) imply some restrictions on the pattern functionals j4[fe(s)] and F[/(s)]. In preparation for this, plain calculations give... 

In what follows we deal everywhere with the primary family of homogeneous conservative schemes (16), (17) and (16 ), (17) as well as with linear nonnegative pattern functionals j4[ (s)] and i [/(s)] still subject to conditions (20) and (21) of second-order approximation. 

When only one coefficient k[x) G is discontinuous, while other coefficients q, f are continuous, any conservative scheme (4) generating a second-order approximation is of second-order accuracy on the sequence of non-equidistant grids w (/ ). This fact is an immediate implication of the expansions, — (feu )i j 2 = 0 h ), valid for the... 

We are now interested in the question concerning the accuracy of scheme (4) with second-order approximation on an arbitrary non-equidistant grid. A discontinuity point a = is free to be chosen for the relevant coefficients ... 

In this way, the third kind difference boundary-value problem (2)-(4) of second-order approximation on the solution of the original problem is put in correspondence with the original problem (1). 

The main idea here is connected with the design of a new difference scheme of second-order approximation for which the maximum principle would be in full force for any step h. The meaning of this property is that we should have (see Chapter 1, Section 1)... 

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

It is worth mentioning here that the sign of r x) has had a significant impact on construction of monotone schemes. One way of providing a second-order approximation and taking care of this sign is connected with a monotone scheme with one-sided first difference derivatives for the equation with perturbed coefficients... 

In the case of the second boundary-value problem with dv/dn = 0, the boundary condition of second-order approximation is imposed on 7, as a first preliminary step. It is not difficult to verify directly that the difference eigenvalue problem of second-order approximation with the second kind boundary conditions is completely posed by... 

Both factorized schemes (44) and (45) generate second-order approximations in T for any a and they are stable under the condition 4cr > 1 -1- e,... 

Observe that provides a second-order approximation at the regular nodes A w — LaU = O(h ), while A m — L,yU = 0(1) at the irregular ones. 

A second-order approximation provided by the difference operator Aq, on a regular pattern... 

Both first and second order approximation statements are here. 

In the first-order approximation (Eq. 9-47a) the magnitude of the inducible dipoles is determined based on the assumption that they cannot interact with each other at all. The second-order approximation from Eq. (9-47b) is able to retain a greater part... 

Christiansen O, Koch H, Jorgensen P (1995) The second-order approximate coupled cluster singles and doubles model CC2. Chem Phys Lett 243 409 t 18... 

In both examples discussed in this section, the second-order approximation to AA turned out to be satisfactory. We, however, do not want to leave the reader with the impression that this is always true. If this were the case, it would imply that probability distributions of interest were always Gaussian. Statistical mechanics would then be a much simpler field. Since this is obviously not so, we have to develop techniques to deal with large and not necessarily Gaussian-distributed perturbations. This issue is addressed in the remainder of this chapter. 

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