Amplitude equation

The CCSD energy is given by the general CC equation (4.53), and amplitude equations are derived by multiplying (4.50) with a singly excited determinant and integrating (analogously to eq. (4.54)). 

As the unstable growth of the fluctuations proceeds, the minimum current starts to increase. According to Section III.l, the unstable component of the asymmetrical concentration fluctuation is provided by the amplitude equation... 

Amplitude equations and fluctuations during passivation, 279 Analytical formulae for microwave frequency effects, accuracy of, 464 Andersen on the open circuit scrape method for potential of zero charge, 39 Anisotropic surface potential and the potential of zero charge, (Heusler and Lang), 34... 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

In this form, the amplitude equations (21) have been previously studied by Kutzelnigg and named the generalized Brillouin conditions [38]. 

The primary questions that remain to be answered are the following to evaluate the energy and amplitude equations (7) and (21), we need to (i) construct H and (ii) have some means of evaluating expectation values of operators O with the reference o-... 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

This modihed amplitude equation does not correspond to the minimization of the energy functional Eq. (7), and thus the generalized Hellmann-Feynman theorem [49] does not apply. 

We also measured the energy contributions from the different classes of excitation operators used in solving the amplitude equations (Section lll.C). 

Owing to its complexity, the CC-R12 method was initially realized in various approximate forms. The first implementation of the CCSD-R12 method including noniterative connected triples [CCSD(T)-R12] was reported by Noga et al. [31,32,57-60] within the SA. The use of the same basis set for the orbital expansion and the RI in the SA rendered many diagrammatic terms to vanish and, thereby, drastically simplified the CCSD-R12 amplitude equations, easing its implementation effort. However, the simplified equations also meant that large basis sets (such as uncontracted quintuple- basis set) were needed to obtain reliable results and, therefore, the SA CCSD-R12 method was useful only in limited circumstances. 

Subsequently, Klopper and coworkers developed the CCSD(R12) and CCSD(T)(R12) methods [61-63] in which the use of the SA was avoided, while maintaining the simplicity of the equations. The "(R12)" approximation retains the terms that are at most linear in ff and thus simplifies the amplitude equations considerably. Equations (20)—(22) are, therefore, replaced by [61]... 

Figure 1 A computational sequence of the CCSD-R12 geminal amplitude equation. For the definitions of symbols, see ref. 33.

It is now seen that assumption (a) is, indeed, valid provided / Aay, and T are slowly varying functions of the energy. That is, by integration over the intermediate states in eq. (2-34) a function is obtained having the same energy dependence as the original (assumed) transition amplitude. Equation (2-39) may be solved for Aga. The result is ... 

Ax2 >, and < Ay2 > are the mean square perpendicular vibrational amplitudes. Equation (12) demonstrates two features a) even for negligible perpendicular amplitudes shrinkage will be observed in the revalue, b) shrinkage also occurs in the harmonic approximation for P(r). 

The fact that these semi-classical results agree in all respects with those obtained from Schrodinger s amplitude equation... 

A time-independent Schrodinger equation, or amplitude equation, is obtained by substituting = ipexp(—iEt/h). 

Madelung made the important observation that, irrespective of the overall validity of the hydrodynamic analogy, all linear combinations of the stationary solutions of the amplitude equation... 

This expression reduces to Schrodinger s amplitude equation (A = ip) by equating k = E — V, the eigenvalues of kinetic energy ... 

Furthermore, for the time-dependent case we have the following CC amplitude equation (Eq. (13-74))... 

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